Stress Analysis of a Singly Reinforced Concrete Beam with Uncertain Structural

ParametersDr.M.V.Rama Rao Dr.M.V.Rama Rao Department of Civil Department of Civil

Engineering,Engineering,Vasavi College of Vasavi College of

EngineeringEngineeringHyderabad-500 031, IndiaHyderabad-500 031, India

Dr.Ing.Andrzej PownukDepartment of Mathematical

Sciences,University of Texas at El Paso

Texas 79968, USADr.Iwona SkalnaDepartment of Applied Computer

Science University of Science and

Technology AGH, ul. Gramatyka 10, Cracow, Poland

Objective

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.

To introduce interval uncertainty in the To introduce interval uncertainty in the stress analysis of reinforced concrete stress analysis of reinforced concrete

flexural membersflexural members

Stress Stress AnalysisAnalysis of RC sections of RC sections based on nonlinear and/or discontinuous based on nonlinear and/or discontinuous

stress-strain relationships - analysis is stress-strain relationships - analysis is difficult to performdifficult to perform

aim of analyzing the beam is toaim of analyzing the beam is to• predict structural behavior in mathematical predict structural behavior in mathematical

termsterms• locate the neutral axis depthlocate the neutral axis depth• find out the stresses and strainsfind out the stresses and strains• compute the moment of resistancecompute the moment of resistance

design is followed by analysis - process of design is followed by analysis - process of iteration.iteration.

design process becomes clear only when design process becomes clear only when the process of analysis is learnt thoroughly.the process of analysis is learnt thoroughly.

A singly reinforced concrete beam subjected to A singly reinforced concrete beam subjected to an interval moment is taken up for analysis. an interval moment is taken up for analysis.

Area of steel reinforcement and the Area of steel reinforcement and the corresponding Young’s modulus are taken as corresponding Young’s modulus are taken as interval valuesinterval values

Moment of resistance of the beam is expressed Moment of resistance of the beam is expressed as a function of interval values of stresses in as a function of interval values of stresses in concrete and steelconcrete and steel

Stress distribution model for the cross section of Stress distribution model for the cross section of the beam is modified for the interval casethe beam is modified for the interval case

Internal moment of resistance is equated to the Internal moment of resistance is equated to the external bending moment arising due to interval external bending moment arising due to interval loads acting on the beam.loads acting on the beam.

Stresses in concrete and steel are obtained as Stresses in concrete and steel are obtained as interval values and combined membership interval values and combined membership functions are plottedfunctions are plotted

Steps involvedSteps involved

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

- design value - characteristic value

ud

m

SS

m

dS

cS

Partial safety factor for materialsaccount 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

- design value - characteristic value

f

d f cF F

dF

cF

Limit state is a function of safety factors

L Q TR D L Q T

0iL

0L Q TR D L Q T

Calibration of safety factors

0iL

0f fP P

fP - probability of failure

Interval limit state

L

1

fP

0L L L

fP

,L L L

0 0L L

Design of structures with interval parameters

A

P AP 0

Safe area

],[ 000

Design of structures with interval parameters

A

P

AP 0

],[ 000

0P

0A

0P],[ 00

PPP

}],,[],,[:{ 000000 APPPPA

More complicated safety conditions

lim it s ta te

uncerta in lim it s tate

1

2

crisp sta te

uncerta in sta te

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 Stress distribution due to a crisp momentmoment

z

fcc

b

Cross section

s

d

As

cyxy

(d-x) Ns=Asfs

Nc

Strains

Stresses

Neutral Axis

Stress-strain curves

Strains

Es

fy

Mild steelfco

Concrete

cocu

Governing Governing equationsequations

cy co cy cof =f for ε =ε

cy cc

yε = ε

x

Compressive strain in concrete

2

cy cycy co

co co

ε εf =f 2 -

ε ε

cy cofor ε ε

Compressive stress in concrete

Governing equationsGoverning equations

21 1

0

y x

c cy cc cc

y

N f bdy C C x

2 23co

co

bfC

1co

co

bfC

and

Tensile stress in steel ( )s s s cc

d xN A E

x

Compressive stress in concrete

where

Equation of longitudinal equilibrium leads to

21 2 0cc s s s sC C x A E x A E d

Governing equationsGoverning equations

Depth of resultant compressive force from the neutral axis is given by

1 2

0

1 2

0

2 33 4

y x

cy ccy

y xcc

cy

y

C Cbf ydy

y xC C

bf dy

( )R c cM N z N y d x Internal resisting moment is given by

For equilibrium RM M

Stress in steel 0.87s s s s cc y

d xf E E f

x

Singly reinforced section with Singly reinforced section with uncertain structural parameters and uncertain structural parameters and

subjected to an interval momentsubjected to an interval moment All the governing equations are expressed in the All the governing equations are expressed in the

equivalent interval form. equivalent interval form. The following are considered as interval valuesThe following are considered as interval values

ccε Interval extreme fiber strain in concrete

ccf Interval extreme fiber stress in concrete

x Interval depth of neutral axis

Interval stress in steelsf

Stress distribution due to an interval Stress distribution due to an interval momentmoment

z

fccb

Cross section

s

d

As

cyxy

(d-x) Ns=Asfs

Nc

Strains

Stresses

Neutral Axis

cc

cy cc

yε = ε

x

SEARCH-BASED ALGORITHM (SBA)SEARCH-BASED ALGORITHM (SBA) Used to compute the interval value of strain in Used to compute the interval value of strain in

concrete as concrete as Mid value M is computed as Mid value M is computed as The interval strain in concrete is initially The interval strain in concrete is initially

approximated as the point interval approximated as the point interval The lower and upper bounds of are obtained asThe lower and upper bounds of are obtained as

where and are the step sizes in strain, where and are the step sizes in strain, where where are multipliersare multipliers

While While are non-zero, interval form of are non-zero, interval form of

is solvedis solved

,cc 2

M MM

,cc cc

1 2,cc cc ccd d d d

1 2 and 1 2 and

21 2 0cc s s s sC C x A E x A E d

SEARCH-BASED ALGORITHM (SBA)SEARCH-BASED ALGORITHM (SBA)….….

1 2 and are incremented till M M is satisfiedR

1 is set to zero if ηR

R

M M

M

2 is set to zero if ηR

R

M M

M

=

=0.

1 2Search is discontinued if and are zero

Sensitivity analysis - Sensitivity analysis - AlgorithmAlgorithm

Sensitivity analysis -Sensitivity analysis - AlgorithmAlgorithm

Interval stress in extreme concrete fiber

min, min, min, min,1, , ,...,cc cc cc ccf f f f

cc cc cc mf f x p p

max, max, max, max,1, , ,...,cc cc cc ccf f f f

cc cc cc mf f x p p

Interval stress in steel

min, min, min, min,1, , ,...,s s s sf f f f

s s cc mf f x p p

max, max, max, max,1, , ,...,s s s sf f f f

s s cc mf f x p p

- Sensitivity cc cc cc cccc

i cc i i i

f f fxf

p p x p p

- Sensitivity s cc s ss

i cc i i i

f f fxf

p p x p p

Example ProblemExample ProblemA 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 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 2External 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 applicationComputations are performed online Computations are performed online using the web application developed using the web application developed by the authorsby the authors Posted at the website of University Posted at the website of University of Texas, El Paso, USA at the URLof Texas, El Paso, USA at the URL http://www.math.utep.edu/Faculty/ampownuk/php/concrete

-beam/

SNAP SHOTS OF RESULTS SNAP SHOTS OF RESULTS OBTAINED ARE PRESENTED OBTAINED ARE PRESENTED IN THE NEXT TWO SLIDESIN THE NEXT TWO SLIDES

Fig. 2 Membership function for bending moment

91

92

93

94

95

96

97

98

99

100

101

102

103

104

105

106

107

108

109

0

0.2

0.4

0.6

0.8

1

90 92 94 96 98 100 102 104 106 108 110

Bending moment (kNm)

Me

mb

ers

hip

va

lue

Figure 3 Membership function for area of steel reinforcement

2799

2946

2961

2975

2990

3005

3020

3034

3049

3064

3079

3093

2813

2828

2843

2858

2872

2887

2902

2917

2931

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

2796 2846 2896 2946 2996 3046 3096Area of steel reinforcement (mm^2)

Mem

ber

ship

val

ue

Figure 4 Membership function for Young's modulus of steel reinforcement

190

191

192

193

194

195

196

197

198

199

200

201

202

203

204

205

206

207

208

209

2100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

190 195 200 205 210Young's modulus (GPa)

Mem

ber

ship

val

ue

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

Combined membership functions

Figure 5 Combined membership function for neutral axis depth(x)

0

0.2

0.4

0.6

0.8

1

260.6 265.6 270.6 275.6 280.6Neutral Axis Depth (mm)

Mem

ber

ship

val

ue

Combinatorial Solution

Search-based algorithm

Figure 6 Membership function for strain in concrete

0

0.2

0.4

0.6

0.8

1

4.50E-04 4.70E-04 4.90E-04 5.10E-04 5.30E-04Strain in extreme concrete fiber(ecc)

Mem

ber

ship

val

ue

Combinatorial SolutionSearch-based algorithm

Figure 7 Combined membership function for stress in extreme concrete fiber

0

0.2

0.4

0.6

0.8

1

5.35 5.45 5.55 5.65 5.75 5.85 5.95 6.05 6.15 6.25Stress in extreme concrete fiber (N/mm^2)

Mem

ber

ship

val

ue

CombinatorialSearch-based algorithm

Figure 8 Combined membership function for stress in steel

0

0.2

0.4

0.6

0.8

1

67 77 87 97Stress in steel reinforcement (N/mm^2)

Mem

ber

ship

val

ue

combinatorialsearch-based algorithm

ConclusionsConclusions Cross section of a singly reinforced Cross section of a singly reinforced

beam subjected to an interval beam subjected to an interval bending moment is analyzed bybending moment is analyzed by search based algorithm, search based algorithm, sensitivity sensitivity analysisanalysis and and combinatorial approachcombinatorial approach..

The results obtained are in excellent The results obtained are in excellent agreement and allow the designer to agreement and allow the designer to have a detailed knowledge about the have a detailed knowledge about the effect of uncertainty on the stress effect of uncertainty on the stress distribution of the beam.distribution of the beam.

ConclusionsConclusions In the present paper, a singly reinforced In the present paper, a singly reinforced

beam with interval values of area of beam with interval values of area of steel reinforcement and interval steel reinforcement and interval Young’s modulus and subjected to an Young’s modulus and subjected to an external interval bending moment is external interval bending moment is taken up.taken up.

The stress analysis is performed by The stress analysis is performed by three approaches viz. a search based three approaches viz. a search based algorithm and sensitivity analysis and algorithm and sensitivity analysis and combinatorial approach.combinatorial approach.

It is observed that the results obtained It is observed that the results obtained are in excellent agreement.are in excellent agreement.

ConclusionsConclusions

These approaches allow the These approaches allow the designer to have a detailed designer to have a detailed knowledge about the effect of knowledge about the effect of uncertainty on the stress distribution uncertainty on the stress distribution of the beam.of the beam.

The combined membership functions The combined membership functions are plotted for neutral axis depth are plotted for neutral axis depth and stresses in concrete and steel and stresses in concrete and steel and are found to be triangular.and are found to be triangular.

ConclusionsConclusions Interval stress and strain are also Interval stress and strain are also

calculated using sensitivity analysis.calculated using sensitivity analysis. Because the sign of the derivatives in the Because the sign of the derivatives in the

mid point and in the endpoints is the mid point and in the endpoints is the same then the solution should be exact.same then the solution should be exact.

More accurate monotonicity test is based More accurate monotonicity test is based on second and higher order derivatives.on second and higher order derivatives.

Results with guaranteed accuracy can be Results with guaranteed accuracy can be calculated using interval global calculated using interval global optimization.optimization.

Extended version of this paper is published Extended version of this paper is published on the web page of the Department of on the web page of the Department of Mathematical ScienceMathematical Sciencess at the University of at the University of Texas at El Paso Texas at El Paso

http://www.math.utep.edu/preprints/2007-05.pdfhttp://www.math.utep.edu/preprints/2007-05.pdf

THANTHANK YOUK YOU