境界面に二等辺三角形の分布荷重を受ける半無限体の平面応力関数

抄録

This is one of the plane stress problems of a semi-infinite body under a load distributed in geometrical figure on its boundary. Starting from the well-known Airy's stress function with α=nπ/a, we have
F=∞Σ1A_n(1+αy)e^-αy·cosαx
for a symmetrically distributed load along the boundary line.
According to Fig. 3, equation for a distributed load in equilateral triangle will be given by
p(x)=p₀(1-x/c)
for a range 0<x<c, where p₀ is the height of the load triangle and 2c is the length of its base line. After expanding p(x) in Fourier's series, we have
p(x)=∞Σ12p₀/ac1/α²(1-cosαc)cosαx
For y=0, the normal stress on the boundary will be given on the other hand by
σ_y=-∞Σ1A_nα²cosαx=-p(x)
After determining constant A_n, we have following stress function in the from of integral instead of series:
F=p₀/πc∫∞02/α⁴(1-cosαc)(1+αy)e^-αycosαx·dα
Stress components are the second derivatives of the stress function F referred to x and y respectively, thus we have
σ_x=p₀/πc[-4r·cosφ·logr+2r·sinφ·φ+2r₁·cosφ₁·logr₁-r₁·sinφ₁·φ₁+2r₂·cosφ₂·logr₂-r₂·sinφ₂·φ₂],
σ_y=p₀/πc[2r·sinφ·φ-r₁·sinφ₁·φ₁-r₂·sinφ₂·φ₂],
τ_xy=-p₀/πc[2r·cosφ·φ+r₁·cosφ₁·φ₁-r₂·cosφ₂·φ₂].
Transformation of coordinates from rectangular to polar is made in the above calculation.