By Wai-Fah Chen, Shouji Toma

ISBN-10: 0849382815

ISBN-13: 9780849382819

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156) and the stress-strain relationships. 61). Lekhnitskii (1981) expresses these in terms of the principal axes x and y of the cross-section. These axes are such that fxdA = fydA A = fxydA A =0 n 157") A where the integration is over the area A of the cross-section. For a simple resultant axial force P acting along the centroidal axis, the line with coordinates (0,0, z), »* = £(%* + \w) "y - j ( | * 3 6 * p A 33^ = —} (ί,,Ζ T + W) Λ + ·>35 S,,X + S,34 y) d·158) [Ch. 154), it can be seen that there is an overall strain εζ equal to PsJA.

12, then [Ch. dxàzdy? 12 Work done in shearing a cuboid. dxdydz Because τ^ and τν are equal, the sum of these two increments of work is f du, du \ bW„ + bW_. dxdydz 'y The total work done by all such increments is 6Ψ = (σχχδε1 + σ Ô V xx yy = x T (ÔY)dF + σ ζ,δε« + VÔV where dFis the volume of the cuboid, dxdydz. If the stress σ„ and the strain t„ are increased in proportion to one another up to theirfinalvalues, then the work done per unit volume is shown by the shaded area in Fig. 13 and is equal to 'Λσ^ ε „ .

It gives all possible states of stress under conditions of plane strain. It is often used to analyse general conditions of plane stress. 119). 74). 121) are additional constraints on the state of plane stress, unless Poisson's ratio is zero. A discussion of the conditions of plane stress will be found in Sokolnikoff (1956) section 67, where they are justified for the mean values of the stresses across the thickness of a plate. Lists of two- and three-dimensional biharmonic functions, suitable for use as χ functions, Airy stress functions, and Galerkin vectors will be found in Appendix A2.

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advanced analysis of steel frames by Wai-Fah Chen, Shouji Toma

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