An analytical study has been carried out to examine the influence of plate thickness on the stress distribution around the crack. The qualitative feature of the three-dimensional solution is first determined by an asymptotic expansion of the stresses and displacements in terms of the cylindrical polar coordinates r, θ, z for small values of r which is referenced from the border of a semi-infinite crack. It is found that the stresses σrr, σθθ, σ zz , and σ rθ are singular of the order r $$r{\text{ }}^{{\text{ - }}\tfrac{1}{2}} $$ , but the transverse shear stresses σ rz and σθz , are bounded for plates under stretching and bending. The intensity of crack-border stress field becomes a function of the thickness coordinate z. Knowing that the problem prohibits any exact analytical solutions of a quantitative nature, the three-dimensional equations of elasticity will be approximated by appealing to minimum principles in the calculus of variations. Guided by the results obtained from the asymptotic expansions, each one of the six stress components is assumed to be the product of two functions, one being assigned to describe the stress distribution in the plane of the plate and the other across the thickness. The z-distribution of the stresses may either be pre-assigned arbitrarily or determined from the plane strain condition ahead of the crack. On the basis of the principle of minimum complementary energy, a system of three simultaneous differential equation in two variables is obtained and solved for the problem of an infinite plate containing a through crack by means of integral representations. Determined in closed elementary form are the detailed structure of the three-dimensional crack-edge stress field. The stress-intensity factor, which varies in the thickness direction, is shown to be a function of the ratio of plate thickness to crack length and is found to increase rapidly in magnitude as the plate thickness is perturbed slightly from zero. The present analysis suggests a method by which the effect of a finite plate thickness can be incorporated into an examination of the fracture toughness of cracked sheet specimens.