2.16 Equilibrium of a Two-Dimensional or Plane Element differential Element
25
In a like manner, it may be shown that
τ yz = τ zy and τ xz = τ zx
Now, from the equilibrium of forces in x-direction, we obtain
−σ x y +
σ x +
∂σ x
∂ x
x
y +
τ yx +
∂τ yx
∂ y
y
x − τ yx x + F x xy = 0
Simplifying, we get
∂σ x
∂ x
+
∂τ yx
∂ y
+ F x = 0
or
∂σ x
∂ x
+
∂τ xy
∂ y
+ F x = 0
A similar expression is written to describe the equilibrium of y forces. The x and
y equations yield the following differential equations of equilibrium.
∂σ x
∂ x
+
∂τ xy
∂ y
+ F x = 0
(2.32a)
or
∂σ y
∂ y
+
∂τ xy
∂ x
+ F y = 0 since τ xy = τ yx
(2.32b)
The differential equations of equilibrium for the case of three-dimensional stress
may be generalized from the above expressions as follows (Fig. 2.12).
∂σ x
∂ x
+
∂τ xy
∂ y
+
∂τ xz
∂z
+ F x = 0
(2.33a)
∂σ y
∂ y
+
∂τ xy
∂ x
+
∂τ yz
∂z
+ F y = 0
(2.33b)
∂σ z
∂z
+
∂τ xz
∂ x
+
∂τ yz
∂ y
+ F z = 0
(2.33c)