Stress Analysis of a Singly Reinforced Concrete Beam

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 n 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 n n based on nonlinear and/or discontinuous stressstrain relationships - analysis is difficult to perform aim of analyzing the beam is to • • n n 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 n n n 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 n n 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 n n - design value - characteristic value

Partial safety factor for materials account for… n n 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 n n - design value - characteristic value

Limit state is a function of safety factors

Calibration of safety factors - probability of failure

Interval limit state

Design of structures with interval parameters

More complicated safety conditions

Advantages of the interval limit state n n n 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 Compressive strain in concrete Compressive stress in concrete

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

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

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

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

n n 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 n While and are the step sizes in strain, where are multipliers are non-zero, interval form of is solved

SEARCH-BASED ALGORITHM (SBA)…. = =0.

Sensitivity analysis - Algorithm

Interval stress in extreme concrete fiber Interval stress in steel

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 mm 2 (6 – 25 Ø TOR 50 bars) Moment = 100 k. Nm Allowable compressive stress in concrete fco = 13. 4 N/mm 2 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 n n Case 1 External moment M= [96, 104] k. Nm Area of Steel reinforcement = 2946 mm 2 Young’s modulus of Steel reinforcement Es= 2× 105 N/mm 2 Case 2 External moment M= [90, 110] k. Nm Area of Steel reinforcement = [0. 9, 1. 1]*2946 mm 2 Young’s modulus of Steel reinforcement = 2× 105 N/mm 2 Case 3 External moment M= [80, 120] k. Nm Area of Steel reinforcement = 2946 mm 2 Young’s modulus of Steel reinforcement = [0. 98, 1. 02]*2× 10 5 N/mm 2 Case 4 External moment M= [90, 110] k. Nm Area of Steel reinforcement As = [0. 98, 1. 02]*2946 mm 2 Young’s modulus of Steel reinforcement Es= [0. 98, 1. 02]*2× 10 5 N/mm 2

Web-based application n. Computations are performed online using the web application developed by the authors n 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

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

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

Conclusions n n 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

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