Stress Analysis of a Singly
Reinforced Concrete Beam
with Uncertain Structural
Parameters
Dr.M.V.Rama Rao
Dr.Ing.Andrzej Pownuk
Department of Civil Engineering, Department of Mathematical Sciences,
Vasavi College of Engineering
University of Texas at El Paso
Hyderabad-500 031, India
Texas 79968, USA
Dr.Iwona Skalna
Department of Applied Computer Science
University of Science and Technology
AGH, ul. Gramatyka 10, Cracow, Poland
Objective
To introduce interval uncertainty in the
stress analysis of reinforced concrete
flexural members

In the present work, a singly-reinforced
concrete beam with interval area of steel
reinforcement and corresponding interval
Young’s modulus and subjected to an
interval moment is taken up for analysis.
Interval algebra is used to establish the
bounds for the stresses and strains in
steel and concrete.
Stress Analysis of RC sections


based on nonlinear and/or discontinuous stressstrain relationships - analysis is difficult to
perform
aim of analyzing the beam is to
•
•
•
•


predict structural behavior in mathematical terms
locate the neutral axis depth
find out the stresses and strains
compute the moment of resistance
design is followed by analysis - process of
iteration.
design process becomes clear only when the
process of analysis is learnt thoroughly.
Steps involved






A singly reinforced concrete beam subjected to an
interval moment is taken up for analysis.
Area of steel reinforcement and the corresponding
Young’s modulus are taken as interval values
Moment of resistance of the beam is expressed as a
function of interval values of stresses in concrete
and steel
Stress distribution model for the cross section of the
beam is modified for the interval case
Internal moment of resistance is equated to the
external bending moment arising due to interval
loads acting on the beam.
Stresses in concrete and steel are obtained as
interval values and combined membership functions
are plotted
IS 456-2000 - Indian Standard Code for
Plain and Reinforced concrete


The characteristic values should be based on statistical
data, if available. Where such data is not available, they
should be based on experience. The design values are
derived from the characteristic values through the use of
partial safety factors, both for material strengths and for
loads. In the absence of special considerations, these
factors should have the values given in this section
according to the material, the type of load and the limit
state being considered. The reliability of design is ensured
by requiring that
Design Action ≤ Design Strength.
Partial safety factors
for materials  m
Sd 


Su
m
Sd - design value
Sc - characteristic value
Partial safety factor for materials
account for…




the possibility of unfavorable deviation of material
strength from the characteristic value.
the possibility of unfavorable variation of member
sizes.
the possibility of unfavorable reduction in member
strength due to fabrication and tolerances.
uncertainty in the calculation of strength of the
members.
Partial safety factors
for loads  f
Fd   f Fc


Fd - design value
Fc - characteristic value
Limit state is a function
of safety factors
 R   D    L L  QQ  T T 


 R   D   L L  QQ  T T   0
L  i   0
Calibration of safety factors
L  i   0
Pf  Pf 0
  Pf
- probability of failure
Interval limit state
L   L    , L   
Pf
L  0  L     0
1
  Pf
L0
L 
L 
L
Design of structures
with interval parameters
P
P  0 A
Safe area
 0  [ 0 ,  0 ]
A
Design of structures
with interval parameters
P  0A
P
P  [ P0 , P0 ]
P0
P0
 0  [ 0 ,  0 ]
A0
A
{ A : P  [ P0 , P0 ],  0  [ 0 ,  0 ], P   0 A}
More complicated
safety conditions
2
uncertain limit state
crisp state
limit state
uncertain state
1
Advantages of the interval limit state



Interval limit state takes into account all
worst case combinations of the values
of loads and material parameters.
Interval limit state has clear
probabilistic interpretation.
Interval methods can be applied
in the framework of existing civil
engineering design codes
Stress distribution due to a crisp moment
fcc
b
cy
x
Neutral Axis
Nc
y
z
d
(d-x)
As
Ns=Asfs
s
Cross section
Stresses
Strains

fy

Strains
Mild steel
fco
Concrete
Es
Stress-strain curves
co
cu
Governing equations
y
ε cy =   ε cc
x
Compressive strain in concrete
Compressive stress in concrete
  ε cy   ε cy 
f cy =f co 2   -  
  ε co   ε co 
f cy =f co for ε cy =ε co
2

 for ε  ε
cy
co

Governing equations
Compressive stress in concrete
yx
Nc 

y 0
2

f cy bdy  C1 cc  C1 cc  x
 bf co 
C1  

  co 
and
where
 bf co 
C2   2 
 3 co 
d x
Tensile stress in steel N s  ( As Es ) cc 

x


Equation of longitudinal equilibrium leads to
C1  C2 cc  x
2
 As Es x  As Es d  0
Governing equations
Depth of resultant compressive force from the neutral axis is given by
yx

2
C
3
C





1
2
y 0 bfcy ydy  3    4   cc 
x
y  yx

C1  C2 cc 

 bfcy dy
y 0
Internal resisting moment is given by
M R  Nc  z  N c  ( y  d  x)
For equilibrium M  M R
Stress in steel f s  Es s  Es  d  x   cc  0.87 f y
 x 
Singly reinforced section with uncertain
structural parameters and subjected to an
interval moment
All the governing equations are expressed in the
equivalent interval form.
The following are considered as interval values
ε cc
Interval extreme fiber strain in concrete
f cc
Interval extreme fiber stress in concrete
x
Interval depth of neutral axis
fs
Interval stress in steel
Stress distribution due to an interval
moment
b
cy
x
Neutral Axis
Nc
y
d
z
(d-x)
As
Cross section
fcc
cc
Ns=Asfs
s
Strains
y
ε cy =   ε cc
x
Stresses





SEARCH-BASED ALGORITHM (SBA)
Used to compute the interval value of strain in concrete
as  cc  ,  
M M
M

Mid value M is computed as
2
The interval strain in concrete is initially approximated
as the point interval  cc ,  cc 
The lower and upper bounds of are obtained as
 cc  cc  1d  ,  cc  2d 
where d  and d  are the step sizes in strain, where
1 and 2 are multipliers
While 1 and 2 are non-zero, interval form of
C1  C2 cc  x
2
 As Es x  As Es d  0
is solved
SEARCH-BASED ALGORITHM (SBA)….
1 and 2 are incremented till M  M Ris satisfied
1 is set to zero if
MR  M
MR
η
=
MR  M
2 is set to zero if
η
=0.
MR
Search is discontinued if 1 and 2 are zero
Sensitivity analysis - Algorithm
Sensitivity analysis - Algorithm
Interval stress in extreme concrete fiber
f  cc f cc x f cc

f cc  cc


- Sensitivity
pi
 cc pi
x pi pi


f cc  f cc  ccmin, fcc , xmin, fcc , p1min, fcc ,..., pmmin, fcc
f cc  f cc
max, fcc
max, fcc
max, fcc
,
x
,
p
,...,
cc
1

pmmax, fcc

Interval stress in steel
f  cc f s x f s

fs  s


pi
 cc pi x pi pi

 f 
- Sensitivity
f s  f s  ccmin, f s , xmin, f s , p1min, f s ,..., pmmin, f s
fs
s
max, f s
max, f s
max, f s
,
x
,
p
,...,
cc
1
pmmax, f s


Example Problem
A singly reinforced beam with the following data is taken up
as an example problem
Breadth = 300 mm Overall depth = 550 mm
Effective depth = 500 mm
As = 2946 mm2 (6 – 25 Ø TOR50 bars) Moment = 100 kNm
Allowable compressive stress in concrete fco = 13.4 N/mm2
Allowable strain in concrete = 0.002
Young’s modulus of steel = 200 GPa
The stress-strain curve for concrete as detailed IS 456-2000 is
adopted
Case studies

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

Case 1
External moment M= [96,104] kNm
Area of Steel reinforcement = 2946 mm2
Young’s modulus of Steel reinforcement Es= 2×105 N/mm2
Case 2
External moment M= [90,110] kNm
Area of Steel reinforcement = [0.9,1.1]*2946 mm2
Young’s modulus of Steel reinforcement = 2×105 N/mm2
Case 3
External moment M= [80,120] kNm
Area of Steel reinforcement = 2946 mm2
Young’s modulus of Steel reinforcement = [0.98,1.02]*2×105 N/mm2
Case 4
External moment M= [90,110] kNm
Area of Steel reinforcement As = [0.98, 1.02]*2946 mm2
Young’s modulus of Steel reinforcement Es= [0.98, 1.02]*2×105 N/mm2
Web-based application
Computations
are performed online using
the web application developed by the
authors
 Posted at the website of University of
Texas, El Paso, USA at the URL
http://www.math.utep.edu/Faculty/ampownuk/php/concrete-beam/
SNAP SHOTS OF RESULTS
OBTAINED ARE PRESENTED
IN THE NEXT TWO SLIDES
1
100
99
Membership value
0.8
98
97
0.6
96
103
105
94
106
93
0.2
102
104
95
0.4
101
107
92
108
91
109
0
90
92
94
96
98
100 102 104 106 108 110
Bending moment (kNm)
Fig. 2 Membership function for bending moment
1
2946
2931
0.9
2917
Membership value
0.8
2902
0.7
2887
0.6
2990
3005
3020
2858
0.4
3034
2843
0.3
0.1
2975
2872
0.5
0.2
2961
3049
2828
3064
2813
0
2799
2796
3079
2846
2896
2946
2996
Area of steel reinforcement (mm^2)
3046
Figure 3 Membership function for area of steel reinforcement
3093
3096
1
200
0.9
199
Membership value
0.8
198
0.7
197
0.6
202
203
196
0.5
204
195
0.4
205
194
0.3
206
193
0.2
207
192
0.1
0
201
208
191
209
190
190
210
195
200
Young's modulus (GPa)
205
210
Figure 4 Membership function for Young's modulus of steel reinforcement
Combined membership functions
Combined membership functions are
plotted for
•Neutral axis depth
•Stress and Strain in extreme concrete fiber
•Stress and Strain in steel reinforcement
using the -sublevel strategy suggested by
Moens and Vandepitte
1
Membership value
0.8
0.6
0.4
Combinatorial Solution
Search-based algorithm
0.2
0
260.6
265.6
270.6
275.6
280.6
Neutral Axis Depth (mm)
Figure 5 Combined membership function for neutral axis depth(x)
Membership value
1
0.8
0.6
0.4
0.2
Combinatorial Solution
Search-based algorithm
0
4.50E-04
4.70E-04
4.90E-04
5.10E-04
5.30E-04
Strain in extreme concrete fiber(ecc)
Figure 6 Membership function for strain in concrete
1
Membership value
0.8
0.6
0.4
Combinatorial
Search-based algorithm
0.2
0
5.35
5.45
5.55 5.65 5.75
5.85 5.95 6.05 6.15 6.25
Stress in extreme concrete fiber (N/mm^2)
Figure 7 Combined membership function for stress in extreme concrete fiber
Membership value
1
combinatorial
search-based algorithm
0.8
0.6
0.4
0.2
0
67
77
87
97
Stress in steel reinforcement (N/mm^2)
Figure 8 Combined membership function for stress in steel
Conclusions


Cross section of a singly reinforced beam
subjected to an interval bending moment
is analyzed by search based algorithm,
sensitivity analysis and combinatorial
approach.
The results obtained are in excellent
agreement and allow the designer to have
a detailed knowledge about the effect of
uncertainty on the stress distribution of the
beam.
Conclusions



In the present paper, a singly reinforced
beam with interval values of area of steel
reinforcement and interval Young’s
modulus and subjected to an external
interval bending moment is taken up.
The stress analysis is performed by three
approaches viz. a search based algorithm
and sensitivity analysis and combinatorial
approach.
It is observed that the results obtained are
in excellent agreement.
Conclusions
These approaches allow the designer
to have a detailed knowledge about
the effect of uncertainty on the
stress distribution of the beam.
 The combined membership functions
are plotted for neutral axis depth
and stresses in concrete and steel
and are found to be triangular.

Conclusions




Interval stress and strain are also calculated
using sensitivity analysis.
Because the sign of the derivatives in the
mid point and in the endpoints is the same
then the solution should be exact.
More accurate monotonicity test is based on
second and higher order derivatives.
Results with guaranteed accuracy can be
calculated using interval global optimization.
Extended version of this paper is
published on the web page of the
Department of Mathematical Sciences
at the University of Texas at El Paso
http://www.math.utep.edu/preprints/2007-05.pdf
THANK
YOU