FINITE ELEMENT ANALYSIS OF MEMBRANE ACTION IN STEEL DECK PLATES AND
FIBROUS CONCRETE SLABS
Introduction
Tests on steel plate bridge decks have shown that, if a load on the deck is increased beyond
the usual wheel load limits, or if the ratio of the deflection to the span of a ribs is relatively large,
internal axial stresses begin to appear in the loaded ribs, in addition to the purely flexural stresses.
An additional tension, due to membrane action of the deck plate, occurs in the directly loaded rib,
and has to be balanced by an equally large compression in the adjoining ribs [8].
As the loads and the corresponding deflection increase, a complete redistribution of the
stresses takes place in the system, and the membrane stresses almost entirely replace the flexural
stresses which are predominant under working loads.
Thus with sufficiently large deflections, a steel plate deck behaves in a manner radically
different from that predicted by the usual flexural theory which disregards the effect of the
deformations of a system on its stresses. Most importantly, its strength has been found to be many
times greater than predicted by the ordinary flexural theory.
The Aim of the Research
The main aim of this research is to use finite element approach in analyzing steel
fiber concrete slabs and steel deck plate allowing for membrane action. Then, study the
effect of number of parameters on the structural behavior of the above structures.
ANALYSIS OF STEEL DECK
PLATES ALLOWING FOR MEMBRANE ACTION
Comparison of small- and large-deflection
theories using an approximate method
3
po 
64D  wmax  8 E h  wmax 




a3  a  3 1   a  a 
Eq.1
An alternate form of Eq.1, obtained by taking =0.3, is:
po a 4 wmax

64Dh
h
2

 wmax  
 
1  0.65
 h  

Eq.2
Eh4
Eh4
The variation of load deflection for a uniformly loaded circular plate with clamped
edge, wmax/h and poa4/Eh4 are plotted in Fig.3. It is observed that the linear theory which,
neglects the membrane action, is satisfactory for wmax<h/2 and the larger deflections
produce greater error. When wmax=h, for example there is a 65 percent error in the load
according to the bending theory alone[78].
h
(a) Clamped edge
h
(b) Simply supported
Fig.1: Maximum Deflections and Stresses Plates Having Large Deflections [78]
Von-Karman assumptions
The Von-Karmen assumptions for large deformation of plates and shells should take the
following forms when applying shells and plates [22]:
1.The magnitude of the deflection (w) is of the order of the thickness (h).
2.The thickness is much less than the length (L); (this hypothesis is not restrictive) since
otherwise the displacements are not large.
3.The slope is small everywhere
4. The tangent displacements (u, v) are small, only nonlinear term, which depend on have to be
retained in the strain-displacement relations.
5.All strain components are small.
General equations for large deflections of plates
The general differential equation for a thin plate, subjected to combined lateral (p) and in-plane
(Nx, Ny, and Nxy) forces [78]:
4w
4w
4w 1 
2w
2w
2w 

 2 2 2  4   p  N x 2  N y 2  2 N xy
x 4
x y
y
D
x
y
xy 
Eq.3
For a thin plate element, the x and y equilibria of direct forces are expressed by:
N x N xy

0
x
y
N xy
x

N y
y
0
Eq.4
The resultant strain components may be expressed as:
u 1  w 
x    
x 2  x 
2
u 1  w 
x    
x 2  x 
2
 xy 
v u w w


x y x y
Eq.5
The in-plane forces can be expressed as [86]:
Eh
Nx 
1  2
 u 1  w 2   v 1  w 2 
           
 x 2  x    y 2  y  
Eq.6a
Eh
Ny 
1  2
 v 1  w 2   u 1  w 2 
           
 y 2  y    x 2  x  
Eq.6b
N xy 
Eh  v u  w w 


 
2(1   )  x y  x y 
Eq.6c
The non-linear stiffness matrix:
K=`K+Ks
Where:
K   B T DBdv
v
Ks   [G ]T [s ][ G ]dv
v
B:strain displacement matrix
D: elasticity matrix
v: volume
Eq.7
Eq.8
Eq.9
[G]: is a matrix with two rows and number of columns equal to the total number of element nodal
variables. The first row contains the contribution of each nodal variable to the local derivative
corresponding shape function derivatives(w` x` ) and the second row contains similar contributions for
(w` y` )
 s x`
and s   

x `y `
 x `y ` 
s y ` 
Eq.10
Material Nonlinearity
The material nonlinearity deals (for steel deck plates) with elasto-plastic behavior and the
anisotropic affect in the yielding behavior.
Flow theory of plasticity
The flow theory of plasticity [44,45] is employed as the nonlinear material model. For anisotropic
material to be considered in this work is a generalization of the Huber-Mises law and can be written in
general form as:
Eq.11
F(s,c)=f(s)-Y(c)
in which f(s) is some function of the deviatoric stress invariant and the yield level Y(c) can be a
function of a hardening parameter, c.
Tangent stiffness matrix
The tangential stiffness matrix of the material can be expressed as:
K
P
s 
  BT
dv   BT Dep Bdv
a v
 a
v
Eq.12
where Dep: is the elasto-plastic rigidity matrix and equal:
DaaT D
Dep  D 
H ` aT Da
Eq.13
a: is the flow vector and
H`: is the hardening parameter
Computer program (ANSYS software)
ANSYS software is a general purpose FE-program for static, dynamic as well as multiphysics
analysis and includes a number of shell elements with corner nodes only and with corner and midside nodes.
From the available element library in ANSYS, SHELL93 element is used in this work.
a=300 mm
a
a
a
16
0
m
m
Point A
Loadin
g
Poi
nt
A
Rigid steel frame
P/
2
10
m
m
P/
2
Fig.3: Representation of Steel
Deck Plate
Point m
Fig.2: Steel Deck Plate
Fig.4: FE-Model of Steel Deck
Plate
300
250
200
central point load (ton)
Case studies
1.Steel deck plate
The results of a full scale
loading test [8] of a 10 mm
thick deck plate supported on
ribs spaced 300 mm o.c. is
analyzed by ANSYS software,
Fig.2. It is represented and
modeled as shown in Figs.3
and 4. The load deflection
curves
for
both
small
displacement analysis and
large displacement analysis
(allowing
for
membrane
action) are explained in Fig.5.
Conservative solution was
obtained
in
small
displacement analysis but
good
agreement
with
experimental tests was gotten
in case of large displacement
analysis. The deflected shape
at ultimate load is listed in
Fig.6.
150
100
experimental
50
large deformation
small deformation
0
0
-20
-40
-60
-80
-100
-120
-140
-160
-180
-200
central deflction (mm)
central
deflection (mm)
Fig.5: Load Deflection Relationships of Steel
Deck Plate
Fig.6: Deflected Shape of Steel Deck Plate
2. Steel deck plate with open ribs
A steel deck plate with open ribs under a concentrated wheel load, which was conducted at the
Technological University in Darmstadt [8] (Germany), is analyzed using both large and small
displacement analysis. The 5-span continuous test panel was a half-scale model of the deck plating
intended for Save River Bridge in Belgrade, Fig.7. Also conservatives solution and good agreement
compared with experimental test was shown in both small and large deformation analysis
respectively, Fig10.
Load P
Point A
P
1480 mm
300 mm
Ribs 70 mm x 4.5 mm
Fig.8: Representation of Steel Deck Plate
with Open Ribs
P
4.5 mm deck plate
PL 70 mm x 4.5 mm
Rubber bed
150 mm
PL 250 mm x 4.5 mm
3750 mm
Fig.7: Steel Deck Plate with Open Ribs
Fig.9: FE-Model of Steel Deck with Open
Ribs Plate in ANSYS
30
25
central point load (ton)
20
15
10
5
experimental
large deformation
small deformation
0
0
-5
-10
-15
-20
-25
-30
-35
-40
-45
-50
centraldeflection
deflction(mm)
(mm)
central
Fig.10: Load Deflection Relationships of Steel Deck Plate with Open Ribs
Fig.11: Deflected Shape of Steel Deck Plate with Open Ribs
5 mm deck plate
Rigid support
Rigid support
Edge beam
3450 mm
1800 mm
P
P
150 mm
Piont A
Edge beam
4000 mm
3 mm longitudinal ribs
3. Steel deck plate with closed ribs
A steel deck plate with closed (torsionally rigid) longitudinal stiffening ribs was tested at
the Technological University in Stuttgart. The dimensions of test panel with trapezoidal ribs,
which may be regarded as a half scale model of an actual bridge penal, are illustrated in Fig.
12. the same previous notes for large and small displacements analysis can be observed in
Fig.15.
Fig. 13: Representation of Steel Deck Plate
with Closed Ribs
85 mm
P
150 mm
Fig.12: Steel Deck Plate with Closed Ribs
Fig.14: FE-Model of Steel Deck with
Closed Ribs
180
160
140
central point load (ton)
120
100
80
60
40
experimental
large deformation
20
small deformation
0
0
-100
-200
-300
-400
-500
-600
-700
-800
centraldeflection
deflction(mm)
(mm)
central
Fig.15: Load Deflection Relationships of Steel Deck Plate with Closed Ribs
Fig.16: Deflected Shape of Steel Deck Plate with Closed Ribs
ANALYSIS OF FIBROUS
CONCRETE SLABS ALLOWING FOR MEMBRANE ACTION
Computer Program
To analyze fibrous concrete slabs, a program which is coded by Hinton and Owen [40] and called
"conshell", is used. The program was modified by Al-Shather [11] to take into account the presence of
steel fiber in concrete slab. This is done by introducing tension stiffening technique. In the present
study the membrane action is introduced, and this done by using geometrical nonlinearity, and
allowing for high strain limit m during the running of the program.
Case Studies
To verify the computer program which is called "conshell program", number of fibrous concrete
slabs, which were tested by Mohammed [61], are analyzed. The material parameters a and b are
assumed to obtain good agreement with the experimental tests.
These specimens are also analyzed by ANSYS software. The so-called "SOLID65 element",
which is favorable with the analysis of reinforced concrete structures, is used. Geometrical and
material nonlinearities are adopted.
The tested slabs [6] had dimensions of 1.0 m x 0.7 m and 25 mm thick with compressive strength
as shown in table 1.
Table 1: Compressive Strength of Fibrous Concrete
Volume fraction Vf %
Compressive strength f'cf (MPa)
no fiber
28.60
0.5
29.31
1.0
30.49
1.5
31.15
The steel fiber aspect ratio (lf/df) was 20. For slabs with reinforcement, they were reinforced with
smooth wires of 2 mm diameter, with yield strength of 380.6 MPa. These steel wire reinforcement
provided at shorter span were 0.076 mm2/m (rx=0.33%) and 0.108 mm2/m (r'x=0.49%) at positive and
negative moment, respectively. Conversely at longer direction an amount of 0.036 mm2/m (ry=0.17%)
and 0.051 mm2/m (r'y=0.23%) at positive and negative moment, respectively.
1. Fibrous concrete slabs axially unrestrained at all edges
a. Without reinforcement
28
uniformly distributed (kN/m^2) 2
distributed load (kN/m )
uniformly
uniformly
distributed (kN/m^2)
load (kN/m2)
distributed
uniformly
30
25
20
15
10
experimental
ANSYS
present (Leonard)
5
present
0
24
20
16
12
experimental
8
ANSYS
present (Leonard)
4
present
0
0
-5
-10
-15
-20
-25
0
central deflection (mm)
-2
-4
Fig.18: Load Deflection Relationships of Fibrous Concrete Slab Axially
Unrestrained with Vf=0.5%
load (kN/m2)
uniformly
uniformlydistributed
distributed (kN/m^2)
200
160
120
80
experimental
ANSYS
present (Leonard)
40
present
0
0
-5
-10
-15
-20
-8
-10
Fig.19: Load Deflection Relationships of Fibrous Concrete Slab Axially
Unrestrained with Vf=1.0%
240
b. With reinforcement
-6
central deflection (mm)
-25
-30
-35
-40
-45
central deflection (mm)
Fig. 20: Load Deflection Relationships of Fibrous Reinforced Concrete Slab
Axially Unrestrained with Vf=0.5%
2. Fibrous concrete slabs axially restrained at all edges
a. Without reinforcement
(kN/m2)
distributed
uniformly
uniformly distributed
load load
(kN/m^2)
(kN/m2)
uniformly
uniformly distributed
distributed loadload
(kN/m^2)
80
experimental
70
ANSYS
present (Leonard)
60
present
50
40
30
20
80
60
40
experimental
ANSYS
20
present (Leonard)
present
10
present (smeared)
0
0
0
-2
-4
-6
-8
-10
-12
-14
0
-16
-5
-10
uniformly distributed
distributed loadload
(kN/m^2)
(kN/m2)
uniformly
-20
-25
-30
Fig.22: Load Deflection Relationships of Fibrous Concrete slab Axially
Restrained with Vf=0.5%
Fig.21: Load Deflection Relationships of Plain Concrete Slab Axially
Restrained80
60
40
experimental
ANSYS
20
present (Leonard)
present
present (smeared)
0
0
-15
central deflection (mm)
central deflection (mm)
-5
-10
-15
-20
-25
-30
central deflection (mm)
Fig.23: Load Deflection Relationships of Fibrous Concrete Slab Axially
Restrained with Vf=1.0%
b. With reinforcement
280
load (kN/m2)
distributed
uniformly
uniformly
distributed (kN/m^2)
load (kN/m2)
distributed
uniformly
uniformly
distributed (kN/m^2)
240
200
160
120
80
experimental
ANSYS
present (Leonard)
40
240
200
160
120
experimental
80
ANSYS
present (Leonard)
40
present
present
0
0
0
-10
-20
-30
0
-10
central deflection (mm)
Fig.24: Load Deflection Relationships of Reinforced Concrete Slab Axially
Restrained
-30
-40
-50
Fig.25: Load Deflection Relationships of Fibrous Reinforced Concrete Slab
Axially Restrained with Vf=0.5%
280
uniformlydistributed
distributed (kN/m^2)
load (kN/m2)
uniformly
280
load (kN/m2)
distributed
uniformly
uniformly
distributed (kN/m^2)
-20
central deflection (mm)
240
200
160
120
experimental
80
ANSYS
present (Leonard)
40
240
200
160
120
experimental
80
ANSYS
present (Leonard)
40
present
present
0
0
0
-10
-20
-30
-40
central deflection (mm)
Fig.26: Load Deflection Relationships of Fibrous Reinforced Concrete Slab
Axially Restrained with Vf=1.0%
0
-10
-20
-30
-40
-50
central deflection (mm)
Fig.27: Load Deflection Relationships of Fibrous Reinforced Concrete Slab
Axially Restrained with Vf=1.5%
3. Fibrous concrete slabs axially restrained at two longer edges
a. Without reinforcement
40
uniformlydistributed
distributed load
(kN/m^2)
(kN/m2)
load
uniformly
(kN/m2)
load
uniformly
uniformlydistributed
distributed load
(kN/m^2)
35
30
25
20
15
experimental
ANSYS
10
present (Leonard)
present
5
present (smeared)
0
35
30
25
20
experimental
15
ANSYS
10
present (Leonard)
present
5
present (smeared)
0
0
-5
-10
-15
-20
-25
0
-5
central deflection (mm)
Fig.28: Load Deflection Relationships of Fibrous Concrete Slab Axially
Restrained at Two Longer Edges with Vf=0.5%
b. With reinforcement
-10
(kN/m2)
load
distributed
uniformly
uniformly
distributed
(kN/m^2)
-20
-25
Fig.29: Load Deflection Relationships of Fibrous Concrete Slab Axially
Restrained at Two Longer Edges with Vf=1.0%
120
80
40
experimental
ANSYS
present (Leonard)
present
0
0
-15
central deflection (mm)
-4
-8
-12
-16
-20
central deflection (mm)
Fig.30: Load Deflection Relationships of Fibrous Reinforced Concrete
Slab Axially Restrained at Two Longer Edges with Vf=0.5%
THE EFFECT OF SOME
PARAMETERS ON THE BEHAVIOR OF STEEL DECK PLATES
The effect of a number of parameters on the behavior of steel deck plates are studied. ANSYS
software is used to perform analyses of the considered steel deck plate, using SHELL93 element to
model the considered problems. Large deformation analysis is adopted to take into account the
membrane action as well as material nonlinearity.
Four types of the structures are studied in this work. These are: steel rectangular plate, steel
deck plate with open rips running in one direction, steel deck plate with open ribs supported by
transverse floor I-beams, and steel deck plate with closed ribs running in one direction. Two types
of loading conditions are used, they are central point load and uniformly distributed load above the
entire structure.
PL 70mm x 5 mm as ribs
spaced 150mm o.c.
1500 mm
2250
mm
150 mm
PL 250 mm x 5 mm.
PL 70 mm x 5 mm as ribs
spaced 150 mm o.c.
100 mm
PL 5 mm
2625 mm
Symmetry
63.1 mm
2250
Symmetrymm
Symmetry
Symmetry
PL 70 mm x 5 mm.
Deck plate 5 mm
thick
2000 mm
Symmetry
Symmetry
Dimensions of deck
plate
Fig. 31: Types of Steel Deck Plates
2250 mm
Dimensions of deck
plate with open ribs
Symmetry
Symmetry
2250 mm
2250 mm
Dimensions of deck plate
with open ribs and
stiffened by floor beams
Dimensions of deck plate
with closed ribs
1. Effect of Changing Thickness of Rectangular or Top Plate of Steel Deck Plate
a. Steel rectangular plate
200
350
thick=5
thick=5
thick=6
thick=6
300
thick=7
160
thick=7
thick=8
thick=8
uniformlay distributed load (kN/m^2)
thick=9
central point load in (ton)
thick=10
thick=11
120
thick=12
thick=13
thick=14
thick=15
80
thick=9
250
thick=10
thick=11
thick=12
200
thick=13
thick=14
thick=15
150
100
40
50
0
0
-50
-100
-150
-200
-250
-300
0
-350
0
central deflection in (mm)
-20
-40
-60
-80
central deflection in (mm)
Fig.32: Load Deflection Relationships of
Rectangular Plate with Varying Plate Thickness
under Central Point Load
Fig.33: Load Deflection Relationships of
Rectangular Plate with Varying Plate Thickness
under Uniformly Distributed Load
b. Steel deck plate with one-way open ribs
1200
100
thick=5
thick=6
1000
thick=7
80
thick=8
uniformly distributed load (kN/m^2)
thick=9
central point load (ton)
thick=10
thick=11
60
thick=12
thick=13
thick=14
thick=15
40
800
thick=5
thick=6
600
thick=7
thick=8
thick=9
thick=10
400
thick=11
thick=12
20
thick=13
200
thick=14
thick=15
0
0
0
-50
-100
-150
-200
-250
cerntraldeflection
deflection (mm)
(mm)
central
Fig.34: Load Deflection Relationships of Steel
Deck Plate with One-Way Open Ribs and Varying
Thickness of the Top Plate under Central Point
Load
0
-200
-400
-600
-800
cerntraldeflection
deflection(mm)
(mm)
central
Fig.35: Load Deflection Relationships of Steel
Deck Plate with One-Way Open Ribs and Varying
Thickness of the Top Plate under Uniformly
Distributed Load
c. Steel deck plate with open ribs and stiffened by transverse floor I-beams
250
100
thick=5
thick=6
thick=7
200
80
thick=8
uniformly distributed load (kN/m^2)
central point load (ton)
thick=9
60
thick=5
thick=6
thick=7
thick=8
40
thick=9
thick=10
thick=10
thick=11
thick=12
150
thick=13
thick=14
thick=15
100
thick=11
thick=12
20
50
thick=13
thick=14
thick=15
0
0
0
-40
-80
-120
-160
0
-200
-20
-40
Fig.36: Load Deflection Relationships of Steel
Deck plate with One-Way Open Ribs Stiffened by
Floor I-Beams and Varying Thickness of the Top
Plate under Central Point Load
-60
-80
-100
-120
central deflection (mm)
central deflection (mm)
Fig.37: Load Deflection Relationships of Steel
Deck Plate with One-Way Open Ribs Stiffened by
Floor I-Beams and Varying Thickness of the Top
Plate under Uniformly Distributed Load
d. Steel deck plate with closed ribs
1000
200
180
thick=5
thick=5
thick=6
thick=6
thick=7
thick=7
160
800
thick=8
thick=8
thick=9
central point load (ton)
140
uniformly distributed load (kN/m^2)
thick=9
thick=10
thick=11
120
thick=12
thick=13
thick=14
100
thick=15
80
60
thick=10
thick=11
600
thick=12
thick=13
thick=14
thick=15
400
200
40
20
0
0
0
-50
-100
-150
-200
-250
-300
cerntral
deflection
(mm)
central
deflection
(mm)
Fig.38: Load Deflection Relationships of Steel
Deck Plate with Closed Ribs and Varying
Thickness of the Top Plate under Central Point
0
-100
-200
-300
-400
-500
-600
cerntral
deflection(mm)
(mm)
central
deflection
Fig.39: Load Deflection Relationships of Steel
Deck Plate with Closed Ribs and Varying
Thickness of the Top Plate under Uniformly
2. Effect of Changing the Length of Thickening Edge Plate
a. Steel rectangular plate
5
100
suport=0.00L
suport=0.05L
suport=0.10L
4
80
suport=0.15L
uniformly distributed load (kN/m^2)
suport=0.20L
central point load (ton)
suport=0.25L
suport=0.30L
3
2
60
40
suport=0.00L
suport=0.05L
suport=0.10L
suport=0.15L
1
20
suport=0.20L
suport=0.25L
suport=0.30L
0
0
0
-50
-100
-150
-200
-250
0
-20
-40
central deflction (mm)
-60
-80
-100
central deflction (mm)
Fig.40: Load Deflection Relationships of Steel
Rectangular Plate with Varying Lengths of
Thickening Plate under Central Point Load
b. Steel deck plate with one-way open ribs
Fig.41: Load Deflection Relationships of Steel
Rectangular Plate with Varying Lengths of
Thickening Plate under Uniformly Distributed
Load
800
40
support=.00L
suport=.00L
support=.05L
700
suport=.05L
support=.10L
suport=.10L
support=.15L
suport=.15L
uniformly distributed load (kN/m^2)
30
600
suport=.20L
central point load (tons)
suport=.25L
suport=.30L
20
support=.20L
support=.25L
support=.30L
500
400
300
200
10
100
0
0
0
0
-50
-100
-150
-200
-250
-100
-200
-300
-400
-500
-600
central deflction (mm)
central deflction (mm)
Fig.42: Load Deflection Relationships of Steel Deck
Plate with One Way-Open Ribs and Varying Lengths
of Thickening Plate under Central Point Load
Fig.43: Load Deflection Relationships of Steel Deck
Plate with One Way-Open Ribs and Varying Lengths
of Thickening Plate under Uniformly Distributed
Load
35
350
30
300
25
250
uniformly distributed load (kN/m^2)
central point load (ton)
c. Steel deck plate with open ribs and stiffened by transverse floor I-beams
20
15
thick=0.00L
thick=0.05L
10
thick=0.10L
200
150
thick=0.00L
thick=0.05L
100
thick=0.10L
thick=0.15L
thick=0.15L
thick=0.20L
5
thick=0.20L
50
thick=0.25L
thick=0.25L
thick=0.30L
thick=0.30L
0
0
0
-40
-80
-120
-160
-200
0
-50
-100
central deflection (mm)
-150
-200
-250
-300
-350
-400
central deflection (mm)
Fig.44: Load Deflection Relationships of Steel Deck
Plate with One Way-Open Ribs Stiffened by Floor IBeams and Varying Lengths of Thickening Plate
under Central Point Load
Fig.45: Load Deflection Relationships of Steel Deck
Plate with One Way-Open Ribs Stiffened by Floor IBeams and Varying Lengths of Thickening Plate
under Uniformly Distributed Load
d. Steel deck plate with closed ribs
1000
30
suport=0.00L
900
suport=0.05L
suport=0.10L
25
800
suport=0.15L
uniformly distributed load (kN/m^2)
suport=0.20L
suport=0.25L
central point load (ton)
20
suport=0.30L
15
10
700
600
500
400
thick=0.00L
thick=0.05L
300
thick=0.10L
thick=0.15L
200
thick=0.20L
5
thick=0.25L
100
thick=0.30L
0
0
0
-20
-40
-60
-80
central deflction (mm)
Fig.46: Load Deflection Relationships of Steel Deck
Plate with Closed Ribs and Varying Lengths of
Thickening Plate under Central Point Load
0
-100
-200
-300
-400
-500
-600
-700
cent ral deflection (mm)
Fig.47: Load Deflection Relationships of Steel Deck
Plate with Closed Ribs and Varying Lengths of
Thickening Plate under Uniformly Distributed Load
3. Effect of Changing the Aspect Ratio of the Steel Rectangular Plate
1000
5
as=1
as=1
as=1.137
as=1.137
as=1.306
as=1.306
4
800
as=1.517
as=1.517
as=1.777
uniformly distributed load (kN/m^2)
as=1.777
central point load (ton)
as=2.1157
as=2.56
3
as=3.16
as=4
2
as=2.1157
as=2.56
as=3.16
600
as=4
400
200
1
0
0
0
-10
-20
-30
-40
0
-50
-4
-8
-12
-16
-20
central deflction (mm)
central deflction (mm)
Fig.48: Load Deflection Relationships of Steel
Rectangular Plate with Varying Aspect Ratios under
Central Point Load
Fig.49: Load Deflection Relationships of Steel
Rectangular Plate with Varying Aspect Ratios under
Uniformly Distributed Load
4. Effect of Changing the Spacing between Ribs with Keeping the Same Total Weight
a. Steel deck plate with one-way open ribs
100
300
space=150
space=150
space=200
space=200
space=250
80
space=250
250
space=300
space=300
space=350
uniformly distributed load (kN/m^2)
space=350
central point load (ton)
space=400
space=450
60
40
space=400
200
space=450
150
100
20
50
0
0
0
-100
-200
-300
-400
-500
central deflection (mm)
Fig.50: Load Deflection Relationship of Steel Deck
Plate with One-Way Open Ribs, Changing the
Spacing between Ribs and Keeping the Total Weight
under Central Point Load
0
-50
-100
-150
-200
-250
central deflection (mm)
Fig.51: Load Deflection Relationship of Steel Deck
Plate with One-Way Open Ribs, Changing the
Spacing between Ribs and Keeping the Total Weight
under Uniformly Distributed Load
b. Steel deck plate with open ribs and stiffened by transverse floor I-beams
80
400
350
300
uniformly distributed load (kN/m^2)
central point load (ton)
60
40
space=150
space=200
space=250
space=300
space=350
20
250
200
space=150
space=200
150
space=250
space=300
space=350
100
space=400
space=400
space=450
space=450
50
space=500
space=500
space=550
space=550
0
0
0
-50
-100
-150
-200
-250
0
-100
-200
central deflection (mm)
-300
-400
-500
central deflection (mm)
Fig.45: Load Deflection Relationship of Steel Deck
Plate with One-Way Open Ribs Stiffened by Floor IBeams, Changing the Spacing between Ribs and
Keeping the Total Weight under Central Point Load
c. Steel deck plate with closed ribs
Fig.46: Load Deflection Relationship of Steel Deck
Plate with One-Way Open Ribs Stiffened by Floor IBeams, Changing the Spacing between Ribs and
Keeping the Total Weight under Uniformly
Distributed Load
1000
100
space=300
900
space=300
space=350
space=350
space=400
space=400
80
800
uniformly distributed load (kN/m^2)
space=500
space=500
space=550
central point load (ton)
space=450
space=450
space=600
60
space=650
40
700
space=550
space=600
600
space=650
500
400
300
200
20
100
0
0
0
0
-40
-80
-120
-160
-200
-100
-200
-300
-400
-500
-600
-700
cerntraldeflection
deflection (mm)
central
(mm)
cerntraldeflection
deflection (mm)
central
(mm)
Fig.47: Load Deflection Relationship of Steel Deck
Plate with Closed ribs, Changing the Spacing
between Ribs and Keeping the Total Weight under
Fig.48: Load Deflection Relationship of Steel Deck
Plate with Closed ribs, Changing the Spacing
between Ribs and Keeping the Total Weight under
Uniformly Distributed Load
5. Effect of Changing the Spacing between Ribs
a. Steel deck plate with one-way open ribs
140
80
space=150
space=150
space=200
space=200
120
space=250
space=250
space=300
space=300
space=350
space=350
uniformly distributed load (kN/m^2)
60
central point load (ton)
space=400
space=450
40
100
space=400
space=450
80
60
40
20
20
0
0
0
-100
-200
-300
0
-400
-40
Fig.52: Load Deflection Relationship of Steel Deck
Plate with One-Way Open Ribs and Changing the
Spacing between Ribs under Central Point Load
-80
-120
central
cerntraldeflection
deflection (mm)
cerntral
deflection (mm)
central deflection
(mm)
Fig.53: Load Deflection Relationship of Steel Deck
Plate with One-Way Open Ribs and Changing the
Spacing between Ribs under Uniformly Distributed
Load
70
350
60
300
50
250
uniformly distributed load (kN/m^2)
central point load (ton)
b. Steel deck plate with open ribs and stiffened by transverse floor I-beams
40
30
s=150
s=200
s=250
20
200
150
space=150
space=200
space=250
100
space=300
s=300
space=350
s=350
space=400
s=400
10
50
space=450
s=450
space=500
s=500
0
0
0
-50
-100
-150
-200
-250
central deflction (mm)
Fig.54: Load Deflection Relationship of Steel Deck
Plate with One-Way Open Ribs Stiffened by Floor IBeams and Changing the Spacing between Ribs
under Central Point Load
0
-50
-100
-150
-200
-250
-300
-350
central deflection (mm)
Fig.55: Load Deflection Relationship of Steel Deck
Plate with One-Way Open Ribs Stiffened by Floor IBeams and Changing the Spacing between Ribs
under Uniformly Distributed Load
c. Steel deck plate with closed ribs
900
80
800
700
uniformly distributed load (kN/m^2)
central point load (ton)
60
40
space=300
space=350
space=400
space=450
600
500
space=300
400
space=350
space=400
300
space=450
space=500
space=500
20
200
space=550
space=550
space=600
space=600
space=650
space=650
100
space=700
space=700
0
0
0
-50
-100
-150
-200
-250
0
-100
-200
central deflection (mm)
-300
-400
-500
-600
-700
-800
central deflection (mm)
Fig.56: Load Deflection Relationship of Steel Deck
Plate with Closed Ribs and Changing the Spacing
between Ribs under Central Point Load
Fig.57: Load Deflection Relationship of Steel Deck
Plate with Closed Ribs and Changing the Spacing
between Ribs under Uniformly Distributed Load
6. Effect of Changing the Support Conditions
a. Steel rectangular plate
500
20
400
unformly distributed load (kN/m^2)
central point load (ton)
16
12
8
200
s.s all eges
100
s.s all eges
4
300
f-short side
f-short side
f-long side
f-long side
f.f. all edges
f.f. all edges
0
0
0
0
-20
-40
-60
-4
-8
-12
-16
-20
central deflction (mm)
central deflction (mm)
Fig.58: Load Deflection Relationships of Steel
Rectangular Plate with Different Edge Conditions
under Central Point Load
Fig.59: Load Deflection Relationships of Steel
Rectangular Plate with Different Edge Conditions
under Uniformly Distributed Load
b. Steel deck plate with one-way open ribs
40
80
70
60
uniformly distributed load (kN/m^2)
central point load (ton)
30
20
s.s
fxsy
10
50
40
30
s.s
fxsy
20
sxfy
sxfy
platefree
platefree
ribfree
ribfree
10
f.f.
f.f.
0
0
0
-100
-200
-300
-400
0
-20
-40
central deflection (mm)
-60
-80
-100
-120
-140
-160
central deflection in (mm)
Fig.60: Load Deflection Relationships of Steel Deck
Plate with One-Way Open Ribs and Different Edge
Conditions under Central Point Load
Fig.61: Load Deflection Relationships of Steel Deck
Plate with One-Way Open Ribs and Different Edge
Conditions under Uniformly Distributed Load
c. Steel deck plate with open ribs and stiffened by transverse floor I-beams
120
50
uniformly distributed load (kN/m^2)
central point load (ton)
40
30
20
80
40
sxsy
sxsy
10
platefree
platefree
ribfree
ribfree
fxfy
fxfy
0
0
0
-40
-80
-120
-160
-200
central deflection (mm)
Fig.62: Load Deflection Relationships of Steel Deck
Plate with One-Way Open Ribs Stiffened by Floor IBeams and Different Edge Conditions under Central
Point Load
0
-20
-40
-60
-80
-100
central deflection (mm)
Fig.63: Load Deflection Relationships of Steel Deck
Plate with One-Way Open Ribs Stiffened by Floor IBeams and Different Edge Conditions under
Uniformly Distributed Load
d. Steel deck plate with closed ribs
120
80
sxsy
fxsy
sxfy
platefree
60
uniformly distributed load (kN/m^2)
ribfree
central point load (ton)
fxfy
40
80
40
sxsy
fxsy
20
sxfy
platefree
ribfree
fxfy
0
0
0
-100
-200
0
-300
-40
-80
Fig.64: Load Deflection Relationships of Steel Deck
Plate with Closed Ribs and Different Edge
Conditions under Central Point Load
-120
-160
-200
central
cerntraldeflection
deflection (mm)
(mm)
cerntral
dflection (mm)
(mm)
central
deflection
Fig.65: Load Deflection Relationships of Steel Deck
Plate with Closed Ribs and Different Edge
Conditions under Uniformly Distributed Load
7. Effect of Changing of the Material Properties
a. Steel rectangular plate
100
50
fy=290
fy=250
fy=345
fy=290
fy=345
fy=414
40
80
fy=448.5
uniformly distributed load (kN/m^2)
fy=621
central point load (ton)
fy=414
fy=448.5
fy=483
fy=690
30
20
fy=483
fy=621
60
fy=690
40
20
10
one quarter of the other plates
0
0
0
-40
-80
-120
-160
-200
central deflection (mm)
Fig.66: Load Deflection Relationships of Steel
Rectangular Plate with Different Types of Steel
Property under Central Point Load
0
-20
-40
-60
-80
central deflection (mm)
Fig.67: Load Deflection Relationships of Steel
Rectangular Plate with Different Types of Steel
Property under Uniformly Distributed Load
b. Steel deck plate with one-way open ribs
80
1000
fy=250
fy=290
fy=345
800
fy=414
60
uniformly distributed load (kN/m^2)
fy=448.5
central point load (ton)
fy=483
fy=621
fy=690
40
600
fy=250
400
fy=290
fy=345
fy=414
20
fy=448.5
200
fy=483
fy=621
fy=690
0
0
0
-50
-100
-150
-200
-250
0
-200
central deflection (mm)
-400
-600
-800
central deflection (mm)
Fig.68: Load Deflection Relationships of Steel Deck
Plate with One-Way Open Ribs and Different Types
of Steel Property under Central Point Load
Fig.69: Load Deflection Relationships of Steel Deck
Plate with One-Way Open Ribs and Different Types
of Steel Property under Uniformly Distributed Load
c. Steel deck plate with open ribs and stiffened by transverse floor I-beams
120
800
fy=250
fy=290
fy=345
fy=414
600
uniformly distributed load(kN/m^2)
fy=448.5
fy=483
80
central point load (ton)
fy=621
fy=690
40
400
fy=250
fy=290
fy=345
fy=414
200
fy=448.5
fy=483
fy=621
fy=690
0
0
0
-50
-100
-150
-200
-250
central deflection (mm)
Fig.70: Load Deflection Relationships of Steel Deck
Plate with One-Way Open Ribs Stiffened by Floor IBeams and Different Types of Steel Property under
Central Point Load
0
-100
-200
-300
-400
-500
central deflection (mm)
Fig.71: Load Deflection Relationships of Steel Deck
Plate with One-Way Open Ribs Stiffened by Floor IBeams and Different Types of Steel Property under
Uniformly Distributed Load
d. Steel deck plate with closed ribs
1000
120
fy=250
fy=290
fy=345
800
fy=414
uniformly distributed load (kN/m^2)
fy=448.5
fy=483
80
central point load (ton)
fy=621
fy=690
40
600
fy=250
400
fy=290
fy=345
fy=414
fy=448.5
200
fy=483
fy=621
fy=690
0
0
0
-40
-80
-120
-160
-200
0
-200
central deflection (mm)
-400
-600
central deflection (mm)
Fig.72: Load Deflection Relationships of Steel Deck
Plate with Closed Ribs and Different Types of Steel
Property under Central Point Load
Fig.73: Load Deflection Relationships of Steel Deck
Plate with Closed Ribs and Different Types of Steel
Property under Uniformly Distributed Load
8. The Effect of Initial Imperfection on the Steel Rectangular Plate
20
40
defl=5
16
defl=7.5
defl=5
defl=10
defl=7.5
defl=12.5
defl=10
defl=15
defl=12.5
30
uniformly distributed load (kN/m^2)
central point load (ton)
defl=17.5
defl=20
12
defl=22.5
defl=25
defl=27.5
defl=30
8
4
defl=15
defl=17.5
defl=20
defl=22.5
defl=25
20
defl=27.5
defl=30
10
0
0
-10
-20
-30
central
cerntraldeflection
deflection (mm)
(mm)
-40
-50
0
0
Fig.74: Load Deflection Relationships of Steel
Rectangular Plate with Initial Imperfection under
Central Point Load
-4
-8
-12
-16
-20
central deflection
(mm)
cerntral
deflection (mm)
Fig.75: Load Deflection Relationships of Steel
Rectangular Plate with Initial Imperfection under
Uniformly Distributed Load
SENSITIVITY ANALYSIS OF SOME
PARAMETERS AFFECTING FIBROUS CONCRETE SLABS
The dimensions of the slab are 4.5 m by 4.5 m and 150 mm thick.
The steel fiber volume fraction (Vf) is 1.0 percent with aspect ratio (lf/df) of 30. For fibrous
reinforced concrete slabs, it is assumed that top and bottom reinforcements in two directions are
used of the following values: the reinforcements in x-direction are 0.076 mm2/m (rx=0.33%) and
0.108 mm2/m (r'x=0.49%) at positive and negative moment, respectively. Conversely at ydirection an amount of 0.036 mm2/m (ry=0.17%) and 0.051 mm2/m (r'y=0.23%) at positive and
negative moment, respectively. The compressive strength is assumed equal to 30.49 MPa and the
other material properties are calculated by using the equations listed previously.
1. The Effect of Changing Thickness of Fibrous Concrete Slabs
a. Unrestrained fibrous concrete slab
b. Restrained fibrous concrete slab
200
c. Restrained fibrous reinforced
concrete slab
600
700
thick15
thick16
600
thick17
500
160
thick18
thick19
500
thick20
120
thick15
thick16
thick17
thick18
80
thick19
400
central point load (kN)
central point load (kN)
central point load (kN)
400
thick15
thick16
thick17
300
thick18
thick19
thick22
thick25
thick22
thick23
thick23
100
thick24
100
thick24
thick25
thick25
0
0
0
-6
thick24
thick21
40
-4
thick23
300
thick20
200
thick21
-2
thick22
200
thick20
0
thick21
-8
-10
0
-10
-20
centraldeflection
deflction (mm)
central
(mm)
-30
-40
-50
-60
-70
-80
-90
-100
0
-110
-20
-40
Fig.76: Load Deflection Relationship of Unrestrained Fibrous
Concrete Slab with Varying Thickness under Central Point
Load
Fig.78: Load Deflection Relationship of Restrained Fibrous
Concrete Slab with Varying Thickness under Central Point
Load
-60
-80
-100
centraldeflection
deflction (mm
(mm))
central
central deflection (mm)
Fig.80: Load Deflection Relationship of Restrained Fibrous
Reinforced Concrete Slab with Varying Thickness under
Central Point Load
180
18
100
160
16
90
14
80
12
thick15
10
thick16
thick17
8
thick18
thick19
6
thick20
thick21
4
70
60
thick15
thick16
50
thick17
thick18
40
thick19
120
thick15
100
thick16
thick17
80
thick18
thick19
60
thick20
thick21
thick20
30
thick22
40
thick21
thick22
thick23
thick22
20
thick23
thick23
thick24
2
uniformly distributed load (kN/m^2)
uniformly distributed load (kN/m^2)
uniformly distributed load (kN/m^2)
140
thick24
20
thick25
thick24
10
thick25
thick25
0
0
0
0
-2
-4
-6
-8
central deflction
(mm)
central
deflection
(mm)
Relationship of
-10
-12
-14
Fig.77: Load Deflection
Unrestrained Fibrous
Concrete Slab with Varying Thickness under Uniformly
Distributed Load
-20
-40
-60
-80
-100
-120
-140
-160
-180
-200
centraldeflection
deflction (mm)
central
(mm)
0
0
-40
-80
-120
-160
-200
central deflection (mm)
Fig.79: Load Deflection Relationship of Restrained Fibrous
Concrete Slab with Varying Thickness under Uniformly
Distributed Load
Fig.81: Load Deflection Relationship of Restrained Fibrous
Reinforced Concrete Slab with Varying Thickness under
Uniformly Distributed Load
2. The Effect of Changing Length of the Edge Thickening Slabs
a. Unrestrained fibrous concrete slab
b. Restrained fibrous concrete slab
160
500
300
thick.00L
thick.00L
thick.05L
140
thick.05L
thick.10L
120
c. Restrained fibrous reinforced
concrete slab
thick.10L
250
400
thick.15L
thick.15L
thick.20L
thick.20L
thick.25L
thick.25L
80
60
central point load (kN)
thick.30L
centeal point load (kN)
central point load(kN)
200
thick.30L
100
150
300
200
thick.00L
thick.05L
100
thick.10L
40
thick.15L
100
thick.20L
50
thick.25L
20
thick.30L
0
0
0
0
-5
-10
-15
0
-20
-20
Fig.82: Load Deflection Relationship of Unrestrained Fibrous
Concrete Slab with Varying Length of Edges Thickening Slab
under Central Point Load
-40
-60
-80
0
-100
-50
Fig.84: Load Deflection Relationship of Restrained Fibrous
Concrete Slab with Varying Length of Edges Thickening
Slab under Central Point Load
80
-100
-150
-200
-250
-300
-350
centraldeflection
deflction (mm)
central
(mm)
central deflection (mm)
central
(mm)
centraldeflection
deflction (mm)
Fig. 86: Load Deflection Relationship of Restrained Fibrous
Reinforced Concrete Slab with Varying Length of Edges
Thickening Slab under Central Point Load
1000
700
thick.00L
thick.05L
70
600
thick.10L
800
thick.15L
thick.25L
thick.30L
50
40
30
uniformly distributed load (kN/m^2)
thick.20L
uniformly distributed load (kN/m^2)
uniformly distributed load(kN/m^2)
60
500
400
300
thick.00L
thick.05L
200
thick.10L
600
400
thick.00L
thick.05L
thick.10L
20
thick.15L
thick.15L
200
100
10
thick.20L
thick.20L
thick.25L
thick.25L
thick.30L
0
thick.30L
0
0
0
-5
-10
-15
centraldeflection
deflction (mm)
central
(mm)
-20
-25
-30
0
-20
-40
-60
-80
central deflection (mm)
Fig.83: Load Deflection Relationship of Unrestrained Fibrous Fig.85: Load Deflection Relationship of Restrained Fibrous
Concrete Slab with Varying Length of Edges Thickening Slab Concrete Slab with Varying Length of Edges Thickening Slab
under Uniformly Distributed Load
under Uniformly Distributed Load
0
-100
-200
-300
-400
-500
-600
central
(mm)
centraldeflction
deflection
(mm)
Fig.87: Load Deflection Relationship of Restrained Fibrous
Reinforced Concrete Slab with Varying Length of Edges
Thickening Slab under Uniformly Distributed Load
3. Effect of Changing Edge Support Conditions
a. fibrous concrete slab
b. Fibrous reinforced concrete slab
350
300
300
250
250
central point load (kN)
central point load (kN)
200
150
200
150
100
100
ss-all edges
ss-all edges
f-short edges
f-short edges
50
50
f-long edges
f-long edges
ff-all edges
ff-all edges
0
0
0
-10
-20
-30
-40
0
-50
-10
-20
-30
Fig.88: Load Deflection Relationship of Fibrous
Concrete Slab with Varying Edge Support
Conditions under Central Point Load
-40
-50
-60
-70
-80
-90
-100
centraldeflection
deflction (mm)
central
(mm)
central deflection (mm)
Fig.90: Load Deflection Relationship of Fibrous
Concrete Reinforced Slab with Varying Edge
Support Conditions under Central Point Load
220
80
200
70
180
uniformly distributed load (kN/m^2)
uniformly distributed load (kN/m^2)
60
50
40
30
160
140
120
100
80
60
20
ss-all edges
ss-all edges
40
f-short edges
f-short edges
f-long edges
10
f-long edges
20
ff-all edges
ss- all edges
0
0
0
0
-10
-20
-30
-40
-50
-60
-70
central deflection (mm)
Fig.89: Load Deflection Relationship of Fibrous
Concrete Slab with Varying Edge Support
Conditions under Uniformly Distributed Load
-20
-40
-60
-80
-100
-120
-140
-160
-180
-200
central deflction
(mm)
central
deflection
(mm)
Fig.91: Load Deflection Relationship of Fibrous
Concrete Reinforced Slab with Varying Edge
Support Conditions under Uniformly Distributed
Load
4. Effect of Initial Imperfection
a. Unrestrained fibrous concrete slab
b. Restrained fibrous concrete slab
400
70
c. Restrained fibrous reinforced
concrete slab
700
defl=25
defl=50
350
60
600
defl=75
defl=100
300
defl=125
500
defl=150
defl=25
40
defl=50
defl=75
defl=100
30
defl=125
defl=150
defl=175
central point load (kN)
central point load (kN)
central point load (kN)
50
250
defl=200
defl=225
defl=250
200
defl=275
defl=300
150
defl=25
400
defl=50
defl=75
defl=100
300
defl=125
defl=150
defl=175
defl=175
200
20
defl=200
defl=200
100
defl=225
defl=225
defl=250
defl=250
10
100
50
defl=275
defl=275
defl=300
defl=300
0
0
0
-1
-2
-3
-4
-5
0
0
-6
-20
-40
Fig.92: Load Deflection Relationship of Unrestrained Fibrous
Concrete Slab with Varying Initial Imperfection under Central
Point Load
-60
-80
-100
-120
-140
-160
0
-20
-40
Fig. 94: Load Deflection Relationship of Restrained Fibrous
Concrete Slab with Varying Initial Imperfection under
Central Point Load
-80
-100
-120
-140
-160
-180
Fig.96: Load Deflection Relationship of Restrained Fibrous
Reinforced Concrete Slab with Varying Initial Imperfection
under Central Point Load
50
10
-60
central
(mm)
centraldeflection
deflction (mm)
central deflection (mm)
central
(mm)
centraldeflection
deflction (mm)
400
defl=25
9
defl=50
defl=75
40
8
defl=100
300
defl=25
6
defl=50
defl=75
5
defl=100
defl=125
4
defl=150
defl=175
3
uniformly distributed load (kN/m^2)
uniformly distributed load (kN/m^2)
uniformly distributed load (kN/m^2)
defl=125
7
defl=150
defl=175
defl=200
30
defl=225
defl=250
defl=275
defl=300
20
defl=200
1
defl=50
defl=75
200
defl=100
defl=125
defl=150
defl=175
defl=200
100
defl=225
2
defl=25
defl=225
10
defl=250
defl=250
defl=275
defl=275
defl=300
defl=300
0
0
0
0
-1
-2
-3
-4
-5
-6
-7
-8
-9
-10
central
(mm)
central deflection
deflction (mm)
Fig.93: Load Deflection Relationship of Unrestrained Fibrous
Concrete Slab with Varying Initial Imperfection under
Uniformly Distributed Load
0
-50
-100
-150
-200
-250
-300
-350
-400
central deflection (mm)
Fig.95: Load Deflection Relationship of Restrained Fibrous
Concrete Slab with Varying Initial Imperfection under
Uniformly Distributed Load
0
-50
-100
-150
-200
-250
central deflction
(mm)
central
deflection
(mm)
Fig.97: Load Deflection Relationship of Restrained Fibrous
Reinforced Concrete Slab with Varying Initial Imperfection
under Uniformly Distributed Load
5. Effect of Changing Strength of Concrete
a. Unrestrained fibrous concrete slab
b. Restrained fibrous concrete slab
70
300
c. Restrained fibrous reinforced
concrete slab
800
700
60
250
600
50
40
30
central point load (kN)
central point load (kN)
central point load (kN)
200
150
500
400
300
100
20
fc=20
fc=20
fc=25
fc=25
fc=30
10
fc=25
fc=30
fc=30
50
fc=35
fc=35
fc=40
fc=40
0
100
-2
-4
-6
-8
-10
fc=35
fc=40
0
0
0
fc=20
200
0
-20
-40
centraldeflection
deflction (mm)
central
(mm)
-60
-80
-100
-120
-140
0
-160
-50
-100
Fig.98: Load Deflection Relationship of Unrestrained Fibrous
Concrete Slab with Varying Ultimate Strength under Central
Point Load
Fig.100: Load Deflection Relationship of Restrained Fibrous
Concrete Slab with Varying Ultimate Strength under Central
Point Load
9
-150
-200
-250
-300
centraldeflection
deflction (mm)
central
(mm)
central deflection (mm)
Fig.102: Load Deflection Relationship of Restrained
Reinforced Fibrous Concrete Slab with Varying Ultimate
Strength under Central Point Load
600
50
8
500
40
6
5
4
3
uniformly distributed load (kN/m^2)
uniformly distributed load (kN/m^2)
uniformly distributed load (kN/m^2)
7
30
20
fc=20
fc=25
10
fc=30
fc=30
fc=35
fc=35
fc=40
fc=40
-2
-4
-6
-8
-10
centraldeflction
deflection
(mm)
central
(mm)
Fig.99: Load Deflection Relationship of Unrestrained Fibrous
Concrete Slab with Varying Ultimate Strength under
Uniformly Distributed Load
fc=30
100
fc=35
fc=40
0
0
0
0
200
fc=25
fc=25
1
300
fc=20
fc=20
2
400
0
-40
-80
-120
-160
-200
central deflection (mm)
Fig.101: Load Deflection Relationship of Restrained Fibrous
Concrete Slab with Varying Ultimate Strength under
Uniformly Distributed Load
0
-100
-200
-300
-400
-500
central
deflection
(mm)
central deflction
(mm)
Fig.103: Load Deflection Relationship of Restrained Fibrous
Reinforced Concrete Slab with Varying Ultimate Strength
under Uniformly Distributed Load
6. Effect of Changing Aspect Ratio of the Steel Fibers
a. Unrestrained fibrous concrete slab
b. Restrained fibrous concrete slab
c. Restrained fibrous reinforced
concrete slab
1000
300
70
900
60
250
800
50
700
40
30
150
100
as=30
20
central point load (kN)
central point load (kN)
central point load (kN)
200
as=30
as=45
as=45
as=60
as=60
as=75
600
500
400
as=30
300
as=45
as=60
200
as=75
as=75
50
10
as=90
as=90
as=105
as=105
0
as=90
100
as=105
0
0
0
-2
-4
-6
-8
-10
-12
0
-20
-40
centraldeflection
deflction (mm)
central
(mm)
-60
-80
-100
-120
-140
0
-160
-50
-100
Fig.104: Load Deflection Relationship of Unrestrained
Fibrous Concrete Slab with Varying Aspect Ratio under
Central Point Load
Fig.106: Load Deflection Relationship of Restrained Fibrous
Concrete Slab with Varying Aspect Ratio under Central Point
Load
8
-150
-200
-250
-300
-350
central
deflction (mm)
central
deflection
(mm)
central deflection (mm)
Fig.108: Load Deflection Relationship of Restrained Fibrous
Reinforced Concrete Slab with Varying Aspect Ratio under
Central Point Load
50
600
7
500
40
5
4
3
as=30
30
20
as=30
as=60
as=90
as=105
as=105
0
0
0
-6
-8
-10
as=75
100
as=90
as=105
-4
as=30
as=45
as=75
as=90
-2
200
10
as=75
0
300
as=60
as=60
1
400
as=45
as=45
2
uniformly distributed load (kN/m^2)
uniformly distributed load (kN/m^2)
uniformly distributed load (kN/m^2)
6
-12
centraldeflction
deflection
(mm)
central
(mm)
Fig.105: Load Deflection Relationship of Unrestrained
Fibrous Concrete Slab with Varying Aspect Ratio under
Uniformly Distributed Load
0
-40
-80
-120
-160
-200
central deflection (mm)
Fig.107: Load Deflection Relationship of Restrained Fibrous
Concrete Slab with Varying Aspect Ratio under Uniformly
Distributed Load
0
-50
-100
-150
-200
-250
-300
-350
-400
central
deflection
(mm)
central deflction
(mm)
Fig.109: Load Deflection Relationship of Restrained Fibrous
Reinforced Concrete Slab with Varying Aspect Ratio under
Uniformly Distributed Load
7. The Effect of Changing Reinforcement Steel Ratio in Restrained Fibrous Reinforced Concrete Slab
a. Changing bottom steel ratio
b. Changing top steel ratio
400
450
400
350
350
300
central point load (kN)
central point load (kN)
300
r
250
r
r
200
r
r
150
250
r
r
200
r
r
150
r
r
r
r
r
100
r
r
100
r
r
r
50
r
50
r
r
0
0
0
-20
-40
-60
-80
-100
-120
0
-140
-20
-40
-60
-80
-100
central deflection (mm)
central deflection (mm)
Fig.110: Load Deflection Relationship of
Restrained Fibrous Reinforced Concrete Slab with
Varying Bottom Steel Ratio under Central Point
Load
Fig.112: Load Deflection Relationship of
Restrained Fibrous Reinforced Concrete Slab with
Varying Top Steel Ratio under Central Point Load
180
140
160
120
uniformly distributed load (kN/m^2)
uniformly distributed load (kN/m^2)
140
100
80
r
r
r
60
r
r
r
40
r
120
r
100
r
r
80
r
r
60
r
r
r
r
40
r
r
20
r
r
20
r
r
0
0
0
-20
-40
-60
-80
-100
-120
-140
central deflection (mm)
Fig.111: Load Deflection Relationship of
Restrained Fibrous Reinforced Concrete Slab with
Varying Bottom Steel Ratio under Uniformly
Distributed Load
0
-20
-40
-60
-80
-100
-120
-140
-160
-180
-200
-220
central deflection (mm)
Fig.113: Load Deflection Relationship of
Restrained Fibrous Reinforced Concrete Slab with
Varying Top Steel Ratio under Uniformly
Distributed Load
Conclusions
1.The results obtained using ANSYS software, in the analysis of steel deck plates adopting large
deformation analysis (allowing for membrane action), are in agreement with the experimental tests.
2.While very stiff solutions are obtained, when using ANSYS software in the analysis of fibrous
reinforced concrete slabs, if compared with the experimental tests.
3.Conshell program gives very stiff solutions when Leonard's material parameters are used. But
agreement with the experimental tests is obtained when the assumed material parameters are
adopted.
4.For fibrous concrete slabs without reinforcement, an equivalent smeared layer for steel fibers shows
good modification in the analysis results compared with other analysis results.
5.Small deformation analysis (membrane action is not allowing) could not give full load deflection
curve. But it fails in a very small load compared with the experimental tests. So, conservative
solution is obtained with small displacement analysis.
6.The stiffness of steel deck plate is affected by increasing the spacing between ribs for the same total
weight. At the beginning of loading, the stiffness increases significantly. This is due to increasing the
moment of inertia of the structure which makes the bending action be dominant. But as the deflection
becomes large, the membrane action will be the dominant. So the changes in the stiffness of the steel
deck plate has restricted values with increasing the spacing between ribs and keeping the same total
weight.
7.Restraining the top plate of the steel deck plate and releasing the ribs give a structure having softening
behavior at the beginning of loading but as the deformation become large the stiffening behavior
appears, since the inducing of the membrane action. A reverse behavior could be gotten when
releasing the top plate and restraining ribs against horizontal movement.
8.When the initial imperfection increases the stiffness of the steel deck plate increases. This is attributed
to increasing the inclination of the induced membrane forces which increase the resisting of these
forces to the applied loads.
9.In fibrous concrete slab, as the initial imperfection increases the strength of unrestrained fibrous
concrete slab and restrained fibrous reinforced concrete slab increase. This is due to increasing the
inclination of the induced membrane forces which increases the vertical components of them. But,
when the initial imperfection increases the strength of restrained fibrous concrete slab reduces. This
may attributed to formation more than one region having high tensile stresses which causes cracks in
concrete slabs and reduce their strength.
THANK YOU
FINITE ELEMENT ANALYSIS OF MEMBRANE ACTION IN STEEL DECK PLATES AND
FIBROUS CONCRETE SLABS