1. Stress Analysis of a Singly Reinforced Concrete Beam with Uncertain Structural Parameters
Dr.M.V.Rama Rao
Department of Civil Engineering,
Vasavi College of Engineering
Hyderabad , India
Dr.Ing.Andrzej Pownuk
Department of Mathematical Sciences,
University of Texas at El Paso
Texas 79968, USA
Dr.Iwona Skalna
Department of Applied Computer Science
University of Science and Technology AGH, ul. Gramatyka 10, Cracow, Poland

2. 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.

3. Stress Analysis of RC sections
based on nonlinear and/or discontinuous stress-strain 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. 3

4. 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 4

5. 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.

6. Partial safety factors for materials
- design value
- characteristic value

7. 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.

8. Partial safety factors for loads
- design value
- characteristic value

9. Limit state is a function of safety factors

10. Calibration of safety factors
- probability of failure

11. Interval limit state

12. Design of structures with interval parameters

13. Design of structures with interval parameters

14. More complicated safety conditions

15. 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

16. Stress distribution due to a crisp moment
fcc b
Cross section s d As
cy x y
(d-x)
Ns=Asfs Nc
Strains
Stresses
Neutral Axis z  
Strains Es fy
Mild steel
fco
Concrete
co
cu
Stress-strain curves

17. Governing equations Compressive strain in concrete
Compressive stress in concrete

18. Governing equations Compressive stress in concrete and
where
and
Tensile stress in steel
Equation of longitudinal equilibrium leads to

19. Governing equations Internal resisting moment is given by
Depth of resultant compressive force from the neutral axis is given by
Internal resisting moment is given by
For equilibrium
Stress in steel

20. 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
Interval extreme fiber strain in concrete
Interval extreme fiber stress in concrete
Interval depth of neutral axis
Interval stress in steel

21. Stress distribution due to an interval moment
fcc
cc As Nc x
cy y
Neutral Axis d z
(d-x)
Ns=Asfs s
Stresses
Cross section
Strains

22. SEARCH-BASED ALGORITHM (SBA)
Used to compute the interval value of strain in concrete as
Mid value M is computed as
The interval strain in concrete is initially approximated as the point interval
The lower and upper bounds of are obtained as
where and are the step sizes in strain, where are multipliers
While are non-zero, interval form of
is solved

23. SEARCH-BASED ALGORITHM (SBA)….=
=0.

24. Sensitivity analysis - Algorithm

25. Sensitivity analysis - Algorithm

26. Interval stress in extreme concrete fiber
Interval stress in steel

27. 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 is adopted

28. 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 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
Young’s modulus of Steel reinforcement = [0.98,1.02]*2×105 N/mm2
Case 4
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

29. 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
SNAP SHOTS OF RESULTS OBTAINED ARE PRESENTED
IN THE NEXT TWO SLIDES 29

30. 30

31. 31

39. 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

44. 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.

45. 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.

46. 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. 46

47. 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.

48. 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

49. THANK YOU