Information Technology Reference
In-Depth Information
Equation (1.75) is a mixed governing differential equation. Note that in contrast to the
governing equations for the displacement and force formulations, Eq. (1.75) for the mixed
method contains no derivatives of the unknowns ( u and σ ) higher than the first and does
not involve derivatives of the material parameters.
1.7
Analysis of Stress
1.7.1 Principal Stresses
Figure (1.6) shows the state of stress at a point P in a body by specifying the stresses acting
on three coordinate planes passing through point P .Define three stress vectors on these
planes as
σ x = σ x e x + τ xy e y + τ xz e z
σ y = τ yx e x + σ y e y + τ yz e z
(1.77)
σ z = τ zx e x + τ zy e y + σ z e z
where e x , e y and e z are the unit vectors along the x, y, and z axes. The vector σ x defines
the stress on the face of the cube whose outward normal is e x , and similarly for σ y and σ z .
We now calculate the stress vector σ a on an arbitrarily oriented plane passing through the
point P . The orientation of this plane is specified by a unit vector a normal to the plane
a
=
a x e x +
a y e y +
a z e z
(1.78)
where a x ,a y , and a z are the direction cosines with respect to the x, y, and z axes of the
normal to the plane. To find σ a , a tetrahedron element of volume
V is isolated from the
body (Fig. 1.13), with three triangular sides parallel to the negative x, y , and z coordinate
y
- e z
a
h
- e x
x
- e y
z
FIGURE 1.13
Tetrahedron of volume
V with sides of areas A, a x A, a y A, and a z A .
Search WWH ::




Custom Search