24
2 Analysis of Stress
Fig. 2.11 Stress components
acting on element
of statics. Fulfilment of these conditions establishes certain relationships, known as
the differential equations of equilibrium. These involve the derivatives of the stress
components.
Assume that σ x , σ y , τ xy and τ yx are functions of x and y but do not vary throughout
the thickness (are independent of z) and that the other stress components are zero.
Also assume that the x and y components of the body forces per unit volume, F x
and F y , are independent of z and that the z component of the body force F z = 0. As
the element is very small, the stress components may be considered to be distributed
uniformly over each face.
Now, taking moments of force about the lower left corner and equating to zero,
− (σ x y)
y
2
+
τ xy y
1
2
−
σ y +
∂σ y
∂ y
y
x
x
2
+
τ yx +
∂τ yx
∂ y
y
xy
−
τ xy +
∂τ xy
∂ x
x
xy +
σ x +
∂σ x
∂ x
x
y
y
2
+ σ y x
x
2
− τ yx x
1
2
+ (F x yx)
y
2
− F y xy
x
2
= 0
Neglecting the higher terms involving Δx and Δy and simplifying, the above
expression is reduced to
τ xy x y = τ yx x y
or
τ xy = τ yx