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structural mechanics §3.3 Bending of a simple beam under uniform load The semi-inverse method will be employed here. The crushing stress is mainly produced by the direct load on the beam. Since the direct load q does not vary with x, we may assume that does not vary with x either and consequently it is only a function of y. §3.3 Bending of a simple beam under uniform load From Eqs.(2.12.12), we have Successive integration with respect to x yields §3.3 Bending of a simple beam under uniform load We substitute the expression for φinto compatibility equation (2.12.11), obtaining where and are arbitrary functions. §3.3 Bending of a simple beam under uniform load This is a quadratic equation of x, it must be satisfied for all values of x between –l and l, as the compatibility requires. This is possible only when the coefficients of x2 and x, as well as the term independent of x, are zero: §3.3 Bending of a simple beam under uniform load Integration the first two equations yields From third equation, we have which, through integration, becomes §3.3 Bending of a simple beam under uniform load The constant term and the linear in y are neglected, because they will not affect the stress. With the function , and known, the stress function becomes §3.3 Bending of a simple beam under uniform load According to Eqs.(2.12.12), the stress components will be §3.3 Bending of a simple beam under uniform load Since the yz plane is a plane of symmetry of the beam and the loading, the stress distribution must be symmetric with respect to the plane. Thus, the expressions for and must be even function of x, while that for must be an odd function of x. This requires §3.3 Bending of a simple beam under uniform load The boundary conditions on the horizontal sides are Substituting Eqs.(3.3.3) and (3.3.4) into these equations and noting that ,then we have §3.3 Bending of a simple beam under unif