Basic Equations: Differential Form - Mechanics of Structures: Variational and Computational Methods - page 21

FIGURE 1.9
Planar element used in the derivation of the differential equations of equilibrium.

in the stress

y in
the coordinate directions have a similar form as i nd ica te d in Fig. 1.9. The prescribed body
forces with units of force/volume are denoted by p Vx , p Vy ∗ . For an element of thickness dz ,
the resultant forces in the x direction must vanish
F x =

σ

x is

∂ σ

x dx . The changes of the shear components

τ

xy ,

τ

yx and the stress

σ

x

dx dy dz dy dx dz σ x + ∂σ x

∂

τ yx + ∂τ yx
∂

0: + − σ x dy dz

x y

− τ yx dx dz + p Vx dx dy dz = 0

or, for each volume element dx dy dz

∂σ x
∂ x + ∂τ yx y +

p Vx = 0 (1.49a)

∂

In the y direction, we find

∂σ y
∂ y + ∂τ xy x +

p Vy = 0 (1.49b)

∂

In matrix form,

∂ x p Vx

p Vy

σ x
σ y
τ xy

0 ∂ y

+

= 0

0 ∂ y ∂ x

(1.50a)

D T

σ + p V = 0

These relations apply for both plane stress and plane strain. For a thin, flat element in which
the stresses are replaced by stress resultants, i.e., integrals of the stresses over the element

∗ Quantities with an overbar are applied (prescribed).