90
9 Plane Problem of Elasticity Theory
Fig. 9.2 Wedge bending by
uniform pressure
σ r =
q
β − tg β
ϑ − β + tg β sin
2 ϑ + sin ϑ cos ϑ
,
σ ϑ =
q
β − tg β
ϑ − β + tg β cos
2 ϑ − sin ϑ cos ϑ
,
τ rϑ =
q sin ϑ
β − tg β
(tg β cos ϑ − sin ϑ) .
(9.10)
9.2 Complex Representation of a Bi-Harmonic Function
Any bi-harmonic function (U ) under the Goursat formula can be expressed through
two analytical functions (ϕ and ) of a complex variable (z = x + iy):
U = Re [zϕ(z) + ] ,
(9.11)
where Re is the real part of the complex function enclosed by brackets and a dash
above means a complex conjugate value (z = x − iy).
Let us consider an arbitrary arch AB in the area occupied by an elastic body; let
S be the arch length counted in the positive direction, which is direction from A to
B. Let us select the direction of the normal line to this arch to the right relative to
the viewer moving from A to B as a positive direction and designate components
of the force acting on the element (ds) of the arch from the external normal line as
follows:
X n ds, Y n ds
(9.12)
will have
X n =
∂ 2 U
∂y 2 cos
nx −
∂ 2 U
∂x∂y
cos
ny,
Y n =
∂ 2 U
∂x 2 cos
ny −
∂ 2 U
∂x∂y
cos
nx.