El-MANSOURA UNIVERSITY
FACULITY OF ENGINEERING
STRUCTURAL ENG. DEPARTMENT




































DESIGN OF REINFORCED CONCRETE
BEAMS WITH OPENINGS
By
Eng. Waleed El-Demerdash El-Demerdash El-Sawi
B. Sc. in Civil Engineering, Mansoura University, 2008
Teaching Assistant, Civil Engrg. Dept., MET Academy.
A thesis Submitted in Partial Fulfillment for
the Requirements of the Degree of Master of Science
IN
STRUCURAL ENGINEERING
Under the Supervision of
Prof. Dr.
Salah El-Din E. El-Metwally
Prof., Structural Engrg. Dept.,
Faculty of Engineering,
Mansoura University
Dr.
Ahmed Amin Ghaleb
Dr.
Mohamed El-Said El-Zoughiby
Associate Prof., Structural Engrg. Dept.,
Faculty of Engineering,
Mansoura University
Associate Prof., Structural Engrg. Dept.,
Faculty of Engineering,
Mansoura University
2013
SUPERVISORS
Thesis Title:
DESIGN OF REINFORCED CONCRETE
BEAMS WITH OPENINGS
Researcher Name:
Waleed El-Demerdash El-Demerdash El-Sawi
Supervisors:
Name
Prof. Dr.
Salah El-Din E. ElMetwally
Asso. Prof. Dr.
Ahmed Amin Ghaleb
Asso. Prof. Dr.
Mohamed El-Said ElZoughiby
Head of the Dept.
Prof. Dr. Ahmed M. Yousef
Position
Signature
Professor,
Structural Engineering Department,
Faculty of Engineering,
Mansoura University.
Associate Professor,
Structural Engineering Department,
Faculty of Engineering,
Mansoura University.
Associate Professor,
Structural Engineering Department,
Faculty of Engineering
Mansoura University.
Faculty Vice Dean
Prof. Dr. Kassem Salah Al-Alfy
Faculty Dean
Prof. Dr. Zaki M. Zeidan
JUDGES
Thesis Title: DESIGN OF REINFORCED CONCRETE
BEAMS WITH OPENINGS
Researcher Name: Waleed El-Demerdash El-Demerdash El-Sawi
Supervisors:
Name
Position
Signature
Prof. Dr.
Salah El-Din E. ElMetwally
Professor,
Structural Engineering Department,
Faculty of Engineering- Mansoura University.
Asso. Prof. Dr.
Ahmed Amin Ghaleb
Associate Professor,
Structural Engineering Department,
Faculty of Engineering- Mansoura University.
Asso. Prof. Dr.
Mohamed El-Said ElZoughiby
Associate Professor,
Structural Engineering Department,
Faculty of Engineering- Mansoura University.
Judges:
Name
Prof. Dr.
Youssef Ibrahim Agag
Prof. Dr.
Salah El-Din E. ElMetwally
Position
Signature
Professor,
Structural Engineering Department,
Faculty of Engineering- Mansoura University.
Professor,
Structural Engineering Department,
Faculty of Engineering- Mansoura University.
Prof. Dr.
Mashhour Ghoneim
Ahmed Ghoneim
Professor,
Structural Engineering Department,
Faculty of Engineering- Cairo University
Asso. Prof. Dr.
Ahmed Amin Ghaleb
Associate Professor,
Structural Engineering Department,
Faculty of Engineering- Mansoura University.
Head of the Dept.
Prof. Dr. Ahmed M. Yousef
Faculty Vice Dean
Prof. Dr. Kassem Salah Al-Alfy
Faculty Dean
Prof. Dr. Zaki M. Zeidan
CONTENTS
Page
ACKNOWLEGEMENTS………………………………………………………………
ix
ABSTRACT ……………………………………………………………………………..
x
CHAPTER 1: INTRODUCTION
1.1 GENERAL ……..……………………………………………………….........
1.2 PROBLEM IDENTIFICATION ………………………………………………
1.3 RESEARCH SIGNIFICANCE …………………………………………….....
1.4 OBJECTIVES AND SCOPE …..……………………………………………..
1.5 THESIS ARRANGEMENT ………………………………………………......
1
3
3
3
4
CHAPTER 2: LITERATURE REVIEW
2.1 INTRODUCTION ……………………………………………………………
2.2 PREVIOUS STUDIES ON ORDINARY BEAMS WITH OPENINGS
2.2.1 Classification of Openings……………………………………………
2.2.1.1 Small Openings…………………………………………..
2.2.1.2 Large Openings ………………………………………….
5
5
5
5
10
2.3 PREVIOUS STUDIES ON SHEAR BEHAVIOR OF SIMPLY
SUPPORTED NORMAL-STRENGTH CONCRETE DEEP BEAMS
WITH AND WITHOUT OPENINGS……………………………………….
16
2.4 PREVIOUS STUDIES ON SHEAR BEHAVIOR OF SIMPLY
SUPPORTED HIGH STRENGTH CONCRETE DEEP BEAMS
WITH AND WITHOUT OPENINGS……………………………………….
17
2.5 PREVIOUS STUDIES ON SHEAR STRENGTH OF NORMAL- AND
HIGH-STRENGTH CONCRETE SOLID CONTINUOUS DEEP
BEAMS ………………………………………………………………………
19
2.6 PREVIOUS STUDIES ON THE SHEAR BEHAVIOR OF
CONTINUOUS DEEP BEAMS WITH WEB OPENINGS……………………
19
i
CHAPTER 3: STRUT-AND-TIE MODELING OF BEAMS WITH OPENINGS
3.1 INTRODUCTION ……………………………………………………………
3.2 OPENINGS IN ORDINARY BEAMS………………………………………..
3.2.1 Modeling …………………………………………………………….
3.2.1.1 The Approach for Developing a STM for Beams with
Openings……………………………………………
3.2.1.2 Case-Study……………………………………………….
3.2.2 Strength Limits of Strut-and-tie Model′s Components…………………
3.2.3 Verification Examples………………………………………………..
21
23
23
3.3 DEEP BEAMS AND OPENINGS IN DEEP BEAMS………………………..
3.3.1 Modeling …………………………………………………………….
3.3.1.1 Case-Study……………………………………………….
3.3.2 Verification Examples …………………………………………….....
3.4 SUMMARY AND CONCLUSIONS………………………………………….
76
76
76
58
93
23
24
55
57
CHAPTER 4: NONLINEAR FINITE ELEMENT ANALYSIS
4.1 INTRODUCTION……………………………………………………………..
4.2 ANSYS′ FINITE ELEMENT MODELS………………………………………...
4.2.1 Element Types ………………………………………………………..
4.2.1.1 Solid65.…………………………………………………….
4.2.1.2 Solid45….………………………………………………….
4.2.1.3 Link8-3D…….……………………………………………..
4.2.2 Material Models……………………………………………………….
4.2.2.1 Concrete in Compression………………………………......
4.2.2.2 Concrete in Tension………………………………………..
4.2.2.3 Reinforcement in Tension…………………………………
4.2.2.4 Bond between Concrete and Reinforcement………………
4.2.3 Solution Strategy………………………………………………………
4.2.3.1 Automatic Time Stepping…………………………………
4.2.3.2 Loading…………………………………………………….
4.2.3.3 Newton-Raphson Method of Analysis……………………
94
94
94
94
97
98
100
103
104
105
105
105
106
106
107
4.3 ANALYSIS OF ORDINARY BEAMS WITH OPENINGS…………………....
4.3.1 Verification Group B: Simple Beams With and Without Circular
Openings………………………………………………………………
4.3.1.1 Model Description and Material Properties..………………
4.3.1.2 Meshing…………………………………………………….
4.3.1.3 Loads and Boundary Conditions…………………………..
4.3.1.4 Finite Element Results……………………………………...
4.3.1.5 Comparison of the Results………………………………….
108
4.3.2 Verification Group C: Simple Beams With and Without Rectangular
Openings………………………………………………………………
4.3.2.1 Model Description and Material Properties……………….
4.3.2.2 Meshing……………………………………………………
4.3.2.3 Loads and Boundary Conditions………………………….
4.3.2.4 Finite Element Results…………………………………….
ii
108
108
109
109
110
111
122
122
122
122
123
4.3.2.5 Comparison of the Results…………………………………
4.3.3 Verification Group D: Simple Beams With Rectangular Openings….
4.3.3.1 Model Description and Material Properties ………………
4.3.3.2 Meshing…………………………………………………….
4.3.3.3 Loads and Boundary Conditions…………………………..
4.3.3.4 Finite Element Results……………………………………..
4.3.3.5 Comparison of the Results………………………………....
123
130
130
130
130
131
131
4.4 ANALYSIS OF DEEP BEAMS WITH OPENINGS…………………………..
4.4.1 Verification Group A: Simple and continuous Beams With
Rectangular Openings………………………………………………...
4.4.1.1 Model Description and Material Properties ………………
4.4.1.2 Meshing……………………………………………………
4.4.1.3 Loads and Boundary Conditions………………………….
4.4.1.4 Finite Element Results…………………………………….
4.4.1.5 Comparison of the Results………………………………..
4.4.2 Verification Group B: Simple Beams With Rectangular Openings….
4.4.2.1 Model Description and Material Properties ………………
4.4.2.2 Meshing…………………………………………………...
4.4.2.3 Loads and Boundary Conditions………………………….
4.4.2.4 Finite Element Results…………………………………….
4.4.2.5 Comparison of the Results………………………………...
133
4.5 CONCLUSIONS……………………………………………………………….
154
134
134
135
135
136
137
146
146
146
146
147
147
CHAPTER 5: DESIGN PROCEDURE, DETAILING, AND DESIGN
RECOMMENDATIONS FOR BEAMS WITH OPENINGS
5.1 INTRODUCTION………………………………………………………………
5.2 SHALLOW (ORDINARY) BEAMS…………………………………………...
5.2.1 General Guidelines……………………………………………………
5.2.2 Design of Reinforced Concrete Beams with Small Openings using
Traditional Approach………………………………………………….
5.2.2.1 Pure Bending………………………………………………
5.2.2.2 Combined Bending and Shear……………………………..
5.2.2.3 Reinforcement Detailing…………………………………..
5.2.2.4 Numerical Example for Beam ND80X350 (Case 2)
with Small Openings……………………………………….
5.2.3 Redesign of Reinforced Concrete Beam ND80X350 using
Strut-and-Tie Method …………………………………………………
5.2.4 Design of Reinforced Concrete Beams with Large Openings using the
Traditional Approach………………………………………………….
5.2.4.1 Available Design Procedures ……………………………...
5.2.4.2 Numerical Example for Case 2-Beam (Group C)
with Large Rectangular Openings 100×300mm……………..
5.2.5 Redesign for the previous Case 2-Beam (Group C) with Large
Rectangular Opening 100×300mm using Strut-and-Tie Method………
iii
155
155
155
156
157
157
161
161
166
170
170
178
183
5.3 DEEP BEAMS…………………………………………………………………..
5.3.1 A general Procedure for Strut-and-Tie Modeling for
Discontinuity Regions…………………………………………………
5.3.2 Example-Design of a RC Deep Beam with Openings using
Strut-and-Tie Method …………………………………………………
5.3.2.1 Geometry and Loads……………………………………….
5.3.2.2 Design Procedure…………………………………………..
5.3.2.3 Design Calculations………………………………………...
5.4 DESIGN RECOMMENDATIONS………………………………………………
187
187
190
190
191
191
198
CHAPTER 6: SUMMARY AND CONCLUSIONS
6.1 INTRODUCTION……………………………………………………………...
6.2 SUMMARY……………………………………………………………………
6.3 CONCLUSIONS……………………………………………………………….
iv
199
599
599
LIST OF TABLES AND FIGURES
LIST OF TABLES
Page
Table 3.1 Details of tested beams (Group A)………………………………………….
Table 3.2 Details of tested beams……………………………………………………..
Table 3.3 Mechanical properties of rebars used in tested beams……………………..
Table 3.4 Details of the tested beams…………………………………………………
Table 3.5 ACI 318M-11Code values of coefficient βs for strut……………………..…
Table 3.6 ACI 318M-11Code values of coefficient βn for nodes..……………….……
Table 3.7 The STM results compared with test results………………………………..
Table 3.8 Calculated member forces for the strut-and-tie model……………………...
Table 3.9 Summary of concrete struts calculations…………………………………….
Table 3.10 Summary of the effective concrete node calculations…………..…………..
Table 3.11 Calculated forces of the strut-and-tie model of Beam IT1…………………..
Table 3.12 Summary of concrete struts calculations…………………………………….
Table 3.13 Summary of effective concrete node calculations…..………………………
Table 3.14 The STM results compared with test results…………………………………
Table 3.15 Concrete properties of the investigated beams Group B……………………
Table 3.16 Calculated member forces for the strut-and-tie model………………………
Table 3.17 Summary of concrete struts calculations…………………………………….
Table 3.18 Summary of critical concrete node calculations………………………..……
Table 3.19 Calculated member forces for proposed simplified the strut-and-tie model…
Table 3.20 Summary of concrete struts calculations……………………………………..
Table 3.21 Summary of critical concrete node calculations……………………………...
Table 3.22 The STM results compared with test results………………………………….
Table 3.23 The STM results compared with test results………………………………….
Table 4.1 Material models for SOLID65, SOLID45 and LINK8 element……………..
Table 4.2 Material properties for concrete and reinforcement………………………….
Table 4.3 First flexural cracking, diagonal cracking, and ultimate loads from ANSYS.
Table 4.4 Comparison of ultimate loads………………………………………………...
Table 4.5 Material properties for concrete and reinforcement…………………………..
Table 4.6 First flexural cracking, diagonal cracking, and ultimate loads from ANSYS.
Table 4.7 Comparison of ultimate loads……………………………………………..….
Table 4.8 Details of the tested specimens……………………………………………….
Table 4.9 Reinforcement properties………………………………………………..……
Table 4.10 First diagonal cracking, flexure cracking, and ultimate loads from ANSYS.
Table 4.11 Comparison of ultimate loads………………………………………………..
Table 4.12 First diagonal cracking, flexural cracking, and ultimate loads from ANSYS.
Table 4.13 Comparison of ultimate loads……………………………………………..….
Table 5.1 STM forces……………………………………………………………………
Table 5.2 STM forces……………………………………………………………………..
v
25
38
39
47
54
57
64
70
71
71
74
75
75
76
88
87
88
88
99
95
19
98
92
599
599
558
558
539
535
535
533
533
536
536
547
547
568
585
LIST OF FIGURES
Page
Figure 1.1
Figure 2.1
Figure 2.2
Figure 2.3
An ordinary beam with circular openings…………………………………..
Opening shapes considered by Prentzas (1968)………………………….....
Definition of small openings according to openings dimensions…….…....
Definition of small openings according to structural response
(beam-type behavior)……………………………………………………….
Figure 2.4 Definition of small and large openings (approximate validity of
Bernoulli’s hypothesis of plane strain distribution)………………………….
Figure 2.5 Typical shear failure of a beam without shear reinforcement……….……....
Figure 2.6 Reinforcement schemes for beams with small openings (Salam, 1977)…….
Figure 2.7 Shear failure of Beam B4 at the throat section (Salam, 1977)…………..…...
Figure 2.8 Beam-type shear failure at small openings……………………..……………
Figure 2.9 Frame-type shear failure at small openings…………………………………..
Figure 2.10 Cracking patterns of beams…………………………………………………..
Figure 2.11 Beams and reinforcement details………………………………………….…
Figure 2.12 Cracking patterns of beams tested by Mansur et al. (1999)..………………...
Figure 2.13 Definition of large openings according to structural response……………....
Figure 2.14 Failure of a beam with multiple rectangular openings……………………….
Figure 2.15 Load versus maximum deflection curves (Mansur et al., 1991)……………..
Figure 2.16 The formation of a mechanism formation containing four hinges in the chords
Figure 2.17 Schematic view of the test setup (Abdalla et al., [1] in 2003)… ……..………
Figure 2.18 Typical specimen details (units are in mm)………………………… ………..
Figure 2.19 Details of reinforcement of the tested continuous deep beams……………….
Figure 3.1 Possible load paths for deep and slender beams……………………………..
Figure 3.2 Examples of D-regions (ACI 318-2011)………………………………………
Figure 3.3 St. Venant’s principle, Brown et al……………………………………………
Figure 3.4 Description of deep and slender beams (ACI 318-2011)……………………..
Figure 3.6 Beam R1………………………………………………………………………
Figure 3.7 Beam R5……………………………………………………………………….
Figure 3.8 Beam R7……………………………………………………………………….
Figure 3.9 Beam R14……………………………………………………………………...
Figure 3.10 Beam B1……………………………………………………………………...
Figure 3.11 Beam B2……………………………………………………………………...
Figure 3.12 Beam B3………………………………………………………………………
Figure 3.13 Beam C1……………………………………………………………………...
Figure 3.14 Beam C3……………………………………………………………………...
Figure 3.15 Beam C5………………………………………………………………………
Figure 3.16 Properties of Beam ND80X350-sd and the type of location of bars…………
Figure 3.17 Beam S………………………………………………………………………..
Figure 3.18 Beam ND80X350……………………………………………………………..
Figure 3.19 Beam ND80X150-S…………………………………………………………..
Figure 3.20 Beam ND80X250-S…………………………………………………………..
Figure 3.21 Beam ND100X350-S…………………………………………………………
Figure 3.22 Solid Beam……………………………………………………………………
Figure 3.23 Beam with opening……………………………………………………………
Figure 3.24 Beam IT1……………………………………………………………………...
vi
1
5
6
6
6
7
8
8
8
9
9
9
59
59
55
58
53
54
58
89
85
88
88
83
86
88
89
35
38
33
34
35
36
37
39
49
45
48
43
44
45
46
48
49
Figure 3.25 Beam IT4………………………………………………………………………
Figure 3.26 Beam IT8………………………………………………………………………
Figure 3.27 Description of the different components of a strut-and-tie model for
a deep beam……………………………………………………………………
Figure 3.28 The width of the tie wt used to determine the dimensions of the node…………
Figure 3.29 Geometric shapes of struts…………………………………………………..…
Figure 3.30 Types of struts…………………………………………………………………..
Figure 3.31 Classification of nodes………………………………………………………....
Figure 3.32 States of stress in hydrostatic and non-hydrostatic nodes, Brown et al..………
Figure 3.33 Solid Beam S…………………………………………………………………...
Figure 3.34 Beam ND80X350……………………………………………………………….
Figure 3.35 Solid beam………………………………………………………………………
Figure 3.36 Beam with rectangular openings (100×300mm)………..………………………
Figure 3.37 Beam IT1………………………………………………………………………..
Figure 3.38 Details of reinforcement (Beam IT1)…………………………………………...
Figure 3.39 Beam DSON3…………………………………………………………………..
Figure 3.40 Beam DSOH10……………………………………………………………….…
Figure 3.41 Beam DCON3…………………………………………………………….…….
Figure 3.42 Beam DCOH2………………………………………………………….……….
Figure 3.43 Beam DCOH8…………………………………………………………………..
Figure 3.44 Beam geometry for Group B……………………………………………….……
Figure 3.45 Web reinforcement patterns for beams Group B…………………………..…...
Figure 3.46 Beams for Group B………………………………………………………….…..
Figure 3.47 Details of the proposed refined strut-and-tie model for Beam DSON3 using
inclined ties……………………………………………………………..………
Figure 3.48 Details of the proposed simplified strut-and-tie model for Beam DSON3
using inclined ties……………………………………………………………….
Figure 3.49 Details of reinforcement for Beam DSON3……………………………………
Figure 3.50 Visualization of strut widths……………………………………………………
Figure 3.51 Alternative proposed refined strut-and-tie model for Beam DSON3
using vertical and horizontal ties………………………………………………..
Figure 3.52 Alternative proposed simplified strut-and-tie model for Beam DSON3
using vertical and horizontal ties…………………………………………..……
Figure 3.53 Visualization of strut widths…………………………………………………...
Figure 4.1 SOLID65 3-D reinforced concrete solid element…………………………..….
Figure 4.2 SOLID65 3-D stress output……………………………………………………
Figure 4.3 SOLID45 3-D element………………………………………………………….
Figure 4.4 SOLID45 3-D stress output…………………………………………………….
Figure 4.5 Link 8-3D element bars………………………………………………………...
Figure 4.6 Models for reinforcement in reinforced concrete elements:
(a) discrete; (b) embedded; and (c) smeared…………………………….
Figure 4.7 Multilinear isotropic stress-strain curve for concrete in compression (Egyptian
Code)……………………………………………………………………………
Figure 4.8 Multilinear isotropic stress-strain curve for concrete in compression
(ACI Code)……………………………………………………………………..
Figure 4.9 Typical stress-strain curves for concrete in compression………………………
Figure 4.10 Idealized stress-strain curve for concrete in compression………………………
Figure 4.11 Idealized stress-strain curve for steel……………………………………………
Figure 4.12 Load steps, substeps, and time…………………………………………………..
Figure 4.13 Incremental Newton-Raphson procedure………………………………………..
vii
59
55
58
53
55
55
56
58
65
65
69
73
74
77
78
79
89
85
88
83
84
85
86
86
87
89
89
99
96
96
96
97
98
99
595
102
104
104
105
106
107
598
Figure 4.14 Traditional Newton-Raphson method vs. arc-length method…………………..
Figure 4.15 Initial-stiffness Newton-Raphson……………………………………………… 598
Figure 4.16 Output of “ANSYS” Program for solid Beam S……………………………….. 556
Figure 4.17 Output of “ANSYS” Program for Beam ND80X350………………………….. 558
Figure 4.18 Output of “ANSYS” Program for Beam ND100X350-S………………………. 585
Figure 4.19 Output of “ANSYS” Program for Solid Beam…………………………………. 586
Figure 4.20 Output of “ANSYS” Program for the beam with rectangular openings……….. 589
Figure 4.21 Output of “ANSYS” Program figures for Beam IT1…………………………… 538
Figure 4.22 Output of “ANSYS” Program figures for Beam DSON3……………………… 545
Figure 4.23 Output of “ANSYS” Program for Beam DCON3…………………………….... 144
Figure 4.24 Output of “ANSYS” Program for Beam NO-0.3/4…………………………….. 150
Figure 4.25 Output of “ANSYS” Program for Beam NW7-0.3/4…………………………… 553
Figure 5.1 Guidelines for the location of web openings (Tan et al., 1996)…………….….. 556
Figure 5.2 Beam with opening under pure bending…………………………………….…. 157
Figure 5.3 The two modes of shear failure around small openings……………………..…. 557
Figure 5.4 Shear resistance, Vs, provided by shear reinforcement at an opening….….…… 559
Figure 5.5 Free-body diagram at beam opening……………………………………..…….. 569
Figure 5.6 Reinforcement details around a small opening…………………………….…… 161
Figure 5.7 Beam ND80X350 and loading…………………………………………….……. 568
Figure 5.8 Reinforcement details of the Beam ND80X350………………………….…….. 566
Figure 5.9 Proposed strut-and-tie model for Beam ND80X350…………………….……… 566
Figure 5.10 Details of the strut-and-tie model for Beam ND80X350……………………….. 167
Figure 5.11 Reinforcement details of the Beam ND80X350 using strut-and-tie method…… 569
Figure 5.12 Beam with an opening under bending and shear. (a) The beam;
(b) Free-body diagram of opening segment; (c) Free-body diagram of the chords. 578
Figure 5.13 Assumed collapse mechanism for a beam with large openings…………………. 578
Figure 5.14 Failure of a beam with multiple rectangular openings separated by
adequately reinforced post………………………………………………………. 574
Figure 5.15 Forces acting on the post between adjacent openings…………………………... 575
Figure 5.16 Idealized model for the estimation of deflection at opening (Barney et al., 1977). 576
Figure 5.17 A suitable reinforcement scheme for the large opening………………………… 577
Figure 5.18 Ductile failure of a beam under combined bending and shear………………….. 578
Figure 5.19 Beam and loading……………………………………………………………….. 578
Figure 5.20 Reinforcement details at opening segment……………………………………… 183
Figure 5.21 Proposed strut-and-tie model……………………………………………………. 183
Figure 5.22 Details of the strut-and-tie model……………………………………………….. 584
Figure 5.23 Reinforcement details at opening segment using STM…………………………. 187
Figure 5.24 Flowchart illustrating STM steps, Brown et al. [9]……………………………… 188
Figure 5.25 Beam geometry and loading……………………………………………………... 595
Figure 5.26 Details of the proposed simplified strut-and-tie model (using inclined ties)
for Beam DSON3………………………………………………………………… 598
Figure 5.27 Alternative proposed simplified strut-and-tie model for Beam DSON3
(using vertical and horizontal ties)…………………………………………….... 593
Figure 5.28 Visualization of struts' widths…………………………………………………… 595
Figure 5.29 Nodal zone N1…………………………………………………………………… 196
Figure 5.30 Final reinforcement detailing according to the strut-and-tie model…………….. 197
viii
ACKNOWLEDGEMENTS
To Allah, everything in life is resumed. In this work he has helped me a lot and
offered me what I did not know and what I have to know. Allah is the first and
the last. Then, those offered by Allah to advise and guide have to be thanked.
It is my pleasure to express deepest gratitude to Prof. Dr. Salah El-Din E.
El- Metwally, who, very kindly, and generously, devoted much of his time and
experience in helping, guiding, and advising me. Indeed this work is the outcome of
his great continuous efforts and wide experience in the field of structural
engineering.
I am especially grateful and specially indebted to Asso. Prof.
Mohamed El-Said El-Zoughiby and Asso. Prof. Dr. Ahmed Amin Ghaleb, for
constructive keen supervision, fruitful criticism, continuous support
encouragement to complete this work. They sacrificed good deal of their
for the accomplishment of this work. I express my thanks for their efforts
help to me, and for the time spent on overcoming any obstacle.
Special
Dr.
their
and
time
and
thanks
and gratitude must be offered to my family, my
father, mother, my wife, daughter Jody, and son Omar for the great support and
encouragement which they have given me during the course of this research. I am
deeply grateful to all of them.
Waleed El-Demerdash
2013
ix
ABSTRACT
Ordinary beams with openings and deep beams with and without openings are
considered disturbed regions where their strains within any section are significantly
nonlinear. Therefore, it is not adequate to design those regions using either bending
theory or conventional shear design equations. Hence, it is essential to rely on a
rational method such as the strut-and-tie model.
The behavior of experimentally tested reinforced normal- and high- strength
concrete simply supported shallow beams (with and without openings) and simple
and continuous deep beams (with openings) was studied. In this study, the Strutand-Tie Models STM for all such selected beams are suggested based on the
available experimental results of crack patterns, modes of failure, and internal
stresses trajectors obtained from elastic finite element analysis. The obtained STM
results are compared with test results.
To draw a complete picture of the response of the studied beams, a 3D
nonlinear finite element analysis is conducted. From which, the output results of
cracking patterns, deflections, failure mode and strain and stress distributions (that
can not be obtained using the strut-and-tie model) are obtained
In addition, a full design procedure along with numerical examples,
reinforcement detailing, and design recommendations for beams with openings only
is presented.
x
CHAPTERT
1
INTRODUCTION
CHAPTER 1
INTRODUCTION
1.1 GENERAL
In practice, transverse openings in Reinforced Concrete, RC, beams are a facility, which allows
the utility line to pass through the structure such as a network of pipes and ducts (which is
necessary to accommodate essential services like water supply, sewage, air-conditioning,
electricity, telephone, and computer network(, Fig. 1.1. Passing utility services through openings
in the floor beam webs minimizes the required story height and encourages the designer to
reduce the height of the structure, which leads to more economical design. Including transverse
openings in the web of a reinforced concrete beam and therefore, the sudden changes in the
dimensions of the cross section of the beam; the corners of the opening would be subjected to
stress concentration and it is possible to induce transverse cracks in the beam. Also, it can reduce
the stiffness, which lead to deformations and excessive deflections under service load and
considerable distribution of forces and internal moments in a continuous beam. So, the effect of
openings on the strength and behavior of reinforced concrete beams must be considered and the
design of these beams needs special consideration. However, current codes of practice for design
of RC structures do not provide provisions for design of RC beams with openings.
Figure 1.1 An ordinary beam with circular openings.
In this research, two types of reinforced concrete beams (ordinary and deep) with and
without openings are studied. Reinforced concrete deep beams have useful applications in tall
buildings, offshore structures, long-span structures (as transfer girders), foundations, and water
tanks [Khalaf, (1986), Mahmoud, (1992)]. Since deep beams usually fail in shear at the ultimate
limit state, their shear capacities have to be accurately understood. In continuous deep beams, the
regions of high shear and high moment coincide and failure usually occurs in these regions. In
simple deep beams, the region of high shear coincides with the region of low moment. Current
codes, e.g. the ACI Code (2011) [3] and the Egyptian Code (2007) [10], define a beam to be deep
when the span-to-overall member depth ratio (L/h) is less than or equal to 4, or the shear span-tooverall member depth ratio (a/h) is less than or equal to 2 and span-to-depth ratio (L/d) is less
than or equal to 4, or the shear span-to-depth ratio (a/d) is less than or equal to 2, respectively. As
a result of its proportions, the strength of a deep beam is usually controlled by shear, rather than
by flexure, provided that normal amounts of longitudinal reinforcement are used. On the other
hand, shear strength of deep beams is significantly greater than that predicted using expressions
developed for shallow (ordinary) beams because deep beams have a more complex and different
behavior in many features in comparison with ordinary beams:
1



in deep beams, the hypothesis of Bernoulli is not valid; i.e., transverse sections which are
plane before bending dose not remain plane after bending,
the neutral axis does not usually lie at mid-depth and moves away from the loaded face of the
member as the span-to-depth ratio decreases, and
flexural stresses and strains are not linearly distributed across the beam depth [(Winter and
Nilson (1978)].
Three design approaches are available for deep beams; namely, a semi-empirical design
approach, a design approach based on stress analysis, and a strut-and-tie modeling. In design
codes, the semi-empirical design approach is based on some empirical shear equations. In this
approach, the concrete and steel reinforcement do not interact with each other, or in other words,
this conventional approach does not give a physical representation of the interplay between
concrete and steel contribution to shear strength. The second approach, based on stress analysis,
involves the use and development of finite element models which consider the effect of cracking
and transverse tensile strains on concrete behavior. Finally, a strut-and-tie model involves the
development and design of an analogous truss. This contains concrete struts, tension ties, and
nodal zones that realistically model the internal load path within the actual structure.
Openings in the web area of deep beams are frequently provided for essential services and
accessibility, for example door openings, windows, ventilating ducts and heating pipes. Such
openings may influence the beam behavior (either the ultimate capacity or the serviceability
requirements) and stress distribution especially when openings are present in the critical shear
zones and in the load path between the loading plate and the end support. The main factors
affecting the behavior and ultimate capacity of deep beams with web openings are as follows:








span-to-depth ratio (L/d),
cross section properties,
amount and location of main longitudinal reinforcement,
amount, type, and location of web reinforcement,
properties of concrete and reinforcements,
shear-span-to-depth ratio (a/d),
type and position of loading, and
size, shape and location of web openings, etc…
Most current codes [3,10,12] give simplified design methods for deep beams without
special consideration to the effect of web openings and no national codes even provide any
guidance for the design of deep beams with openings. These design methods are based mainly on
tests of deep beams constructed from Normal-Strength Concrete, NSC, with design compressive
strength generally less than 50MPa. There have been extensive experimental and analytical
investigations of simply supported deep beams with and without web openings. Very few tests of
continuous deep beams constructed from NSC with and without web openings have been
reported, while tests on reinforced High Strength-Concrete HSC continuous deep beams with
web openings have been rarely reported.
2
1.2 PROBLEM IDENTIFICATION
The use of reinforced concrete beams with openings is necessary to pass various services, and the
analysis and design of such beams need special consideration. Conventional methods of analysis
and design of solid beams cannot be used for beams with openings. As a result of the nonlinear
character of the strains in openings' segments in shallow- and deep-beams as D-regions, a more
suitable design method is needed. So, the strut-and-tie model has been used in order to obtain the
optimal model of the load path. This can be achieved with the aid of linear elastic finite element
analysis to obtain the stress trajectories.
The strut-and-tie model has been proved to be a useful and consistent method for the
analysis and design of structural concrete including, of course, D-regions. This model is an
extension of the so-called truss analogy, and gives a physical representation of the actual stress
fields resulting from applied loads and support conditions. In this method the flow of forces in a
structural member is approximated by the use of struts to represent the flow of compressive
stresses and ties to represent the flow of tensile stresses.
1.3 RESEARCH SIGNIFICANCE
This thesis aims not to present but to revise the design and detailing of reinforced concrete
(shallow and deep) beams with and without openings, utilizing both of the strut-and-tie model
and a 3-D nonlinear finite element analysis. The concept of strut-and-tie model is introduced and
extended to include several types of reinforced concrete simple and continuous shallow and deep
beams that are subject to top point loads. The effect of the shear span-to-depth ratio, the concrete
strength, opening size and shape, the variation of mechanisms between shallow and deep beams,
and simple versus continuous beams, are considered in design.
In addition, the finite element package (ANSYS-12) [5] is used to carry-out the 3-D
nonlinear finite element analysis. This finite element analysis, on one hand, is used to check the
output results that are obtained using the strut-and-tie model and completes, on the other hand,
the understanding of the behavior of the considered reinforced concrete beams (shallow and
deep). The verification process of both the strut-and-tie results and the finite element model is
also achieved using the experimental data available in literature. The key features (the shear
span-to-depth ratio, span-to-depth ratio, concrete strength, opening size and shape, load type, and
web reinforcement) that affect the design and detailing of normal- and high-strength reinforced
concrete beams utilizing the 3-D nonlinear finite element analysis are presented. Proposed design
procedure for beams with openings and design recommendations are introduced.
1. 4 OBJECTIVES AND SCOPE
The purpose of this study is to investigate the effect of web openings on the behavior of
reinforced concrete shallow and deep beams, simple and continuous considering the effects of
different parameters such as size, shape and location of openings, web reinforcement, shear spanto-depth ratio and concrete strength. This study focuses on the design and detailing of Reinforced
Concrete, RC, shallow and deep beams with and without openings utilizing both:
 the Strut-and-Tie Models STM and
 a 3-D nonlinear finite element analysis using ANSYS-package [5].
3
The verification process of both the strut-and-tie results and the finite element model is also
achieved using the experimental data available in literature.
1. 5 THESIS ARRANGEMENT
Chapter 2 presents a literature review of the previous research work in the field of shallow and
deep beams with and without web openings.
Chapter 3 is devoted entirely to a detailed presentation and development of the strut-and-tie
model approach and its applications, modeling and analysis using verification examples. The
output results of the strut-and-tie model are compared with test data.
Chapter 4 presents a three-dimensional nonlinear finite element analysis of reinforced concrete
beams (shallow and deep), using ANSYS program. The element types and material models of
both reinforcing steel and concrete are presented. Simple and continuous reinforced concrete
beams (shallow and deep), with and without web openings are then modeled and analyzed using
the 3-D nonlinear finite element package. Finally, the output results of the finite element model
are compared with both of test data and some proposed strut-and-tie models.
Chapter 5 presents a design procedure with numerical examples, reinforcement detailing, and
design recommendations for (shallow and deep) beams with openings.
Chapter 6 presents a summary and the main conclusions of this thesis.
4
CHAPTERT
2
LITERATURE REVIEW
CHAPTER 2
LITERATURE REVIEW
2.1 INTRODUCTION
In modern buildings, transverse openings are often provided in beams in order to pass the
mechanical and electrical services, ducts and pipes for air conditioning, water supply and sewage
below the floor ceiling. So we usually end up having regular openings in every beam below the
slab. The effect of these openings on the strength and behavior of beams has to be considered in
the analysis. With the exception of nonlinear Finite Element Method, FEM, there is no detailed
method available to calculate the majority of cases. With this in mind, the design of such beams
with openings needs special consideration.
2.2 PREVIOUS STUDIES ON ORDINARY BEAMS WITH OPENINGS
2.2.1
Classification of Openings
In his extensive experimental study, even circular and rectangular openings are the most common
ones in practice, Prentzas (1968) considered openings of circular, rectangular, diamond, triangular,
trapezoidal and even irregular shapes, as shown in Fig. 2.1. When the size of opening is concerned,
many researchers have used the terms small and large. From a survey of available literature,
however, it has been noted (Mansur and Tan, 1999) that the essence of such classification lies in
the structural response of the beam. When the opening is small enough to maintain the beam-type
behavior or, in other words, if the usual beam theory applies, then the opening may be termed as a
small opening. When beam-type behavior ceases to exist due to the provision of openings, then
the opening may be classified as a large opening thus, beams with small and large openings need
separate treatments in design.
Figure 2.1 Opening shapes considered by Prentzas (1968).
2.2.1.1 Small Openings
According to Somes and Corley [42], an opening may be considered as small when its depth d or
diameter D is less than or equal to 0.25 times the depth of the beam h and its length is less than or
equal to its depth d, Fig. 2.2. In such a case, as shown in Fig. 2.3, the beam action may be assumed
to prevail. Therefore, the analysis and design of a beam with small openings may follow the
similar course of action as that of a solid beam. Small openings are, thus, defined as openings
which are small enough and located in such a way that a STM is able to jump over the openings
without causing additional vertical or horizontal struts in the chords above and below the openings
[39].
5
h
Figure 2.2 Definition of small openings according to openings dimensions.
Figure 2.3 Definition of small openings according to structural response
(beam-type behavior).
The assumption that a strut-and-tie model is able to pass by openings without any
additional struts or ties in the bottom- or top-chord might be interpreted in this case as an
indication of almost linear strain distribution in the cross-section of the opening, Fig. 2.4a [38].
The presence of openings, however, produces discontinuities or disturbances in the normal flow of
stresses, thus leading to stress concentration and early cracking around the opening region. Similar
to any discontinuity, special reinforcement, enclosing the opening close to its periphery, should
therefore be provided in sufficient quantity in order to control crack widths and prevent possible
premature failure of the beam.
N.A
(a)
(b)
- Bernoulli's-hypothesis is approximately valid
even in opening area
- Strut and tie model can pass by the opening
- Bernoulli's-hypothesis is not valid
- Additional strut and tie for top and
bottom chords are necessary
Figure 2.4 Definition of small and large openings (approximate validity of Bernoulli’s
hypothesis of plane strain distribution) [39].
6
Hanson [17], Somes, and Corley [42] explored a series of longitudinally reinforced Tbeams showing a typical joist floor with square and circular small openings in the web and
concluded that a small opening located near the support does not reduce the strength, Fig.
2.5. As the opening is moved away from the support, gradual reduction in strength occurs until
it levels off to a constant value. The analysis revealed that the vertical position of opening has no
significant effect, while an increase in the size of opening leads to an almost linear reduction in
strength. However, it appears to be a size of opening below which no reduction in shear strength
occurs. This size corresponds to about 20% of the beam depth for square openings and 30% for
circular openings. They have also noted that the strength of such a beam may be fully restored
providing stirrups on either side of the opening.
Figure 2.5 Typical shear failure of a beam without shear reinforcement.
Salam [35] carried out a study on perforated beams of rectangular cross-section tested
under two symmetrical point loads. His study aimed at developing a suitable reinforcement
scheme in order to restore the strength to the level of a corresponding solid beam. The
reinforcement schemes and the beam details employed are shown in Fig. 2.6. Salam found that, in
addition to the longitudinal reinforcement above and below the opening and full depth stirrups by
its sides, short stirrups in the members both above and below the opening (as in Beam B6) are
necessary to eliminate the weakness due to the provision of openings. In his study, Salam
(1977) also noted that when sufficient reinforcement (as in Beam B4) is provided to prevent
failure along a diagonal crack passing through the center of opening and traversing the entire
depth (see Fig. 2.5), failure is then precipitated at the minimum section. In such a case, formation
of two independent diagonal cracks in the members above and below the opening splits the beam
into two separate segments, thus leading to the final failure. Figure 2.7 shows the cracking pattern
of Beam B4 that failed in this manner.
7
Figure 2.6 Reinforcement schemes for beams with small openings (Salam, 1977).
Figure 2.7 Shear failure of Beam B4 at the throat section (Salam, 1977).
In beams, shear is always associated with bending moment, except for the section at
inflection point. When a small opening is introduced in a region subjected to predominant
shear and the opening is enclosed by reinforcement, as shown by solid lines in Fig. 2.8, test
data reported by Hanson (1969), Somes and Corley (1974), Salam (1977), and Weng (1998)
indicated that the beam may fail in two distinctly different modes. The first type is typical of
the failure commonly observed in solid beams except that the failure plane passes through the
center of the opening, Fig. 2.8. In the second type, formation of two independent diagonal cracks,
one in each member bridging the two solid beam segments, leads to the failure (Fig. 2.9). Labeled
respectively as beam-type failure and frame-type failure (Mansur, 1998), these modes of failure
require separate treatment.
Figure 2.8 Beam-type shear failure at small openings.
8
Figure 2.9 Frame-type shear failure at small openings.
The Architectural Institute of Japan (AIJ) [6] has adopted a different approach [18] based
on the truss model, but this also lacks adequate experimental substantiation. Kiang-Hwee et al.,
[24] explored the adequacy of the ACI Code approach for shear design of a beam with circular
openings. Seven T-beams with circular web small openings were designed from moderate to high
shear force. They were tested in an inverted position to simulate the conditions that exist in the
negative moment region of a continuous beam, Fig. 2.10. The results of the test indicated that
crack control and preservation of ultimate strength are achievable through providing reinforcement
around the opening. It was found that diagonal bars decrease the high stress in the compression
chord and avoid premature crushing of the concrete. However, the provision of transverse openings
alters the simple beam behavior into a more complex behavior (Kiang-Hwee et al., 2001). The
provision of openings produces discontinuity in the normal flow of stresses and results in stress
concentration and early cracking around the opening. The ultimate strength of the beam may also
be seriously affected. Hence, special reinforcement should be provided around the opening to
maintain the width of cracks and to prevent possible premature failure of the beam. They
concluded that diagonal bars are essential for achieving adequate crack control and the amount of
diagonal bars should be sufficient enough to carry at least 50% of the applied shear with a shear
concentration factor of at least 1.3. Figure 2.10 shows the cracking patterns of the tested beams and
Fig. 2.11 shows the beams reinforcement details.
Figure 2.10 Cracking patterns
of beams [24].
Figure 2.11 Beams and reinforcement details [24].
9
Nine beams were fabricated and tested, as shown in Fig. 2.12, by Mohamed A. Mansur,
Kiang-Hwee Tan, and Weng Wei [30] with circular small openings through the web to simulate
drilling of holes in an existing beam either for the passage of service ducts or for the determination
of in-place concrete strength. The major parameters considered were the size and location of
openings. Such beams were tested to failure.
Figure 2.12 Cracking patterns of beams tested by Mansur et al. (1999).
The test results indicated that a large opening, when created near the concentrated load
region of an existing beam, can seriously impair the safety and serviceability of the structure. Also,
filling an opening by non-shrink grout, as is usually done for openings created by removing of
concrete cores for the determination of in-place concrete strength of an old building, is not
adequate to restore the original strength and stiffness. The risk may, however, be minimized by
limiting the size of opening or drilling the opening without cutting any stirrup. In any case, the
designer must carefully analyze and assess the situation. Unless a larger than usual factor of safety
is incorporated in the original design or suitable measures to strengthen the beam are undertaken,
no opening should be created in an existing beam. Although limited in the number of tests, this
study reveals that the weakness introduced in terms of cracking, deflection, and ultimate strength
by creating an opening in existing beams can be completely eliminated by strengthening the
opening region of the beam using a suitable method, such as the use of externally bonded Fiber
Reinforced Polymer FRP plates.
2.2.1.2 Large Openings
In analogy to a beam with small openings, large openings can be defined as an opening that
requires additional vertical and horizontal struts in the chords above and below the opening [39].
Bernoulli’s hypothesis of plane strain distribution is invalid concerning the whole cross-section
through a large opening as in Fig. 2.4b [39]. An opening may be considered as large when its depth
d or diameter D is greater than 0.25 times the depth of the beam h and its length ℓ is greater than its
depth d because the introduction of such openings reduces the strength of the beam and the beamtype behavior will be as shown in Fig. 2.13, according to Somes and Corley [42].
ℓ>d
d > 0.25h
h
Figure 2.13 Definition of large openings according to structural response.
01
Kiang-Hwee Tan, Mohamed A. Mansur, and Loon-Meng Huang [23] tested fifteen
specimens (each simulating either the negative or positive moment regions of a reinforced concrete
continuous T-beam) with large openings through the web, to failure, Fig. 2.14. The test results
indicated that the presence of web openings leads to a reduction in both the cracking and ultimate
strength as well as the post cracking stiffness. For the same passageway, beams with multiple
openings were found to perform better in terms of strength and serviceability than those with
a single opening. Test results confirmed the vierendeel panel behavior at the opening segment of
the beam. An analytical procedure for both the ultimate strength and service load analysis
incorporating the use of equivalent stiffnesses for the opening segment was found to predict the
test results well.
Besides, the following conclusions can be drawn from the above-mentioned study:
1. Performance of a beam with multiple openings is more desirable in terms of strength and
serviceability. The thickness of the post between adjacent openings should not be less than
one-half the overall beam depth, and the post should be adequately reinforced to avoid
premature failure.
2. It was confirmed that a continuous T-beam containing a large rectangular opening behaves
similarly to a vierendeel panel at opening segment. Under combined bending and shear, the
chord members bend in double curvature with contra-flexure points located approximately at
their mid-span.
3. Total applied shear may be apportioned between the top and bottom chords in accordance with
their flexure stiffnesses, based on either gross or cracked transformed sections. This
distribution applies at both service load and ultimate conditions irrespective of whether the
opening is located within the positive or negative moment.
Figure 2.14 Failure of a beam with multiple rectangular openings.
4.
Analytical method incorporating equivalent stiffnesses for the opening segment was found to
give good predictions of the ultimate strength, failure mode, and service load deflections of the
continuous beams.
Mansur et al., [27] carried out an experimental study to investigate eight reinforced concrete
continuous beams, each containing a large transverse opening. All the beams contained the same
amount and arrangement of longitudinal reinforcement and were rectangular in cross-section. The
major variables were the number of spans and the size of opening and its location along the span.
They pointed out that the existing methods for ultimate strength design can be applied only to
simply supported beams where bending moment and shear force distributions are uniquely defined.
Considering continuous beams, reduction in stiffness due to the presence of openings causes a
00
redistribution of internal forces and moments, the amount of which needs to be properly evaluated
to achieve a satisfactory design. Moreover, the calculation of deflections is influenced by such
redistributions. Figure 2.15 a typical curve, shows four distinctly different stages of behavior.
Figure 2.15 Load versus maximum deflection curves (Mansur et al., 1991).
Furthermore, Mansur et al., (1991) made an attempt to identify a hinge on the basis of
visual observation of the extent of cracking, measured steel strains, and the load-curvature
relationships obtained for the critical sections. Due to the rapid growth of cracks at the formation
of a plastic hinge, visual observation provides an approximate indication of hinge formation.
Generally, yielding of steel provides the attainment of ultimate moment capacity at which the
corresponding load-curvature relationship becomes flat. In fact, openings along the span which are
located at a relatively high-moment region lead to a smaller collapse load. The test results showed
that the depth of the opening influenced the behavior and strength of the beams to some extent
similar to the effect of the opening length. Deeper openings led to both early cracking and early
yielding of reinforcement; hence, the stiffness of the beam decreased and Vierendeel action
became more pronounced. All beams failed in a similar mode; however, the load at collapse
decreased as the opening depth was increased. The results indicated that an increase in the opening
depth from 140mm to 220mm led to a reduction in collapse load from 240kN to 180kN. The
following conclusions can be drawn from their study:
1. The final failure of a continuous beam happens by the formation of a mechanism.
2. The two opening ends are the most vulnerable locations for the development of plastic hinges.
3. The increase of length or depth of the opening leads to early cracking of beam, produces larger
deflections, or gives more pronounced Vierendeel action. In general, by the increase of opening
size, the load at collapse decreases but it does not affect the collapse mode.
4. The location of openings along the span has very little effect on the cracking load; however,
openings located in a relatively high-moment region lead to larger deflections and smaller
collapse loads. The mode of collapse and deflections due to Vierendeel action remain virtually
unaffected by the location of opening.
02
Mansur, [28] examined the influence of introducing a transverse opening on the behavior
and strength of reinforced concrete beams under predominant shear. Siao and Yap's [40] tests
indicated that the beams fail prematurely by sudden formation of a diagonal crack in the
compression chord when no additional reinforcement is given to the members above and below the
opening (chord members). The experimental observations of the effects of introducing an opening
on the overall response of a beam are summarized as follows:
1. Introduction of an opening in the web of a beam causes early diagonal cracking and the load at
first crack decreases with an increase in either the length or depth of the opening. Unless
additional reinforcement is provided to restrict the growth of cracks, the opening corners have
tendency to show wide cracking. Provided that the same amount and scheme of reinforcement is
employed, an increase in the opening size decreases the strength as well as the stiffness of the
beam. However, the eccentricity of the opening has a very little influence on both the strength
and the stiffness of the beam.
2. The chord members above and below the opening behave to some extent similar to the chords
of a Vierendeel panel with contra-flexure points located approximately at the mid-span of the
chords. As it is shown in Fig. 2.16, the final failure occurs by the formation of a mechanism
containing four hinges in the chords, one at each corner of the opening.
Figure 2.16 The formation of a mechanism formation containing four hinges in the chords
(Mansur, [28]).
Based on Fig. 2.16, it can be pointed out that these hinges form in the chord members at a
distance ht/2 from the vertical faces of the opening, where ht and hb are the overall depths of top
and bottom chords, respectively. Hence, large and small openings can be defined as follows:
• An opening is considered as small when the length of the opening, L0, is less than or equal to
hmax , where hmax is the larger of ht and hb.
• An opening is large when L0 > hmax.
In these two definitions, the members above and below the opening are assumed to have adequate
depth to accommodate the reinforcement scheme. In the case of circular openings, the circle should
be replaced by an equivalent square for the determination of the value of hmax.
In an experimental study, Abdalla et al., [1] employed Fibre Reinforced Polymer FRP
sheets to strengthen the opening region. Several design parameters including opening width and
depth, and amount and configuration of the FRP sheets in the vicinity of the opening were
considered. In their experimental study, they tested 10 reinforced concrete beams, five of them
were strengthened with FRP sheets around the opening, four were tested without strengthening,
and the remaining beam was solid without opening considered as a control beam. Abdalla et al.,
[1] examined the influence of this strengthening technique on deflection, strain, cracking, and
ultimate load. Figure 2.17 represents the schematic view of the test setup conducted by Abdalla et
01
al. In general, the openings in such beams give rise to excessive stresses that may be detrimental
unless properly assessed and designed which is considered in the design review of openings in the
web by Amiri et al., [4]. Practical and experimental experiences have shown that inclined and
vertical cracks develop frequently at the corners of the opening at the service load stage. Such
cracks reduce the load carrying capacity of the beam. Fiber Reinforced Polymer sheets are
becoming widely used to strengthen the reinforced concrete structures.
Figure 2.17 Schematic view of the test setup (Abdalla et al., [1] in 2003).
Abdalla et al., [1] also explored the efficiency of employing FRP sheets to prevent the local
cracks around the openings. They drew the following conclusions:
• The provision of an un-strengthened opening in the shear zone of a reinforced concrete beam
reduces its ultimate capacity. An un-strengthened opening with height of 0.6 of the beam
depth can decrease the beam capacity by 75%.
•
The use of FRP sheets for strengthening the area around the openings can retrieve the full
capacity of the beam for relatively small openings.
• The shear failure at the opening chords of strengthened openings occurs due to a combination
of shear cracking of concrete and bond failure of the FRP sheets glued to the concrete.
Allam [2] examined the process of strengthening RC beams with large openings in the shear
zone. In his experimental study, he tested nine reinforced concrete beams in order to explore the
efficiency of external strengthening of such beams when provided with large openings within their
shear zones. These beams were externally strengthened with steel plates or sheets Carbon Fiber
Reinforced Plastics (CFRP) along the opening edges. It was concluded that both types of material
used for strengthening and its configuration scheme significantly affect the efficiency of
strengthening in terms of beam deflection, steel strain, cracking, ultimate capacity and failure
mode of the beam. Allam [2] believed that previous research studies revealed that the external
strengthening of the beams could significantly increase their shear and flexural strength. Moreover,
such strengthening enhances beam stiffness and controls the propagation of cracks. However,
debonding of the externally bonded materials was one of the disadvantages of external
strengthening. In general, Allam's [2] experimental study aimed to:
1. Examine the behavior of reinforced concrete beams provided with openings within the
shear zone,
2. Explore the efficiency of internal strengthening of beams with openings using internal steel
reinforcement around the opening, and
3. Identify the reliability of external strengthening of beams with openings using either steel
plates or CFRP sheets.
01
With increasing the applied load, the cracks propagated towards the load and the support.
Moreover, more cracks were observed at the opposite opening corners and some shear cracks were
also visible at the lower chord of the opening. As the applied load was enhanced, two main
diagonal shear cracks were observed at the upper and lower chords of the opening and more
flexural cracks were formed within the middle part of the beam. The main diagonal crack in the
lower chord started very near to the inner vertical edge of the opening and extended towards the
lower edge of the lower chord approximately along the diagonal line of the lower chord. Finally,
the beam failed in a shear mode at the opening along the two previously formed diagonal cracks at
its chords. Allam's [2] results indicated that the presence of an opening within the shear zone not
only decreased the ultimate load capacity of the beam but also altered the failure mode from a
flexural mode to a shear mode of failure. Due to the presence of a 150mm×450mm opening
located within the shear span, the reduction in the ultimate failure load of the beam was
about 37%.
According to Allam's [2] study, the following conclusions may be drawn:
• The presence of an opening in a reinforced concrete beam within its shear zone reduces its
ultimate strength. Due to the stress concentration, the cracks appeared around the opening
corners and diagonal cracks are formed along its upper and lower chords.
• External strengthening of the beam opening employing steel plates or CFRP sheets is more
effective than internal strengthening of the opening using internal steel reinforcement.
Furthermore, external strengthening of the beams with opening in shear zone could enhance
its shear strength.
• The use of steel plates is more effective than the use of CFRP sheets for external
strengthening of the beam.
• The use of external steel plates to strengthen the internal and external sides of the beam
opening is more effective than external strengthening of external sides only.
• The use of steel bolts for enhancing the bond between the steel plates used for external
strengthening and the concrete has a very little effect on the beam strength. However, it
decreases the beam stiffness due to the holes drilled in the concrete.
• External strengthening of the beam opening using inside U-shape CFRP sheets
perpendicular to the opening edges in addition to outside CFRP sheets parallel to the
opening edges is more efficient than using outside CFRP sheets parallel to the opening
edges only.
• Based on the experimental and theoretical analyses presented in the study, it was
recommended that the engineer should take into consideration the following when
designing external strengthening of the beam with opening: (i) providing enough shear
strength to the chords of the opening; and (ii) extending the strengthening material along
and beyond the opening corners to overcome the stress concentration and to prevent
formation of plastic hinges at corners.
• For internally strengthened beams with openings, the contribution of the concrete to the
total ultimate shear force decreases significantly after concrete cracking.
Amiri and Masoudnia [4] simulated a three-dimensional nonlinear finite element model of
simply supported concrete beams consisting of circular openings with varying diameters.
Moreover, the effects of circular opening size on the behavior of such beams were investigated.
Subsequently, numerous models of simply supported reinforced concrete rectangular beams with
circular opening were loaded monotonically with two concentrated loads. The beams were
simulated to obtain the load-deflection behavior and compared with the solid concrete beam. All
beams had an identical cross section of 100mm×250mm and 2000mm in length with the circular
opening in seven diameters: 150mm, 130mm, 120mm, 110mm, 100mm, 80mm and 60mm. The
05
results showed that the performance of the beams with circular openings with diameter less than
0.48h (h is depth of the beam web) has no effect on the ultimate load capacity of the RC
rectangular beams. On the other hand, introducing the circular opening with diameter more than
0.48h reduces the ultimate load capacity of the RC rectangular beams at least 26%.
2.3 PREVIOUS STUDIES ON SHEAR BEHAVIOR OF SIMPLY SUPPORTED NORMALSTRENGTH CONCRETE DEEP BEAMS WITH AND WITHOUT OPENINGS
Smith, and Vantssiotis, [41] performed a comparison between test data obtained from an
experimental investigation of 52 simply supported deep beams and mathematical models employed
by present building codes (ACI 318-77 and CAN3-A23.3-77) and proposed revisions by the ACIASCE joint Committee 426. This study indicated that considerable increase in load-carrying
capacity occurs with increasing concrete strength and decreasing shear span-to-effective depth
ratio. Although concrete is capable of carrying significantly higher shear in deep beams, upon
comparing design codes it seems that they excessively attribute this excess capacity to the
contribution of web reinforcement. Test results showed that, ultimate shear strengths calculated for
deep beams without web reinforcement were about 40 to 45 percent of the observed ultimate shear
strength.
Farahat, [14] studied the effects of creating an opening in existing RC beams by testing 18
simply supported RC deep beams with web openings. It covered the influence of opening size,
opening location along the span as well as across the height, beam span-to-depth ratio, type and
position of load, column width to span ratio and concrete strength. The analysis is carried-out by
simplifying the material non-linearity by bi-linear stress-strain relationships. The reduction of
stiffness due to concrete cracking is also considered. Test results indicate that openings represent
potential source of weakness in beams. Increasing the opening size, span-to-depth ratio reduces the
initial cracking load, ultimate load decreases, leads to an increase in the deflection values.
Hamdy [16] studied the effects of reinforcement detailing in the form of main steel and
web reinforcements, on the behavior and strength of reinforced concrete deep beams, having
symmetrical openings along their shear span and loaded symmetrically by two concentrated loads
located at their top chord. Their program included 47 deep beams divided into 17 groups. Test
results showed that difference in values of main steel yield strength cross sectional area anchorage
length or distributions of steel into two layers does not affect the cracking load or the cracking
pattern where, cracking depends mainly on concrete grading.
Moussa, et al., [33] tested eleven deep beams, in order to establish the effect of presence of
the openings on behavior of deep beams as well as the nature and magnitude of stress distribution.
The test results showed that increasing the size of the opening leads to an increase in the top
deflections. Moving the opening upwards lead to an increase in the top deflection and a decrease in
the bottom one due to the deviation of arch action. For solid beams and beams with openings not
intercepting the load path, the shear capacity predicted by both the Egyptian and ACI Codes are
quite close.
Maxwell [31] investigated the use of the strut-and-tie model to predict the behavior of
experimentally tested specimens with a large opening near the support. The study revealed that the
specimens performed very well following the predictions of the strut-and-tie model. However,
those tested beams were of normal strength concrete and relatively thin deep beams with large web
openings.
06
2.4 PREVIOUS STUDIES ON SHEAR BEHAVIOR OF SIMPLY SUPPORTED HSC DEEP
BEAMS WITH AND WITHOUT OPENINGS
Hai Tan, et al. [15] studied the effect of effective span and shear span variations of HSC deep
beams by testing nineteen reinforced concrete deep beams with compressive strength in the range
of 41 to 59MPa. All the beams were singly reinforced with main steel percentage of 1.23 and with
nominal percentage of shear reinforcement of 0.48 percent. The beams were tested for seven shear
span-to-depth ratios ranging from 0.27 to 2.70, and four effective span-to-depth ratios ranging
from 2.15 to 5.38. The test results indicated that the effective span-to-depth ratio has minor
influence on the magnitude of the failure load. But for beams with shear span-to-depth ratio greater
than 1.0, the flexural failure mode becomes dominant with increasing the effective span-to-depth
ratio. The test results are compared with predictions based on the current ACI Building Code. The
comparisons reported in the study provided an added assurance to designers that the deep beams
provisions in the ACI Code, though essentially based on concrete strengths of less than 41MPa,
will insure safe designs for higher strength deep beams. The shear span-to-depth ratio has a
significant influence on the ultimate strength but only a marginal influence on the diagonal
cracking strength. For all beams considered, the diagonal cracking strength is between 20 to 35
percent of the ultimate strength.
Ozcebe, et al. [34] studied the evaluation of minimum shear reinforcement requirements
for HSC given in the ACI, Canadian, and Turkish codes. Thirteen beams having the minimum
shear reinforcement (required by ACI 318-83 and the Turkish code) were tested. Concrete strength
varied between 60 and 80MPa. Test results showed that, when all full diagonal crack developed
crack width was approximately 0.7mm in the ACI beams at this stage the cracks width in beams
was approximately 0.1 mm this difference in crack width is very significant and raises serious
questions about adequacy of ACI 318-83.
Keun Oh, et al. [21] studied the effect of shear strength of HSC Deep beams by testing
fifty three reinforced concrete deep beams with compressive strength in the range of 23MPa to
74MPa to determine their diagonal cracking and ultimate shear capacities symmetrically under two
point loading. The effective span-to-depth ratio was varied from 3.0 to 5.0 and shear span-toeffective depth ratio from 0.5 to 2.0. All the beams were singly reinforced with ρν ranges from 0 to
0.0034 and ρh ranges from 0 to 0.0094, respectively. Test results showed that, with their increase of
the load, additional flexural crack were developed in the mid-span, and new flexural cracks were
formed on the shear span between the loading point and support. The flexural cracks formed on the
early stage new diagonal cracks formed in the shear span joined loading point and support. All the
beams without web reinforcement showed abrupt shear failure without any caution regardless of
the shear span-to-depth ratio. It was observed that web reinforcement showed restrained the sudden
shear failure the beams failed due to crushing of the concrete strut.
Yousef, [46] studied the effect of using minimum web reinforcement in high-strength
concrete deep beams by testing 9 simply supported HSC deep beams without web reinforcement.
The main parameters investigated are the minimum vertical web reinforcement ρν and maximum
vertical web reinforcement spacing Sv. The tested beams were considered to be deep beams
according to the definitions of both the ACI 318-95 and NZS 3101-89. The test results showed that
the vertical web reinforcement is more effective than the horizontal web reinforcement comprising
both vertical and horizontal reinforcements is the most efficient in increasing the beam stiffness,
restricting the diagonal crack width development and enhancing the ultimate shear strength.
Yousef, et al. [47] studied the effect of shear behavior of High Strength Fiber Reinforced
Concrete HSFRC deep beams by testing fourteen HSC simply supported deep beams without web
reinforcement under two point symmetric top loading. The primary variables considered in this
study were the characteristic compressive strength, the amount of steel fibers and shear span-todepth ratios. Comparing the behavior of identical beams with and without steel fibers showed that
07
High-Strength Fiber Reinforced Concrete HSFRC deep beams. Increasing the amount of steel
fibers considerably reduces the crack width and increases the shear strength. Depending on the
results of this investigation, an empirical equation (similar to the equation used by the Egyption
code for predicting the shear strength of deep beams without steel fibers) was proposed for
predicting the shear strength of HSFRC deep beams.
Sallam, [36] conducted an experimental and analytical investigation on reinforced HSC
deep beams with openings by testing sixteen reinforced concrete deep beams to investigate the
effect of the opening presence on the behavior of the deep beams as well as concrete compressive
strength. All tested specimen had the same geometry and main longitudinal top and bottom
reinforcement. The specimens were cast of concrete with compressive strengths of 25, 40 or
70Mpa. The location of centerlines of loads and supports were the same for all beams. Test results
showed that, increasing the concrete compressive strength leads to an increase in the cracking and
failure loads. With increasing the opening size, cracking and ultimate loads were than the beam
with relatively small opening. Presence of web opening on the load path between the load zone and
supports led to reduction in the cracking and ultimate strength of deep beams. The increase in shear
span-to-depth ratio showed an increase in cracking load and mid-span deflection.
Mahmoud, [25] conducted an experimental program including testing of five simply
supported deep beams HSC with or without openings loaded at the top, at beam mid-height and at
bottom, and supported at bottom at the Advanced Composite Materials Research Laboratory,
University of Zagazig, Faculty of Engineering at Shoubra [25].
Keun-Hyeok Yang, et al. [22] studied the influence of web openings on reinforced
concrete deep beams. Thirty-two reinforced high-strength concrete deep beams, as shown in Fig.
2.18, with and without openings were tested under two-point top loading. Test variables included
concrete strength, shear span-to-depth ratio, and the width and depth of the opening.
Figure 2.18 Typical specimen details (units are in mm).
2.5 PREVIOUS STUDIES ON NSC AND HSC SOLID CONTINUOUS DEEP BEAMS
Ashour, [8] tested eight reinforced concrete continuous deep beams. The main parameters
considered were the shear span-to-depth ratio, amount and type of web reinforcement, and amount
of main longitudinal reinforcement. Vertical web reinforcement had more influence on shear
capacity than horizontal web reinforcement. Failure is initiated by a major diagonal crack in the
intermediate shear span between the edges of the load and intermediate support plates.
Comparisons between test results and current codes of practice were done. Test results showed
that, all beams exhibited the same type of failure. A major diagonal crack in the intermediate shear
span ran between the edges of the load and the intermediate support plates. The recorded strains on
the concrete surface in the failure zone are much higher than those recorded elsewhere.
08
Farag, [13] studied the behavior of bottom loaded continuous deep beams by testing six
large-scale continuous deep beams up to failure in a test setup that was especially constructed for
this investigation. The variables considered in the experimental program were the shear span-todepth ratio, vertical and horizontal web reinforcement ratios. A formula for predicting the ultimate
capacity of bottom loaded continuous deep beams has been derived based on a shear friction truss
model for the behavior of such beams. The proposed model is checked against the experimental
results and a reasonably good agreement has been obtained. Test results showed that, loading
direction has a major effect on the behavior and the ultimate capacity of continuous deep beams.
Bottom loaded continuous deep beams fail in a fairly ductile manner in which, yielding of both the
longitudinal bottom reinforcement and the web reinforcement takes place before crushing of
concrete in web.
Subedi, [43] described the structural behavior of two equal span continuous reinforced
concrete deep beams with clear shear span-to-depth ratios less than 1.30 and loaded symmetrically
from the top. The main mode of failure of such beams is characterized by the formation of two
symmetric inclined cracks emanating from the edges of the loading plates. The crushing of the
concrete near the edges of the plates completes the failure mechanism. Four beams were tested to
failure and five others tested by pervious researchers are discussed. A simple method of analysis
for the prediction of the ultimate load of such beams is put forward. The comparison of results
between the analysis and test shows good agreement. Test results showed that, the splitting cracks
extended from the edge of loading to the edge of support and the mechanism is completed by
crushing of the concrete at the edges of the support and loads.
2.6 PREVIOUS STUDIES ON THE SHEAR BEHAVIOR OF CONTINUOUS DEEP
BEAMS WITH WEB OPENINGS
Ashour, et al. [7] tested 16 reinforced concrete two-span continuous deep beams with web
openings. All test specimens had the same geometry and main longitudinal top and bottom
reinforcement. The main parameters considered were the size and position of the web openings,
and web reinforcement arrangement. All tested beams failed due to diagonal cracking. The failure
mode depends mainly on the position of web openings. On the other hand, for beams having
openings within interior shear spans the end support reaction were increased compared with that of
their companion solid beams. Significant influence on the capacity of deep beams web openings
within interior shear spans caused more reduction on the beam capacity than those within exterior
shear spans. Beams having small web openings within exterior shear spans had the highest
capacity in each group. Beams having small web openings within exterior shear spans showed very
close load deflection behavior to their companion solid beams.
El-Azab, [11] studied the behavior of reinforced concrete deep beams with web openings
by testing fourteen reinforced concrete 2-spans continuous deep beams with web reinforcement
divided into 2 groups; the first group consists of 10 beams constructed from HSC while the second
group consists of 4 NSC. All the tested deep beams had the same rectangular cross-section of
80mm wide and 400mm total height. All the specimens had the same clear span-to-depth ratio 1.94
and the same shear span measured from the center of the applied load-to-depth ratio 0.97.
Additional special reinforcement was provided at supports and under loads to avoid local failure.
The details of reinforcement of the tested deep beams are shown in Fig. 2.19. The bottom and top
bars of all beams were four deformed bars with diameter 16mm (ρ = 2.87%) placed on two layers
and were chosen to ensure shear failure of the tested beams. Adequate anchorage was provided to
the longitudinal bars.
09
The skin bars (horizontal web reinforcement) covers the minimum requirements of the
studied codes and are identical for all the tested beams. The main studied parameters were concrete
strength, size and location of opening, spacing between the vertical web reinforcement and two
different diameters of stirrups.
Figure 2.19 Details of reinforcement of the tested continuous deep beams.
21
CHAPTERT
3
STRUT-AND-TIE MODELING
OF BEAMS WITH OPENINGS
CHAPTER 3
STRUT-AND-TIE MODELING OF BEAMS WITH OPENINGS
3.1 INTRODUCTION
In a structure, forces tend to follow the shortest possible path to transfer loads. In a beam subjected
to concentrated loads, the shortest paths to transfer loads are the straight lines connecting the points
of loading and the supports. For deep beams, those shortest paths are possible paths, Fig. 3.1a. The
load is directly transferred to the supports through compression struts with reasonable inclination.
For ordinary (slender) beams, however, those shortest paths are not possible paths as shown in
Fig. 3.1b. Since, the compression struts would be very flat. In order to develop a vertical
component that is large enough to equilibrate the applied force, the actual force in the strut will be
too large to cause concrete crushing. Vertical web reinforcement (ties) provides possible paths, as
shown in Fig. 3.1d, since it increases the inclination of the struts.
Figure 3.1 Possible load paths for deep and slender beams.
In the design of concrete structures, it is necessary to distinguish between two regions, namely; the
main regions and the local regions. In the main regions, often denoted as B-regions (where B
stands for beam, Bernoulli, or bending theory), stresses and strains are distributed so regularly that
they can be easily expressed mathematically. That is, in these regions, stresses and strains are
governed by simple equilibrium and compatibility conditions. In contrast, the stresses and strains
in a local region, denoted as D-regions (where D stands for discontinuity, disturbance, or detail),
such as the ends of a beam or a column, the beam-column connections, the deep beams, and the
region adjacent to a concentrated load or a transverse opening (see Fig. 3.2), are so disturbed and
irregular that they are not amenable to mathematical formulation using the basic requirements of
equilibrium, compatibility, and material laws. As a result, the design of a D-region is usually based
on simplified modeling using equilibrium conditions alone; the strain conditions in the composite
(both steel and concrete) are neglected. One such model is based on the assumption that in a
cracked reinforced concrete member, concrete in between the cracks carries direct compression
and steel carries axial tension. The load-carrying mechanism of the member can then be idealized
as that of a truss comprising a series of concrete struts, steel ties, and nodal zones is known as
‘strut-and-tie model’ (MacGregor, 2011). The use of strut-and-tie models are an excellent
analysis tool for many design problems in concrete structures and especially for discontinuity
12
regions (D-regions), which contain a non-linear stress-strain distribution. The subject was
presented by Professor Schlaich et al. (1987) and also contained in the texts by Collins and
Mitchell (1991) and MacGregor (1992). Finally, it was included in the ACI 318M-02, Appendix A.
Figure 3.2 Examples of D-regions (ACI 318M-2011).
St. Venant’s principle indicates that the stress due to axial load and bending approaches a
linear distribution at a distance approximately equal to the maximum cross-sectional dimension of
a member, h, in both directions, away from a discontinuity. Figure 3.3 shows an illustration of St.
Venant’s principle. For this reason discontinuities are assumed to extend a distance h from the
section where the load or change in geometry occurs. Figure 3.2 illustrates examples of
discontinuities with the resulting D-regions shaded.
Figure 3.3 St. Venant’s principle, Brown et al. [9].
11
As per the ACI 318M-11 code, each shear span (av) of the beam in Fig. 3.4a, where av < 2h,
is a D-region. If two D-regions overlap or meet as shown in Fig. 3.4b, they can be considered as a
single D-region for design purposes. The maximum length-to-depth ratio of such a D-region would
be approximately 2. Thus, the smallest angle between the strut and the tie in a D-region is arctan
(1/2) = 26.5 degrees, rounded to 25 degrees. If there is a B-region between the D-regions in a shear
span, as shown in Fig. 3.4c, the strength of the shear span is governed by the strength of the Bregion if the B- and D-regions have similar geometry and reinforcement. This is because the shear
strength of a B-region is less than the shear strength of a comparable D-region.
Figure 3.4 Description of deep and slender beams (ACI 318-2011).
In this chapter, the development of strut-and-tie models of available tested reinforced
ordinary- and deep-beams with openings from literature are suggested based on experimental
results for crack patterns, modes of failure, and an elastic stress analysis using linear finite element
analysis, which produces the stress trajectories (tension and compression stress trajectories).
3.2 OPENINGS IN ORDINARY BEAMS
3.2.1 Modeling
The modeling of openings in B-regions on the basis of the same criteria as for strut-and-tie models
for D-regions will mostly lead to a standard truss model. This section introduces Strut-Tie-Models,
STM, for reinforced concrete beams with openings. The effect of opening location, form and size,
and cross section properties on the load path will be considered. Finally, the ultimate capacity will
be calculated.
3.2.1.1 The Approach for Developing a STM for Beam with Openings
The procedure for developing a Strut-and-Tie Model, STM, for a whole beam with opening can be
described in the following consequence:
 Use the equilibrium conditions for the whole beam to find the external reactions:
Find the “load path, “i.e. the flow of the external loads from their acting positions to the
supports.” The load path principle results in the position and the orientation of the main tension
and compression stresses taking into account the resulting transverse stresses due to load path
deviation.
 Following the load path, the STM can be constructed:
For additional verification, a stress analysis can be performed using linear finite element, which
produces the stress trajectories (tension and compression stress trajectories). It is also possible
to consider the nonlinear behavior and the cracking of concrete in the finite element analysis.
Following the stress trajectories for both tension and compression stresses, the position and
12
orientation of struts and ties can be traced; thus, the STM is constructed. Generally the strut
direction should be within ±15o of the direction of the compressive and tensile stress
trajectories, respectively.
 Check the external and internal equilibrium of the model. In the first, the equilibrium conditions
of the applied loads with the external reactions for the whole structure or the part around the
opening are fulfilled. The latter is satisfied through the fulfillment of equilibrium conditions at
each node. Through the last step the member forces are calculated.
 Diagonal struts are generally oriented parallel to the expected axis of cracking.
 Struts must not cross or overlap each other; otherwise, the overlapping parts of the struts would
be overstressed. The widths of struts are chosen to carry the forces in the struts using the
effective strength of the concrete in the struts.
 Ties are permitted to cross struts or other ties.
 If photographs of test specimens are available, the crack pattern may assist in selecting the best
strut-and-tie model, the location of the struts fall between cracks. Struts should not cross
cracked regions.
 The model with the least number and the shortest ties is likely the best.
 The angle, θ, between the axis of any strut and any tie entering a single node shall not be taken
less than 25 degrees.
3.2.1.2 Case-Study
In the following, four groups of tested reinforced concrete ordinary beams are introduced, linearly
analyzed using the finite element program ANSYS [5], and modeled using a STM.
Group A: Beams tested by Mansur, et al. [29]:
Practical studies were done by Mansur, et al. [29], to analyze ordinary reinforced concrete beams
at service load with large rectangular openings in the web. The details of the tested beams are
shown in Table 3.1.
13
Table 3.1 Details of tested beams (Group A) [29].
L* = 3m
for Beams R1 through R14; L= 4m for beams in Series B and C.
Beam
R1:
D = beam height, 400mm for all beams.
** Service = Experimental load/1.7
X = distance from concentrated load to the left support (shear span)
ℓo = opening width
dt = height of top chord
db = height of bottom chord
do = opening height
Case 1: Development of a STM for Beam R1
The beam R1 in Fig. 3.6a (tested by Mansur, et al. [29]) was linearly analyzed using the finite
element program ANSYS [5], under a concentrated load at a distance of 1.0m from the left support
and considering the own weight of the beam. From the obtained elastic principal stress trajectories
in Fig. 3.6b and stress diagram in Fig. 3.6c for Beam R1, the strut-and-tie model shown in Fig.
3.6d is proposed. From the elastic principal stress trajectories in Fig. 3.6b, it could be noticed that
the opening affects the beams stress trajectories drastically, where zones of tension stresses are
formed around the left-upper corner of the opening (load side) and the corner on the same diagonal
and compression zones are formed around the two other corners. This justifies the formation of Dregions around openings, as expected. Finally, the tension and compression stress trajectories in
Fig. 3.6b were followed to develop the STM, shown in Fig. 3.6d. The compression stress
trajectories are replaced by compression elements (Struts) and the tension stress trajectories are
replaced by tension elements (Ties).
14
(a)
(b)
(c)
σy
σx
(d)
(e)
Figure 3.6 Beam R1 a) Concrete dimensions and location of opening [29],
b) Principal stress trajectories from FEM (ANSYS prog.),
c) Stress diagram from FEM,
d) Proposed strut-and-tie model, and
e) Part elevation, STM of opening region.
15
Case 2: Development of a STM for Beam R5
The tested Beam R5 in Fig. 3.7a (tested by Mansur, et al. [29]) was linearly analyzed using
the finite element program ANSYS [5], under the same loading conditions of Beam R1. From the
obtained elastic principal stress trajectories in Fig. 3.7b for Beam R5, the strut-and-tie model
shown in Fig. 3.7c is proposed. From the elastic principal stress trajectories in Fig. 3.7b, it could be
noticed that the opening affects the beams stress trajectories drastically, where zones of tension
stresses are formed around the left-upper corner of the opening (load side) and the corner on the
same diagonal and compression zones are formed around the two other corners. This justifies the
formation of D-regions around openings, as expected. Finally, the tension and compression stress
trajectories in Fig. 3.7b were followed to develop the STM shown in Fig. 3.7c. The compression
stress trajectories are replaced by compression elements (Struts) and the tension stress trajectories
are replaced by tension elements (Ties). In the typical STM of Fig. 3.7c the inclination of the
support strut or (load strut) is shown to be much steeper than inclination of the interior struts. This
can be proven by the principle of minimum potential energy if models with different strut
inclinations are compared.
Case 3: Development of a STM for Beam R7
The tested Beam R7 in Fig. 3.8a (tested by Mansur, et al. [29]) was linearly analyzed using the
finite element program ANSYS [5], under the same loading conditions for previous beams. From
the obtained elastic principal stress trajectories in Fig. 3.8b for Beam R7, the strut-and-tie model
shown in Fig. 3.8c is proposed. From the elastic principal stress trajectories in Fig. 3.8b, it could be
also noticed that the opening affects the beam′s stress trajectories drastically, where zones of
tension stresses are formed around the left-upper corner of the opening (load side) and the corner
on the same diagonal and compression zones are formed around the two other corners. This
justifies the formation of D-regions around openings, as expected. Finally, the tension and
compression stress trajectories in Fig. 3.8b were followed to develop the STM shown in Fig. 3.8c.
The compression stress trajectories are replaced by compression elements (Struts) and the tension
stress trajectories by tension elements (Ties). In the typical STM of Fig. 3.8c the inclination of the
support strut or (load strut) is shown to be much steeper than inclination of the interior struts. This
can also be proven by the principle of minimum potential energy if models with different strut
inclinations are compared.
16
(a)
(b)
(c)
(d)
Figure 3.7 Beam R5 a) Concrete dimensions and location of opening [29],
b) Principal stress trajectories from FEM,
c) Proposed strut-and-tie model, and
d) Part elevation, STM of opening region.
17
(a)
(b)
(c)
(d)
Figure 3.8 Beam R7 a) Concrete dimensions and location of opening [29],
b) Principal stress trajectories from FEM,
c) Proposed strut-and-tie model, and
d) Part elevation, STM of opening region.
18
Case 4: Development of a STM for Beam R14
The tested Beam R14 in Fig. 3.9a (tested by Mansur, et al. [29]) was linearly analyzed using
the finite element program ANSYS [5], under the same loading conditions for previous beams.
From the obtained elastic principal stress trajectories in Fig. 3.9b for Beam R14, the strut-and-tie
model shown in Fig. 3.9c is proposed. From the elastic principal stress trajectories in Fig. 3.9b, it
could be noticed that the opening affects the beams stress trajectories drastically, where zones of
tension stresses are formed around the left-upper corner of the opening (load side) and the corner
on the same diagonal and compression zones are formed around the two other corners. This
justifies the formation of D-regions around openings, as expected. Finally, the tension and the
compression stress trajectories in Fig. 3.9b were followed to develop the STM shown in Fig. 3.9c.
The compression stress trajectories are replaced by compression elements (Struts) and the tension
stress trajectories by tension elements (Ties). In the typical STM of Fig. 3.9c the inclination of the
support strut or (load strut) is shown to be much steeper than inclination of the interior struts. This
can be proven by the principle of minimum potential energy if models with different strut
inclinations are compared.
Case 5: Development of a STM for Beam B1
The tested continuous Beam B1 in Fig. 3.10a (tested by Mansur, et al. [29]) was linearly
analyzed using the finite element program ANSYS [5], under the same loading conditions for
previous beams. From the obtained elastic principal stress trajectories in Fig. 3.10b for Beam B1,
the strut-and-tie model shown in Fig. 3.10c is proposed. From the elastic principal stress
trajectories in Fig. 3.10b, it could be noticed that the opening affects the beams stress trajectories
beam drastically, where zones of tension stresses are formed around the left-upper corner of the
opening (load side) and the corner on the same diagonal and compression zones are formed around
the two other corners. This justifies the formation of D-regions around openings, as expected.
Finally, the tension and the compression stress trajectories in Fig. 3.10b were followed to develop
the STM shown in Fig. 3.10c. The compression stress trajectories are replaced by compression
elements (Struts) and the tension stress trajectories by tension elements (Ties). In the typical STM
of Fig. 3.10c the inclination of the support strut or (load strut) is shown to be much steeper than
inclination of the interior struts. This can be proven by the principle of minimum potential energy
if models with different strut inclinations are compared.
23
(a)
(b)
(c)
(d)
Figure 3.9 Beam R14 a) Concrete dimensions and location of opening [29],
b) Principal stress trajectories from FEM,
c) Proposed strut-and-tie model, and
d) Part elevation, STM of opening region.
22
(a)
(b)
Tension
Compressio
n
(c)
(d)
Figure 3.10 Beam B1 a) Concrete dimensions and location of opening [29],
b) Principal stress trajectories from FEM,
c) Proposed strut-and-tie model, and
d) Part elevation, STM of opening region.
21
Case 6: Development of a STM for Beam B2
The tested continuous Beam B2 in Fig. 3.10a (tested by Mansur, et al. [29]) was linearly analyzed
using the finite element program ANSYS [5], under the same loading conditions for previous
beams. From the obtained elastic principal stress trajectories in Fig. 3.11b for Beam B2, the strutand-tie model shown in Fig. 3.11c is proposed. From the elastic principal stress trajectories in Fig.
3.11b, it could be noticed that the opening affects the beams stress trajectories drastically, where
zones of tension stresses are formed around the left-upper corner of the opening (load side) and the
corner on the same diagonal and compression zones are formed around the two other corners. This
justifies the formation of D-regions around openings, as expected. Finally, the tension and the
compression stress trajectories in Fig. 3.11b were followed to develop the STM shown in Fig.
3.11c. The compression stress trajectories are replaced by compression elements (Struts) and the
tension stress trajectories by tension elements (Ties).
(a)
(b)
Tension
Compressio
n
(c)
(d)
Figure 3.11 Beam B2
a) Concrete dimensions and location of opening [29],
b) Principal stress trajectories from FEM,
c) Proposed strut-and-tie model, and
d) Part elevation, STM of opening region.
22
Case 6: Development of a STM for Beam B2
(a)
(b)
Tension
Compressio
n
(c)
(d)
Figure 3.12 Beam B3
a) Concrete dimensions and location of opening [29],
b) Principal stress trajectories from FEM,
c) Proposed strut-and-tie model, and
d) Part elevation, STM of opening region.
23
Case 8: Development of a STM for Beam C1
(a)
(b)
(c)
Figure 3.13 Beam C1 a) Concrete dimensions and location of opening [29],
b) Principal stress trajectories from FEM, and
c) Proposed strut-and-tie model.
24
Case 9: Development of a STM for Beam C3
(a)
(b)
(c)
Figure 3.14 Beam C3 a) Concrete dimensions and location of opening [29],
b) Principal stress trajectories from FEM, and
c) Proposed strut-and-tie model.
25
Case 10: Development of a STM for Beam C5
(a)
(b)
(c)
Figure 3.15 Beam C5
a) Concrete dimensions and location of opening,
b) Principal stress trajectories from FEM,
c) strut-and-tie models.
26
Group B: Beams tested by Javed Vaseghi Amiri and Morteza Hosseinalibygie [19]
An experimental study was carried out by Javed Vaseghi Amiri and Morteza Hosseinalibygie [19],
14 ordinary RC beams, 9 of them were made of normal concrete and 5 of them were made of high
strength concrete. In the group of beams made of normal strength concrete one solid beam was
used as a reference to be compared with beams with an opening. The tested beams have been
loaded with two concentrated symmetrical loads. The effect of concrete strength depended on
parameters such as diameter and the position of opening. In order to control the cracks and restrain
their width, it is better to use diagonal shear reinforcement. For increasing the ultimate shear
strength of the beam usage of diagonal reinforcement and stirrups in top and bottom of opening is
recommended. The length of beams was 1600mm which were loaded over support with span
length of 1400mm. The cross section of these beams was 125mm width, 250mm height, 217mm
effective depth and the distance of compression fiber to compression reinforcement 33mm. For
tensile bars, two bars with 14mm diameter and for compression reinforcement two bars with 6mm
diameter were used. Stirrups also had a diameter of 6mm and apart from the location of opening,
are placed in the shear span, with 100mm distance from each other. Symmetrical and concentrated
loads, which were placed with a 400mm distance from each other, were used. The value of shear
span was equal to 500mm and the shear span-to-depth ratio was 2.3. The structural design and the
choice of value and the pattern of shear and flexural reinforcement of these beams were selected in
a way that the test beam reaches the shear and flexural failure simultaneously. Table 3.2 shows the
details of tested beams, which includes the concrete strength, diameter and the distance of opening
from nearest support. In order to make it easy to recognize the properties of each beam,
abbreviation of words have been used. So "N" indicates beams made of normal concrete and "H"
indicates beams with high strength concrete, and the number after "D" shows the diameter of
opening in millimeter and the number after "X" shows the distance from the center of the circular
opening to the nearest support which is also in mm, and also "s" shows the small stirrups on top
and bottom of opening and "d" shows the presence of diagonal reinforcement around the opening
for example the abbreviation of ND80X350-sd means a beam made of normal concrete, the
diameter of opening is 80mm and the distance from its center to the nearest support is 350mm,
and small stirrups are added at both bottom and top of opening, and diagonal reinforcement have
been used around the opening. Figure 3.16 for example shows the properties of ND80X350-sd
beam.
Table 3.2 Details of tested beams [19].
Beams
S
ND80X350
ND80X150-s
ND80X250-s
ND80X350-s
ND80X350-D
ND80X350-sd
ND60X350-s
ND100X350-s
HD80X150-s
HD80X250-s
HD80X350-s
HD60X350-s
HD100X350-s
Diameter of opening, D,
mm
Opening distance from
support, X, mm
Concrete strength, fc',
MPa
--80
80
80
80
80
80
60
100
80
80
80
60
100
--350
150
250
350
350
350
350
350
150
250
350
350
350
31
29.5
29.2
30.2
29.9
30.7
30.1
29.6
30.4
71.2
70.1
68.7
69.9
70.3
27
Figure 3.26 Properties of Beam ND80X350-sd and the type of location of bars.
All beams were tested at 28 days after the casting of concrete, under two symmetric concentrated
loads that were applied gradually up to failure. In each loading stage the values of strains,
deflection at center of the beam, and the induced cracks were recorded.
Table 3.3 Mechanical properties of rebars used in tested beams [19].
Bars
Tension
reinforcement
Compression
reinforcement
Large Stirrups
Small Stirrups
Bar in top and
bottom of
opening
Diagonal
reinforcement
Modulus of
Elasticity ×105
(MPa)
Ultimate
stress
(MPa)
Yield stress
(MPa)
Diameter of
bar
(mm)
2
600
450
14
2
435
250
6
2
2
435
435
250
250
6
4
2
435
250
6
2
435
250
6
Case 1: Development of a STM for Beam S
The tested Beam S in Fig. 3.17a was linearly analyzed using the finite element program ANSYS12 [5], under the loading conditions of two concentrated loads at a distance of 0.5m from the left
and right supports and considering the own weight of the beam. From the obtained elastic principal
stress trajectories in Fig. 3.17b for Beam S, the strut-and-tie model shown in Fig. 3.17c is
proposed. The compression stress trajectories are replaced by compression elements (Struts) and
the tension stress trajectories are replaced by tension elements (Ties).
28
(a)
(b)
(c)
Figure 3.27 Beam S
a) Concrete dimensions [19],
b) Principal stress trajectories from FEM, and
c) Proposed strut-and-tie model.
33
Case 2: Development of a STM for Beam ND80X350
The tested Beam ND80X350 in Fig. 3.18a was linearly analyzed using the finite element program
ANSYS [5]. From the obtained elastic principal stress trajectories in Fig. 3.18b of the finite
element analysis for Beam ND80X350, the strut-and-tie model shown in Fig. 3.18c is proposed.
From the elastic principal stress trajectories in Fig. 3.18b, it could be noticed that the opening
affects the beams stress trajectories drastically, where zones of tension stresses are formed around
the upper corner of the circular opening from load side and the corner on the same diagonal and
compression zones are formed around the two other corners. The compression stress trajectories
are replaced by compression elements (Struts) and the tension stress trajectories by tension
elements (Ties).
(a)
ND80X350
(b)
(c)
Figure 3.28 Beam ND80X350
a) Concrete dimensions and location of opening [19],
b) Principal stress trajectories from FEM, and
c) Proposed strut-and-tie model.
32
Case 3: Development of a STM for Beam ND80X150-s
(a)
ND80X150-s
(b)
(c)
Figure 3.29 Beam ND80X150-s
a) Concrete dimensions and location of opening [19],
b) Principal stress trajectories from FEM, and
c) Proposed strut-and-tie model.
31
Case 4: Development of a STM for Beam ND80X250-s
(a)
ND80X250-s
(b)
(c)
Figure 3.20 Beam ND80X250-s a) Concrete dimensions and location of opening [19],
b) Principal stress trajectories from FEM, and
c) Proposed strut-and-tie model.
32
Case 5: Development of a STM for Beam ND100X350-s
(a)
ND100X350-s
(b)
(c)
Figure 3.21 Beam ND100X350-s a) Concrete dimensions and location of opening [19],
b) Principal stress trajectories from FEM, and
c) Proposed strut-and-tie model.
33
Group C: Beams tested by Abdalla et al. [1]
Practical studies were done by Abdalla et al. [1], to analyze two RC beams. The first beam was a
RC solid beam without openings and the ultimate load obtained from the experimental test was
83kN. The second beam was a RC beam with a rectangular opening (opening height = 100mm and
opening width = 300mm). The ultimate load obtained from the experimental test for the second
beam was 41kN. The beams full-size was 100mm×2050mm×250mm. The span between the two
supports was 2.0m, Fig. 3.22a.
Case 1: Development of a STM for a solid beam
(a)
(b)
(c)
Figure 3.22 Solid Beam
a) Concrete dimensions [1],
b) Principal stress trajectories from FEM, and
c) Proposed strut-and-tie model.
34
Case 2: Development of a STM for a beam with a rectangular opening (100×300mm)
(a)
(b)
(c)
Figure 3.23 Beam with opening
a) Concrete dimensions and location of opening [1],
b) Principal stress trajectories from FEM, and
c) Proposed strut-and-tie model.
35
Group D: Beams tested by Kiang-Hwee Tan, et al. [23]
Experimental investigations were done by Kiang-Hwee Tan, et al. [23] to study the effect of
openings (in reinforced concrete beams) on the working and ultimate loads. In these tests the effect
of a rectangular openings size and location were investigated. It was noticed that the size and
location of the opening have a big impact on both the ultimate capacity and the behavior of such
beams. Table 3.4 shows the details of the tested beams.
Table 3.4 Details of the tested beams [23].
ho,
mm
fc′,
MPa
Ec,
GPa
**Service
load, KN
Cracking
load, KN
IT1
400 200
IT4 1000 200
IT8* 400 200
36.2
36.8
33.6
28.5
28.7
27.4
185.1
90.6
150.6
30
20
35
Beam
ℓo ,
mm
Maximum
crack width,
mm
0.31
0.88
0.54
Maximum
deflection,
mm
5.66
7.42
5.52
*IT8 contains two openings.
**Service load = Experimental collapse load/1.7
Case 1: Development of a STM for Beam IT1
The tested Beam IT1 in Fig. 3.24a was linearly analyzed using the finite element program ANSYS
[5]. From the elastic principal stress trajectories in Fig. 3.24b, which obtained from the finite
element analysis of the beam, it could be noticed that the stress trajectories are not uniformly
distributed, as a result of the opening zones of tension stresses formed around the upper corner of
the opening from the load side and the lower corner on the same diagonal, while zones of
compression formed around the two other corners. The tension and the compression stress
trajectories were followed in the modeled beam, Fig. 3.24b, and hence the STM, as shown in Fig.
3.24c, is proposed.
36
(a)
(b)
(c)
Figure 3.24 Beam IT1
a) Concrete dimensions and location of opening [23],
b) Principal stress trajectories from FEM, and
c) Proposed strut-and-tie model.
37
Case 2: Development of a STM for a Beam IT4
(a)
(b)
(c)
Figure 3.25 Beam IT4
a) Concrete dimensions and location of opening [23],
b) Principal stress trajectories from FEM, and
c) Proposed strut-and-tie model.
38
Case 3: Development of a STM for a Beam IT8
(a)
(b)
(c)
(d)
Figure 3.26 Beam IT8
a) Concrete dimensions and location of opening [23],
b) Failure of a beam,
c) Principal stress trajectories from FEM, and
d) Proposed strut-and-tie model.
43
3.2.2 Strength Limits of Strut-and-tie Model′s Components
Utilizing a strut-and-tie model, the applied loads on the beam are transmitted to the supports by
means of a system of tension ties and compressive struts, provided by the steel reinforcement and
concrete, respectively, and they are interconnected at the nodes. An illustration of the different
components using a deep beam example is shown in Fig. 3.27.
Figure 3.27 Description of the different components of a strut-and-tie model
for a deep beam.
The strut-and-tie model shown in Fig. 3.27 can fail in one of three ways:



The tension tie could yield, or anchorage failure of the ties.
One of the struts could crush when the stress in the strut exceeds the effective
compressive strength of concrete.
A nodal zone could fail by being stressed greater than the effective compressive
strength of concrete.
For the proposed strut-and-tie model, the strengths of ties, concrete struts, and nodal zones are
discussed in the following.
Reinforced Ties
Ties are STM members that are subjected to tensile forces. In this study, the contribution of tensile
strength of a concrete tie is ignored and normally tie forces are carried by reinforcement.
The tie cross-section is constant along its length and is obtained from the tie force and the yield
stress of the steel. The nominal strength of a tie
shall be taken as
(3.1)
where
is the cross section of area of steel and
is the yield stress of steel.
The width of the tie is to be determined to satisfy safety for compressive stresses at nodes.
Depending on the distribution of the tie reinforcement, the effective tie width
may vary
between the following values but with an upper limit given afterwards.
15

In case of using one row of bars without sufficient development length beyond the nodal
zones (Fig. 3.28a):
(3.2a)

In case of using one row of bars and providing sufficient development length beyond
the nodal zones for a distance not less than , where is the concrete cover (Fig. 3.28b):
(3.2b)
where
is the bar diameter.
 In case of using more than one row of bars and providing sufficient development length
beyond the nodal zones for a distance not less than , where is the concrete cover (Fig.
3.28c):
(
)
(3.2c)
where is number of bars and the clear distance between bars.
In the three cases, in Fig. 3.28 the development length according to the Egyptian code [10] and
ACI code [3] is equal to Lbd and Lanc, respectively. Where the development length Lanc begin at
intersect the center of tie with extended nodal zone.
)
)
(
)
Figure 3.28 The width of the tie
)
used to determine the dimensions of the node.
The upper limit is established as the width corresponding to the width in a hydrostatic nodal zone,
calculated as
(3.3)
⁄(
)
where
is the applicable effective compressive strength of a nodal zone and is computed from
[3,10] as
(3.4)
or
15
The
stands for a cylinder concrete compressive strength and
compressive strength,
is the effectiveness factor for nodal zones, and
beam.
for a cube concrete
is the breadth of the
Concrete Struts
The shape of a strut is highly dependent upon the force path from which the strut arises and the
details of any tension reinforcement connected to the tie. As discussed by Schlaich and Schäfer
[37], there are three major geometric shape classes for struts: prismatic, bottle-shaped, and
compression fan, as shown in Fig. 3.29.
Prismatic struts are the most basic type of struts, and they are typically used to model the
compressive stress block of a beam element as shown in Fig. 3.29a.
Bottle-shaped struts are formed when the geometric conditions at the end of the struts are
well defined, but the rest of the strut is not confined to a specific portion of the structural element.
The geometric conditions at the ends of bottle-shaped struts are typically determined by the details
of bearing pads and/or the reinforcement details of any adjoined steel. The best way to visualize a
bottle-shaped strut is to imagine forces dispersing as they move away from the ends of the strut as
shown in Fig. 3.29b. The bulging stress trajectories cause transverse tensile stresses to form in the
strut which can lead to longitudinal cracking of the strut. Appropriate crack control reinforcement
should always be placed across bottle-shaped struts to avoid premature failure. For this reason,
most design specifications require minimum amounts of crack control reinforcement in regions
designed with STMs.
The last major type of struts is the compression fan, which is formed when stresses flow
from a large area to a much smaller area. Compression fans are assumed to have negligible
curvature and, therefore, they do not develop transverse tensile stresses. The simplest example of a
compression fan is a strut that carries a uniformly distributed load to a support reaction in a deep
beam as shown in Fig. 3.29c.
Figure 3.29 Geometric shapes of struts.
15
The strength of the concrete in compression stress fields depends to a great extent on the
multi- axial state of stress and on the disturbances from cracks and reinforcement. The effective
compressive strength of the concrete in a strut
may be obtained from [3, 10]:
or 0.67
(3.5)
where
is the effectiveness factor for concrete struts, takes into account the stress conditions,
strut geometry and the angle of cracking surrounding the strut. The value of
according to the
ACI 318M-11 code [3] is adopted in this investigation, Table 3.5.
The nominal compressive strength of a strut without longitudinal reinforcement
taken the smaller value of:
shall be
(3.6)
at the two ends of the strut, where
the smaller of:
is the cross-sectional area at one end of the strut, and

The effective compressive strength of the concrete in the strut.

The effective compressive strength of the concrete in the nodal zone.
is
The design of struts shall be based on
(3.7)
In another form
(
)
(3.8)
where
is the largest factored force acting in a strut and obtained from the applicable load
combinations and the factor is 0.75 for ties, struts, and nodes, ACI-318M-11 [3].
Table 3.5 ACI 318M-11Code values of coefficient
for struts.
Strut condition
A strut with constant cross-section along its length (for example a strut equivalent to
the rectangular stress block in a compression zone in a beam).
For struts located such that the width of the midsection of the strut is larger than the
width at the nodes (bottle-shaped struts):
a) With reinforcement normal to the center-line of the strut to resist the
transversal tensile force.
b) Without reinforcement normal to the center-line of the strut
1.0
0.75
0.60λ
For struts in tension members, or the tension flanges of members, for example, to
compression struts in a strut-and-tie model used to design the longitudinal and
transverse reinforcement of the tension flanges of beams, box girders, and walls.
0.40
For all other cases applies to strut applications not included in above cases (struts in a
beam web compression field in the web of a beam where parallel diagonal cracks are
likely to divide the web into inclined struts, and struts are likely to be crossed by
cracks at an angle to the struts, Figs. 3.30a and 3.30b respectively.
0.60λ
λ is a modification factor to account for the use of lightweight concrete. λ = 0.85 for sandlightweight concrete and 0.75 for all-lightweight concrete and λ = 1.0 for normal weight concrete.
15
(a) Struts in a beam web with inclined cracks parallel to struts.
(b) Struts crossed by skew cracks.
Figure 3.30 Types of struts.
Nodal Zones
The compressive strength of concrete of the nodal zone depends on many factors including the
tensile straining from intersecting ties, confinement provided by compressive reactions and
confinement provided by transverse reinforcement.
To distinguish between the different straining and confinement conditions for nodal zones, it is
helpful to identify these zones as follows, Fig. 3.31:




C-C-C nodal zone bounded by compression struts only (hydrostatic node);
C-C-T nodal zone bounded by compression struts and one tension tie;
C-T-T nodal zone bounded by a compression strut and two tension ties; and
T-T-T nodal zone bounded by tension ties only.
Figure 3.31 Classification of nodes.
11
As discussed by Brown et al. in 2006 [9], nodes can be detailed to be either hydrostatic or
non-hydrostatic in theory. For a hydrostatic node, the stress acting on each face of the node is
equivalent and perpendicular to the surface of the node. Because stresses are perpendicular to the
faces of hydrostatic nodes, there are no shear stresses acting on the face of a hydrostatic node.
However, achieving hydrostatic nodes for most STM geometric configurations is nearly
impossible and usually impractical. For this reason, most STMs utilize non-hydrostatic nodes. For
non-hydrostatic nodes, Schlaich et al. in 1987 [38] suggested that the ratio of maximum stress on a
face of a node to the minimum stress on a face of a node should be less than 2. The states of stress
in both hydrostatic and non-hydrostatic nodes are shown in Fig. 3.32.
Figure 3.32 States of stress in hydrostatic and non-hydrostatic nodes, Brown et al. [9].
In this thesis, the effective compressive strength of the concrete in a nodal zone
from [3, 10]:
can be obtained
or
(3.9)
where
is the effectiveness factor of a nodal zone and it is assumed as given in Table 3.6
according to the ACI 318M-11 code [3]. For safety purposes, for C-C-C nodes, the value 1.10
suggested by Schliach et al. in 1991 [37] is reduced to 1.0. Also, in order to appropriately consider
the effect of the tensile strain on the concrete compressive strength, the value of 0.80 taken by
Schliach et al. in 1991 [37] for other nodes is replaced by 0.80 for C-C-T nodes, 0.60 for C-T-T
nodes (two or more ties), and 0.40 for T-T-T nodes.
The nominal compressive strength of a nodal zone,
, shall be
(3.10)
where
is the effective compressive strength of the concrete in the nodal zone and
smaller of:
15
is the


The area of the face of the nodal zone on which
acts, taken perpendicular to the line
of action of the strut force .
The area of a section through the nodal zone, taken perpendicular to the line of action of
the resultant force on the section.
Table 3.6 ACI 318M-11 Code values of coefficient
for nodes.
Nodal zone
Compression-Compression-Compression, C-C-C
1.00
Compression-Compression-Tension, C-C-T
0.80
Compression-Tension-Tension, C-T-T*
0.60
Tension-Tension-Tension, T-T-T
0.40
*In nodal zones anchoring two or more ties with the presence of one strut
In smeared nodes, where the deviation of forces may be smeared or spread over some length, the
check of stress is often not critical and it is only required to check the anchorage of the reinforcing
bars. On the other hand, singular or concentrated nodes have to be carefully checked.
3.2.3 Verification Examples
To illustrate how to model and analyze reinforced ordinary beams with/without openings using a
strut-and-tie method, three groups of simple ordinary beams with/without openings are chosen and
examined. These groups are: Group B tested by Javed Vaseghi Amiri and Morteza
Hosseinalibygie [19], Group C tested by Abdalla, et al. [1], and Group D tested by Kiang-Hwee
Tan, et al [23]. The details of Group B are shown in Tables 3.2 and 3.3 and that for Group D are
shown in Table 3.4.
Capacity of Beams:
Here in, the capacity of each beam with/without openings is determined using the strut-and-tie
model. The capacities were determined using an “iterative procedure”. These predicted capacities
were then compared to the actual failure load.
Group B [19]: Normal- and High-Strength Concrete Simple Ordinary Beams

Case 1: Beam S
Figure 3.33a shows a simple reinforced concrete ordinary beam (Beam S), with two top point
loads along with the proposed strut-and-tie model. The model has four compression struts (S1 to
S4), three tension ties (T1 to T3), and four nodes (N1 to N4). The top point loads are applied at
nodes N4. The tension ties T1 and T3 represent the main longitudinal reinforcement and the vertical
reinforcement is represented by tie T2.
15
(a) Details of the strut-and-tie model.
mm
(b) Visualization of strut widths.
α
(c) Nodal zone N1
(d) Nodal zone N4
Figure 3.33 Solid Beam S.
Numerical scheme for Beam S:
1. Input data:
Beam size: h = 250mm, d = 217mm, b = 125mm, and b1 = b2 = 100mm.
Shear span-to-depth ratio: a = 500mm, and (a/d) = (500/217) = 2.3
Materials: = 30.40MPa (NSC),
= 450MPa,
= 250MPa, and As = 214 (307.88mm2), and
Asv =141.37mm2 per tie (2-legs).
where
is the cylinder compressive strength of concrete,
is the yield stress for longitudinal
steel,
is the yield stress for vertical stirrups, As is the area of main reinforcement, and Asv is the
area of vertical stirrups (2-legs per tie).
15
2. The internal lever arm, Ld :
The term a1 (height of node N1) can be computed from
(
)
(3.11)
where is the number of steel layers,
is the longitudinal steel diameter,
concrete cover, and the clear distance between bars.
The widths of struts and
from equation (3.12).
are assumed to be the same (
is the clear
) and can be computed
(3.12)
Then,
and thus Ld = h – 0.5 (a1 + a2)
= 250 – 0.5 (66 + 42.89) =195.60mm
(3.13)
3. Width of struts:
The widths of struts S1 to S4 can be calculated (based on bearing plates dimensions and the widths
of the tie T1 and the top strut S3) as follows:
(3.14)
(3.15)
4. STM forces:
Assuming that the reinforcing bars (tension ties T2 and T3) will reach their yield strength and from
equilibrium of the model joints, the following relations can be written:
kN
(3.16)
kN
(3.17)
Try
kN
is the nominal strength capacity of the tie when reaching its yield strength.
From equilibrium,
kN
(3.18)
kN
(3.19)
kN which is less than
kN
kN (Okay).
(3.20)
kN
kN
Finally,
and
kN
kN
15
(3.21)
(3.22)
(3.23)
(3.24)
5. Checking of stress limits:
a. Concrete Struts:
It is noted that the strut width changes linearly between the two nodes. However, in this thesis the
cross-sectional area of a strut is assumed constant along its length (
).
Knowing that = 30.40MPa, the term (
) will be:
MPa
for Strut Sj (j = 1, 2, 3 and 4) (3.25)
MPa
for Node Ni (i = 1 and 2)
(3.26)
MPa
for Node N3
(3.27)
MPa
for Node N4
(3.28)
Upon substituting in
obtained:
and comparing the results with
kN >
kN >
kN >
kN >
kN
kN
kN
kN
, the following is
(3.29)
(3.30)
(3.31)
(3.32)
b. Nodes:
The capacity of a node is calculated by finding the product of the limiting compressive stress in the
node region and the cross sectional area of the member at the node interface.
Node N1:
Knowing that:
MPa
(at Node N1)
(
)
or
(3.33)
(
)
or
(3.34)
(
)
or
(3.35)
Node N4:
Knowing that:
(
(
(
(
MPa
)
)
)
)
(at Node N4)
or
or
or
(3.36)
(3.37)
(3.38)
or
(3.39)
It is noted that nodes N2 and N3 are smeared nodes; and therefore, their check is not necessary.
Based on the recommendations of past researchers [45], it was determined that it is unnecessary to
apply the bonding stresses from a developed bar to the back face of a CCT node. Therefore, only
directly applied stresses, such as those due to bearing of a plate or to an external indeterminacy, are
applied to the back face of CCT nodes and checked with the 0.80 effectiveness factor.
Therefore, the nominal shear force is
kN and
kN
To sum up:
⁄
1.
2. Failure occurred due to the yielding of the tension ties.
56
Case 2: Beam ND80X350
Figure 3.34a shows a simple reinforced concrete ordinary beam (Beam ND80X350), with two top
point loads along with the proposed strut-and-tie model. The model has six compression struts (S1
to S6) four tension ties (T1 to T4), and six nodes (N1 to N6). The loads are applied at nodes N4. The
tension ties T1 and T3 represent the main longitudinal reinforcement and the vertical reinforcement
is represented by tie T2.
(a) Details of the strut-and-tie model.
(b) Visualization of strut widths.
Figure 3.34 Beam ND80X350.
Numerical scheme for Beam ND80X350:
1. Input data:
Beam size: h = 250mm, d = 217mm, b = 125mm, and b1 = b2 = 100mm.
Diameter of opening = 80mm.
Shear span-to-depth ratio: a = 500mm, and (a/d) = (500/217) = 2.3
Materials:
= 28.93MPa (NSC),
= 450MPa,
= 250MPa, and As = 214 (307.88 mm2),
= 26 and Asv =141.37mm2 per tie (2-legs).
where
is the cylinder compressive strength for concrete,
is the yield stress for longitudinal
steel,
is yield stress for vertical stirrups, As is the area of main steel,
is the area of
secondary steel, and Asv is the area of vertical stirrups (2-legs) per tie.
2. The internal lever arm, Ld:
The term a1 (height of node N1) can be computed from
(
55
)
(3.11)
The widths of struts and
from equation (3.12).
are assumed to be the same (
) and can be computed
(3.12)
Then,
and thus Ld = h – 0.5 (a1 + a2) = 194.45mm
3. Width of struts:
The widths of struts S1, S2 and S3 can be calculated (based on bearing plate's dimensions and the
widths of the tie T1 and the top strut S3) as follows:
The other strut widths S4, S5, and S6 can be determined by developing a realistic geometry of the
struts as they extend from the Nodes.
Try:
4. STM forces:
Assuming that the reinforcing bars (tension ties T2 and T3) will reach their yield strength and from
equilibrium of the model joints, the following relations can be written:
kN
kN
Try
kN
is the nominal strength capacity of the tie when reaching its yield strength.
From equilibrium:
Node 2:
kN
kN
Node 1:
kN
kN
Node 3:
……… (1)
55
……… (2)
Solving Eqs. (1) and (2), we get:
Node 4:
Node 6:
Finally,
kN and
kN
5. Checking of stress limits:
a. Concrete Struts:
Knowing that
= 28.93MPa, the term (
Upon substituting in
obtained:
) will be:
MPa for Strut Sj (j = 1, 2, 3,4,5 and 6)
MPa
for Node Ni (i = 1,2,3,5 and 6)
MPa
for Node N4
and comparing the results with
kN >
kN >
kN >
kN >
kN >
kN >
, the following is
kN
kN
kN
kN
kN
kN
b. Nodes:
The capacity of a node is calculated by finding the product of the limiting compressive stress in the
node region and the cross sectional area of the member at the node interface.
55
Node N1:
Knowing that:
(
)
(
MPa
(at Node N4)
or
)
or
Node N4:
Knowing that:
(
(
(
(
(
(at Node N1)
or
)
(
MPa
)
)
)
)
)
or
or
or
or
or
It is noted that nodes N2, N3, N5, and N6 are smeared nodes; and therefore, their check is not
necessary. Based on the recommendations of past researchers [45], it was determined that it is
unnecessary to apply the bonding stresses from a developed bar to the back face of a CCT node.
Therefore, only directly applied stresses, such as those due to bearing of a plate or to an external
indeterminacy, are applied to the back face of CCT nodes and checked with the 0.80 effectiveness
factor.
Therefore, the nominal shear force is
kN and
kN
To sum up:
⁄
1.
2. Failure occurred due to the yielding of the tension ties.
Upon following the previous numerical scheme, the failure load and failure mode of all other
beams have been obtained as shown in Table 3.7. A comparison between the results of the
proposed strut-and-tie model and the test data is shown in the table. The strut-and-tie approach
gives a mean value of 0.651 of the experimental ultimate load.
Table 3.7 The STM results compared with test results.
No.
Beam
PEXP, kN
1
2
3
4
5
6
7
S
ND8X35
ND8X15-s
ND8X25-s
ND10X35-s
HD8X15-s
HD10X35-s
120
100
100
110
105
115
115
Failure Mode,
Exp.
Shear
Shear
Shear
Shear
Shear
Shear
Shear
Mean
55
PSTM, kN
70.68
70.70
70.70
70.83
70.53
70.80
70.55
Failure Mode,
PSTM / PEXP
STM
Tension
0.600
Tension
0.707
Tension
0.707
Tension
0.644
Tension
0.672
Tension
0.616
Tension
0.613
0.651
Group C [1]: Normal Strength Concrete NSC Simple Ordinary Beams
Case 1: Solid Beam
Figure 3.35a shows a simple reinforced concrete ordinary solid beam, with two top point loads
along with the proposed strut-and-tie model. The model has six compression struts S1 to S6, five
tension ties T1 to T5, and six nodes N1 to N6. Two external top point loads are applied at nodes N6.
The tension ties T1, T3, and T4 represent the main longitudinal reinforcement and the vertical
reinforcement is represented by ties T2 and T5.
(a) Details of the strut-and-tie model.
(b) Visualization of strut widths.
Figure 3.35 Solid beam.
Numerical scheme for solid Beam:
1. Input data:
Beam size: h = 250mm, d = 210mm, b = 100mm, and b1 = b2 = 100mm.
Shear span-to-depth ratio: a = 670mm, and (a/d) = (670/210) = 3.2
Materials:
= 49MPa,
= 400MPa,
= 240MPa, and As = 410 (314.16 mm2), and Asv =
50.27 × 2stirrups × 2-legs = 201.08mm2 per tie.
2. The internal lever arm, Ld :
The term a1 (height of node N1) can be computed from:
(
(
From
Then,
and thus Ld = h – 0.5 (a1 + a2) = 194.92mm
51
)
)
3. Width of struts:
The widths of struts S1, S2, S4, S5, and S6 can be calculated (based on bearing plate's dimensions
and the widths tie T1 and the top strut S3) as follows:
The other strut width S3 was determined by developing a realistic geometry of the struts as they
extend from the Nodes. Try: w3 = 100mm.
4. STM forces:
Assuming that the reinforcing bars (tension ties T2, T5, and T3) will reach their yield strength and
from equilibrium of the model joints, the following relations can be written:
kN
kN
kN
From equilibrium:
Node 2:
kN
kN
Node 1:
kN
kN
Node 3:
kN
kN
Node 4:
kN
Node 5:
kN
kN which is greater than
kN,
kN (Not Okay). Try
recalculate the forces :
Try from equilibrium:
Node 2:
kN
55
and
kN
Node 1:
kN
kN
Node 3:
kN
kN
Node 4:
kN
Node 5:
kN
kN (Okay)
Finally,
kN and
kN
5. Checking of stress limits:
a. Concrete Struts:
Knowing that = 49MPa, the term (
Upon substituting in
obtained:
) will be:
MPa
for Strut Sj (j = 1, 2, 3,4,5 and 6)
MPa
for Node Ni (i = 1, 2 and 4)
MPa
for Node Ni (i = 3 and 5)
MPa
for Node N6
and comparing the results with
kN >
kN >
kN >
kN >
kN >
kN >
, the following is
kN
kN
kN
kN
kN
kN
b. Nodes:
The capacity of a node is calculated by finding the product of the limiting compressive stress in the
node region and the cross sectional area of the member at the node interface.
55
Node N1:
Knowing that:
(
)
(
(at Node N1)
MPa
(at Node N6)
or
)
or
)
or
(
MPa
Node N6:
Knowing that:
(
(
(
)
)
)
or
or
or
(
)
or
therefore the nominal shear force is
kN and
To sum up:
1.
⁄
2. Failure occurred due to the yielding of the tension tie T3.
Case 2: Beam with rectangular opening (100×300mm) [1]
Figure 3.36a shows a simple reinforced concrete ordinary beam with rectangular openings
(100×300mm), with two top point loads along with the proposed strut-and-tie model. The model
has thirty-six compression struts S1 to S36, thirty-seven tension ties T1 to T37, and fifty-four nodes
N1 to N54. The tension ties T1, T14 to T21, and T37 represent the main longitudinal reinforcement
and the vertical reinforcement is represented by the other ties.
55
(a) Details of the strut-and-tie model.
(b) Strut labels for strut-and-tie model.
(c) Tie labels for strut-and-tie model.
Figure 3.36 Beam with rectangular openings (100×300mm).
55
Numerical scheme for beam with rectangular openings (100×300mm):
Input data:
Beam size: h = 250mm, d = 210mm, b = 100mm, and b1 = b2 = 100mm.
Shear span-to-depth ratio: a = 670 mm, and (a/d) = (670/210) = 3.2
Materials:
= 52MPa,
= 400MPa,
= 240MPa, and As = 410 (314.16 mm2), and Asv =
50.27 x 2stirrup x 2-legs = 201.08 mm2 per tie.
The internal lever arm, Ld :
(
(
)
)
and thus Ld = h – 0.5 (a1 + a2) = 195.79mm
Width of struts:
The widths of struts S1 to S35 are calculated based on bearing plates dimentions and the widths of
the ties and are as shown in Table 3.9.
STM forces:
The forces in all members are calculated form static and are as shown in Table 3.8. The summary
of concrete struts calculations are shown in Table 3.9.
Table 3.8 Calculated member forces for the strut-and-tie model.
T
Model Force,
Model Force, T or Model Force, T or Model Force, T or
or
Label
kN
Label
kN
C
Label
kN
C
Label
kN
C
C
1
25.97 C
20
15.72
C
3
39.70
T
22
10.68
T
2
17.35 C
21
7.74
C
4
21.42
T
23
8.95
T
3
33.30 C
22
42.91
C
5
6.84
T
24
27.49
T
4
27.49 C
23
33.09
C
6
24.54
T
25
17.65
T
5
24.30 C
24
23.27
C
7
37.85
T
26
7.80
T
6
10.74 C
25
10.99
C
8
18.41
T
27
2.02
T
7
32.42 C
26
1.17
C
9
3.84
T
88
11.87
T
8
13.67 C
27
2.29
C
10
31.15
T
89
25.00
T
9
15.10 C
28
12.12
C
11
22.61
T
03
13.13
T
10
6.99
C
29
24.39
C
12
14.07
T
03
28.30
T
11
22.24 C
30
34.22
C
13
3.39
T
08
18.48
T
12
13.70 C
31
44.04
C
14
4.23
T
00
8.66
T
13
5.15
C
32
53.86
C
15
12.77
T
04
9.82
T
48.51
14
17.65 C
33
C
16
21.32
T
05
7.53
T
15
7.82
C
34
2.05
C
17
31.99
T
06
17.35
T
16
16.41 C
35
23.78
C
18
40.54
T
07
60.57
T
17
18.57 C
06
60.57
C
19
49.08
T
18
21.46 C
1
15.94
T
20
57.62
T
19
13.89 C
2
36.00
T
21
66.16
T
T = Tension (Tie) and C = Compression (Strut)
56
Table 3.9 Summary of concrete struts calculations.
Max.
Actual
Max.
Actual
Strut
Model
strut
Strut
Model
Strut
strut
Strut
width,
Okay
Okay
Label MPa
capacity, force,
Label MPa width capacity, force,
mm
kN
kN
kN
kN
1
44.20 92.91 410.66
25.97
yes
19
44.20 20.00
88.40
13.89
yes
2
44.20 20.00
88.40
17.35
yes
20
44.20 20.00
88.40
15.72
yes
3
44.20 28.43 125.66
33.30
yes
21
44.20 20.00
88.40
7.74
yes
4
44.20 20.00
88.40
27.49
yes
22
44.20 20.00
88.40
42.91
yes
5
44.20 28.43 125.66
24.30
yes
23
44.20 20.00
88.40
33.09
yes
6
44.20 20.00
88.40
10.74
yes
24
44.20 99.35
88.40
23.27
yes
7
44.20 20.00
88.40
32.42
No
25
44.20 20.00
88.40
10.99
yes
8
44.20 20.00
88.40
13.67
yes
26
44.20 20.00
88.40
1.17
yes
9
44.20 28.43 125.66
15.10
yes
27
44.20 20.00
88.40
2.29
yes
10
44.20 20.00
88.40
6.99
yes
28
44.20 20.00
88.40
12.12
yes
11
44.20 28.43 125.66
22.24
yes
29
44.20 20.00
88.40
24.39
yes
12
44.20 20.00
88.40
13.70
yes
30
44.20 20.00
88.40
34.22
yes
13
44.20 28.43 125.66
5.15
yes
31
44.20 20.00
88.40
44.04
yes
14
44.20 28.43 125.66
17.65
yes
32
44.20 20.00
88.40
53.86
yes
48.51
15
44.20 28.43
125.66
7.82
yes
33
44.20 20.00
88.40
yes
16
44.20 28.43
125.66
16.41
yes
34
44.20 20.00
88.40
2.05
yes
17
44.20 20.00
88.40
18.57
yes
35
44.20 20.00
88.40
23.78
yes
18
44.20 28.43
125.66
21.46
yes
36
44.20 20.00
88.40
60.57
yes
Finally,
kN and
kN
The summary of concrete node calculations are shown in Table 3.10.
Table 3.10 Summary of critical concrete node calculation.
Model
Label Type
Node
at
CCT
support
Node
under
load
CCC
βn
0.80
0.80
0.80
1.00
1.00
1.00
1.00
Surrounding
Forces,
kN
25.97
15.94
20.38
20.38
60.57
23.78
48.51
C/T
Available
width, mm
MPa
C
T
C
C
C
C
C
92.91
80.00
50.00
100.0
28.43
99.35
20.00
35.36
35.36
35.36
44.20
44.20
44.20
44.20
The safe solution yields, therefore the nominal shear force is
kN and
⁄
55
Max.
capacity,
kN
328.53
282.88
176.80
448233
385266
409230
88243
Actual
force,
kN
25.97
15.94
20.38
20.38
60.57
23.78
48.51
Okay
yes
yes
yes
yes
yes
yes
yes
Group D [23]: Normal Strength Concrete, NSC, Simple Ordinary Beams
Case 1: Beam IT1
Figure 3.37a shows a simple reinforced concrete ordinary beam with rectangular openings, with
one top point load along with the proposed strut-and-tie model. The model has eighteen
compression struts S1 to S18, seven tension ties T1 to T7, and twenty-two nodes N1 to N22. One
external top point load applied at node N16. The tension ties T5 to T7 represent the main
longitudinal reinforcement and the vertical reinforcement is represented by ties T1 and T2.
Numerical scheme for Beam IT1 (Inverted T-beam):
Input data:
Beam size: h = 500mm, d = 450mm, b = 200mm, b1 = 100mm, and b2 = 100mm. Opening size is
200×400mm.
Shear span-to-depth ratio: a = 1500mm, and (a/d) = (1500/450) = 3.33
Materials:
= 36.20MPa,
= 538.8MPa (Ø10),
= 513.6MPa (Ø13),
= 355.2MPa
2
(Ø6),
= 321.8MPa (Ø8), and As = 413 (530.93 mm ).
The internal lever arm, Ld :
The term a1 (height of node N1) is taken equal to the height of flange. Thus,
From equilibrium,
and thus Ld = h – 0.5 (a1 + a2) = 427.85mm
Width of struts:
The widths of struts S1, S3, S5, S6, S8 and S9 were calculated based on bearing plates dimensions
and the widths of the tie T8 and the top strut S8 as follows:
(from geometry of node N16)
For other struts, the widths were determined by developing a realistic geometry of the struts as
they extend from the Nodes.
STM forces:
[
]
[
kN
]
kN
[
]
kN
[
]
kN
[
]
kN
The struts, ties, and nodes are labeled as in Fig. 3.37. The forces in all members are calculated form
static and are as shown in Table 3.11.
55
(a) Details of the strut-and-tie model.
(b) Strut labels for strut-and-tie model.
(c) Tie labels for strut-and-tie model.
55
(d) Node labels for strut-and-tie model.
Figure 3.37 Beam IT1.
Figure 3.38 Details of reinforcement (Beam IT1).
Table 3.11 Calculated forces of the strut-and-tie model of Beam IT1.
Model Label
Force, kN T or C Model Label
1
134.32
C
17
2
63283
C
18
3
53293
C
1
4
67296
C
2
5
90296
C
3
6
389289
C
4
7
39264
C
5
8
390203
C
6
9
33623
C
7
10
36263
C
8
11
84287
C
9
12
98244
C
10
13
44236
C
11
14
58250
C
12
15
0823
C
13
16
47234
C
14
T = Tension (Tie) and C = Compression (Strut)
55
Force, kN
57234
88208
152.75
124.89
99.94
125.0
84.03
433.14
233.27
148.15
135.20
90.12
90.12
85.12
45.10
0
T or C
C
C
T
T
T
T
T
T
T
T
T
T
T
T
T
T
Checking of stress limits:
a. Concrete Struts:
Knowing that = 36.20MPa, the term (
) will be:
MPa, for Strut Sj (j = 1 to 18)
The maximum strut capacity is
. Table 3.12 summarizes the calculations
performed for all struts.
Table 3.12 Summary of concrete struts calculations.
Model
Label
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
MPa
30.77
30.77
30.77
30.77
30.77
30.77
30.77
30.77
30.77
30.77
30.77
30.77
30.77
30.77
30.77
30.77
30.77
30.77
Strut width,
mm
140.88
32.00
44.31
20.00
44.31
44.31
25.00
44.31
95.00
40.00
21.00
20.00
18.00
18.00
18.00
18.00
15.00
15.00
Max. strut
capacity, kN
866.98
196.93
272.68
123.08
272.68
272.68
153.85
272.68
584.63
246.16
129.23
123.08
110.77
110.77
110.77
110.77
92.31
92.31
Actual Strut
force, kN
134.32
63283
53293
67296
90296
389289
39264
390203
33623
36263
84287
98244
44236
58250
0823
47234
57234
88208
Okay
yes
yes
yes
yes
yes
yes
yes
yes
yes
yes
yes
yes
yes
yes
yes
yes
yes
yes
b. Nodes:
The capacity of a node is calculated by finding the product of the limiting compressive stress in the
node region and the cross sectional area of the member at the node interface. The maximum node
capacity is
where
. Table 3.13 summarizes the calculations performed for
the effective concrete nodes.
Model
Label Type
1
CCT
2
CCT
16
CCC
Table 3.13 Summary of effective concrete node calculations.
Surrounding
Available
Max.
βn
Forces,
C/T
width,
capacity,
MPa
kN
mm
kN
0.80
134.32
C
140.88
24.62
693.69
0.80
148.15
T
100.00
24.62
492.40
0.80
125.0
C
100.00
24.62
492.40
0.80
134.32
C
28.43
44.20
251.32
0.80
158.97
C
99.35
44.20
878.25
0.80
14.86
C
28.43
44.20
251.32
0.80
135.20
T
100.0
44.20
884.00
1.00
193.01
C
95.00
30.77
584.63
1.00
286.08
C
44.31
30.77
272.68
1.00
250.0
C
200.0
30.77
1230.8
51
Actual
force,
kN
134.32
148.15
125.0
134.32
158.97
14.86
135.20
193.01
286.08
250.0
Okay
yes
yes
yes
yes
yes
yes
yes
yes
No
yes
The nominal shear force is
and
⁄
Upon following the previous numerical scheme, the failure load and failure mode of all other
beams have been obtained as shown in Table 3.14. A comparison between the results of the
strut-and-tie model and the test data is shown in the table. The strut-and-tie approach gives a
mean value of 0.760 of the experimental ultimate load.
Table 3.14 The STM results compared with test results.
No.
Beam
PEXP, kN
1
2
3
IT1
IT4
IT8
314.67
154.02
256.02
Failure Mode,
Exp.
Tension
Tension
Tension
Mean
PSTM, kN
134.83
98.46
212.83
Failure Mode,
PSTM / PEXP
STM
Tension
0.800
Tension
0.640
Tension
0.831
0.760
3.3 DEEP BEAMS AND OPENINGS IN DEEP BEAMS
3.3.1 Modeling
The modeling of deep beams will follow the same rules used, in previous, for shallow beams.
3.3.1.1 Case-Study
In the following, two groups of tested reinforced concrete deep beams will be introduced, linearly
analyzed using a finite element program ANSYS [5], and modeled using a STM.
Group A: Tested Beams Carried out by El-Azab, M. F. [11]
An experimental investigation was done by Mohamed Fawzy El-Azab [11] to study the shear
behavior of reinforced High- and Normal-Strength Concrete simply supported and continuous
deep beams with and without web openings considering different parameters such as size and
location of openings, vertical web reinforcement, and shear span-to-depth ratio.
Case 1: Development of a STM for Beam DSON3
The tested Beam DSON3 in Fig. 3.39a was linearly analyzed using a finite element program,
ANSYS [5]. From the obtained elastic principal stress trajectories in Fig. 3.39b for Beam DSON3,
the strut-and-tie model shown in Fig. 3.39c was proposed. From the elastic principal stress
trajectories in Fig. 3.39b, it could be noticed that openings affect the beam′s stress trajectories
drastically, where zones of tension stresses are formed around the left-upper corner of the opening
(load side) and the corner on the same diagonal and compression zones are formed around the two
other corners. This justifies the formation of D-regions around openings, as expected. Finally, the
tension and compression stress trajectories in Fig. 3.39b were followed to develop the STM shown
in Fig. 3.39c. The finite elements with compression stress trajectories will be replaced by
compression elements (Struts) and the finite elements with tension stress trajectories will be
replaced by tension elements (Ties).
55
Case 1: Beam DSON3
(a)
(b)
(c)
Figure 3.39 Beam DSON3
a) Concrete dimensions and location of opening [11],
b) Principal stress trajectories from FEM and,
c) Proposed strut-and-tie models.
55
Case 2: Development of a STM for Beam DSOH10
(a)
(b)
(c)
Figure 3.40 Beam DSOH10
a) Concrete dimensions and location of opening [11],
b) Principal stress trajectories from FEM,
c) Proposed strut-and-tie models.
55
Case 3: Development of a STM for Beam DCON3
(a)
(b)
(c)
Figure 3.41 Beam DCON3 a) Concrete dimensions and location of opening [11],
b) Principal stress trajectories from FEM,
c) Proposed strut-and-tie models.
55
Case 4: Development of a STM for Beam DCOH2
(a)
(b)
(c)
Figure 3.42 Beam DCOH2 a) Concrete dimensions and location of opening [11],
b) Principal stress trajectories from FEM,
c) Proposed strut-and-tie models.
56
Case 5: Development of a STM for Beam DCOH8
(a)
(b)
(c)
Figure 3.43 Beam DCOH8 a) Concrete dimensions and location of opening [11],
b) Principal stress trajectories from FEM,
c) Proposed strut-and-tie models.
55
Group B: Tested Beams Carried out by Mohammad, Kh. I. [32]
The experimental investigations done by Khalaf Ibrahem Mohammad [32] were nonlinearly
analyzed using the finite element method to predict the ultimate load and mode of failure for
reinforced Normal-Strength Concrete NSC simply supported deep beams with web openings.
Beam NO-0.3/4 having no web reinforcement, as shown in Fig. 3.44. The differences between
these five beams were in the web reinforcement patterns, Fig. 3.45, and concrete properties, Table
3.15. All beams, which are identical in geometry and longitudinal reinforcement, had a thickness
of 100mm and dimensions as shown in Fig. 3.46a. The longitudinal reinforcement consisted of one
20mm diameter deformed bar of 430MPa yield strength, anchored to external steel blocks at the
ends. Web reinforcement consisted of 10mm diameter deformed bar stirrups of 450MPa yield
strength.
Table 3.15 Concrete properties of the investigated beams of Group B [32].
MPa
Beam No.
NO-0.3/4
NW1-0.3/4
NW2-0.3/4
NW3-0.3/4
NW7-0.3/4
43.7
36.8
43.4
46.2
42.9
MPa
4.09
3.94
3.43
3.80
3.74
Figure 3.44 Beam geometry Group B [32].
55
Figure 3.45 Web reinforcement patterns for beams Group B [32].
55
Case 1: Development of a STM for tested beams of Group B [32]
(a)
(b)
(c)
Figure 3.46 Beams Group B a) Concrete dimensions and location of opening [32],
b) Principal stress trajectories from FEM,
c) Proposed strut-and-tie models.
55
3.3.2 Verification Examples
To illustrate how to model and analyze reinforced concrete deep beams with openings using a
strut-and-tie method, two groups of simple and continuous deep beams with openings are chosen
and examined. Beams of Group A were tested by Mohamed Fawzy El-Azab [11]. Beams of
Group B were tested by Khalaf Ibrahem Mohammad [32]. Finally, the obtained strut-and-tie
results are compared with the experimental results.
Capacity of Beams
Herein, the capacity of each beam with openings was determined using the strut-and-tie model.
The capacities were determined using the “iterative method”. These predicted capacities were then
compared to the actual failure load.
Group A [11]: Normal- and High-Strength Concrete Simple and Continuous Deep Beams
Case 1: Beam DSON3

Figure 3.47 shows a simple reinforced concrete deep beam with rectangular openings, with one top
point load along with the proposed refined strut-and-tie model.
Figure 3.47 Details of the proposed refined strut-and-tie model for Beam DSON3 using
inclined ties.
Figure 3.48 shows a simple reinforced concrete deep beam with rectangular openings, with one top
point load along with the proposed simplified strut-and-tie model. The model has five compression
struts S1 to S5, five tension ties T1 to T5, and six nodes N1 to N6. One external top point load is
applied at node N4. The tension ties T1 and T2 represent the main longitudinal reinforcement and
the vertical and horizontal reinforcement is represented by ties T3 and T4.
51
S3
N4
S4
S2
T4
N3
N6
N2
T5
S1
T3
S5
T2
N5
T1
N1
Figure 3.48 Details of the proposed simplified strut-and-tie model for Beam DSON3
using inclined ties.
Numerical scheme for Beam DSON3:
Input data:
Beam size: h = 400mm, d = 360mm, b = 80mm, and b1 = b2 = 100mm.
Opening size is 80 × 180mm.
Shear span-to-depth ratio: a = 400mm, and (a/d) = (400/360) = 1.11
Materials: = 30.45MPa, = 410MPa (Ø16),
= 244.5MPa (Ø6),
2
= 410, and As = 416 (804.25 mm ).
= 260.2MPa (Ø8),
Sh
Where
is the cylinder compressive strength for concrete, is the yield stress for longitudinal
steel,
is the yield stress for vertical stirrups,
is the yield stress for horizontal stirrups
is the area of secondary steel, and As is the area of main steel.
Sh = 100mm
Sv = 200mm
Sv
Figure 3.49 Details of reinforcement for Beam DSON3.
55
For Beam DSON3, to simplify the visualization of strut widths and geometry of nodes, use the
simplified strut-and-tie model shown in Fig. 3.48.
Width of struts:
The strut widths were determined by developing a realistic geometry of the struts as they extend
from the nodes shown in Fig. 3.50.
(
(
)
)
Figure 3.50 Visualization of strut widths.
STM forces:
The forces in all members are determined from statics and their magnitudes in kN are as indicated
in Table 3.16. The struts, ties, and nodes are labeled as in Fig. 3.48.
kN
is the nominal strength capacity of the tie
when reaching its yield strength.
Table 3.16 Calculated member forces for the strut-and-tie model.
Model
Force
Model
Force
T or C
T or C
Label
(kN)
Label
(kN)
S1
64.14
C
T1
61.59
T
S2
58.04
C
T2
4.27
T
S3
75.42
C
T3
46.86
T
S4
48.44
C
T4
37.04
T
S5
31.34
C
T5
22.46
T
T = Tension (Tie) and C = Compression (Strut)
Finally,
kN
55
Checking of stress limits:
a. Concrete Struts:
Knowing that
= 30.45MPa, the term (
) will be:
MPa, for Strut Sj (j = 2 to 5)
MPa, for Strut Sj (j = 1)
Maximum strut capacity,
for all struts.
. Table 3.17 summarizes the calculations performed
Table 3.17 Summary of concrete struts calculations.
Model
Label
1
2
3
4
5
βs
0.60
1.00
1.00
1.00
1.00
Strut
MPa width, mm
15.53
52.00
25.88
53.00
25.88
43.00
25.88
42.00
25.88
52.00
Max. strut
capacity, kN
64.60
109.73
89.03
86.96
107.66
Actual Strut
force, kN
64.14
58.04
75.42
48.44
31.34
Okay
yes
yes
yes
yes
yes
b. Nodes:
The capacity of a node is calculated by finding the product of the limiting compressive stress in the
node region and the cross sectional area of the member at the node interface. The maximum node
capacity is
where
. Table 3.18 summarizes the calculations performed
for the effective critical nodes N1, N3, and N4.
Table 3.18 Summary of critical concrete node calculation.
Model
Label Type
Surrounding T Available
βn
Forces,
or
width,
kN
C
mm
0.80
64.14
C
94.00
64.00
C
100.00
1
CCT 0.80
0.80
4.27
T
80.00
1.00
75.42
C
43.00
4
CCC 1.00
48.44
C
42.00
1.00
128.0
C
100.0
0.60
31.34
C
52.00
0.60
46.86
T
55.00
3
CTT
0.60
4.27
T
69.00
0.60
61.59
T
69.00
T = Tension (Tie) and C = Compression (Strut)
kN
⁄
55
MPa
20.71
20.71
20.71
25.88
25.88
25.88
15.53
15.53
15.53
15.53
Max.
capacity,
kN
155.71
165.68
132.54
89.03
86.96
207.04
63.77
68.33
85.73
85.73
Actual
force,
kN
64.14
64.00
4.27
75.42
48.44
128.0
31.34
46.86
4.27
61.59
Okay
yes
yes
yes
yes
yes
yes
yes
yes
yes
yes
The alternative proposed strut-and-tie model for Beam DSON3 using vertical and horizontal ties as
shown in Fig. 3.51 can be used and the capacity of the beam can be recalculated.
Figure 3.51 Alternative proposed refined strut-and-tie model for Beam DSON3
using vertical and horizontal ties.
S3
N4
S N5
T6 S4
T5 T5 5 T7 N7 N6
S6
S2
S1
N3
T3
T4
T3
S7
S8
T1
N8
N2
N9
N10
T2
N1
Figure 3.52 Alternative proposed simplified strut-and-tie model for Beam DSON3
using vertical and horizontal ties.
55
Table 3.15 Calculated member forces for proposed simplified the strut-and-tie model.
Model
Force,
Model
T or C
Label
kN
Label
S1
66.29
C
T1
S2
75.25
C
T2
S3
91.81
C
T3
S4
88.11
C
T4
S5
37.58
C
T5
S6
43.00
C
T6
S7
65.27
C
T7
S8
12.83
C
-T = Tension (Tie) and C = Compression (Strut)
Finally, based on Table 3.19,
Force,
kN
72.39
13.00
27.00
17.33
53.27
6.47
6.47
--
T or C
T
T
T
T
T
T
T
--
kN
Checking of stress limits:
a. Concrete Struts:
Knowing that
= 30.45MPa, the term (
) will be:
MPa, for Strut Sj (j = 1, 3, 4, 5, 7 and 8)
MPa, for Strut Sj (j = 2, and 5)
The maximum strut capacity is
performed for all struts.
. Table 3.20 summarizes the calculations
Figure 3.53 Visualization of strut widths.
56
Table 3.20 Summary of concrete struts calculations.
Model
Label
1
2
3
4
5
6
7
8
βs
1.00
0.60
1.00
1.00
0.60
1.00
1.00
1.00
MPa
25.88
15.53
25.88
25.88
15.53
25.88
25.88
25.88
Strut
width
113.0
62.00
59.00
52.00
36.00
47.00
51.00
62.00
Max. strut
capacity, kN
233.96
77.03
122.15
107.66
44.73
97.31
105.59
128.36
Actual strut
capacity, kN
66.29
75.25
91.81
88.11
37.58
43.00
65.27
12.83
Okay
yes
yes
yes
yes
yes
yes
yes
yes
b. Nodes:
The capacity of a node is calculated by finding the product of the limiting compressive stress in the
node region and the cross sectional area of the member at the node interface. The maximum node
capacity is
where
. Table 3.21 summarizes the calculations performed
for the effective nodes N1, N5, and N8.
Table 3.21 Summary of critical concrete node calculations.
Model
Label Type
1
5
8
βn
0.80
CCT 0.80
0.80
1.00
CCC 1.00
1.00
0.80
0.80
CCT 0.80
0.80
0.80
Which is safe,
Surrounding
Forces,
C/T
kN
65.00
C
66.29
C
13.00
T
130.0
C
88.11
C
37.58
C
43.00
C
12.83
C
65.27
C
24.92
T
10.13
T
Available
width,
mm
100.00
113.0
80.00
100.0
52.00
36.00
47.00
62.00
51.00
51.00
63.00
,
MPa
20.71
20.71
20.71
25.88
25.88
25.88
20.71
20.71
20.71
20.71
20.71
Max.
Actual
capacity, capacity, Okay
kN
kN
165.70
65.00
yes
187.22
66.29
yes
yes
13.00
132.54
207.04
130.0
yes
107.66
88.11
yes
74.53
37.58
yes
77.87
43.00
yes
102.72
12.83
yes
84.50
65.27
yes
84.50
24.92
yes
104.40
10.13
yes
kN
⁄
The model with vertical and horizontal ties (Fig. 3.52) is better than that with inclined ties (Fig.
3.48). This is because it gives larger capacity.
55
Upon following the previous numerical scheme, the failure load and failure mode of all other
beams have been obtained as shown in Table 3.22. A comparison between the results of the strutand-tie model and the test data is shown in table. The strut-and-tie approach gives a mean value of
0.785 of the experimental ultimate load.
Table 3.22 The STM results compared with test results.
No.
1
2
3
4
5
Beam
DSON3
DSOH10
DCON3
DCOH2
DCOH8
PEXP, kN
140
110
220
360
290
Failure Mode,
Exp.
Opening Failure
Opening Failure
Opening Failure
Opening Failure
Opening Failure
Mean
PSTM, kN
130.0
98.56
176.43
280.12
150.31
Failure Mode,
STM
Opening Failure
Opening Failure
Opening Failure
Opening Failure
Opening Failure
PSTM / PEXP
0.930
0.896
0.801
0.778
0.518
0.785
Group B [32]: Normal Strength Concrete NSC Simple Deep Beams
For verification purposes, the failure load and failure mode of all beams in Group B [32] have
been obtained as shown in Table 3.23. A comparison between the results of the strut-and-tie model
and the test data is shown in table. The strut-and-tie approach gives a mean value of 0.732 of the
experimental ultimate load.
Table 3.23 The STM results compared with test results.
No.
Beam
PEXP, kN
1
2
3
4
5
NO-0.3/4
NW1-0.3/4
NW2-0.3/4
NW3-0.3/4
NW7-0.3/4
240
420
580
620
720
Failure Mode,
Exp.
Opening Failure
Opening Failure
Opening Failure
Opening Failure
Opening Failure
Mean
PSTM, kN
212.32
305.16
317.45
490.18
510.47
Failure Mode,
STM
Opening Failure
Opening Failure
Opening Failure
Opening Failure
Opening Failure
PSTM / PEXP
0.885
0.727
0.547
0.791
0.709
0.732
The absence of web reinforcement in Beam NO-0.3/4 caused a reduction in load-carrying capacity
and ductility where the failure is brittle and sudden. Improvements were significantly noted on
response of the beams with different web reinforcement patterns. The best advantage could be
gained by reinforcing the beam above and below the opening.
55
3.4 SUMMARY AND CONCLUSIONS
In summary, this chapter introduces an approach for how to develop design models for RC beams
with and without openings using the strut-and-tie method. This approach is based on the force
tracing of the elastic analysis results using FEM and/or following the load path method. This
approach helps the design Engineer to develop the design models, which suit the design
conditions. Also in this chapter verification examples were done by STM to predict ultimate loads
and compare then with experimental loads.
From this study, the following conclusions can be drawn:

The strut-and-tie approach gives freedom to designer to choose the suitable model,
according to elastic principal stress trajectors from finite element analysis and practice.

The Strut-and-Tie Model gives reasonable lower bound estimate of the load carrying
capacity of the chosen RC beams when compared with the experimental failure loads.

Strut-and-Tie approach gives mean values ranges from 0.65 to 0.75 % of the experimental
ultimate loads for all chosen beams from literature.

The proposed Strut-and-Tie approach is a powerful tool to predict the ultimate strength and
behavior of reinforced concrete (ordinary or deep) beams with and without openings.

Small openings are defined as openings which are small enough, when the depth (or
diameter) of the opening is less than or equal to (0.25 - 0.40) times the overall beam depth
and located in such a way that a Strut-and-Tie model in a RC beams is able to jump over
the openings without causing additional vertical or horizontal struts in the chords above
and below the openings.

Large openings are defined as openings which are large enough and located in such a way
that a strut-and-tie model in a RC beams is causing additional vertical or horizontal struts
in the chords above and below the openings.
55
CHAPTERT
4
NONLINEAR FINITE ELEMENT
ANALYSIS
CHAPTER 4
NONLINEAR FINITE ELEMENT ANALYSIS
4.1 INTRODUCTION
The possibility of implementing nonlinear finite element analysis as a preliminary design tool,
especially for the cases where a clear force path in the STM is not obvious, is examined in this
chapter. In order to predict the complete response of reinforced concrete beams such as;
displacements, strains and stresses distributions, ultimate shear loads and failure modes, and
cracking patterns, a three dimensional nonlinear finite element model using “ANSYS-12” package
[5] is utilized. In this model, nonlinear constitutive models of concrete and reinforcement are
considered. Concrete is modeled using a three dimensional reinforced concrete element named
SOLID65, which is capable of cracking in tension and crushing in compression. The main and
web reinforcements are modeled using LINK8-3D bar element within the concrete SOLID65 one.
In this Chapter, the material model for concrete and reinforcement are first reviewed. The
input data (geometry, meshing, loads, and boundary conditions) are then defined for the analyzed
reinforced concrete beams. Finally, the obtained results are discussed and compared with the
experimental data and strut-and-tie model results.
4.2 ANSYS' FINITE ELEMENT MODELS
The finite element method (using ANSYS-12 package) can be used to closely predict the behavior
of reinforced concrete beams subjected to in-plane forces if proper care is taken in modeling the
material characteristics. Using the finite element method, the load deflection behavior, crack
pattern, and failure load and failure mode can be predicted with an accuracy that is acceptable for
engineering purposes. Furthermore, the program accounts for; (1) material nonlinearity of both
concrete and steel, (2) biaxial failure surface of concrete, (3) nonlinear stress-strain curve of steel,
and (4) concrete cracking and crushing.
4.2.1 Element Types
Two element types are used in ANSYS program, namely; SOLID65 for concrete, SOLID45 for
steel plates and supports, and LINK8-3D for steel reinforcement.
4.2.1.1 Solid65
A concrete solid element SOLID65 is used to model the three-dimensional behavior of concrete
with or without reinforcing bars (rebars). The solid element is capable of cracking in tension and
crushing in compression. The element is defined by eight nodes each having three translational
degrees of freedom (ux, uy, and uz).
49
The following basic assumptions are considered in the element formulation:










Cracking is permitted in three orthogonal directions at each integration point.
If cracking occurs at an integration point, the cracking is modeled through an adjustment of
material properties which effectively treats cracking as a smeared band of cracks, rather
than discrete cracks.
The concrete material is assumed to be initially isotropic.
Whenever the reinforcement capability of the element is used, the reinforcement is
assumed to be discrete throughout the element, see Fig. 4.6.
The element may be numbered either as shown in Fig. 4.1 or may have the planes IJKL
and MNOP interchanged.
All elements must have eight nodes.
The element is nonlinear and requires an iterative solution.
The Load must be applied slowly to prevent possible fictitious crushing of the concrete
before proper load transfer can occur through a closed crack.
The lost shear resistance of cracked and/or crushed elements cannot be transferred to the
rebars, which have no shear stiffness.
For convergence purposes, large strains and large deflections are not considered.
The element geometry, node locations, and the element coordinate system are shown in Fig.
4.1. The element is defined by eight nodes and the isotropic material properties. The element has
one solid material and up to three rebar materials. Rebar specifications include the material
number, the volume ratio, and the orientation angles. The volume ratio is defined as the rebar
volume divided by the total element volume. The orientation is defined by two angles from the
element coordinate system. Additional concrete material data, such as the shear transfer
coefficient, tensile stresses, and compressive stresses are input in the data. Typical shear transfer
coefficient ranges from zero to one. With the zero represents a smooth crack, which means a
complete loss of shear transfer, and the one represents a rough crack which means no loss of shear
transfer. The output results associated with the element SOLID65 are the nodal displacements and
element stresses. SOLID65 stress output is shown in Fig. 4.2. The element stress directions are
parallel to the element coordinate system. Nonlinear material printout appears only if nonlinear
properties are specified. Rebar printout appears only for the rebar defined. If cracking or crushing
is possible, printout for the concrete is also available at the integration points, since cracking or
crushing may occur at any integration point.
45
Figure 4.1 SOLID65 3-D reinforced concrete solid element.
Figure 4.2 SOLID65 3-D stress output.
49
4.2.1.2 Solid45
In order to avoid stress concentration problems, the steel plates at the supports and at loading
points of the beam are modeled using SOLID45, Fig. 4.3. This element has eight nodes with three
degrees of freedom at each node; translations in the nodal x, y, and z directions, which handles
plasticity, creep, swelling, stress stiffening, and large deflection and strain. SOLID45 stress output
is shown in Fig. 4.4. The element stress directions are parallel to the element coordinate system.
The surface stress outputs are in the surface coordinate systems and are available for any face.
Figure 4.3 SOLID45 3-D element.
Figure 4.4 SOLID45 3-D stress output.
49
4.2.1.3 Link8-3D
A LINK8-3D element was used to model steel reinforcement. This element (a 3-D spar element) is
a uniaxial tension–compression element with three translational degrees of freedom at each node;
translations in the x, y, and z directions. As in a pin-jointed structure, no bending of the element is
considered; plasticity, creep, swelling, and stress stiffening capabilities are included. The spar
element assumes a straight bar, axially loaded at its ends and of uniform properties from end to
end. The length of the spar must be greater than zero; Nodes I and J must not coincide and the
cross-sectional area must be greater than zero. The temperature is assumed to vary linearly along
the length of the spar. The geometry, node locations, and coordinate system for this element are
shown in Fig.4.5. The element is defined by two nodes. The element x-axis is oriented along the
length of the element from Node I toward Nodes J.
Figure 4.5 LINK8-3D element bars.
Finite Element Modeling of Steel Reinforcement in Concrete Elements:
Tavarez in 2001 [44] discussed the three techniques that exist to model steel reinforcement in
finite element models for reinforced concrete, Fig. 4.6. The three techniques are; (1) the discrete
model, (2) the embedded model, and (3) the smeared model. The reinforcement in the discrete
model, Fig. 4.6a, uses bar or beam elements that are connected to concrete mesh nodes. Therefore,
the concrete and the reinforcement mesh share the same nodes and concrete occupies the same
regions occupied by the reinforcement. A drawback to this model is that the concrete mesh is
restricted by the location of the reinforcement and the volume of the mild-steel reinforcement is
not deducted from the concrete volume.
49
(a) Discrete model
(b) Embedded model
(c) Smeared model
Figure 4.6 Models for reinforcement in reinforced concrete elements:
(a) discrete; (b) embedded; and (c) smeared.
The embedded model (Fig. 4.6b) overcomes the concrete mesh restriction(s) because the
stiffness of the reinforcing steel is evaluated separately from the concrete elements. The model is
built in a way that keeps reinforcing steel displacements compatible with the surrounding concrete
elements. When reinforcement is complex, this model is very advantageous. However, this model
increases the number of nodes and degrees of freedom in the model, therefore, increasing the run
time and computational cost.
The smeared model (Fig. 4.6c) assumes that reinforcement is uniformly spread throughout
the concrete elements in a defined region of the finite element mesh. This approach is used for
large-scale models where the reinforcement does not significantly contribute to the overall
response of the structure.
Fanning in 2001 modeled the response of the reinforcement using the discrete and the
smeared models for reinforced concrete beams. It was found that the best modeling strategy was
to use the discrete model when modeling reinforcement. Hence, the discrete modeling is
considered in all of the analyses presented in this thesis.
44
4.2.2 Material Models
Parameters needed to define the material models can be found in Table 4.1. As seen in Table 4.1,
there are multiple parts of the material model for each element.
Table 4.1 Material models for SOLID65, SOLID45 and LINK8 element.
Material
Model
No.
Element
Type
Material Properties
Linear Isotropic
Elasticity Modulus, EX, is equal to
at point 1 on the curve
Poisson’s Ratio, PRXY, is equal to 0.20
Multilinear Isotropic
Five coordinates are needed to represent the stress-strain curve for
concrete, Figs. 4.7 and 4.8.
1
SOLID65
Concrete
Open Shear Transfer Coeff.
Closed Shear Transfer Coeff.
Uniaxial Cracking Stress
(Modules of rupture)
Uniaxial Crushing Stress
Biaxial Crushing Stress
Hydrostatic Pressure
Hydro Biax Crush Stress
Hydro Uniax Crush Stress
Tensile Crack Factor
2
3
SOLID45
LINK8
0.3
1
The concrete tensile strength
is typically 8% - 15% of the
compressive strength or equal
√ [42].
to
-1
This value means that the
crushing stress value is taken
from the stress-strain curve
0
0
0
0
0
Linear Isotropic
Elasticity Modulus, EX, is equal to
Poisson’s Ratio PRXY, is equal to 0.30
Linear Isotropic
Elasticity Models, EX, is equal to
Poisson’s Ratio PRXY, is equal to 0.30
Bilinear Isotropic
Yield Stresses, follow the design material properties used for the
experimental investigation
Tangent Modulus, Tang Mod, is given from 10 to 20N/mm2
111
The SOLID65 element requires linear isotropic and multilinear isotropic material properties to
properly model concrete. The multilinear isotropic material uses the Von-Mises failure criterion
along with the Willam and Warnke (1974) model to define the failure of the concrete. EX is the
initial tangent modulus of elasticity of the concrete (Ec) and PRXY is the Poisson’s ratio (ν). The
elasticity modulus was based on the following equation [3],
√
MPa
(4.1)
with a value of
equal to a cylinder compressive strength in MPa units. Poisson’s ratio was
assumed to be 0.2 for concrete. If data from test is not available, the compressive uniaxial stressstrain relationship for the concrete model was obtained using the following equation [10] to
compute the multilinear isotropic stress-strain curve for the concrete.
[
(
)
(
) ]
(4.2)
Where is the stress at any strain
and
is the cube compressive strength. Strains were
selected and the stress was calculated for each strain from equation 4.2.
Figure 4.7 Multilinear isotropic stress-strain curve for concrete
in compression (Egyptian Code [10]).
111
There is also another equation to compute the multilinear isotropic stress-strain curve for
the concrete in compression (ACI code, MacGregor 2011) and this equation will be used in this
chapter:
(4.3)
( )
where
(4.4)
and
(4.5)
where is the stress at any strain
and
is the strain at the cylinder compressive strength
The multilinear isotropic stress-strain curve, requires the first point of the curve to be defined by
the user It must satisfy Hooke’s Law, that is,
(4.6)
The multilinear curve is used to help for the convergence of the nonlinear solution algorithm.
Figure 4.8 Multilinear isotropic stress-strain curve for concrete
in compression (ACI Code [3]).
Figure 4.8 shows the stress-strain relationship which is based on a work done by Kachlakev, et al.
[20]. Point 1, defined as 0.3 , is calculated in the linear range using Equation 4.5 to get strain at
this point. Points 2 and 3 are calculated from Equation 4.3 with
obtained from Equation 4.4.
Strains were selected and the stress was calculated for each strain. Point 4 is defined at
and
obtained from Equation 4.4 or about 0.002 and Point 5 is defined at
and of 0.003 indicating
111
traditional crushing strain for unconfined concrete. The behavior is assumed to be perfectly plastic
after Point 4.
Implementation of the Willam and Warnke (1974) material model in ANSYS requires to define
nine constants. These nine constants are as follows:
1.
2.
3.
4.
5.
6.
7.
8.
9.
Shear transfer coefficients for an open crack;
Shear transfer coefficients for a closed crack;
Uniaxial tensile cracking stress;
Uniaxial crushing stress (positive);
Biaxial crushing stress (positive);
Ambient hydrostatic stress state for use with constants 7 and 8;
Biaxial crushing stress (positive) under the ambient hydrostatic stress state (constant 6);
Uniaxial crushing stress (positive) under the ambient hydrostatic stress state (constant 6);
Stiffness multiplier for cracked tensile condition.
Typical shear transfer coefficients range from 0.0 to 1.0, with 0.0 representing a smooth
crack (complete loss of shear transfer) and 1.0 representing a rough crack (no loss of shear
transfer). The shear transfer coefficients for open and closed cracks were determined using the
work of Kachlakev et al. [20] as a basis. Convergence problems occurred when the shear transfer
coefficient for the open crack dropped below 0.2. No deviation of the response occurs with the
change of the coefficient. Therefore, the coefficient for the open crack was set to 0.3.
The SOLID45 element is being used for the steel plates at loading points and supports on the
beam in order to avoid stress concentration problems. Therefore, this element is modeled as a
linear isotropic element with a modulus of elasticity for the steel (Es = 2×105MPa) and Poisson’s
ratio (ν = 0.3).
The LINK8 element is being used for all the steel reinforcement in the beam and it is
assumed to be bilinear isotropic and identical in tension and compression as shown in Fig. 4.11.
Bilinear isotropic material is also based on the Von-Misses failure criteria. For bilinear
isotropic hardening model of LINK8 element, the specified yield stress, the stress-strain curve of
reinforcement continues along the second slope defined by the tangent modulus. It also
experienced that for tangent modulus a small value of 10 to 20N/mm2 shall be used to avoid loss
of stability upon yielding. Elastic modulus and yield stress for the steel reinforcements used in
the finite element modeling will follow the same design material properties used for the
experimental investigation.
4.2.2.1 Concrete in Compression
Since concrete is used mostly in compression, its compressive typical stress-strain curve is nonlinear. Various strengths with a descending branch after reaching the maximum stress are shown
in Fig. 4.9. All curves have somewhat similar characteristics. They consist of an initial relatively
straight elastic portion in which stresses and strains are closely proportional, then begin to curve to
reach a maximum value at a strain of 0.002 to 0.003. There is a descending branch after the peak
stress is reached.
111
Figure 4.9 Typical stress-strain curves for concrete in compression.
Concrete stress, fc
For computational purposes, mathematical representation of the stress-strain curves of
concrete in compression are available. For example, the stress-strain curve shown in Fig. 4.10 may
be used [10]. The curve consists of a parabola up to a strain of 0.002 and straight horizontal line up
to a strain of 0.003. Such a curve has been used widely in research purposes.
Concrete strain, εc
Figure 4.10 Idealized stress-strain curve for concrete in compression.
4.2.2.2 Concrete in Tension
The low tensile strength of concrete, which is generally about one tenth of its compressive
strength, is one of its most important properties. In this study, concrete is assumed to behave as a
linear elastic brittle material in tension, and this is a major factor causing the nonlinear behavior.
Cracks are assumed to form in planes perpendicular to the direction of maximum principal tensile
stress as soon as this reaches the specified concrete tensile strength.
119
4.2.2.3 Reinforcement in Tension
The mechanical properties of steel are well-known and understood. Steel is homogeneous and has
usually the same yield strength in tension and compression. In the present study reinforcing steel is
modeled as a bilinear elasto-plastic material using the stress-strain curve shown in Fig. 4.11.
Figure 4.11 Idealized stress-strain curve for steel.
4.2.2.4 Bond between Concrete and Reinforcement
One of the fundamental assumptions of reinforced concrete design is that at the interface of the
concrete and the steel bars, perfect bonding exists and no slippage occurs. For “ANSYS-12”
package, the command "Merge Items" merges separate entities that have the same location. These
items will then be merged into single entities. Caution must be taken when merging entities in a
model that has already been meshed because the order in which merging occurs is significant.
Merging key points before nodes can result in some of the nodes becoming “orphaned”; that is, the
nodes lose their association with the solid model. The orphaned nodes can cause certain
operations (such as boundary condition transfers, surface load transfers, and so on) to fail. Care
must be taken to always merge in the order that the entities appear. All precautions were taken to
ensure that everything was merged in the proper order. Also, the lowest number was retained
during merging.
4.2.3 Solution Strategy
The procedure used for modeling and analyzing any structure consists of three main steps: (1)
building the model, (2) applying loads and obtaining the solution, and (3) reviewing and
interpreting the results. In the first step, the used element type, element real constants, material
properties, and model geometry must be defined. In the second step, the applied loads, load step
options, and analysis type are defined. Then the finite element solution is initiated. In the last step,
the obtained results are analyzed and compared with practice and experimental results. In the
following sections, load time curve, automatic time stepping, and nonlinear options will be defined
in brief.
115
4.2.3.1 Automatic Time Stepping
Automatic time stepping (also called time step optimization in a transient analysis) attempts to
adjust the integration time step during solution based on the response frequency and the effects of
nonlinearities. The main benefit of this feature is that the total number of substeps can be reduced,
resulting in computer resource savings. Also, the number of times that you might have to rerun the
analysis (adjusting the time step size, nonlinearities, and so on) is greatly reduced. If nonlinearities
are present, automatic time stepping gives the added advantage of incrementing the loads
appropriately and retreating to the previous converged solution (bisection) if convergence is not
obtained. Automatic time stepping allows ANSYS to determine the size of load increments
between substeps. It also increases or decreases the time step size during solution, depending on
how the model responds. There are two features of the automatic time stepping. The first feature
concerns the ability to estimate the next time step size based on the current and past analysis
conditions and make proper load adjustments. The second feature is referred to as the time step
bisection component. Its purpose is to decide whether or not reduce the present time step size, and
redo the substep with a smaller step size. In automatic time stepping, the program calculates an
optimum time step at the end of each substep, based on the response of the structure or component
to the applied loads. When used in a nonlinear static (or steady-state) analysis, for most problems,
you should turn on automatic time stepping and set upper and lower limits for the integration time
step.
4.2.3.2 Loading
The primary objective of the finite element analysis is to examine the response of the structure or
its components to certain loading conditions. To make a proper closed relation between loaddeflection curves, the applied loads are divided into substeps up to failure, the substeps are the
points within a load step at which solutions are calculated, Fig.4.12. They are used in nonlinear
static to apply the loads gradually so that an accurate solution can be obtained. At each substep, the
program will perform a number of equilibrium iterations to obtain converged solution. The
program uses the time as a tracking parameter.
Figure 4.12 Load steps, substeps, and time.
119
4.2.3.3 Newton-Raphson Method of Analysis
ANSYS employs the "Newton-Raphson" approach to solve nonlinear problems. In this approach,
the load is subdivided into a series of load increments, Fig. 4.13. The load increments can be
applied over several load steps. Before each solution, the Newton-Raphson method evaluates the
out-of-balance load vector, which is the difference between the restoring forces (the loads
corresponding to the element stresses) and the applied loads. The program then performs a linear
solution, using the out-of-balance loads, and checks for convergence. If convergence criteria are
not satisfied, the out-of-balance load vector is reevaluated, the stiffness matrix is updated, and a
new solution is obtained. This iterative procedure continues until the problem converges. A
number of convergence-enhancement and recovery features, such as line search, automatic load
stepping, and bisection, can be activated to help the problem to converge. If convergence cannot
be achieved, then the program attempts to solve with a smaller load increment.
In some nonlinear static analyses, if you use the Newton-Raphson method alone, the
tangent stiffness matrix may become singular (or non-unique), causing severe convergence
difficulties. Such occurrences include nonlinear buckling analyses in which the structure either
collapses completely or "snaps through" to another stable configuration. For such situations, you
can activate an alternative iteration scheme, the arc-length method, to help avoid bifurcation points
and track unloading.
The arc-length method causes the Newton-Raphson equilibrium iterations to converge
along an arc, thereby often preventing divergence, even when the slope of the load vs. deflection
curve becomes zero or negative. This iteration method is represented schematically in Fig. 4.14.
Convergence
Solution
Divergence
Solution
Figure 4.13 Incremental Newton-Raphson procedure.
119
Figure 4.14 Traditional Newton-Raphson method vs. arc-length method.
Figure 4.15 Initial-stiffness Newton-Raphson.
4.3 ANALYSIS OF ORDINARY BEAMS WITH OPENINGS
4.3.1 Verification Group B: Simple Beams With and Without Circular Openings.
4.3.1.1 Model Description and Material Properties
A nonlinear finite element analysis has been performed for seven simple ordinary beams, one
beam without openings and six beams with openings. The tested beams have been simply loaded
with two symmetrical concentrated loads. The details of the beams are shown in Tables 3.2 and
3.3. The beams, loading plates, and supports were modeled as volumes. The length of beams is
1600mm and loading span is1400mm. The width of beam section is 125mm, the height of beam
section is 250mm, the effective depth is 217mm, and the distance from compression fiber to
compression reinforcement is 33mm. Regarding reinforcement, two bars with 14mm diameter
were used for the main reinforcement and two bars with 6mm diameter were used for the
secondary reinforcement. Stirrups have a diameter of 6mm and are placed in the shear span, with
100mm distance from each other. The finite element mesh for the Beam S model is shown in Fig.
4.16a.
119
The finite element models for other beams are shown Figs. 4.17b, and 4.18b. The material
properties for concrete and reinforcement for all beams are summed in Table 4.2.
The concrete compressive cylinder strength (fc ), the concrete tensile strength ( ), and the
modulus of elasticity (Ec) are as shown in Table 4.2. The concrete material properties are
numbered as a material number 1. The steel yield strength (f y) and the steel modulus of elasticity
(Es) are shown in Table 4.2. The reinforcement material properties are numbered as materials 2
and 3, and steel plate material properties are numbered as material 4.
Table 4.2 Material properties for concrete and reinforcement.
No.
Beam
1
2
3
4
5
6
7
S
ND80X350
ND80X150-S
ND80X250-S
ND100X350-S
HD80X150-S
HD100X350-S
(MPa)
30.40
28.93
28.65
29.62
29.81
69.82
68.94
(MPa)
3.04
2.89
2.87
2.96
2.98
6.98
6.89
Ec(MPa)
25914
25280
25157
25579
25661
39272
39024
Vertical Web
reinforcement
Large*
Small**
10 ф 6
10 ф 6
10 ф 6
10 ф 6
10 ф 6
10 ф 6
10 ф 6
--3ф4
3ф4
3ф4
3ф4
3ф4
Yield strength, fy
(MPa)
Top
Bott. Stirrups
250
250
250
250
250
250
250
450
450
450
450
450
450
450
250
250
250
250
250
250
250
Modulus of
Elasticity, Es
(MPa)
2×105
2×105
2×105
2×105
2×105
2×105
2×105
* Stirrups in full depth
** Stirrups in above and below of openings
4.3.1.2 Meshing
The beams are modeled using the nonlinear solid element SOLID65. To obtain good results from
the SOLID65 element, the use of a rectangular mesh is recommended. Therefore, the mesh was set
up such that square or rectangular elements were created (Fig. 4.16a). The volume sweep
command was used to mesh the steel plates and supports. This properly sets the width and length
of elements in the plates to be consistent with the elements and nodes in the concrete portions of
the model. The overall mesh of the concrete, plates, and supports volumes is shown in Fig. 4.16a
for Beam S and other beams as shown in Figs. 4.17b and 4.18b. The meshing of the reinforcement
is a special case compared to the volumes. No mesh of the reinforcement is needed because
individual elements were created in the modeling through the nodes created by the mesh of the
concrete volume. However, the necessary mesh attributes as described above need to be set before
each section of the reinforcement is created. For these seven simple ordinary beams the cross
section is divided into four vertical strips according to the main bottom, top, and web
reinforcement, as shown in Fig. 4.16. The position of reinforcement lies at intersection of mesh
nodes. The Strip S1 is 33mm thick and represents the cover for bottom reinforcement. The second
Strip S2 is 200mm thick 50mm×4 and represents the part of concrete between the top and bottom
reinforcement. The third Strip S3 is 33mm thick and represents the cover for the top reinforcement.
4.3.1.3 Loads and Boundary Conditions
Displacement boundary conditions are needed to constrain the model to get a unique solution. To
ensure that the model acts the same way as the experimental beam, boundary conditions need to be
applied at points of symmetry, and where the supports and loadings exist. Four stiff reinforced
concrete solid elements SOLID45 are meshed and used to model the two supports and the two
point loads, as shown in Figs. 4.16c, 4.17d and 4.18d. Hinges are put at the lower parts of the
reinforced concrete elements that represent the one support and other supports represent the
114
rollers. The point load acting at the steel plate is applied across the entire centerline of the plate
(Fig. 4.16c). The point load is acting at five nodes on the plate. The finite element model for this
analysis is a simple beam under transverse loading. For the purposes of this model, the Static
Analysis Type is utilized. The Restart command is utilized to restart an analysis after the initial run
or load step has been completed The Sol’n Controls command dictates the use of a linear or nonlinear solution for the finite element model. In the particular case considered in this thesis the
analysis is of the small displacement and static type. The time at the end of the load step refers to
the ending load per load step. The numbers of sub-steps 50 are set to indicate load increments used
for this analysis, Fig. 4.12.
The analysis process for the finite element analysis of the model was set-up to examine
three different behaviors: (1) initial cracking of the beam, (2) yielding of the steel reinforcement,
and (3) the strength limit state of the beam. The Newton-Raphson method of analysis was used to
compute the nonlinear response. The application of the loads up to failure was done incrementally
as required by the Newton-Raphson procedure, Fig. 4.13. After each load increment was applied,
the restart option was used to go to the next step after convergence.
The steps taken to the initial cracking of the beam can be decreased to one load increment
to model/capture initial cracking. Once initial cracking of the beam has been passed, the load
increments increased slightly until subsequent cracking of the beam. Once the yielding of the
reinforcing steel is reached, the load increments must be decreased again further because
displacements are increasing more rapidly. Eventually, the load increment size is decreased to
capture the failure of the beam. Failure of the beam occurs when convergence fails, with this very
small load increment.
4.3.1.4 Finite Element Results
Behavior at first cracking, the analysis of the linear region can be based on the design for flexure
given in MacGregor for a reinforced concrete beam. The cracking pattern(s) in the beam can be
obtained using the Crack/Crushing plot option in ANSYS. Vector Mode plots must be turned on to
view the cracking in the model. This first crack occurs in the constant moment region, and is a
flexural crack, Fig. 4.16f. Behavior beyond first cracking, in the non-linear region of the
response, subsequent cracking occurs as more load is applied to the beam. Cracking increases in
the constant moment region, and the beam begins cracking out towards the supports, Fig. 4.16h.
Also, diagonal tension cracks are beginning to form in the model, Fig. 4.16i. This cracking
increased after yielding of reinforcement, the predicted and experimental cracking patterns of the
beams at failure are shown in Fig. 4.16j, and the occurrence of smeared cracks is indicated by
short lines, whereas discrete cracks to indicate crushed concrete is indicated by gray spots.
Generally, for all specimens, at about 19 percent of the ultimate load, the first vertical
flexural crack was formed in the region of the maximum bending moment. At about 40 percent of
the ultimate load, a sudden major inclined tension crack was formed almost in the middle part of
the shear span. With increasing the load, the inclined cracks propagated backwards until it reached
the beam bottom at the support blocks edges, as shown in Fig. 4.18i. In the mean time, the cracks
propagated above openings to point load and down opening to supports.
With further increase in the applied load, the existing vertical flexural and inclined shear
cracks were formed parallel to the original inclined cracks in the shear span as shown in Fig. 4.18j.
At about 99 percent of the ultimate load, cracks at the corner of opening to point load and support
increased and failure occurred in opening region.
111
Table 4.3 shows the finite element results for the first flexural and diagonal cracking load
and ultimate load. Figures 4.16, 4.17 and 4.18 show output of “ANSYS” package for some tested
beams. For the solid Beam S, the normally expected distribution of principal stresses has been
predicted. Compression stresses are concentrated along the load path. The tensile stresses are
eliminated in the non-linear analysis producing cracks in concrete, while tensile stresses
transferred to steel bars crossing this zone as shown in Fig. 4.16e. For beams having openings, the
load will not cross the opening instead it deviates and pass around it, and as a result the stress
redistribution increases as shown in Fig. 4.17f, 4.18e. Increasing the concrete strength tends to
increase the load capacity. The higher compressive stresses exist at nodal zones (point loads and
supports), while a reduction in the compressive stresses takes place in the inclined struts joining
the point loads and supports. This reduction is due to the diagonal cracks and the web opening
across the load path.
4.3.1.5 Comparison of the Results
To examine the accuracy of the nonlinear finite element approach, the obtained results are
compared with test results of Group B by (Javed Vaseghi Amiri and Morteza Hosseinalibygie
[19]). A comparison between the recorded experimental ultimate failure load Vu,Exp and the
predicted failure load for the tested simple ordinary beams calculated from the finite element
model Vu,FEM is given in Table 4.4. The mean value of the ratio Vu,FEM to Vu,Exp for NSC and HSC
ordinary beams is 0.99, respectively, which demonstrates that the nonlinear finite element model
provides accurate prediction of the ultimate load for the tested NSC and HSC ordinary beams.
Clear that the adopted nonlinear finite element model provides useful tool in understanding the
behavior of simple NSC and HSC ordinary beams with and without openings.
111
Table 4.3 First flexural cracking, diagonal cracking, and ultimate loads from ANSYS.
No.
Beam
First
Cracking Loads
Flexure Shear
2Vcrf
2Vcrs
(kN)
1
2
3
4
5
6
7
S
ND80X350
ND80X15-S
ND8X25-S
ND10X35-S
HD8X15-S
HD10X35-S
Analytical
Ultimate Load
2Vu, FEM
(kN)
(kN)
21.00
45.50
19.40
41.00
19.80
41.67
20.00
42.30
20.00
39.00
22.00
44.30
22.23
44.15
Average
117.22
100.00
98.45
108.64
105.00
114.20
114.86
0.18
0.19
0.20
0.18
0.19
0.19
0.19
0.19
0.39
0.41
0.42
0.39
0.37
0.39
0.38
0.40
Table 4.4 Comparison of ultimate loads.
No.
1
2
3
4
5
6
7
Beam
S
ND8X350
ND80X150-S
ND80X250-S
ND100X350-S
HD80X150-S
HD100X350-S
Experimental
Ultimate Load
2Vu, Exp
Analytical
Ultimate Load
2Vu, FEM
(kN)
(kN)
120.00
100.00
100.00
110.00
105.00
115.00
115.00
117.22
100.00
98.45
108.64
105.00
114.20
114.86
Average
0.98
1.00
0.98
0.99
1.00
0.99
0.99
0.99
111
Concrete Beam
Steel Loading Plate
S3
2ф6
S2
Stirrups ф6
S1
2ф14
Steel Support
Meshing of cross section
(a) Meshing and cross section of Beam S model.
2ф6
10ф6/m
2ф14
(b) Reinforcement configurations.
111
Loading Applied on
the Plate at each Node
Hinged Support
X- and Y- Constraints
Roller Support
Y- Constraints
(c) Applied loads and boundary conditions.
(d) Deformed shape.
(e) Vector plots of principal stresses.
119
(f) 1st cracks for flexure at load of 21kN.
(g) Flexure cracks pattern.
(h) 1st Cracks for shear at load of 45.50kN
(i) Diagonal cracks pattern.
(j) Cracks pattern at failure load of 117.22kN.
Figure 4.16 Output of “ANSYS” Program for solid Beam S.
115
Concrete Beam
Steel Loading Plate
Steel Support
(a) Beam model (Volumes Created in ANSYS).
S3
2ф6
S2
Stirrups ф6
S1
2ф14
Meshing of cross section
(b) Meshing and cross section of beam model.
(c) Reinforcement configurations.
119
Loading Applied on
the Plate at each Node
Hinged Support
X- and Y- Constraints
Roller Support
Y- Constraints
(d) Applied loads and boundary conditions.
(e) Deformed shape.
(f) Vector plots of principal stresses.
119
(g) 1st cracks for flexure at load of 19.40kN.
(h) Flexure cracks pattern.
(i) 1st Cracks for shear at load of 41kN.
(j) Diagonal cracks pattern.
(k) Cracks pattern at failure load 100kN.
Figure 4.17 Output of “ANSYS” Program for Beam ND80X350.
119
Concrete Beam
Steel Loading Plate
Steel Support
(a) Beam model (Volumes Created in ANSYS program).
(b) Meshing of beam.
(c) Reinforcement configurations.
114
Loading Applied on
the Plate at each Node
Hinged Support
X- and Y- Constraints
Roller Support
Y- Constraints
(d) Applied loads and boundary conditions
(e) Vector plots of principal stresses
(f) 1st cracks for flexure at load of 20kN
(g) Flexure cracks pattern
111
(h) 1st Cracks for shear at load 39kN
(i) Diagonal cracks pattern.
(j) Cracks pattern at failure load 105kN
Figure 4.18 Output of “ANSYS” Program for Beam ND100X350-S.
111
4.3.2 Verification Group C−Simple Beams With and Without Rectangular Openings.
4.3.2.1 Model Description and Material Properties
A nonlinear finite element analysis has been performed for two simple an ordinary beams. The
first beam was a RC solid beam without openings and the ultimate load obtained from the
experimental test was 83,000N. The second beam was a RC beam with a rectangular opening
(opening height = 100mm and opening width = 300mm). The ultimate load obtained from the
experimental test for the beam was 41,000N. The beam(s) full-size was 100mm × 2050mm ×
250mm. The span between the two supports was 2000mm, Fig. 3.22a. For tensile bars, four bars
with 10mm diameter and for compression reinforcement two bars with 10mm diameter were
used. Stirrups also have a diameter of 8mm, with 146mm distance from each other. The finite
element mesh for the solid beam is shown in Fig. 4.19b and that for the beam with two rectangular
openings is shown in Fig. 4.20b.
The concrete compressive cylinder strength for the RC solid beam without openings and
that with openings are 49MPa and 52MPa, respectively. The concrete tensile strength
is
typically 8% - 15% of the compressive strength (Shah, et al; 1995). Poisson’s ratio ν for concrete
was assumed to be 0.2 (Bangash, 1989) and 0.3 for the steel plates (Gere and Timoshenko,
1997). The modulus of elasticity Ec is taken from equation 4.1. The concrete material properties
are numbered as a material number 1. The steel yield strength fy for longitudinal reinforcements is
400MPa and yield strength of steel stirrups
is 240MPa, and the steel modulus of elasticity Es
5
equals to 2×10 MPa. The reinforcement material properties are numbered as materials 2 and 3 and
the steel plate material properties are numbered as material 4.
4.3.2.2 Meshing
The two beams are modeled using nonlinear solid element SOLID65. The overall mesh of the
concrete, plate, and support volumes is shown in Fig. 4.19b for solid beam and that for the beam
with rectangular openings is shown in Fig. 4.20b. The necessary element divisions are noted. The
meshing of the reinforcement is a special case compared to the volumes. No mesh of the
reinforcement is needed because individual elements were created in the modeling through the
nodes created by the mesh of the concrete volumes. However, the necessary mesh attributes as
described above need to be set before each section of the reinforcement is created. As Fig. 4.19b
indicates, the cross section is divided into four vertical strips (the main bottom, top, and web
reinforcement). The reinforcement is positioned at intersections of mesh nodes.
4.3.2.3 Loads and Boundary Conditions
Displacement boundary conditions are needed to constrain the model to get a unique solution. To
ensure that the model acts the same way as the experimental beam, boundary conditions need to be
applied at points of symmetry, and where the supports and loadings exist. Four stiff reinforced
concrete solid elements SOLID45 are meshed and used to model the two supports and the two
point loads, as shown in Fig. 4.19c, and Fig. 4.20c. Hinges are put at the lower parts of the
reinforced concrete elements that represent a one support and the other support is represented by a
roller. The point load applied at the steel plate is applied across the entire centerline of the plate.
The point load acting at each node (five nodes) on the plate is half of the actual force applied. This
beams group C needed to work of the twelve load step to access the fast collapse load to be closer
than the fast collapse load of experimental. The number of sub steps 70 is set to indicate load
increments used for this analysis.
111
4.3.2.4 Finite Element Results
For the two specimens and at about 20-35 percent of the ultimate load, the first vertical flexural
cracks were formed in the region of the maximum bending moment. At about 40-70 percent of the
ultimate load, a sudden major inclined tension crack was formed almost in the middle part of the
shear span. With increasing the load, the inclined cracks propagated backwards until it reached the
beam bottom at the support blocks edges, as shown in Fig. 4.19i. In the mean time, the cracks
propagated above openings to the point load, and down opening to the supports. With further
increase in the applied load, the existing vertical flexural and inclined shear cracks were formed
parallel to the original inclined cracks in the shear span, as shown in Fig. 4.19j. At about 97 to 100
percent of the ultimate load, cracks at the corner of opening to point load and support increased
and failure occurred in opening region.
The finite element results for the first flexural and diagonal cracking load and ultimate load
for the solid beam are 17.87kN, 34kN and 80.53kN, respectively. The finite element results for the
first flexural and diagonal cracking load and ultimate load for the beam with rectangular openings
are 14.86kN, 33.40kN and 42kN, respectively. Figure 4.19 and Fig. 4.20 show output of
“ANSYS” package for the two tested beams. For the solid beam, the normally expected
distribution of principal stresses has been predicted. Compression stresses are concentrated along
the load path. The tensile stresses are eliminated in the non-linear analysis producing cracks in
concrete, while tensile stresses transferred to steel bars crossing this zone as shown in Fig. 4.19f.
For the beam with rectangular openings the load path is deviated around the opening, the concrete
stresses are forced to deviate through a narrow load path, and as a result the stress redistribution
increases as shown in Fig. 4.20f. Increasing the concrete strength tends to increase the load
capacity. The higher compressive stresses exists at nodal zones (point loads and supports), while a
reduction in the compressive stresses takes place in the inclined struts joining the points load and
supports. This reduction is due to the diagonal cracks and the web opening in the load path.
4.3.2.5 Comparison of the Results
To examine the accuracy of the nonlinear finite element approach, the obtained results are
compared with the test results of the two beams Group C (Abdalla et al.) [1]. The recorded
experimental ultimate failure load Vu,Exp and the predicted failure load for the tested simple
ordinary solid beam calculated from the finite element model Vu,FEM are 83kN and 80.53kN,
respectively . The recorded experimental ultimate failure load Vu,Exp and the predicted failure load
for the tested simple ordinary beam with rectangular openings calculated from the finite element
model Vu,FEM are 41kN and 42kN, respectively. The mean value of the ratio Vu,FEM to Vu,Exp for
the two ordinary beams is 0.985, which demonstrates that the nonlinear finite element model
provides accurate prediction of the ultimate load for the tested two ordinary beams. Clear that the
adopted nonlinear finite element model provides useful tool in understanding the behavior of
simple ordinary beams with and without openings.
111
Concrete Beam
Steel Loading Plate
Steel Support
(a) Beam model (Volumes Created in ANSYS program).
Elements intersect at reinforcement
Reinforcement
(b) Meshing of beam.
Loading Applied on
the Plate at each Node
Hinged Support
X- and Y- Constraints
Roller Support
Y- Constraints
(c) Applied loads and boundary conditions.
119
Steel in two rows
2ф10
0
Stirrups Ф8
4ф10
(d) Reinforcement configurations.
(e) Deformed shape.
(f) Vector plots of principal stresses.
115
(g) 1st cracks for flexure at load 17.87kN.
(h) Flexure cracks pattern.
(i) 1st Cracks for shear at load 34kN.
(j) Diagonal cracks pattern.
(k) Cracks pattern at failure load 80.53kN.
Figure 4.19 Output of “ANSYS” Program for Solid Beam.
119
Concrete Beam
Steel Loading Plate
Steel Support
(a) Beam model (Volumes Created in ANSYS).
Elements intersect at reinforcement
Reinforcement
(b) Meshing of beam.
Loading Applied on
the Plate at each Node
Hinged Support
X- and Y- Constraints
Roller Support
Y- Constraints
(c) Applied loads and boundary conditions.
119
Steel in two rows
2ф10
0
Stirrups Ф8
4ф10
(d) Reinforcement configurations.
(e) Deformed shape.
(f) Vector plots of principal stresses.
119
(g) 1st cracks for flexure at load of 14.86kN.
(h) Flexure cracks pattern.
(i) 1st Cracks for shear at load of 33.40kN.
(j) Diagonal cracks pattern.
(k) Cracks pattern at failure load of 42kN.
Figure 4.20 Output of “ANSYS” Program
for the beam with rectangular openings.
114
4.3.3 Verification Group D: Simple Beams with Rectangular Openings.
4.3.3.1 Model Description and Material Properties
A nonlinear finite element analysis has been performed for three simple supported ordinary beams
with rectangular openings. The beams' full-size was 200/700mm×3300mm×500mm. The span
between the two supports was 3000mm, Fig. 3.24a. The tested beams have been loaded as simple
beams with one concentrated load. The details of reinforcement are as shown in Fig. 3.38. The
finite element mesh for the beam model having rectangular openings is shown in Fig. 4.21a.
The concrete compressive cylinder strength (fc ), the concrete tensile strength ( ), and the
modulus of elasticity (Ec), are as shown in Table 4.5. The concrete material properties are
numbered as a material number 1. The steel yield stress (fy) and the steel modulus of elasticity
(Es) are as shown in Table 4.5. The reinforcement material properties are numbered as materials 2,
3, 4 and 5, and the steel plate material properties are numbered as a material number 6.
Table 4.5 Material properties for concrete and reinforcement.
No. Beam
(MPa)
(MPa)
Ec (MPa)
Vertical web
reinforcement
Large* Small**
1
2
3
IT1
IT4
IT8
36.20
36.80
33.60
3.62
3.68
3.36
28500
28700
27400
6
6
6
8
8
8
Yield strength, fy (MPa)
13
513.6
513.6
513.6
10
538.8
538.8
538.8
8
6
321.8 355.2
321.8 355.2
321.8 355.2
Modulus
of
elasticity,
Es (MPa)
2x105
2x105
2x105
* Stirrups in full depth
** Stirrups in above and below of openings
4.3.3.2 Meshing
The three beams are modeled using a nonlinear solid element SOLID65. The overall mesh of the
concrete, plate, and support volumes is shown in Fig. 4.21a for Beam IT1. The necessary element
divisions are clearly noted. The meshing of the reinforcement is a special case compared to the
volumes. No mesh of the reinforcement is needed because individual elements were created in the
modeling through the nodes created by the mesh of the concrete volumes. However, the necessary
mesh attributes as described above need to be set before each section of the reinforcement is
created. The reinforcement is positioned at the intersections of mesh nodes.
4.3.3.3 Loads and Boundary Conditions
Displacement boundary conditions are needed to constrain the model to get a unique solution. To
ensure that the model acts the same way as the experimental beam, boundary conditions need to be
applied at points of symmetry and where the supports and loadings exist. Hinges are put at the
lower parts of the reinforced concrete elements that represent a one support and the other support
is represented by the roller applied at the nodes of the concrete flange. The point load applied at
the concrete support is applied across the entire centerline of the support. The point load is applied
at each node (nineteen nodes) on the concrete support. The beams in Group D need to work
through the twelve load steps to access the fast collapse load to be closer than that of the collapse
load of experimental. The number 100 of sub-steps is set to indicate the number of the load
increments used for this analysis.
111
4.3.3.4 Finite Element Results
For the three specimens and at about 23 percent of the ultimate load, the first vertical flexural
cracks were formed in the region of the maximum bending moment. At about 40 percent of the
ultimate load, a sudden major inclined tension crack was formed almost in the middle part of the
shear span. With increasing the load, the inclined cracks propagated backwards until it reached the
beam bottom at the support blocks edges. In the mean time, the cracks propagated above openings
to the point load, and down opening to the supports. With further increase in the applied load, the
existing vertical flexural and inclined shear cracks were formed parallel to the original inclined
cracks in the shear span. At about 93 percent of the ultimate load, cracks at the corner of opening
to point load and support increased and failure occurred in opening region. Table 4.6 shows the
finite element results for the first flexural and diagonal cracking load and ultimate load. Fig. 4.21
shows output of “ANSYS” Program figures for tested Beam IT1.
4.3.3.5 Comparison of the Results
To examine the accuracy of the nonlinear finite element approach, the obtained results are
compared with the test results of beams in Group D (Kiang-Hwee Tan, Mohamed A. Mansur, and
Loon-Meng Huang) [23]. The recorded experimental ultimate failure load Vu,Exp and the predicted
failure load for the tested simple ordinary beams calculated from the finite element model Vu,FEM
are given in Table 4.7. The mean value of the ratio Vu,FEM to Vu,Exp for ordinary beams is 0.93,
which demonstrates that the nonlinear finite element model provides accurate prediction of the
ultimate load for the tested ordinary beams. Clear that the adopted nonlinear finite element model
provides useful tool in understanding the behavior of simple ordinary beams with openings.
Table 4.6 First flexural cracking, diagonal cracking, and ultimate loads from ANSYS.
No.
Beam
First
Cracking Loads
Flexure Shear
2Vcrf
2Vcrs
(kN)
1
2
3
IT1
IT4
IT8
Analytical
Ultimate Load
2Vu, FEM
(kN)
(kN)
62.91
122.20
29.30
58.51
61.46
85.51
Average
305.12
130.16
245.23
0.21
0.23
0.25
0.23
0.40
0.45
0.35
0.40
Table 4.7 Comparison of ultimate loads.
No.
1
2
3
Beam
IT1
IT4
IT8
Experimental
Ultimate Load
2Vu, Exp
Analytical
Ultimate Load
2Vu, FEM
(kN)
(kN)
314.67
154.02
256.02
Average
305.12
130.16
245.23
111
0.97
0.85
0.96
0.93
Concrete support
Concrete Beam
(a) Meshing and Beam IT1 model.
Loading Applied at
each Node
Hinged Support
X- and Y- Constraints
Roller Support
Y- Constraints
(b) Applied loads and boundary conditions.
(c) Reinforcement configurations.
Figure 4.21 Output of “ANSYS” Program figures for Beam IT1.
111
4.4 ANALYSIS OF DEEP BEAMS WITH OPENINGS
4.4.1 Verification Group A: Simple and continuous Beams with Rectangular Openings.
4.4.1.1 Model Description and Material Properties
In this section, a nonlinear finite element analysis has been performed for five reinforced concrete
deep beams with openings (NSC and HSC), which were tested by Mohamed Fawzy El-Azab [11].
Two of them are simple span and three are two-span continuous deep beams. All tested beams
have the same length and thickness. The location of center lines of loads and supports were the
same for all specimens. Figures 3.39a to 3.43a show the dimensions and locations of the web
openings for all of the selected tested beams. The details of reinforcement for each beam are
shown in Table 4.8. The beams, plates, and supports were modeled as volumes in ANSYS
package. The finite element mesh for the simple beam model DSON3 is shown in Fig. 4.22b, and
that for the continuous beam model DCON3 is shown in Fig. 4.23b.
The concrete compressive cylinder strength ( ), the concrete tensile strength ( ), and the
modulus of elasticity (Ec), are as shown in Table 4.8. The concrete material properties are
numbered as a material number 1. The steel yield stress (fy) and the steel modulus of elasticity
(Es) are as shown in Table 4.9. The reinforcement material properties are numbered as materials
number 2 to 5. The steel plate material properties are numbered as material 6.
Table 4.8 Details of the tested specimens.
No.
Beam
(MPa)
(MPa)
Ec (MPa)
Web reinforcement
Vertical
1
2
3
4
5
DSON3
DSOH10
DCON3
DCOH2
DCOH8
30.45
69.60
30.30
70.90
70.90
3.04
6.96
3.03
7.09
7.09
25935
39211
25871
39575
39575
Main reinforcement
Horizontal
6
6
6
6
6
8
8
8
8
8
Bottom
4
4
4
4
4
16
16
16
16
16
Table 4.9 Reinforcement Properties.
Material
No.
iameter, mm
2
3
4
5
16
10
6
8
Area, mm2
201.06
78.540
28.270
50.270
111
Yield strength,
fy (MPa)
Modulus of
elasticity, Es
410.00
407.60
244.50
260.20
2x105
2x105
2x105
2x105
(MPa)
Top
2
2
4
4
4
10
10
16
16
16
4.4.1.2 Meshing
The beams are modeled using a nonlinear solid element SOLID65. To obtain good results from the
SOLID65 element, the use of a rectangular mesh is recommended. Therefore, rectangular elements
were created (Figs. 4.22b and 4.23b). The Volume Sweep command was used to mesh the steel
plates and supports. This properly sets the width and length of elements in the plates to be
consistent with the elements and nodes in the concrete portions of the model. The overall mesh of
the concrete, plate, and support volumes is shown in Figs. 4.22b and 4.23b for Beam DSON3 and
Beam DCON3, respectively. All other beams such this beams. The meshing of the reinforcement
is a special case compared to the volumes. No mesh of the reinforcement is needed because
individual elements were created in the modeling through the nodes created by the mesh of the
concrete volumes. However, the necessary mesh attributes as described above need to be set
before each section of the reinforcement is created. The beams have concrete material type
(material no. 1) and reinforcement materials are numbered from 2 to 5. For all specimens (simple
and continuous deep beams) the cross section is divided into three vertical strips according to the
main bottom, top, and web reinforcement, Figs. 4.22b and 4.23b. The reinforcement is positioned
at the intersections of mesh nodes.
4.4.1.3 Loads and Boundary Conditions
Displacement boundary conditions are needed to constrain the model to get a unique solution. To
ensure that the model acts the same way as the experimental beams, boundary conditions need to
be applied at points of symmetry, and where the supports and loadings exist. Four stiff reinforced
concrete solid elements SOLID45 are used to model the two supports and the two point loads,
Figs. 4.22c and 4.23c. Hinges are put at the lower parts of the reinforced concrete elements that
represent a one support and the other support is represented by a roller. The point load acting at the
steel plate is applied across the entire centerline of the plate (Fig. 4.22c). The point load acting at
each node of the four nodes on the plate is half the actual applied force. For this model, the Static
Analysis type is utilized. The Restart command is utilized to restart an analysis after the initial run
or load step has been completed The Sol’n Controls command dictates the use of a linear or nonlinear solution for the finite element model. In the particular case considered in this thesis the
analysis is small displacement and static. The time at the end of the load step refers to the ending
load per load step. The beams' Group A need to work through the ten load steps to access the fast
collapse load to be closer than the fast collapse load of experimental. The number 100 of sub-steps
is set to indicate the number of load increments used for this analysis.
The analysis process for the finite element analysis of the model was set up to examine
three different behaviors: (1) initial cracking of the beam, (2) yielding of the steel reinforcement,
and (3) the strength limit state of the beam. The Newton-Raphson method of analysis was used to
compute the nonlinear response. The application of the loads up to failure was done incrementally
as required by the Newton-Raphson procedure, Fig. 4.13. After each load increment was applied,
the restart option was used to go to the next step after convergence.
The steps taken to the initial cracking of the beam can be decreased to one load increment
to model/capture initial cracking. Once initial cracking of the beam has been passed, the load
increments increased slightly until subsequent cracking of the beam. Once the yielding of the
reinforcing steel is reached, the load increments must be decreased again further because
displacements are increasing more rapidly. Eventually, the load increment size is decreased to
119
capture the failure of the beam. Failure of the beam occurs when convergence fails, with this very
small load increment.
4.4.1.4 Finite Element Results
Behavior at first cracking (at corner of opening). It could be noticed that the opening affects the
beams stress trajectories drastically, where zones of tension stresses are formed around the leftupper corner of the opening (load side) and the corner on the same diagonal, so first cracking
occur at this corner of opening and is a shear crack, see Figs. 4.22g and 4.23g. Inversly in ordinary
beams first crack occurs in the constant moment region, and is a flexural crack (vertical cracks).
The cracking pattern(s) of the beam can be obtained using the Crack/Crushing plot option in
ANSYS. Vector Mode plots must be turned on to view the cracking in the model. Behavior
beyond first cracking. In the non-linear region of the response, subsequent cracking occurs as
more load is applied to the beam. Cracking increases out towards the supports and the beams'
flexural cracking (vertical cracks) begins in the constant moment region, Fig. 4.22h. Also,
diagonal tension cracks start to form in the model, Fig. 4.22i. This cracking increased after
yielding of reinforcement, the predicted and experimental cracking patterns of the beams at failure
are shown in Fig. 4.22j. The occurrence of smeared cracks is indicated by short lines, whereas
discrete cracks (indicator of crushed concrete) are indicated by gray spots.
Generally, for all specimens at about 43 percent of the ultimate load, the first vertical
flexural cracks were formed in the region of the maximum bending moment. At about 38 percent
of the ultimate load, a sudden major inclined tension crack was formed almost in the middle part
of the shear span. With increasing the load, the inclined cracks propagated backwards until it
reached the beam bottom at the support blocks edges, Fig. 4.22i. In the mean time, the cracks
propagated above openings to the point load, and down openings to supports. With further increase
in the applied load, the existing vertical flexural and inclined shear cracks were formed parallel to
the original inclined cracks in the shear span, Fig. 4.22j. At about 96 percent of the ultimate load,
cracks at the corner of opening to point load and at support increased and failure occurred in
opening region.
Table 4.10 shows the finite element results for the first diagonal and flexural cracking and
ultimate loads. Figures 4.22 and 4.23 show the output of “ANSYS” package for Beams DSON3
and DCON3, respectively. For those beams, the normally expected distribution of principal
stresses has been predicted. Compression stresses are concentrated along the load path. The tensile
stresses are eliminated in the non-linear analysis producing cracks in concrete, while tensile
stresses transferred to steel bars crossing this zone. For beams having openings the load path is
deviated around the opening, the concrete stresses are forced to deviate through a narrow load
path, and as a result the stress redistribution increases, Figs. 4.22f and 4.23f. Increasing the
concrete strength tends to increase the load capacity. The higher compressive stresses exists at
nodal zones (point loads and supports), while a reduction in the compressive stresses takes place in
the inclined struts joining the point loads and supports. This reduction is due to the diagonal cracks
and the web opening in the load path.
115
4.4.1.5 Comparison of the Results
To examine the accuracy of the nonlinear finite element approach, the obtained results are
compared with test results of beams from Group A (Mohamed Fawzy El-Azab) [11]. A
comparison between the recorded experimental ultimate load Vu,Exp and the predicted failure load
for the tested (simple and continuous) deep beams using the finite element modeling Vu,FEM is
given in Table 4.11. The mean value of the ratio Vu,FEM to Vu,Exp for NSC and HSC deep beams is
0.96, which demonstrates that the nonlinear finite element model provides accurate prediction of
the ultimate load for the tested simple and continuous NSC and HSC deep beams. Clear that the
adopted nonlinear finite element model provides useful tool in understanding the behavior of
simple and continuous NSC and HSC deep beams with openings.
Table 4.10 First diagonal cracking, flexure cracking, and ultimate loads from ANSYS.
No.
Beam
First
Cracking Loads
Shear Flexure
2Vcrs
2Vcrf
(kN)
1
2
3
4
5
DSON3
DSOH10
DCON3
DCOH2
DCOH8
Analytical
Ultimate Load
2Vu, FEM
(kN)
(kN)
36.70
45.45
29.20
35.60
107.30 109.80
158.70 172.50
115.00 135.80
Average
130.00
105.00
218.58
345.00
285.00
0.28
0.28
0.49
0.46
0.40
0.38
0.35
0.34
0.50
0.50
0.48
0.43
Table 4.11 Comparison of ultimate loads.
No.
1
2
3
4
5
Beam
DSON3
DSOH10
DCON3
DCOH2
DCOH8
Experimental
Ultimate Load
2Vu, Exp
Analytical
Ultimate Load
2Vu, FEM
(kN)
(kN)
140
110
220
360
290
130.00
105.00
218.58
345.00
285.00
Average
0.93
0.95
0.99
0.96
0.98
0.96
119
Steel Loading Plate
Concrete Beam
Steel Support
(a) Beam model (Volumes Created in ANSYS).
2ф10
ф6
2ф8
2ф8
4ф16
Meshing of
cross section
Elements intersect at reinforcement
Reinforcement
(b) Meshing and cross section of beam model.
119
Loading Applied at
each Node
Hinged Support
X- and Y- Constraints
Roller Support
Y- Constraints
(c) Applied loads and boundary conditions.
Steel in two rows
2ф10
0
2ф8
Stirrups ф6
4ф16
(b) Reinforcement configurations.
119
(e) Deformed shape.
(f) Vector plots of principal stresses.
114
(g) 1st Cracks for shear at load 36.7kN.
(h) 1st cracks for flexure at load 45.45kN.
(i) Diagonal cracks pattern.
191
(j) Cracks pattern at failure load 130kN.
Figure 4.22 Output of “ANSYS” Program figures for Beam DSON3.
-----------------------------------------------------------------------------------------------------------------------
Steel Loading Plate
Concrete Beam
Steel Support
(a) Beam model (Volumes Created in ANSYS Program).
191
4ф16
ф6
2ф8
2ф8
4ф16
Meshing of
cross section
Elements intersect at reinforcement
Reinforcement
(b) ) Meshing and cross section of beam model.
Loading Applied at
each Node
Roller Support
X- and Y- Constraints
Roller Support
Y- Constraints
Hinged Support
X- and Y- Constraints
(c) Applied loads and boundary conditions.
191
Steel in two rows
2ф8
4ф16
0
Stirrups ф6
4ф16
(d) Reinforcement configurations.
(e) Deformed shape.
191
(f) Vector plots of principal stresses.
(g) 1st Cracks for shear at load 107.30kN.
(h) 1st cracks for flexure at load 109.80kN.
(i) Diagonal cracks pattern.
(j) Cracks pattern at failure load 218.58kN.
Figure 4.23 Output of “ANSYS” Program for Beam DCON3.
199
4.4.2 Verification Group B: Simple Beams with Rectangular Openings.
4.4.2.1 Model Description and Material
In this section, a nonlinear finite element analysis has been performed for five reinforced Normal
Strength Concrete NSC simply supported deep beams with web openings, which were tested by
Khalaf Ibrahem Mohammad [32]. All beams, which are identical in geometry and longitudinal
reinforcement, had a thickness of 100mm and other dimensions as shown in Fig. 3.46a. The
differences between these five beams were in the web reinforcement patterns, Fig. 3.45, and
concrete properties, Table 3.14. The longitudinal reinforcement consisted of one 20mm diameter
deformed bar of 430N/mm2 yield strength, anchored to external steel blocks at the ends. Web
reinforcement consisted of 10mm diameter deformed bar stirrups of 450N/mm 2 yield strength.
Beam NO-0.3/4 having no web reinforcement, Fig. 3.44. The beams, plates, and supports were
modeled as volumes created in ANSYS. The finite element mesh for beam model NO-0.3/4 is
shown in Fig. 4.24b and the finite element mesh for beam model NW7-0.3/4 is shown in Fig.
4.25a.
The concrete compressive cylinder strength ( ) and the concrete tensile strength ( ) are
as shown in Table 3.14. The modulus of elasticity (Ec) is computed from Eq. 4.1. The concrete
material properties are numbered as a material number 1. The steel modulus of elasticity Es equals
to 2×105MPa. The reinforcement material properties are numbered as materials 2 and 3 and steel
plate material properties are numbered as material 4.
4.4.2.2 Meshing
The beams are modeled using a nonlinear solid element SOLID65. The overall mesh of the
concrete, plate, and support volumes for Beams NO-0.3/4 and NW7-0.3/4 is shown in Figs. 4.24b
and 4.25a, respectively. The necessary element divisions are clearly noted. The meshing of the
reinforcement is a special case compared to the volumes. No mesh of the reinforcement is needed
because individual elements were created in the modeling through the nodes created by the mesh
of the concrete volumes. However, the necessary mesh attributes as described above need to be
set before each section of the reinforcement is created. For the deep Beam NO-0.3/4 in Group B,
the cross section is divided into two vertical strips according to the main bottom, top, and web
reinforcement, Fig. 4.24b. For the other deep beams in Group B, the cross section is divided into
foure vertical strips according to the main bottom, top, and web reinforcement, Fig. 4.25a. The
reinforcement is positioned at the intersection of mesh nodes.
4.4.2.3 Loads and Boundary Conditions
Displacement boundary conditions are needed to constrain the model to get a unique solution. To
ensure that the model acts the same way as the experimental beam, boundary conditions need to be
applied at points of symmetry and where the supports and loadings exist. Four stiff reinforced
concrete solid elements SOLID45 are used to model the two supports and the two point loads,
Figs. 4.24c and 4.25b. Hinges are put at the lower parts of the reinforced concrete elements that
represent a one support and the other support is represented by a roller. The point load acting at the
steel plate is applied across the entire centerline of the plate. The point load acting at each node of
the three nodes of Beam NO-0.3/4 and the five nodes of the other beams on the plate is ½ the
actual force. The beams in Group B need to work through the twelve load steps to access the fast
collapse load to be closer than the fast collapse load of experimental. The number 100 of sub-steps
is set to indicate the number of load increments used for this analysis.
195
4.4.2.4 Finite Element Results
Behavior at first cracking (at corner of opening) - It could be noticed that the opening affects the
beams stress trajectories drastically, where zones of tension stresses are formed around the leftupper corner of the opening (load side) and the corner on the same diagonal, so first cracking
occur at this corner of opening and is a shear crack, see Figs. 4.24g and 4.25f. Inversely, in
ordinary beams first crack occurs in the constant moment region, and is a flexural crack (vertical
cracks). The cracking pattern(s) in of the beam can be obtained using the Crack/Crushing plot
option in ANSYS. Vector Mode plots must be turned on to view the cracking in the model.
Behavior beyond first cracking- In the non-linear region of the response, subsequent cracking
occurs as more load is applied to the beam. Cracking increases out towards the supports and the
beams' flexural cracking (vertical cracks) begins in the constant moment region; Figs. 4.24f and
4.25g. Also, diagonal tension cracks are start to form in the model. This cracking increased after
yielding of reinforcement. The predicted and analytical cracking patterns of the beams at failure
are shown in Figs. 4.24j and 4.25h. The occurrence of smeared cracks is indicated by short lines,
whereas discrete cracks (indicator for of crushed concrete) are indicated by gray spots.
Generally, for all specimens at about 52 percent of the ultimate load, the first vertical
flexural cracks were formed in the region of the maximum bending moment. At about 50 percent
of the ultimate load, a sudden major inclined tension crack was formed almost in the middle part
of the shear span. With increasing the load, the inclined cracks propagated backwards until it
reached the beam bottom at the support blocks edges. In the meantime, the cracks propagated
above openings to point load, and down opening to supports. With further increase in the applied
load, the existing vertical flexural and inclined shear cracks were formed parallel to the original
inclined cracks in the shear span. At about 98 percent of the ultimate load, cracks at the corner of
opening to point load and at support increased and failure occurred in opening region.
Table 4.12 shows the finite element results for the first diagonal and flexural cracking and
ultimate loads. Figures 4.24, and 4.25 show the output of “ANSYS” package for Beams NO-0.3/4
and NW7-0.3/4, respectively. For those beams, the normally expected distribution of principal
stresses has been predicted.
4.4.2.5 Comparison of the Results
To examine the accuracy of the nonlinear finite element approach, the obtained results are
compared with the test results of the beams in Group B (Khalaf Ibrahem Mohammad) [32]. The
recorded experimental ultimate failure load Vu,Exp and the predicted failure load for the tested
simple deep beams calculated from the finite element model Vu,FEM are given in Table 4.13. The
mean value of the ratio Vu,FEM to Vu,Exp for NSC deep beams is 0.98, which demonstrates that the
nonlinear finite element model provides accurate prediction of the ultimate load for the tested
simple NSC deep beams. Clear that the adopted nonlinear finite element model provides useful
tool in understanding the behavior of simple NSC deep beams with openings.
199
Table 4.12 First diagonal cracking, flexural cracking, and ultimate loads from ANSYS.
No.
Beam
First
cracking loads
Shear Flexure
2Vcrs
2Vcrf
(kN)
1
2
3
4
5
NO-0.3/4
NW1-0.3/4
NW2-0.3/4
NW3-0.3/4
NW7-0.3/4
Analytical
ultimate load
2Vu, FEM
(kN)
(kN)
133.80 135.40
246.00 254.20
280.77 286.50
282.90 307.50
260.00 282.50
Average
238.00
410.00
573.00
615.00
685.00
0.56
0.60
0.49
0.46
0.38
0.50
0.57
0.62
0.50
0.50
0.41
0.52
Table 4.13 Comparison of ultimate loads.
No.
1
2
3
4
5
Beam
NO-0.3/4
NW1-0.3/4
NW2-0.3/4
NW3-0.3/4
NW7-0.3/4
Experimental
ultimate load
2Vu, Exp
Analytical
ultimate load
2Vu, FEM
(kN)
(kN)
240
420
580
620
720
238.00
410.00
573.00
615.00
685.00
Average
0.99
0.98
0.99
0.99
0.95
0.98
199
Steel Loading Plate
Concrete Beam
Steel Support
(a) Beam model (Volumes Created in ANSYS Program).
1ф6
1ф20
Meshing of
cross section
(b) ) Meshing and cross section of beam model.
Loading Applied at
each Node
Hinged Support
X- and Y- Constraints
Roller Support
Y- Constraints
(c) Applied loads and boundary conditions.
199
(d) Reinforcement configurations.
(e) Deformed shape.
(e) Vector plots of principal stresses.
194
(g) 1st Cracks for shear at load of 133.8kN.
(h) Flexure cracks at load of 135.40kN.
Experimental crack patterns
at failure load 240kN
(j) Cracks pattern at failure load of 238kN.
Figure 4.24 Output of “ANSYS” Program for Beam NO-0.3/4.
151
2ф6
Sti ф10
10
2ф10
2ф10
1ф20
Meshing of
cross section
(a) ) Meshing and cross section of beam model
Loading Applied at
each Node
Hinged Support
X- and Y- Constraints
Roller Support
Y- Constraints
(b) Applied loads and boundary conditions
151
2ф6
Stirrups ф10
1ф20
(c) Reinforcement configurations.
(d) Deformed shape.
(e) Vector plots of principal stresses.
151
2ф10
(f) 1st Cracks for shear at load of 260kN.
(g) Flexure cracks pattern after appearance shear cracks.
Experimental crack patterns
at failure load 720kN
(h) FE Cracks pattern at failure load of 685kN.
Figure 4.25 Output of “ANSYS” Program for Beam NW7-0.3/4.
151
4.5 CONCLUSIONS
This chapter aims to verify the results of some available experimental beams (shallow and deep),
with a three dimensional nonlinear finite element analysis, using ANSYS-12 package.
The main conclusions drawn from this study can be summarized as follows:

The 3-D nonlinear finite element analysis of simple Normal- and High-Strength Concrete
ordinary (shallow and deep) beams with and without openings yields accurate predictions
of both the ultimate load and the complete response.

The finite element solutions show that the increase of the concrete strength results in an
increase in the cracking strength and ultimate strength.

For ordinary (shallow) beams with and without openings loaded by two symmetrical point
loads, the first crack occurs in the constant moment region. It, in general, happens in the
mid-span of beam and it is a flexural crack (vertical crack). Cracking increases with
increasing the loads in the constant moment region, and cracking propagates towards the
supports (diagonal cracks).

For deep beams with openings, the openings affect the beams' stress trajectories drastically,
where zones of tension stresses are formed around the upper corners of the openings
(nearest to the load) and around the lower corner (nearest to the supports) on the same
diagonal. These shear cracks occur, in general, in the shear-span of beam. Cracking
increases with increasing the load in the shear-span and propagates towards in the mid-span
(flexural cracks, vertical cracks).

For all considered ordinary (shallow) beams without openings, a flexural failure occurred
in mid-span of beam, after the yielding of the longitudinal rebars.

For all considered ordinary (shallow) beams with openings, a diagonal shear failure
occurred at corner of tension zones of opening, before the yielding of the longitudinal
rebars.

For all considered deep beams with and without openings, a diagonal shear failure occurred
on the shear-span or at the upper corners of the openings (nearest to the load) and around
the lower corner (nearest to the supports) on the same diagonal before yielding of the
longitudinal bars.

The mean value of the ratio Vu,FEM to Vu,Exp for NSC and HSC beams (shallow and deep)
ranging from 0.95 to 0.99, which demonstrates that the nonlinear finite element model
using ANSYS package provides accurate prediction of the ultimate load for the tested
simple and continuous NSC and HSC beams (shallow and deep). Clear that the adopted
nonlinear finite element model provides useful tool in understanding the behavior of simple
and continuous NSC and HSC beams (shallow and deep) with and without openings.

The failure modes of HSC beams (shallow and deep) with openings are similar to that of
NSC beams.
159
CHAPTERT
5
DESIGN PROCEDURE,
DETAILING, AND DESIGN
RECOMMENDATIONS FOR
BEAMS WITH OPENINGS
CHAPTER 5
DESIGN PROCEDURE, DETAILING, AND DESIGN RECOMMENDATIONS FOR
BEAMS WITH OPENINGS
5.1 INTRODUCTION
The aim of this chapter is to introduce the design steps through which the amount and details of
reinforcement around openings in reinforced concrete (shallow and deep) beams can be
determined.
5.2 SHALLOW (ORDINARY) BEAMS
In this chapter, two types of reinforced concrete (shallow and deep) beams with openings are
studied. Some of the existing design codes; e.g., the ACI 318M-11 Code [3] and the Egyptian
Code (2007) [10], define a beam to be shallow when:
1. ACI 318M-11 Code [3]:
2. Egyptian Code (2007) [10]:

ℓn /h > 4 or

L/d > 4 or

a/h > 2.

a/d > 2.
where L and ℓn is the effective span and clear span, respectively.
5.2.1 General Guidelines
A review to the literature on the behavior and strength of beams with web openings and refereeing
to Fig. 5.1 the following guidelines can be used to facilitate the selection of the size and location
of web openings:
For Tee beams, openings should preferably be positioned flushed with the flange for ease in
construction. In the case of rectangular beams, openings are commonly placed at mid-depth
of the section, but they may be placed eccentrically with respect to the depth if situation
dictates. Care must be exercised to provide sufficient concrete cover to the reinforcement for
the chord member above and below the opening. The compression chord should also have
sufficient concrete area to develop the ultimate compression block in flexure and have
sufficient depth to provide effective shear reinforcement.
Openings should not be located closer than half the beam depth, 0.5h, to the supports to avoid
the critical region for shear failure and reinforcement congestion. Similarly, positioning of an
opening closer than 0.5h to any concentrated load should be avoided.
The depth of openings should be limited to 50% of the overall beam depth.
The factors that limit the length of an opening are the stability of the chord member, (in
particular the compression chord), and the serviceability requirement of deflection. It is
preferable to use multiple openings providing the same passageway instead of using a single
long opening.
When multiple openings are used, the width of post separating two adjacent openings should
not be less than 0.5h or 100mm, whichever is larger, to ensure that each opening behaves
independently.
511
Figure 5.1 Guidelines for the location of web openings (Tan et al., 1996).
5.2.2 Design of Reinforced Concrete Beams with Small Openings Using the Traditional
Approach
Openings that are circular, square, or nearly square in shape may be considered as small openings
provided that the depth (or diameter) of the opening is in a realistic proportion to the beam size,
say, about less than or equal to 0.40 times the overall beam depth. In such a case, the beam action
may be assumed to prevail. Therefore, analysis and design of a beam with small openings may
follow the similar course of action as that of a solid beam. The provision of openings, however,
produces discontinuities or disturbances in the normal flow of stresses, thus leading to stress
concentration and early cracking around the opening region. Similar to any discontinuity, special
reinforcement, enclosing the opening close to its periphery, should therefore be provided in
sufficient quantity to control crack widths and prevent possible premature failure of the beam.
5.2.2.1 Pure Bending
In the case of pure bending, placement of an opening completely within the tension zone does not
change the load-carrying mechanism of the beam because concrete there would have cracked
anyway in flexure at ultimate, see Fig. 5.2. Mansur and Tan [26] have illustrated this through
worked out examples, supported by test evidence. Thus, the ultimate moment capacity of a beam
is not affected by the presence of an opening as long as the minimum depth of the compression
chord,
is greater than or equal to the depth of the ultimate compressive stress block, a, that is,
when
(5.1)
in which
is the area of tensile reinforcement, is the yield strength of tensile reinforcement,
is the cylinder compressive strength of concrete, and is the width of the compression zone.
However, due to reduced moment of inertia at a section through the opening, cracks will initiate at
an earlier stage of loading. In spite of this, the effects on maximum crack widths and deflection
under service load have been found to be only marginal, and may safely be disregarded in design.
511
(a) Beam with small openings.
(b) Condition through the opening at collapse.
Figure 5.2 Beam with opening under pure bending.
5.2.2.2 Combined Bending and Shear
According to the traditional design philosophy, bending moment and shear force are treated
separately. A section subject to combined bending and shear, therefore, is designed first for
bending, and second for shear. In a beam, shear is always associated with bending moment,
except for the section at inflection point. When a small opening is introduced in a region
subjected to predominant shear and the opening is enclosed by reinforcement, as shown by solid
lines in Fig. 5.3, test data reported by Hanson (1969), Somes and Corley (1974), Salam (1977),
and Weng (1998) indicate that the beam may fail in two distinctly different modes. The first type
is typical of the failure commonly observed in solid beams except that the failure plane passes
through the center of the opening (Fig. 5.3a). In the second type, formation of two independent
diagonal cracks, one in each member bridging the two solid beam segments, leads to the failure
(Fig. 5.3b). Labeled respectively as beam-type failure and frame-type failure (Mansur, 1998),
these modes of failure require separate treatment.
Figure 5.3 The two modes of shear failure around small openings.
511
Similar to the traditional shear design approach, it may be assumed in both of the two cases that
the nominal shear resistance,
is provided partly by the concrete,
and partly by the shear
reinforcement crossing the failure plane,
That is,
Design for Bending
It is preferred to put the opening close to the support where the bending moment is small
designing for bending may be carried out independently, in the usual manner, and they combined
the results to the shear design solutions. Design for bending will be for two sections; the first is at
maximum bending at mid-span of beam using an effective depth , and the second section is
through the center of opening or the flexural design of the section at the center of opening always
requires a similar amount of the tension reinforcement in the mid-span of the beam.
Shear Design for Beam-type Failure
In designing for beam-type failure, a
inclined failure plane, similar to a solid beam may be
assumed, the plane being traversed through the center of the opening as shown in Fig. 5.4.
Following the simplified approach of the ACI Code [3], the shear resistance by the concrete for
solid beams follows the following simple equation:
√
When the beam contains a small opening, Mansur (1998) proposed that the term
replaced by the net depth,
, so the shear resistance is given by:
in Eq. (5.3) be
√
in which
is the concrete cylinder compressive strength in MPa,
is the width of the web in
mm, is the effective depth in mm, and
is the diameter (or depth) of opening in mm. Equation
(5.4) is applicable for a beam made up of normal-weight concrete. For light-weight concrete
beams, an average reduction factor of 0.8 may be assumed, as suggested by the ACI Code [3].
For the contribution of the shear reinforcement, , reference may be made to Fig. 5.4. It
may be seen that the stirrups available to resist shear across the failure plane are those by the sides
of the opening within a distance
, where
is the distance between the top and bottom
longitudinal rebars, and
is the diameter (or depth) of opening, as shown in the figure. The
contribution of diagonal reinforcement, if any, intercepted by the failure plane may also be taken
into account in the calculation of shear resistance. Thus,
511
in which
and
are the contribution of vertical and diagonal reinforcement, respectively;
is the area of vertical legs of stirrups per spacing across the failure plane;
is the total area of
diagonal reinforcement through the failure surface; is the inclination of diagonal reinforcement;
is the yield strength of stirrups; and
is the yield strength of diagonal reinforcement.
Figure 5.4 Shear resistance,
provided by shear reinforcement at an opening.
Knowing the values of
and
the required amount of web reinforcement to carry the
factored shear through the center of the opening may be calculated in the usual way. This amount
should be contained within a distance
or preferably be lumped together on either
side of the opening. Other restrictions applicable to the usual shear design procedure of solid
beams must also be strictly adhered to.
Shear Design for Frame-type Failure
Frame-type failure occurs due to the formation of two independent diagonal cracks, one on each of
the chord members above and below the opening, as shown in Fig. 5.3b. It appears that each
member behaves independently similar to the members in a framed structure. Therefore, each
chord member requires independent treatment, as suggested by Mansur (1998).
In order to design reinforcement for this mode of failure, let us consider the free-body
diagram at beam opening, Fig. 5.5. Clearly, the applied factored moment,
at the center of the
opening from the global action is resisted by the usual bending mechanism, that is, by the couple
formed by the compressive and tensile stress resultants,
, in the members above and below the
opening. These stress resultants may be obtained from:
*
+
subject to the restrictions imposed by Eq. (5.1). In this equation, d is the effective depth of the
beam, a is the depth of equivalent rectangular stress block, and the subscripts t and b denote the
top and bottom cross members of the opening, respectively.
511
(a) Beam and loading.
(b) Beam segment A-B.
Figure 5.5 Free-body diagram at beam opening.
The applied shear, , may be distributed between the two members in proportion to their crosssectional areas according to Nasser et al., (1967). Thus,
[
]
and
Knowing the factored shear and axial forces, each member can be independently designed
for shear by following the same procedure as for conventional solid beams with axial compression
for the top chord and axial tension for the bottom chord.
511
5.2.2.3 Reinforcement Detailing
The reinforcement details for the solid segments of the beam should follow the normal detailing
procedure for simple and continuous beams. The opening segment requires additional
reinforcement and it should be detailed carefully keeping in mind the strength and crack control
requirements.
Consideration of beam-type failure will require long stirrups to be placed on either side of
the opening, while that of the frame-type failure will need short stirrups above and below the
opening. For anchorage of short stirrups, nominal bars must be placed at each corner, if none is
available from the design of solid segments. This will ensure adequate strength. For effective
crack control, nominal bars should also be placed diagonally on either side. The resulting
arrangement of reinforcement around the opening is shown in Fig. 5.6. Under usual
circumstances, introduction of a small opening with proper detailing of reinforcement does not
seriously affect the service load deflection. However, in case of any doubt one can follow the
procedure described for beams with large openings to calculate the service load deflections and
check them against the permissible values.
Figure 5.6 Reinforcement details around a small opening.
5.2.2.4 Numerical Example for Beam ND80X350 (Case 2) with Small Openings
A simply supported rectangular beam 125mm wide and 250mm height carries two concentrated
and symmetrical factored loads 50kN on a span of 1.60m, Fig. 5.7. The beam contains a 80mm
diameter circular opening located at mid-depth of the beam and at a distance of 350mm from the
left support. It is required to design the shear reinforcement for the opening segment of the beam.
Given
, and
515
Pu = 50kN
Pu = 50kN
Vu
S.F.D
B.M.D
Mu
Figure 5.7 Beam ND80X350 and loading.
Solution:
(a) Analysis
At the center of opening, we get, Vu = 50kN and Mu = 17.50kN-m
Since the beam is subjected to combined bending and shear, Eqs. (5.7a) and (5.7b) will be used
and give:
[
]
[
]
(b) Design for flexure
Flexural design of the section at mid-span:
*
+
*
511
+
The flexural design of the section at the center of opening always requires a similar amount of the
tension reinforcement in the mid-span of the beam, 2 14. Provide 2ф6 bars on the compression
side of the beam for anchorage of stirrups.
(c) Shear design for beam-type failure
Assuming 20mm clear concrete cover and 6mm bars for stirrups,
Check the section adequacy:
Maximum shear. In order to ensure yielding of steel reinforcement when the failure strength in
shear is reached and, hence, avoid web crushing failure, the upper limit of the factored shear is
given by:
√
In the ACI [3] Strength Design Method for shear, it is required that
where
is the factored shear force and
is the strength reduction factor (0.85 for shear).
From Eq. (5.10) we get,
√
which is greater than
therefore, the section is adequate.
Design of full-depth stirrups:
Maximum spacing of stirrups. For shear reinforcement to function effectively without any
localized failure before the load-carrying mechanism is fully established, some limitations on
maximum spacing of stirrups must be followed. Assuming that
is given by Eq. (5.4), ACI Code
[3] limits the stirrup spacing to:
511
=65.24kN, hence shear reinforcement is
required,
and
Assuming that the shear resistance of the steel is provided by vertical stirrups only and that twolegged ф6mm stirrups (2-legs) are used, the required number of stirrups, n, is given by:
[
(
]
)
using of two full-depth stirrups on either side of the opening at a spacing of 50mm and positioning
small stirrups above and below the opening, ,
would satisfy the requirements of
maximum spacing. In addition, nominal diagonal bars for crack control to be provided.
(d) Shear design for fram-type failure
Member below the opening (tension member):
For this section,
therefore,
√
Which is greater than
compression failure.
and, thus, the section is adequate to avoid diagonal
Neglecting the contribution of concrete and using two-legged stirrups of ф6mm bars will yield
(
)
As
is less than
the maximum is
which is quite small.
Considering the difficulty in achieving proper compaction of concrete and keeping in mind that
diagonal bars for crack control would resist part of the applied shear, it is decided to use 5 short
stirrups below the opening in between the full-depth stirrups, which gives a spacing of about
15mm. Provide two nominal ф6mm longitudinal bars just below the opening for anchorage of
stirrups.
511
Member above the opening (compression member):
Since the section for this member having identical dimension to that of the section below, and it is
subjected to axial compression, the same spacing of stirrups can used and it would provide a
conservative design and avoid any confusion during construction.
(e) Design for crack control
The reinforcement designed as above would ensure adequate strength. However, due to sudden
reduction in beam cross section, stress concentration occurs at the edge of the opening. Adequate
reinforcement with proper detailing should therefore be provided to prevent wide cracking under
service load conditions.
In the case of small openings, the reinforcement requirements for crack control are quite
less. Since full-depth stirrups are already provided by the sides of the opening to ensure adequate
strength, provision of diagonal reinforcement may be considered to restrict the growth of cracks
along the failure plan. An amount of diagonal reinforcement that is sufficient to carry the total
shear along the
failure plane (beam-type failure) has recently been recommended by Mansur
(1998). Thus, the total area of diagonal reinforcement,
through the failure surface (Fig. 5.4) is
in which
is the inclination of diagonal reinforcement and
is the yield strength of diagonal
reinforcement. This amount should be distributed equally on either side of the opening and be
placed perpendicular to this reinforcement to avoid confusion during construction and to take care
of any possible load reversal.
For this example, use diagonal reinforcement to avoid crack control only under service load
condition. Using Eq. (5.15), and assuming
the required area of diagonal
reinforcement is
Use 4 8 diagonal bars in each direction.
511
(f) Reinforcement details
The final arrangement of reinforcement in the opening region of the beam is shown in Fig.5.8.
2ф6
2Ф8
2Ф8
2Ф8
2Ф8
Stirrups
ф6
80
2ф6
2ф6
2ф14
Figure 5.8 Reinforcement details of the Beam ND80X350.
5.2.3 Redesign of Reinforced Concrete Beam ND80X350 using Strut-and-Tie Method
The first step in this method is to visualize the flow of forces from the applied loads to the
supports. This is accomplished from the obtained elastic principal stress trajectories from finite
element analysis. The finite elements with compression stress trajectories will be replaced by
compression elements (Struts) and the finite elements with tension stress trajectories will be
replaced by tension elements (Ties). Once the model is obtained, the forces in the struts and ties
can be calculated from statics. The required area of tension tie reinforcement is then chosen.
(a) Develop the strut-and-tie model
A Strut-and-Tie Model for the Beam ND80X350 in this example is shown in Fig. 5.9. Here, the
compressive struts are shown in dotted lines while tension ties are shown in solid lines.
Figure 5.9 Proposed strut-and-tie model for Beam ND80X350.
511
Figure 5.10 Details of the strut-and-tie model for Beam ND80X350.
(b) STM forces
The forces in all members are determined from statics and their magnitudes in kN are as indicated
in Table 5.1. The struts, ties, and nodes are labeled as in Fig. 5.10. From equilibrium,
*
+
*
*
+
Node 1:
kN
kN
Node 2:
kN
kN
Node 3:
……… (1)
511
+
……… ( )
Solving Eqs. 1 and 2 yields:
Node 6:
Solving Eqs. 1 and 2 gives:
Table 5.1 STM forces.
Force,
kN
70.71
50.00
127.40
46.20
53.86
50.64
50.00
50.00
127.40
18.40
Member
S1
S2
S3
S4
S5
S6
T1
T2
T3
T4
(c) Reinforcement sizes:
a- Longitudinal reinforcement
Bottom chord member
For the bottom reinforcement
511
Since curtailment of reinforcement is not possible before reaching the throat section, the same
reinforcement is continued throughout the length of the bottom chord of opening segment.
Top chord member
Assuming that concrete will carry all of the compressive force, use compression reinforcement
only to anchor vertical stirrups.
b- Transverse reinforcement
Design for crack control
In this example, add diagonal reinforcement for crack control only under service load condition.
Using Eq. (5.15) and assuming that
the required area of diagonal reinforcement is
Use 4 8 diagonal bars in each direction.
Figure 5.11 Reinforcement details of the Beam ND80X350 using strut-and-tie method.
511
5.2.4 Design of Reinforced Concrete Beams with Large Openings Using the Traditional
Approach
Openings that are circular, square, or nearly square in shape may be considered as large openings
provided that the depth (or diameter) of the opening is in a realistic proportion to the beam size,
say, about greater than 0.40 times the overall beam depth. The introduction of a large opening in a
reinforced concrete beam would normally reduce its load-carrying capacity considerably.
However, it is possible to reinforce such a beam, restoring its strength to that of a similar solid
beam.
5.2.4.1 Available Design Procedures
In the next proposed design procedure, the ACI code [3] has been followed throughout unless
otherwise stated. In general, the design of reinforced concrete structures involves:
Structural analysis, whereby the structure is analyzed to determine the distribution of shear
forces and moments due to ultimate loads. All possible loading combinations are considered,
and the bending moment and shear forces envelopes are determined accordingly.
Strength design, wherein the critical sections are designed for ultimate strength in bending,
shear, and torsion. The strength requirements are fulfilled throughout the whole structure.
Serviceability design, to ensure that the structure performs its intended functions satisfactorily
under working loads.
Structural Analysis
In the case of a statically determinate beam, the shear force and bending moment envelopes can be
obtained from statics.
Bending moment and shear force envelopes
The beam can be analyzed for all possible load combinations by any finite element elastic method
to obtain the shear force and bending moment envelopes.
Design for Strength
Knowing the bending moment and shear force envelopes, the solid segments of the beam can be
designed in the usual manner. The recommended design process for the opening segment is based
on the observed Vierendeel truss behavior of chord members at an opening. That is, consistent
with test results, contraflexure points are assumed at midspan of chord members, for which the
axial load is obtained by dividing the beam moment at the center of the opening by the distance
between the plastic centroids of the chord members. The shear force acting at the center of the
opening is distributed between the chord members according to their relative flexural stiffnesses.
Such an assumption has been found to give a realistic distribution of the applied shear (Barney et
al., 1977) and simplifies the calculation. Methods used in the design; plastic hinge method and
strut-and-tie method. The steps involved are summarized as follows:
511
Step 1: Forces and moments in chord members
In practice, beams are usually subject to combined bending and shear. Figure 5.12 shows a
simply-supported reinforced concrete beam with an opening, subjected to a uniform load. The
free-body diagram at the beam opening can be represented as in Fig. 5.12b, and the free-body
diagrams of the chord members above and below the opening as in Fig. 5.12c. It is observed that
the unknown action effects at the center of the opening are the axial forces (Nt and Nb), the
bending moments (Mt and Mb), and the shear forces (Vt and Vb) in the chord members. There are
three
equilibrium
equations
relating
these
six
unknowns.
These
are:
In which Mm and Vm are the applied moment and shear force, respectively, at the center of the
opening. Determine the ultimate design bending moment, Mm, and shear force, Vm, at the middle
of the opening segment from bending moment and shear force envelopes obtained from the global
action, and calculate axial forces Nt and Nb (positive for compression) acting in the top and bottom
chords, respectively, as:
Where Z is the distance between the plastic centroids of the top and bottom chords. Distribute the
applied shear between the top and bottom chords in proportion to their gross flexural stiffnesses
as:
(
)
(
)
515
Figure 5.12 Beam with an opening under bending and shear. (a) The beam; (b) Free-body
diagram of opening segment; (c) Free-body diagram of the chords.
Where and are the shear forces carried by the top and bottom chords, respectively.
Calculate moments at the ends of the top and bottom chords from statics (refer to Fig. 5.12):
Where W is the uniformly distributed load acting directly on top chord and M is the moment. The
subscripts 1, 2, 3, and 4 designate the opening corners as shown in Fig. 5.12. The collapse
mechanism consistent with experimental observations, the assumed mechanism consists of four
hinges in the chord members, with one at each corner of the opening, as shown in Fig. 5.13
Figure 5.13 Assumed collapse mechanism for a beam with large openings.
511
When the chord members are symmetrically reinforced then the moments at two ends of each
chord member (potential hinge location) are numerically the same at plastic collapse. That is, M1
= M2 and M3 = M4. From the free-body diagram of the chord members (Fig. 5.12c), it may be
readily shown that the contraflexure points occur at midpoint of the chord members. This means
that Mt = 0 and Mb = 0.
Step 2: Stability of compression chord
When the section being analyzed is a T-beam, the effective width of flange in determining the
properties and capacities of the compressive strut should not exceed the limits set by the ACI
Code [3]. Where the opening segment is subject to positive bending (for example, in the mid span
region of a continuous beam), the compression (top) chord will be restrained by the continuity of
the slab and, thus, may be considered as a non-sway frame member for which, according to ACI
Code, the effect of slenderness may be neglected when:
in which the effective length factor K is taken as 1,
is the unsupported length of the
compression chord, and r is the radius of gyration. The value of
can be taken as M3
and M1, respectively, with the signs as directed by ACI Code. According to the Code, r can be
taken as:
where dc can be taken as the depth of the compression chord. However, when an opening segment
is subject to negative bending (for example, in between the inflection points and the support of a
continuous beam), the compression (bottom) chord should be considered as a non-sway frame
member for which, according to ACI Code, the effects of slenderness may be neglected when:
If Eq. (5.13) or Eq. (5.15) is not satisfied, the moment magnification method as described in ACI
Code may be used to design the compression chord. However, it is suggested that the dimensions
of the compression chord be revised so as to eliminate the effects of slenderness.
Step 3: Design of longitudinal reinforcement for chord members
The longitudinal reinforcement in the top and bottom of the solid section adjacent to the opening
should be continued throughout the opening segments. Additional reinforcement required to resist
the combined moment and axial force in each chord member is designed and, as a trial, it could be
such that each chord is symmetrically reinforced. Use the same amount and arrangement at its
bottom as additional reinforcement required to restore the strength and avoid brittle failure of the
beam due to the provision of openings. With the reinforcement for the chord members so decided,
the corresponding idealized column interaction diagrams can be constructed by the method of
strain compatibility. The critical combinations of bending moment and axial load for the chord
members as determined earlier are then plotted in the interaction diagrams. If all the combinations
fall within the appropriate interaction diagrams, the reinforcement provided will be sufficient.
511
Otherwise, a revision of reinforcement is necessary. Also, the flexural capacity of the top chord
should be sufficient to support any direct external loading.
Step 4: Design of shear reinforcement for chord members
The shear forces carried by the top and bottom chords are given by Eqs. (5.7a) and (5.7b),
respectively. Knowing these forces, the required amount of reinforcement can be designed in a
manner similar to reinforced concrete beams and slabs. However, according to ACI Code, the
effects of axial forces in the chord members must be accounted for in design. For a T-beam where
the opening is placed flushed with the flange, the top chord can be considered as a slab. Although
the flange may be too shallow for effective placement of shear reinforcement, the shear stresses
are usually low and, consequently, shear reinforcement would not usually be necessary in the top
chord.
Step 5: Multiple openings and design of post between openings
When multiple openings are placed close to each other in a beam, the element between two
adjacent openings is known as a post. Proper design and detailing for the post should be provided.
Tests carried out at the laboratory of Portland Cement Association (ACI-ASCE, 1973) have
indicated that closely spaced multiple openings can be placed in a beam if each opening has
adequate side reinforcement. Specimens with multiple circular and oval holes failed in the chord
members when the width of the post was equal to or greater than 3/8 the depth of the web. Figure
5.14 shows an inverted T-beam with multiple rectangular openings separated by adequately
reinforced posts after it has been tested to failure.
Figure 5.14 Failure of a beam with multiple rectangular openings separated by adequately
reinforced post.
To ensure that the posts behave rigidly, Barney et al. (1977) recommended that adjacent
openings should be separated by posts having overall width-to-height ratios of at least 2.0 where
the width of the posts is the distance between adjacent stirrups. It was also suggested that nominal
design shear stress for the posts be limited to
(MPa).
√
When two openings are placed close to each other, it is evident from the free-body diagram
shown in Fig. 5.15 that a horizontal shear, Vp, compression force, Np, and bending moment, Mp,
act on the post between the openings. Assuming that points of contraflexure occur at the midlength of the chord members in each opening, equilibrium of forces gives:
511
Vp = T2 – T1
Np = Vb1 – Vb2
(
)
(
)
(
)
Where
= width of post, taken as the distance between vertical stirrups in post adjacent to the
sides of the two openings; T = tensile force acting on the bottom chord member of opening;
=
vertical shear force acting on the bottom chord member of opening; e = eccentricity of opening; d p
= depth of bottom chord member;
= length of opening, taken as the distance between the
vertical stirrups adjacent to the two sides of the opening; do = depth of opening; and subscripts 1
and 2 refer to the openings to the left and right of post, respectively. Knowing the values of Vp, Np,
and Mp, the required reinforcement can be obtained by designing the post as a short, braced
column.
Figure 5.15 Forces acting on the post between adjacent openings.
Step 6: Design for serviceability
The two important serviceability requirements to be met are cracking and deflection.
a- Cracking
Assuming that the crack control requirements of the solid segments are met either by proper
reinforcement detailing or by physical calculation, the following crack control provisions are
recommended for the critical sections at corners of the opening. At each vertical edge of the
opening, a combination of vertical stirrups and diagonal bars would be used with a shear
concentration factor, , of 2 such that at least 50% of the shear resistance is provided by the
diagonal bars (Tan, 1982). Thus, for each side of the opening, the required area of vertical stirrups,
Av, is given by:
In which V,
and
are the design shear, capacity reduction factor, and yield stress of stirrups,
respectively. The vertical stirrups should be placed as close to the edge of the opening as permitted
by the required concrete cover. The required area of diagonal reinforcement, Ad, is given as
511
Where
is the yield stress and is the angle of inclination of the diagonal bars to the beam axis.
To avoid confusion during construction and to account for any possible load reversal, the same
amount of diagonal reinforcement should be provided both at the top and bottom corners of the
opening.
b- Deflection
The indirect way of satisfying the serviceability requirement of deflection by limiting the spaneffective depth ratio is not valid for a beam with openings. Therefore, an estimate of the actual
service load deflection is necessary. For this purpose, the method used for the analysis of the beam
at ultimate load may be used. Since the reinforcement details are fully known, a conservative
estimate of service load deflection may be obtained and checked against code requirements by
using the cracked moment of inertia of various segments.
The model shown in Fig. 5.16 considers that the chord members act as struts framing into
rigid abutments on each side of the opening. The effective length,
of the struts is
conservatively taken as the distance between the full-depth stirrups on each side of the opening.
To reflect the Vierendeel truss action observed in the tests, points of contraflexure are assumed at
the mid-length of each strut. Thus each half of the chords bends as a cantilever, as shown.
Denoting the moments of inertia for the top and bottom struts as It and Ib, respectively, the relative
displacement of one end of the opening with respect to the other under the action of V may be
obtained as
Where Ec is the modules of elasticity of concrete. Under service load, It may be based on gross
concrete section while Ib can be conservatively based on a fully cracked section.
Figure 5.16 Idealized model for the estimation of deflection at opening (Barney et al., 1977).
511
The maximum deflection of the beam can be calculated as
Where w is the maximum deflection in the absence of opening. A more rigorous method to
calculate deflections that entails an elastic analysis is also available (Mansur et al., 1992). In the
method, the beam is treated as a structural member with several segments constituting the portions
with solid beam sections and those with sections traversed by the opening. An equivalent stiffness
is adopted for the latter segments and the beam can be analyzed using methods such as the Direct
Stiffness Method to obtain the maximum beam deflection under service load.
Step 7: REINFORCEMNT DETAILING
The reinforcement details for the solid segments of the beam should follow the normal detailing
procedure for simple and continuous beams. The opening segment requires additional
reinforcement, and it should be detailed carefully keeping in mind the strength and crack control
requirements.
Similar to a beam with small openings, incorporation of a large opening in the pure
bending zone of a beam will not affect its moment capacity provided that the depth of the
compression chord hc is greater than or equal to the depth of ultimate compressive stress block a,
and that instability failure of the compression chord is prevented by limiting the length of the
opening (Mansur and Tan, 1999).
In practice, openings are located near the supports where shear is predominant. In such a
case, tests have shown that a beam with insufficient reinforcement and improper detailing around
the opening region fails prematurely in a brittle manner (Siao and Yap, 1990). When a suitable
scheme consisting of additional longitudinal bars near the top and bottom faces of the bottom
and top chords, to resist the combined moment and axial force in each chord member is designed
and, as a trial, it could be such that each chord is symmetrically reinforced, as shown in Fig. 5.17.
Short stirrups in both the chords, as shown in Fig. 5.17, to resist the shear forces carried
by the top and bottom chords, then the chord members behave in a manner similar to a Vierendeel
panel and failure occurs in a ductile manner. The failure of such a beam is shown in Fig. 5.18.
Clearly, the failure mechanism consists of four hinges, one at each end of the top and bottom
chords.
The critical section for cracking at corners of the opening, at each vertical edge of the
opening, a combination of vertical stirrups and diagonal bars would be used. At least 50% of
the shear resistance is provided by the diagonal bars (Tan, 1982).
Figure 5.17 A suitable reinforcement scheme for the large opening.
511
Figure 5.18 Ductile failure of a beam under combined bending and shear.
5.2.4.2 Numerical Example Case 2- Beam (Group C) with Large Rectangular Openings
100×300mm
A simply supported rectangular beam 100mm wide and 250mm height carries two concentrated
and symmetrical factored loads 20.5kN on a span of 2.0m is shown in Fig. 5.19. The beam having
a rectangular openings 100×300mm, located at the shear-span of the beam. Provide a suitable
design for the beam with particular emphasis on the opening segment of the beam. Given
, and
Pu = 20.5kN
Pu = 20.5kN
Vm = 20.5kN
S.F.D
Mm = 7.38kN.m
B.M.D
Figure 5.19 Beam and loading.
511
Solution:
1. Structural Analysis
(a) Bending moment and shearing force diagrams
Bending moment and shearing force diagrams for this beam are shown in Fig. 5.19.
2. Design for Strength
(a) Solid section
The flexural design of the solid section at mid-span of the beam is as follows:
*
+
*
+
(b) Opening segment
For the opening segment, the axial loads and shear forces in chord members (refer to Fig. 5.12) are
evaluated from the bending moment
and shear force
at the center of opening using Eqs.
(5.19) to (5.22) with
The secondary moments of the critical end sections, calculated by Eqs. 5.23 to 5.26 are as follows:
and
(
)
(
)
511
Since the top chord of the opening segment is under compression, it may be considered a nonsway frame member for which, according to ACI Code, the effect of slenderness may be neglected
when
Since
Hence, the compression chord is satisfactory with regarding stability.
Longitudinal reinforcement: The solid section adjacent to the opening has
bottom and
top rebars. Because Contraflexure points occur at midspan, symmetrical arrangements of
reinforcement to be provided to top of the bottom chord (
and
to the bottom of the
top chord. This is the amount of reinforcement to be checked using the interaction diagrams for
the chord members. If all points fall within the respective interaction diagrams, the amount of
reinforcement provided is satisfactory.
Shear reinforcement: The shear at opening center is 20.5kN. Therefore, the design shear force is
The top chord is subject to combined bending and axial compression, thus the shear strength of
concrete is (according to the ACI Code [3], sec. 11.2.1.2) as follows:
(
) √
But not less than zero, where λ =
for s nd-lightweight concrete and 0.75 for all-lightweight
⁄ shall be expressed in MPa.
concrete,
is positive for compression
511
(
√
)
The top chord is treated as a beam, therefore a minimum amount of links must be provided. The
maximum spacing limit is ⁄
. Therefore, provide stirrups of diameter 8mm
spaced at 30mm.
The bottom chord is subject to combined bending and axial tension, thus the shear strength of
concrete is (according to the ACI Code [3], sec. 11.2.2.3) as follows:
(
Where
is negative for tension
(
) √
⁄
shall be expressed in MPa.
)
√
2. Design for Serviceability
(a) Cracking
Assuming that the crack control requirements of the solid segments are met either by proper
reinforcement detailing or by physical calculation. For the opening segment, the maximum shear
is Vm = 20.5kN.
Assuming  = 2, α =
and 75% of total shear is carried by the
diagonal bars, the required areas of vertical stirrups and diagonal bars (refer to Eqs. 5.33 and 5.34)
are:
(b) Deflections
The service load is
, and the corresponding shear force is
12.1kN. The effective length of chord members for deflection calculations is taken as
515
Based on gross section properties, the moment of inertia of the chord members is equal to
For the estimation of deflection, conservatively assume
Also, the modulus of elasticity of concrete is
√
√
Hence, from Eq. (5.35), the deflection due to shear force at the opening is
For the load arrangement shown in Fig. 5.19, the midspan deflection for a beam without openings,
w is
w
Hence, the total midspan deflection of the beam can be calculated as
3. Details of Reinforcement
The reinforcement details for the solid segments of the beam should follow the normal detailing
procedure for simple beams. The opening segments require additional reinforcement, and it should
be detailed carefully keeping in mind the strength and crack control requirements. Figure 5.20
shows the final arrangement of reinforcement for the opening segment.
511
Stirrups ф8mm @ 30mm
2Ф10
1ф6
Vertical Stirrup
2Ф10
2×2Ф10
Diagonal bars
2×2Ф10
Diagonal bars
3Ф10
1ф6
Vertical Stirrup
3Ф10
Stirrups ф8mm @ 30mm
Figure 5.20 Reinforcement details at opening segment.
5.2.5 Redesign for the previous Case 2-Beam (Group C) with Large Rectangular Opening
100×300mm using Strut-and-Tie Method
The first step in this method is to visualize the flow of forces from the applied loads to the
supports. This is accomplished from the obtained elastic principal stress trajectories from the 3D
nonlinear finite element analysis. Compression stress trajectories will be replaced by compression
elements (Struts) and tension stress trajectories will be replaced by tension elements (Ties). Once
the model is established, the forces in the struts and ties can be calculated from statics. The
required reinforcements of the tension ties can then be chosen.
(a) Developing the strut-and-tie model
A strut-and-tie model for the considered beam is shown in Fig. 5.21. Here, the compressive struts
are shown in dotted lines while tension ties are shown in solid lines.
Figure 5.21 Proposed strut-and-tie model.
511
(a) Strut labels for strut-and-tie model.
(b) Tie labels for strut-and-tie model.
Figure 5.22 Details of the strut-and-tie model.
511
(b) STM forces
The forces in all members are determined from statics and their magnitudes in kN are as indicated
in Table 5.2. The struts, ties, and nodes are labeled as in Fig. 5.22.
Table 5.2 STM forces.
T
Model Force, T or Model
or
Label
kN
C
Label
C
1
25.97 C
20
15.72
C
3
2
17.35 C
21
7.74
C
4
3
33.30 C
22
42.91
C
5
4
27.49 C
23
33.09
C
6
5
24.30 C
24
23.27
C
7
6
10.74 C
25
10.99
C
8
7
32.42 C
26
1.17
C
9
8
13.67 C
27
2.29
C
10
9
15.10 C
28
12.12
C
11
10
6.99
C
29
24.39
C
12
11
22.24 C
30
34.22
C
13
12
13.70 C
31
44.04
C
14
13
5.15
C
32
53.86
C
15
48.51
14
17.65 C
33
C
16
15
7.82
C
34
2.05
C
17
16
16.41 C
35
23.78
C
18
17
18.57 C
03
60.57
C
19
18
21.46 C
1
15.94
T
20
19
13.89 C
2
36.00
T
21
T = Tension (Tie) and C = Compression (Strut)
Model Force,
Label
kN
Force,
kN
T or
C
39.70
21.42
6.84
24.54
37.85
18.41
3.84
31.15
22.61
14.07
3.39
4.23
12.77
21.32
31.99
40.54
49.08
57.62
66.16
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
Model Force, T or
Label
kN
C
22
23
24
25
26
27
82
82
03
03
08
00
03
03
03
03
-
10.68
8.95
27.49
17.65
7.80
2.02
11.87
25.00
13.13
28.30
18.48
8.66
9.82
7.53
17.35
60.57
-
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
-
(c) Rebar's Sizes
Longitudinal reinforcement
Bottom chord member
For the top reinforcement, the maximum tensile force is 39.70kN. Therefore, the required area is
Therefore, provide
bars at the top of the bottom chord.
For the bottom reinforcement, the maximum tensile force is 60.57kN. Hence, the required area is
Therefore, provide
bars at the top of the bottom chord.
Top chord member
The maximum compressive force in strut S36 is 60.57kN. Referring to the prismatic section on the
right of the beam under the same moment and shear force, the concrete may be assumed to take
67.90kN, no remaining force to be resisted by compressive steel reinforcement. So concrete
511
carried all compressive force, so use compression reinforcement of
the top of the top chord member to anchor vertical stirrups.
(as a minimum) only at
The maximum tensile force is 28.30kN. Hence, the required area of bottom steel is
Therefore, provide
bars at the bottom of the top chord.
Transverse reinforcement
Top and bottom chords
The maximum tensile force in the vertical Tie T22 is 10.68kN. Therefore,
(
)
(
Therefore, provide
)
stirrups (2-legs) spaced at 40mm.
Sides of opening
Left side
The maximum tensile vertical force in Tie T2 is 36kN. Therefore, the area of shear reinforcement
required at the low-moment end of the left side of opening is
Therefore, provide
stirrups (2-legs full depth, 201.06mm2) to the left of opening. Also
use
stirrups (2-legs full depth) to resist the force in tie T7 which is 37.85kN.
Right side
The maximum tensile vertical force in Tie T29 is 25kN. Therefore, the area of shear reinforcement
required at the high-moment end of the right side of opening is
Therefore, provide
stirrups (2-legs full depth, 113.10mm2) to the right of opening. Also
use
stirrups (2-legs full depth) to resist the force in Tie T24 which is 27.49kN.
(d) Design for crack control
In this example, use diagonal reinforcement to avoid cracking only under service load conditions.
The factored shear at the center of opening is
and the design shear (assuming a
shear concentration factor of 2) is 2
Therefore, provide additional diagonal
bars at 450 to resist the remaining shear. The required area is
511
Stirrups ф6mm @ 40mm
2Φ6
2ф6 Vertical Stirrup
2ф6 Vertical Stirrup
2Φ8
2×2Ф8
Diagonal bars
2×2Ф8
Diagonal bars
2Φ8
2ф8 Vertical Stirrup
2ф8 Vertical Stirrup
2Φ10
Stirrups ф6mm @ 40mm
Figure 5.23 Reinforcement details at opening segment using STM.
5.3 DEEP BEAMS
Some of the existing design codes; e.g., the ACI 318M-11 Code [3] and the Egyptian Code (2007)
[10], define a beam to be deep when:
1. ACI 318 M-11 Codes [3]:
2. Egyptian Code (2007) [10]:

ℓn /h ≤ 4 or

L/d ≤ 4 or

a/h ≤ 2.

a/d ≤ 2.
5.3.1 A general Procedure for Strut-and-Tie Modeling for Discontinuity Regions
The process used in the developing STM models for discontinuity regions is illustrated in Figure
5.24.
STEP 1 – DISTINGUISH THE D-REGIONS
As discussed in previous, using St Ven nt’s principle, D-region is assumed to extend a distance
equal to the largest cross-sectional dimension of the member away from a geometrical
discontinuity or a large concentrated load. The determined B-/D-region interface is the assumed
location where the stress distribution becomes linear again. Using this basic assumption, the Dregions can be described. The entire deep beam usually is a disturbed region.
STEP 2 – DETERMINE THE BOUNDARY CONDITIONS OF THE D-REGIONS
Once the extent of a D-region has been determined, the bending moments, shear forces, and axial
forces must be determined at the B-/D-region interface from analysis of the B-region and are then
used to determine the stress distributions at the B-/D-region interface. These stress distributions
can then be modeled as equivalent point loads having locations and magnitudes which can be
determined directly from the stress distributions. When determining the boundary conditions on
the B-/D-region interface, it is essential that equilibrium be maintained on the boundary between
B-and D-regions. If the bulk of the structure falls into a D-region it may be expedite to use a
global model of the structure and use the external loads and reactions as the boundary conditions.
511
Distinguish D-regions
from B-regions
Determine the Boundary
Conditions of the D-region
Sketch the Flow of Forces
Through the D-region
Develop a STM that is Compatible
with the Flow of Forces
Calculate Strut and Tie Forces
Select Steel for Ties and
Determine its Location
Yes
No
Steel Fits in Assumed
STM Geometry
Change Location of
Ties and Modify STM
Check Stress Levels in
Strut and Nodes
Yes
No
Stress levels
Okay
Modify STM by Changing
Tie Locations or Increasing
Bearing Areas or Increasing
Member Geometry
Detail Steel Anchorage
and Required Crack
Control Steel
Figure 5.24 Flowchart illustrating STM steps, Brown et al. [9].
511
STEP 3 – SKETCH THE FLOW OF FORCES
After the stress distributions acting on the B-region/D-region interface have been modeled as
equivalent point loads, the flow of forces through the D-region should be determined. For most
design cases, the flow of forces can easily be seen and sketched by the designer. When the flow of
forces becomes too complex to be approximated with a sketch, a finite element analysis can be
used to determine the flow of forces through a reinforced concrete structural member. For most
D-regions, such efforts are unwarranted since the stress paths can be estimated easily. Another
method used to determine the flow of forces is the load path method as proposed by Schlaich et al.
(1987).
STEP 4 – DEVELOP A STM
A STM should be developed to model the flow of forces through the D-region determined in the
previous step. When developing a STM, try to develop a model that follows the most direct force
path through the D-region. Also, avoid orienting struts at small angles when connected to ties.
According to Collins and Mitchell, as the angle between a strut and tie decreases, the capacity of
the strut also decreases (1986). For this reason, many design specifications specify a minimum
angle between struts and ties of about 25 degrees. It should be noted that the AASHTO LRFD
provisions do not specify a minimum angle between struts and ties; however, the limiting strut
compressive stress equation defined in the specification is a function of the angle between the strut
and tie and decreases as the angle between the strut and tie decreases.
A D-region may be subjected to more than one type of loading. It is imperative that a
STM be developed and analyzed for each different loading case. On a similar note, for a given
load case for a D-region, more than one STM can be developed. Schlaich and schäfer [37]
suggested that models with the least and shortest ties are the best. In addition, Schlaich and
schäfer also suggested that two simple models can sometimes be superimposed to develop a more
sophisticated model that better models the flow of forces through a D-region.
Also, Brown et al. [9], expl ined th t “it is prefer ble to h ve model th t is st tic lly
determin nt ” St tic lly determin nt models require no knowledge of the member stiffnesses
which makes it simple to calculate member forces. Conversely, statically indeterminant structures
require that member stiffnesses be estimated. Estimating the member stiffnesses of a STM is
often difficult because the true geometry of the struts are too difficult to be accurately determined.
STEP 5 – CALCULATE THE FORCES IN THE STRUTS AND TIES
The strut and tie forces can be calculated knowing the geometry of the developed STM and the
forces acting on the D-region. It is desirable to use a computer program to calculate the forces
because, often times, the geometry of the STM may need to be modified during the design process
which will require the forces in the struts and ties to be recalculated.
STEP 6 – SELECT STEEL AREA FOR THE TIES
The required amount of reinforcement for each tie can easily be determined by dividing the force
in the tie by the product of the yield stress of the steel and resistance factor specified by a design
specification. The reinforcement chosen to satisfy the steel requirements must be placed so that
the centroid of the reinforcement coincides with the centroid of the tie in the STM. If
reinforcement chosen to satisfy the tie requirements cannot fit in the assumed location of the tie,
the location of the tie in the STM needs to be modified, and the member forces need to be
calculated again.
511
STEP 7 – CHECK STRESS LEVELS IN THE STRUTS AND NODES
The stress levels in all of the struts and nodes must be compared to the allowable stress limits
given in design specifications. In order to determine the stress levels in the struts and nodes, the
geometry of the struts and nodes must first be estimated. The geometry of the struts and nodes
can be determined based on the dimensions of bearing pads and the details of reinforcement
connected to the struts and nodes.
Accurately determining the geometry of internal struts and nodes not attached to bearing
pads and reinforcement is more difficult than finding the geometry of struts and nodes directly in
contact with the boundary of the D-region. In the case of internal nodes and struts, it may not be
possible to precisely define the strut and node geometry. Brown et al. [9] explained that this
uncertainty is acceptable because force redistribution can take place for internal struts and nodes.
When stresses in struts and nodes are found to be larger than permissible stresses, bearing
areas, the reinforcement details, or the overall member geometry of the member can be modified
in an effort to increase the overall geometry of the strut and/or node. When changing any or all of
these items, the STM will likely need to be modified. If the STM is modified, the member forces
need to be calculated again, the ties may need to be redesigned, and then, the stresses in the struts
and nodes can be checked again. The concrete strength can be increased if modifying the
geometry of the STM or the member itself is not possible.
STEP 8 – DETAIL REINFORCEMENT
Once all the steel chosen for the ties in the STM has been finalized, the anchorage of the
reinforcement must be properly detailed in order for it to reach its yield stress prior to leaving
nodal zones. In addition, appropriate crack control should be placed in areas that are expected
to be subjected to cracking. Most design specifications specify a minimum amount of crack
control that must be placed in a D-region that has been designed with a STM.
5.3.2 Example-Design of a RC Deep Beam with Openings using the Strut-and-Tie Method
This example Beam DSON3 in Group A shows how you can use the strut-and-tie method for
designing a RC deep beam with openings.
5.3.2.1 Geometry and Loads
Figure 5.25 shows a simply supported rectangular beam 80mm wide and 400mm height carries
one concentrated factored load of 140kN on a span of 800mm. The beam contains two rectangular
openings 80×180mm, located in the shear-span of the beam. Given
1
,
= 244.5MPa (Ø6),
= 260.2MPa (Ø8).
511
2Vu = 98kN
Vu = 49kN
Vu = 49kN
Figure 5.25 Beam geometry and loading.
5.3.2.2 Design Procedure
A deep beam is entirely a disturbed region because of loading and geometric discontinuities. The
beam will be designed using the strut-and-tie method according to ACI 318-11. The step-by-step
procedure is as follows:
Step 1: Check bearing capacity at loading and support locations.
Step 2: Establish the strut-and-tie model and determine the required model forces.
Step 3: Select the tie reinforcement.
Step 4: Check the capacity of the struts.
Step 5: Design the nodal zones and check the anchorages.
Step 6: Calculate the minimum reinforcement required for crack control.
Step 7: Arrange the reinforcement.
5.3.2.3 Design Calculations
Step 1: Check bearing capacity at loading and support locations
The area of bearing plate is 80×100mm. The bearing stresses are:
The nodal zone beneath the loading locations is a C-C-C Node. The effective compressive strength
of this node ( =1.0) is limited to:
515
The nodal zone over the support location is a C-C-T Node. The effective compressive strength of
this node ( = 0.8) is limited to:
According to ACI 318-11, the following equation shall be satisfied
Where
is a strength reduction factor, equals to 0.75 for all elements.
Since the applied bearing stresses (12.25MPa and 6.125MPa) are less than the limiting values at
both the loading location and the supports, respectively, the provided area of contact (bearing
plates) is adequate.
Step 2: Establish the strut-and-tie model and determine the required model forces
Based on the tension and compression stress trajectories (finite element analysis), the strut-and-tie
model shown in Fig. 5.26 is proposed for Beam DSON3. Elements with compression stress
trajectories will be replaced by compression elements (Struts) and elements with tension stress
trajectories will be replaced by tension elements (Ties). Here, the compressive struts are shown in
dotted lines while tension ties are shown in solid lines.
2Vu = 98kN
S3
S4
S2
N4
T4
N3
N6
N2
T5
S1
T3
T2
S5
N5
T1
N1
Figure 5.26 Details of the proposed simplified strut-and-tie model
(using inclined ties) for Beam DSON3.
511
The reinforcement required for the inclined ties in Fig. 5.26 can be resolved into horizontal and
vertical reinforcement, and the STM can thus, be adjusted as shown in Fig. 5.27. This model is
better because it gives larger capacity. In this model, according to ACI 318-11, the smallest angle
permitted between a strut and a tie in a D-region of 25 degrees is satisfied. The struts, ties, and
nodes are labeled as in the Fig. 5.27. Referring to Chapter 3 (Sec. 3.3.2), the maximum nominal
capacity of Beam DSON3 is 130kN. However, designing the beam according to the ACI 318-11,
the ultimate load will be
2Vu = 98kN
S3
N4
S N5
T6 S4
T5 T5 5 T7 N7 N6
S6
S2
S1
N3
T3
T4
T3
S7
S8
T1
N8
N2
N9
N10
T2
N1
Vu = 49kN
Vu = 49kN
Figure 5.27 Alternative proposed simplified strut-and-tie model for Beam DSON3
(using vertical and horizontal ties).
The forces in all members are determined from statics and their magnitudes in kN are as in Table
5.3. The table also gives the inclination of each strut member (angle from the horizontal).
Table 5.3 STM forces for the strut-and-tie model in Fig. 5.27.
Model
Label
Angle
(Degree)
Force,
kN
T or C
Model
Label
S1
80
49.97
C
S2
59
56.73
C
S3
25
69.21
C
S4
16
66.42
C
S5
80
28.33
C
S6
61
32.41
C
S7
25
49.20
C
S8
0
9.70
C
T = Tension (Tie) and C = Compression (Strut)
511
T1
T2
T3
T4
T5
T6
T7
--
Force,
kN
T or C
54.59
9.80
18.78
20.36
34.43
7.64
22.40
--
T
T
T
T
T
T
T
--
Step 3: Select the Tie Reinforcement
The required amount of reinforcement for a tie will be determined as follows:
Where
is the ultimate tensile force for the tie and
0.75 for all the STM elements.
is a strength reduction factor and equals
Table 5.4 Tie reinforcements.
Ties Force,
Label
(kN)
T1
T2
T3
T4
T5
T6
T7
54.59
9.80
18.78
20.36
34.43
7.64
22.40
Yield
Strength,
(MPa)
410
410
260.2
244.5
260.2
244.5
260.2
177.528
31.870
96.233
111.029
176.428
41.663
114.783
Diameter
(mm)
No. of
Bars
Distribution
12
12
2
3
8
6
8
2 bars
2 bars
2 bars
2 stirrups
4 bars
1 stirrups
4 bars
212
212
2ф
2ф6
4ф
1ф6
ф
Step 4: Check the capacity of the struts
In order to check the capacities of the struts, the area of the struts must be first determined. The
struts' areas were calculated by finding the product of the widths and depths of each strut. For the
strut width ws, refer to the dimensions of the struts in Fig. 5.28.
Knowing that f c = 30.45MPa, the term
will be:
MPa, for Strut Sj (j = 1, 3, 4, 5, 7 and 8)
MPa, for Strut Sj (j = 2 and 5)
Maximum nominal strut capacity,
and ultimate strut capacity,
Table 5.5 summarizes the calculations performed for each strut.
511
Figure 5.28 Visualization of struts' widths.
Table 5.5 Summary of concrete struts calculations.
Model
Label
βs
,
MPa
Strut
width
1
2
3
4
5
6
7
8
1.00
0.60
1.00
1.00
1.00
0.60
1.00
1.00
25.88
15.53
25.88
25.88
25.88
15.53
25.88
25.88
113.0
62.00
59.00
52.00
36.00
47.00
51.00
62.00
Ultimate strut
capacity
(kN)
175.47
57.77
91.62
80.75
55.90
43.79
79.19
96.27
Actual Strut
capacity
Okay
(kN)
49.97
Yes
56.73
yes
69.21
yes
66.42
yes
28.33
yes
32.41
yes
49.20
yes
9.70
yes
Step 5: Design the Nodal Zones and Check the Anchorages
The capacity of a node is calculated by finding the product of the limiting compressive stress in
the node region and the cross-sectional area of the member at the node interface. The maximum
nominal node capacity is
and
and the ultimate node capacity is
Table 5.6 summarizes the calculations performed for the critical nodes N1, N5 and N8.
511
Table 5.6 Summary of critical nodes calculations.
Model
Label Type
1
5
8
βn
0.80
CCT 0.80
0.80
1.00
CCC 1.00
1.00
0.80
0.80
CCT 0.80
0.80
0.80
Surrounding
Forces,
C/T
kN
49.00
49.97
9.80
98.00
66.42
28.33
32.41
9.70
49.20
18.78
7.64
Available
width,
mm
C
C
T
C
C
C
C
C
C
T
T
100.00
113.0
80.00
100.0
52.00
36.00
47.00
62.00
51.00
51.00
63.00
MPa
20.71
20.71
20.71
25.88
25.88
25.88
20.71
20.71
20.71
20.71
20.71
Ultimate
Actual
node
capacity, Okay
capacity,
kN
kN
124.26
49.00
yes
140.42
49.97
yes
yes
9.80
99.41
155.28
98.00
yes
80.75
66.42
yes
55.90
28.33
yes
58.40
32.41
yes
77.04
9.70
yes
63.38
49.20
yes
63.38
18.78
yes
78.30
7.64
yes
According to the ACI 318-11, the 90o standard hook is used to anchor the ties T1 and T2. The
required anchorage length is
√
√
Where
represents the correction factor for excess of reinforcement. ACI
318-11 requires that this development length start at the point where the centroid of the
reinforcement in a tie leaves the extended nodal zone and enters the span. As shown in Fig. 5.29,
the available development length is 181mm. Because this is greater than 14mm, the anchorage
length is adequate.
S1
212
Figure 5.29 Nodal zone N1.
511
Step 6: Calculate the Minimum Reinforcement Required for Crack Control
According to ACI 318-11, the provided vertical web reinforcement must be at least:
According to ACI318-11, the provided horizontal web reinforcement provided must be at least:
Where and is the vertical and horizontal spacing between web reinforcement, respectively and
shall not exceed the smaller of d/5 and 300 mm.
For vertical web reinforcement, use Ø6-50mm on each side,
.
For horizontal web reinforcement use Ø8-50mm on each side,
.
⁄
⁄
⁄
⁄
Step 7: Arrange the Reinforcement
The details of reinforcement are as shown in Fig. 5.30
Horizontal Web
Rft ф @ mm
× ф
ф6
Closed stirrups
1ф6
Closed stirrups
Ф1
Framing bars
Ф1
Vertic l Web Rft ф
@50mm
Bearing Plate
80×100mm
Figure 5.30 Final reinforcement detailing according to the strut-and-tie model.
511
5.4 DESIGN RECOMMENDATIONS
The following recommendations (which are not covered in the present study) are suggested for
future researches:

Based on the finite element output, future studies considering different types of top and
bottom loadings (concentrated and distributed at various locations) to be done for simple
shallow beams with openings.

Continuous shallow beams with distributed loads, having openings with different shapes,
sizes, and locations need more studies.

The location of opening is a major factor influencing the strength of the beam. The effect
of opening location on the strength and behavior of such beams have to be studied. This
may yield the optimum location of openings in Beams.
511
CHAPTER
6
SUMMARY
AND
CONCLUSIONS
CHAPTER 6
SUMMARY AND CONCLUSIONS
6.1 INTRODUCTION
In this chapter, a brief summary of the thesis is presented and the results obtained from the
application of both the strut-and-tie approach and the 3-D nonlinear finite element analysis using
ANSYS-12 package on reinforced concrete (shallow and deep) beams with and without openings
are discussed.
6.2 SUMMARY
In this thesis, the strut-and-tie approach is used to verify the capacity of reinforced concrete
(shallow and deep) beams with and without openings subjected to different loading and boundary
conditions. Verification examples (12 simple reinforced concrete shallow beams, and 10 simple
and continuous reinforced concrete deep beams) that had been tested in previous by other
researchers (with different dimensions, materials, loads, web reinforcement, concrete strengths
and boundary conditions) have been modeled and analyzed using the proposed strut-and-tie
approach utilizing elastic principal stress trajectories from finite element analysis. Finally, the
obtained results are compared with the test results.
A 3-D nonlinear finite element analysis has been conducted in order to predict the ultimate
capacity of simple and continuous reinforced concrete (shallow and deep) beams with and without
openings. To evaluate and verify the output results obtained from the finite element modeling,
groups of reinforced concrete (shallow and deep) beams with and without openings (that had been
tested by other researchers) have been modeled using this finite element model, and the obtained
results have been compared with the experimental results. The results obtained from both the
strut-and-tie method and the nonlinear finite element analysis have shown good agreement with
tests results.
In addition, a full design procedure with numerical examples, reinforcement detailing for
(shallow and deep) beams with openings, along with design recommendations have been
presented.
6.3 CONCLUSIONS
Based on the proposed STM approach and for the range of studied factors, the following
conclusions can be drawn:

The strut-and-tie approach gives freedom to designer to choose the suitable model,
according to elastic principal stress trajectors from finite element analysis and practice.

The Strut-and-Tie Model gives reasonable lower bound estimate of the load carrying
capacity of the chosen RC beams when compared with the experimental failure loads.

The proposed Strut-and-Tie approach is a powerful tool to predict the ultimate strength and
behavior of reinforced concrete (ordinary or deep) beams with and without openings.
911

Small openings are defined as openings which are small enough, when the depth (or
diameter) of the opening is less than or equal to (0.25 - 0.40) times the overall beam depth
and located in such a way that a Strut-and-Tie model in RC beams is able to jump over the
openings without causing additional vertical or horizontal struts in the chords above and
below the openings.

Large openings are defined as openings which are large enough and located in such a way
that a strut-and-tie model in RC beams is causing additional vertical or horizontal struts in
the chords above and below the openings.
Based on the analytical study using the 3D-nonlinear finite element analysis and for the
range of studied factors, the following conclusions can be drawn:

The 3-D nonlinear finite element analysis of simple Normal- and High-Strength Concrete
ordinary (shallow) and deep beams with and without openings yields accurate predictions
of both the ultimate load and the complete response.

For ordinary (shallow) beams with and without openings loaded by two symmetrical point
loads, the first crack occurs in the constant moment region. It happens, in general, in the
mid-span of beam, and it is a flexural crack (vertical crack). Cracking increases with
increasing the load in the constant moment region, and cracking propagates towards the
supports (diagonal cracks).

For deep beams with openings, the opening affects the beams' stress trajectories
drastically, where zones of tension stresses are formed around the upper corners of the
openings (nearest to the load) and around the lower corner (nearest to the supports) on the
same diagonal. These shear cracks occur, in general in the shear-span of beam. Cracking
increases with increasing the load in the shear-span and propagates towards the mid-span
(flexural cracks and vertical cracks).

For all considered ordinary (shallow) beams without openings, a flexural failure occurred
in mid-span of beam, after the yielding of the longitudinal rebars.

For all considered ordinary (shallow) beams with openings, a diagonal shear failure
occurred at corners of tension zones of opening, before the yielding of the longitudinal
rebars.

For all considered deep beams with and without openings, a diagonal shear failure
occurred in the shear-span or at the upper corners of the openings (nearest to the load) and
around the lower corner (nearest to the supports) on the same diagonal before yielding of
the longitudinal rebars.
022

The failure modes of HSC (shallow and deep) beams with openings are similar to that of
NSC beams.

The two modes of shear failure for shallow beams with small openings are: (1) beam-type
failure through the center of opening and (2) frame-type failure through the edge of
opening.

The finite element solutions show that the increase of the concrete strength results in an
increase in the cracking strength and ultimate strength.

The vertical web reinforcement has more influence on the ultimate capacity of deep beams
with openings than horizontal web reinforcement.

In general, for a strut-and-tie model, when the opening is located in the compression
transfer zone, the load transfer path is re-routed around the sides of the opening and, thus,
introducing tension ties that connect the compression zones above and below the opening.

Generally the strut-and-tie model uses vertical and horizontal ties to facilitate the position
of reinforcement in vertical and horizontal directions and this model is lightly better than
that having inclined ties where the capacity increases.

The obtained numerical results of reinforced concrete beams demonstrate that the variation
in the opening size has a significant effect on the ultimate load capacity.

For ordinary beams, the finite element stress trajectories along the longitudinal
reinforcement occurred at the point of load application and at the edges of the opening. It is
found that the rebars reach their yielding stresses at these locations.

For ordinary beams, the numerical results show that the load at which the first flexural
crack began does not depend on the presence or not of an opening and its situation. Shear
cracks around the opening will induce sooner than that around the similar area in solid
beams.

For ordinary beams made of NSC with circular openings, the increase of the diameter of
opening will change the pattern of cracks and the type of failure from flexural failure to
frame-type or beam-type shear failure.

For opening located completely outside the shear region, the beam with a web opening
may be assumed to be a solid web beam.

The failure plane always passes through the center of opening, except when the opening is
very close to the support.

For ordinary beams, the presence of longitudinal bars on top and bottom of the opening is
necessary to control the cracks and flexural strains around the opening.

For ordinary beams, the installation of diagonal bars and small stirrups on top and bottom
of the opening will increase the ultimate strength of beams with opening.

The most favorable opening shape is the circular, because the deviation of the stress
trajectories is a minimum.
029
REFERENCES
REFERENCES
1. Abdalla, H.A., A.M. Torkeya, H.A Haggagb and A.F. Abu-Amira., “Design Against
Cracking at Openings in Reinforced Concrete Beams Strengthened with Composite Sheets”.
Composite Structures, 2003, (60), 197-204.
2. Allam, S.M., “ Strengthening of RC Beams with Large Openings in the Shear Zone, ”
Alexandria Engineering Journal, 2005, 44 (1), PP. 59-78.
3. American Concrete Institute, “Building Code Requirements for Reinforced Concrete”,
Detroit, ACI-381M-11, (2011).
4. Amiri, S., R. Masoudnia and M.A Ameri., “A Review of Design Specifications of Opening in
Web for Simply Supported RC Beams,” Journal of Civil Engineering and Construction
Technology, 2011, 2 (4), pp. 82-89.
5. ANSYS Release 12.0, 2009, SAS IP, Inc.
6. Architectural Institute of Japan., “AIJ Structural Design Guidelines for Reinforced
Concrete Buildings,” 1994, pp. 113-118.
7. Ashour, A., and Rishi, G., “Tests of Reinforced Concrete Continuous Deep Beams with Web
Openings,” ACI Structural Journal, May-June 2000, V. 97, No. 3, pp. 418-426.
8. Ashour, A. F., “Tests of Reinforced Concrete Deep Beams,” ACI Structural Journal, Jan.-Feb.
1997, V. 94, No. 1, pp. 3-12.
9. Brown, M. D., Bayrak, O. “Minimum Transverse Reinforcement for Bottle-Shaped Struts.”
ACI Structural Journal. V. 103, No. 6, November-December, 2006, pp 813-821.
10. Egyptian Code for The Design and Construction of Reinforced Concrete structure, Cairo,
2007, Ministry of Housing and Development of New Communities, Cairo, Egypt.
11. El-Azab, M. F., “Behavior of Reinforced High Strength Concrete Deep Beams with Web
Openings,” M. Sc. Thesis in Structural Engineering, Faculty of Engineering, El-Mansoura
University, 2007.
12. European Committee for Standardization, Euro code 2: “Design of Concrete StructuresPart 1 – Final Draft,” Brussels, October, 2001.
13. Farag, A. A., “Behavior of Bottom Loaded Continuous Deep Beams,” Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy Thesis in Structural Engineering, Faculty
of Engineering, Cairo University, 2000.
14. Farahat, A. M., “Behavior of Reinforced Concrete Deep Beams with Web Openings,” M. Sc.
Thesis in Structural Engineering, Faculty of Engineering, Cairo University, 1987.
I
15. Hai Tan, K, Kew Kong, F., Teng, S., and Guan,L., “High Strength Concrete Deep Beams
with Effective Span and Shear Span Variations,” ACI Structural Journal, July 1995, V. 92, No.
4, pp. 395-405.
16. Hamdy, E. M., “Effect of Reinforcement Detailing on the Behavior and Strenth of R.C. Deep
Beams With Openings,” M. Sc. Thesis in Structural Engineering, Faculty of Engineering, Cairo
University, 1996.
17. Hanson, J.M., “Square Openings in webs of Continuous Joists,” Portlant Cement
Association, 1969, PP. 1-14.
18. Ichinose, T., and Yokoo, S., “A Shear Design Procedure of Reinforced Concrete Beams
with Web Openings,” Summaries of Technical Papers of Annual meeting, Architectural
Institute of Japan, Japan, 1990, pp. 319-322.
19. Javed Vaseghi Amiri and Morteza Hosseinalibygie “Effect of Small Circular Opening on the
Shear and Flexure Behavior and Ultimate Strength of Reinforced Concrete Beams using Normal
and High Strength Concrete,” Vancouver, B.C., Canada, August 2004, Paper No. 3239.
20. Kachlakev, D.I.; Miller, T.; Yim, S.; Chansawat, K.; Potisuk, T., “Finite Element Modeling
of Reinforced Concrete Structures Strengthened With FRP Laminates,” California Polytechnic
State University, San Luis Obispo, CA and Oregon State University, Corvallis, Oregon
Department of Transportation, May, 2001.
21. Keun Oh, J. and Woo Shin, S., “Shear Strength of Reinforced High Strength Concrete Deep
Beams,” ACI Structural Journal, March 2001, V. 98, No.2, pp. 164-173.
22. Keun-Hyeok Yang, Hee-Chang Eun, Heon-Soo Chung, “The Influence of web Openings on
the Structural Behavior of Reinforced High-strength Concrete Deep Beams,” South Korea,
Engineering Structures 28 (2006), pp. 1825-1834.
23. Kiag-Hwee Tan, Mohamed A. Mansur, and Loon-Meng Huang., “Reinforced Concrete TBeams with Large Web opening in Positive and Negative Moment Regions”, An Experimental
Study, ACI Structural Journal, May-June 1996,V. 93, No. 3.
24. Kiang-Hwee, T., M.A. Mansur and Wei. Weng , “Design of Reinforced Concrete Beams
with Circular Openings,” ACI Structural Journal, 2001, 98 (No. 3).
25. Mahmoud, A. A., “Behavior of High Strength Reinforced Concrete Deep Beams with and
without Openings Under Eccentric Vertical loads,” Faculty of Engineering, Zagazig University,
Egypt.
26. Mansur, M.A., and Tan, K.H., “Concrete Beams With Openings. Analysis and Design,”
National University of Singapore, Singapore, 1999, pp. 1-70.
27. Mansur, M.A., Y.F. Lee, K.H. Tan and S.L. Lee, “Test on RC Continuous Beams with
Openings,” Journal of Structural Engineering, 1991, 117(6), pp. 1593-1605.
28. Mansur, M.A., “Effect of Openings on the Behavior and Strength of R/C Beams in Shear,”
Cement and Concrete Composites, 1998, 20, pp. 477-486.
II
29. Mansur M. A., L. M. Huang, K. H. Tan and S. L. Lee “Deflection of Reinforced Concrete
Beams with Web Openings,” ACI Structural Journal, July-August 1992, V. 89, No. 4.
30. Mansur M. A., Kiang-Hwee Tan and Weng Wei, “Effect of Creating an Opening in Existing
Beams,” ACI Structural Journal, November-December 1999, V. 96, No. 6.
31. Maxwell, B. S. & Breen, “Experimental Evaluation of Strut-and-Tie Model Applied to Deep
Beams with Opening,” ACI Structural Journal, J. E. 2000, Vol. 97, pp. 142-149.
32. Mohammad, Kh. I., “Prediction of Behavior of Reinforced Concrete Deep Beams with Web
Openings Using Finite Elements,” Al-Rafidain Engineering, Feb. 2007, V. 15, No. 4.
33. Moussa, A., Mahmoud, A., Abdel-Fattah, W., and Abu-Elmagd, S., “Behavior of R.C. Deep
Beams with and without Openings,” Proceedings of the 5th Alexandria International Conference
on Structural and Geotechnical Engineering, Structural Engineering Department, Faculty of
Engineering, Alexandria University, 20-22 December 2003, pp. CRI85-CR202.
34. Ozcebe, G., Erosy, U., and Tankut, T., “Evaluation of Minimum Shear Reinforcement
Requirements for Higher Strength Concrete,” ACI Structural Journal, May 1999, V. 96, No. 3,
pp. 361-368.
35. Salam, S.A., “Beams with Openings under Different Stress Conditions,” Conference on Our
World in Concrete and Structures, Singapore, 25-26 Aug, 1977, pp. 259-267.
36. Sallam, M. A., “Experimental and Analytical Investigation on Reinforced High Strength
Concrete Deep Beams with Openings,” M. Sc. Thesis in Structural Engineering, Faculty of
Engineering, Tanta University, 2004.
37. Schlaich, J. and Schäfer, K. “Design and Detailing of Structural Concrete Using Strut-and-Tie
Models.” The Structural Engineer. V. 69, No. 6, May-June, 1991, pp 113-125.
38. Schlaich, J., Schaefer, K., and Jennewein, M. “Towards a Consistent Design of Structural
Concrete,” PCI Journal, V. 32, No. 3, May-June, 1987, pp. 74-150.
39. Schlaich, J. and Schäfer, K. “The Design of Structural Concrete,” IABSE WORKSHOP, New
Delhi February, 1993.
40. Siao, W.B. and Yap, S.F., 1990, “Ultimate Behavior of Unstrengthen Large Openings Made in
Existing Concrete Beams. Journal of the Institution of Engineers, 30 (3), pp. 51-57.
41. Smith, N. K., and Vantsiotis, S. A., “Deep Beam Test Results Compared with Present
Building Code Models,” ACI Journal, July 1982, V. 79, No. 3, pp. 280-287.
42. Somes, N.F. and W.G. Corley, “Circular Openings in Webs of Continuous Beams,”
American Concrete Institute, 1974, Detroit, MI, pp. 359-398.
43. Subedi, N. K., “Reinforced Concrete Two-Span Continuous Deep Beams,” Proceedings of the
Institution of Civil Engineers: Structural & Building Journal, February 1998, V. 128, pp. 12-25.
III
44. Tavarez, F.A., “Simulation of Behavior of Composite Grid Reinforced Concrete Beams Using
Explicit Finite Element Methods,” Master’s Thesis, University of Wisconsin-Madison,
Madison, Wisconsin, 2001.
45. Thompson, M. K., M. J. Young, J. O. Jirsa, J. E. Breen, and R. E. Klingner, 2003,
“Anchorage of Headed Reinforcement in CCT Nodes,” Research Report 1855-2, Austin, TX,
Center for Transportation Research, University of Texas at Austin.
46. Yousef, A. M., “Minimum Web Reinforcement in High-Strength Concrete Deep Beams,”
Proceeding of the Ninth International Colliquium on Structural and Geotechnical Engineering,
Faculty of Engineering, Ain Shams University, Egypt, 10-12 April 2001, Paper No. RC13.
47. Yousef, A. M. and Agag, I. Y., “Shear Behavior of High Strength Fiber Reinforced Concrete
Deep Beams,” Mansoura Engineering Journal, (MEJ), June 2001, V. 26, No. 2, pp. 28-42.
IV
‫وينقسم البحث الي عذة فصول تحتوي علي الموضوعاث التاليت‪:‬‬
‫الفصل األول‪ٚ :‬تٗ ِمذِح ٌٍّ‪ٛ‬ض‪ٛ‬ع ‪ٚ‬ششغ ٌٍّشىٍح األساسيح اٌّطٍ‪ٛ‬ب دساسر‪ٙ‬ا ‪ٚ‬اٌ‪ٙ‬ذف ِٓ اٌثؽس ‪ٚ‬ذشذية‬
‫اٌثؽس‪.‬‬
‫الفصل الثاني‪ :‬يرضّٓ إسرؼشاضا ً ِ‪ٛ‬ظضا ً ٌألتؽاز اٌساتمح اٌّراؼح ‪ٚ‬اٌّرؼٍمح تّعاي اٌثؽس‪.‬‬
‫الفصل الثالث‪ :‬يؽر‪ٛ‬ي ػٍي ذ‪ٛ‬ضيػ ٌٍفشق تيٓ اٌىّشاخ اٌؼاديح ‪ٚ‬اٌىّشاخ اٌؼّيمح ِٓ ؼيس األتؼاد ‪ ِٓٚ‬ؼيس‬
‫شىً ِساساخ االظ‪ٙ‬اداخ اٌذاخٍيح‪ٚ ،‬أيضا ً يؽر‪ٛ‬ي ػٍي ششغ ِفصً ٌطشيمح ّٔ‪ٛ‬رض اٌضاغظ ‪ٚ‬اٌشذاد ‪ٚ‬ويفيح‬
‫ػًّ ّٔارض اٌضاغظ ‪ٚ‬اٌشذاد ٌعّيغ اٌىّشاخ اٌّخرثشج ِؼٍّيا ً اٌري ذُ ذعّيؼ‪ٙ‬ا ِٓ االتؽاز اٌساتمح‪ .‬في ٘زا‬
‫اٌفصً أيضا ً ذُ إلرشاغ طشيمح ّٔ‪ٛ‬رض اٌضاغظ ‪ٚ‬اٌشذاد ٌؽساب اٌمذسج اٌرؽّيٍيح اٌمص‪ٛ‬ي ٌٍىّشاخ ‪ٚ‬ذّد‬
‫ِماسٔح إٌرائط اٌّسرخشظح تإٌرائط اٌّؼٍّيح‪.‬‬
‫الفصل الرابع‪ :‬يؽر‪ٛ‬ي ػٍي ششغ ٌطشيمح اٌؼٕصش اٌّؽذد ‪ٚ‬اٌرؽٍيً غيش اٌخطي في اٌفشاؽ ‪ٚ‬ششغ ٌٍؼٕاصش‬
‫ا ٌّسرخذِح في اٌثشٔاِط‪ .‬ذُ ذطثيك ٘زٖ اٌطشيمح ػٍي ظّيغ اٌىّشاخ اٌري ذُ ذعّيؼ‪ٙ‬ا ‪ٚ‬اٌّخرثشج ِؼٍّيا ً ِٓ‬
‫أظً سسُ ص‪ٛ‬سج واٍِح ٌسٍ‪ٛ‬ن اٌىّشاخ ذؽد اٌذساسح‪ ،‬ذُ ذؽٍيً إٌرائط اٌّرؽصً ػٍي‪ٙ‬ا ‪ٚ‬ػًّ ِماسٔح تيٕ‪ٙ‬ا‬
‫‪ٚ‬تيٓ إٌرائط اٌّؼٍّيح‪.‬‬
‫الفصل الخامس‪ :‬يؽر‪ٛ‬ي ػٍي ششغ خط‪ٛ‬اخ ذصّيُ اٌىّشاخ اٌؼاديح ا‪ ٚ‬اٌؼّيمح راخ اٌفرؽاخ ِغ اسرخذاَ‬
‫أِصٍح ػذديح ِطثمح ػٍي اٌىّشاخ اٌري ذُ دساسر‪ٙ‬ا في ٘زا اٌثؽس‪.‬‬
‫الفصل السادس‪ :‬يشرًّ ػٍي ٍِخص ٌٍثؽس ‪ٚ‬اٌر‪ٛ‬صياخ اٌّسرٕرعح ِٓ ٘زا اٌثؽس ‪ٚ‬اٌّرؽصً ػٍي‪ٙ‬ا ِٓ‬
‫وال ً ِٓ طشيمح ّٔ‪ٛ‬رض اٌضاغظ ‪ٚ‬اٌشذاد ‪ٚ‬طشيمح اٌؼٕصش اٌّؽذد ‪ٚ‬اٌرؽٍيً غيش اٌخطي في اٌفشاؽ‪.‬‬
‫ملخص البحث‬
‫يّىٓ ا ػرثاس اٌىّشاخ اٌؼاديح راخ اٌفرؽاخ ‪ٚ‬اٌىّشاخ اٌؼّيمح (راخ أ‪ ٚ‬تذ‪ ْٚ‬فرؽاخ) وّٕاطك ػذَ اسرّشاسيح‬
‫(ا ظ‪ٙ‬اداخ غيش ِٕرظّح ‪ٚ‬االٔفؼاالخ غيش خطيح)‪ٌ .‬زٌه‪ ،‬فّٓ غيش إٌّاسة أْ يرُ ذصّيُ ذٍه إٌّاطك‬
‫تا سرخذاَ فش‪ٚ‬ض ٔظشيح االٔؽٕاء أ‪ِ ٚ‬ؼادالخ اٌمص اٌرمٍيذيح‪ٚ .‬تاٌراٌي واْ التذ ِٓ اٌثؽس ػٓ طشيمح أوصش‬
‫‪ٚ‬الؼيح وطشيمح " ّٔ‪ٛ‬رض اٌضاغظ ‪ٚ‬اٌشذاد"‪.‬‬
‫في ٘زا اٌثؽس ذّد دساسح اٌسٍ‪ٛ‬ن إٌظشي ٌىّشاخ خشسأيح ِسٍؽح ػاديح ‪ٚ‬ػاٌيح اٌّما‪ِٚ‬ح اٌّخرثشج ِؼٍّيا ً‬
‫ِٓ إٌ‪ٛ‬ع اٌضؽً اٌثسيظ االسذىاص (راخ ‪ٚ‬تذ‪ ْٚ‬فرؽاخ) ‪ٚ‬أخشي ِٓ إٌ‪ٛ‬ع اٌؼّيك اٌثسيظ ‪ٚ‬اٌّسرّش (راخ‬
‫اٌفرؽاخ) تطشيمح ّٔ‪ٛ‬رض اٌضاغظ ‪ٚ‬اٌشذاد‪ .‬ؼيس أفرشضد ّٔارض اٌضاغظ ‪ٚ‬اٌشذاد ٌىً اٌىّشاخ إػرّادا ً‬
‫ػٍي ِخشظاخ إٌرائط اٌرعشيثيح اٌّراؼح (ّٔارض ذ‪ٛ‬صيغ اٌشش‪ٚ‬ؾ‪ ،‬أّٔاط االٔ‪ٙ‬ياس) ‪ِٚ‬ساساخ ذ‪ٛ‬صيغ االظ‪ٙ‬اداخ‬
‫اٌذاخٍيح ِٓ اٌرؽٍيً اٌّشْ تطشيمح اٌؼٕصش اٌّؽذد‪ .‬ذّد ِماسٔح إٌرائط اٌّسرخشظح ِٓ طشيمح ّٔ‪ٛ‬رض‬
‫اٌضاغظ ‪ٚ‬اٌشذاد تإٌرائط اٌّؼٍّيح‪.‬‬
‫ِٓ أظً سسُ ص‪ٛ‬سج واٍِح ٌسٍ‪ٛ‬ن اٌىّشاخ ذؽد اٌذساسح‪ ،‬ذُ ػًّ ذؽٍيً شالشي األتؼاد غيش خطي تإسرخذاَ‬
‫ٔظــــــشيح اٌؼٕصش اٌّؽذد‪ٚ .‬اٌزي س اُ٘ تذ‪ٚ‬سٖ في اٌؽص‪ٛ‬ي ػٍي إٌرائط اٌري يرؼزس ذؽذيذ٘ا تإسرخذاَ طشيمح‬
‫" ّٔ‪ٛ‬رض اٌضاغظ ‪ٚ‬اٌشذاد" وّٕ‪ٛ‬رض اٌرششؾ‪ ،‬اٌرشخيُ‪ّٔ ،‬ظ االٔ‪ٙ‬ياس ‪ٚ‬ذ‪ٛ‬صيغ االظ‪ٙ‬اد ‪ٚ‬االٔفؼاي‪.‬‬
‫إضافح اٌي ِاسثك‪ ،‬ذُ اسرمذاَ إظشاء ذصّيّي واًِ (ِذػّا ً تأِصٍح ػذديح‪ ،‬ذفصيالخ اٌرسٍيػ‪ ،‬ذ‪ٛ‬صياخ‬
‫ٌٍرصّيُ) ٌٍىّشاخ راخ اٌفرؽاخ فمظ‪.‬‬
‫لجنة المناقشة و الحكم‬
‫عُىاٌ انرصبنت‪:‬‬
‫حصًٍى انكًراث انخرصبٍَت انًضهحه راث انفخحبث‬
‫إصى انببحـــــث‪:‬‬
‫ونٍذ انذيرداط انذيرداط انصبوي‬
‫تحت إشراف‪:‬‬
‫اإلسم‬
‫الوظيفة‬
‫أصخبر دكخىر‬
‫أستاذ بقسم الهندسة اإلنشائية‬
‫كلية الهندسة ‪ -‬جامعة المنصورة‬
‫صالح انذٌٍ انضعٍذ انًخىنً‬
‫أحمد أمين غـــــــالب‬
‫أستاذ مساعد بقسم الهندسة اإلنشائية‬
‫كلية الهندسة ‪ -‬جامعة المنصورة‬
‫دكخىر‬
‫محمد السعيد محمد الزغيبى‬
‫أستاذ مساعد بقسم الهندسة اإلنشائية‬
‫كلية الهندسة ‪ -‬جامعة المنصورة‬
‫دكخىر‬
‫التوقيع‬
‫لجنة المناقشة و الحكم‪:‬‬
‫اإلسم‬
‫الوظيفة‬
‫أصخبر دكخىر‬
‫أستاذ بقسم الهندسة اإلنشائية‬
‫كلية الهندسة ‪ -‬جامعة المنصورة‬
‫يوسف إبراهيم عجاج‬
‫أصخبر دكخىر‬
‫صالح انذٌٍ انضعٍذ انًخىنً‬
‫أستاذ بقسم الهندسة اإلنشائية‬
‫كلية الهندسة ‪ -‬جامعة المنصورة‬
‫مشهور غنيم أحمد غنيم‬
‫أستاذ بقسم الهندسة اإلنشائية‬
‫كلية الهندسة ‪ -‬جامعة القاهرة‬
‫أحمد أمين غـــــــالب‬
‫أستاذ مساعد بقسم الهندسة اإلنشائية‬
‫كلية الهندسة ‪ -‬جامعة المنصورة‬
‫أصخبر دكخىر‬
‫دكخىر‬
‫رئٍش انقضى‬
‫أ‪.‬د‪ .‬أحمد محمد يوسف محمد‬
‫التوقيع‬
‫وكٍم انكهٍت نهذراصبث انعهٍب‬
‫أ‪.‬د‪ .‬قاسم صالح عبدالوهاب اآللفي‬
‫عًٍذ انكهٍت‬
‫أ‪.‬د‪ .‬زكي محمد زيدان‬
‫المشرفون‬
‫عُىاٌ انرصبنت‪:‬‬
‫حصًٍى انكًراث انخرصبٍَت انًضهحت راث انفخحبث‬
‫إصـــــى انببحث‪:‬‬
‫ونٍذ انذيرداط انذيرداط انصبوي‬
‫ححج إشراف‪:‬‬
‫اإلسم‬
‫الوظيفة‬
‫أصخبر دكخىر‬
‫أستاذ بقسم الهندسة اإلنشائية‬
‫كلية الهندسة ‪ -‬جامعة المنصورة‬
‫صالح انذٌٍ انضعٍذ انًخىنً‬
‫أحمد أمين غـــــــالب‬
‫أستاذ مساعد بقسم الهندسة اإلنشائية‬
‫كلية الهندسة ‪ -‬جامعة المنصورة‬
‫دكخىر‬
‫محمد السعيد محمد الزغيبى‬
‫أستاذ مساعد بقسم الهندسة اإلنشائية‬
‫كلية الهندسة ‪ -‬جامعة المنصورة‬
‫دكخىر‬
‫رئٍش انقضى‬
‫أ‪.‬د‪ .‬أحمد محمد يوسف محمد‬
‫وكٍم انكهٍت نهذراصبث انعهٍب‬
‫أ‪.‬د‪ .‬قاسم صالح عبدالوهاب اآللفي‬
‫التوقيع‬
‫عًٍذ انكهٍت‬
‫أ‪.‬د‪ .‬زكي محمد زيدان‬
‫جبيعت انًُصىرة‬
‫كهٍت انهُذصــــــت‬
‫قضى انهُذصت اإلَشبئٍت‬
‫حصًٍى انكًراث انخرصبٍَت انًضهحت راث انفخحبث‬
‫‪‬‬
‫رصبنت عهًٍت يقذيت يٍ‬
‫انًهُذس‪ /‬ونٍذ انذيرداط انذيرداط انصبوي‬
‫بكبنىرٌىس انهُذصت انًذٍَت – جبيعت انًُصىرة ‪2002‬و‬
‫يعٍذ بًعهذ يصر انعبنً نههُذصت وانخكُىنىجٍب ببنًُصىرة‬
‫حىطئت نهحصىل عهً‬
‫درجت انًبجضخٍر‬
‫فً‬
‫انهُذصت اإلَشبئٍت‬
‫ححج إشراف‬
‫األصخبر انذكخىر‬
‫صالح انذٌٍ انضعٍذ انًخىنى‬
‫أصخبر بقضى انهُذصت اإلَشبئٍت‬
‫كهٍت انهُذصت‪ -‬جبيعت انًُصىرة‬
‫انذكخىر‬
‫انذكخىر‬
‫أحًذ أيٍٍ غبنب‬
‫يحًذ انضعٍذ يحًذ انزغٍبى‬
‫أصخبر يضبعذ بقضى انهُذصت اإلَشبئٍت‬
‫كهٍت انهُذصت‪ -‬جبيعت انًُصىرة‬
‫أصخبر يضبعذ بقضى انهُذصت اإلَشبئٍت‬
‫كهٍت انهُذصت‪ -‬جبيعت انًُصىرة‬
‫‪2013‬‬
‫‪View publication stats‬‬