Keywords: trapezoidal cross sections, Eurocode 2, reinforced concrete, determination of required reinforcement.

The uniaxial bending analysis of trapezoidal cross sections and the bending analysis of rectangular cross sections are both based on the assumption of Bernoulli's-hypothesis (all adjacent plane cross sections remain plane during loading). However, the analysis procedures for trapezoidal cross sections deviate from the analysis of a rectangular cross section due to a linear change in the width of the cross section.

The paper presents a detailed analysis of a trapezoidal cross section implementing the bilinear design stress-strain relationship for the concrete. The paper covers both possibilities that result from the value of the maximal strain in concrete that appears in the more strongly compressed edge of the compressive zone. If the absolute maximal strain in the concrete is smaller than , the stresses in the concrete are assumed to have only a linear distribution, which represents the first case. If the compressive stresses in the top compressed edge exceed the value of , the stress-strain relation is linear for the strains below , and stresses have a constant value for strains over . The margin between the two mathematical descriptions is thus .

For both cases mentioned, the paper presents the development of the analytical expressions for coefficients (the average value of concrete stresses in the bending compression zone, related to the design value of concrete strength ) and (the related distance of the concrete compression force from the stronger compressed edge of the compressive cross section). These analytical expressions are necessary for the computation of the required cross section of the reinforcements and (if the compression reinforcement is required). The procedure for determination of required cross section, based on equations resulting from equilibrium conditions, is also briefly described.

The derived coefficients and , implemented with the iterative procedure described, allow the reliable analysis of the trapezoidal cross sections as all the parameters required are determined from equilibrium equations. As all both expressions are given in closed analytical form they replace all charts and diagrams and also represent a very suitable form for computer programs.

1

ENV 1992-1-1: Eurocode 2: Design of concrete structures. Part 1: General rules and rules for buildings, 1991.

2

K.J. Schneider, "Bautabellen f�r Ingenieure", 12. Auflage, Chapter 5, Stahlbeton (EC2). D�sseldorf: Werner-Verlag, Germany, 1996.

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