easily incorporated in MATLAB to provide numerical solution to problems of interest (see Exercise 3-2).
3.4
Principal Stresses We can again use the previous developments from Section 1.6 to discuss the issues of principal stresses and directions. It is shown later in the chapter that the stress is a symmetric tensor. Using this fact, appropriate theory has been developed to identify and determine principal axes and values for the stress. For any given stress tensor we can establish the principal value problem and solve the characteristic equation to explicitly determine the principal values and directions. The general characteristic equation for the stress tensor becomes det[sij sdij ] ¼ s3 þ I1 s2 I2 s þ I3 ¼ 0
(3:4:1)
where s are the principal stresses and the fundamental invariants of the stress tensor can be expressed in terms of the three principal stresses s1 , s2 , s3 as I1 ¼ s1 þ s2 þ s3 I2 ¼ s1 s2 þ s2 s3 þ s3 s1
(3:4:2)
I3 ¼ s1 s2 s3 In the principal coordinate system, the stress matrix takes the special diagonal form 2
s1 sij ¼ 4 0 0
0 s2 0
3 0 05 s3
(3:4:3)
A comparison of the general and principal stress states is shown in Figure 3-5. Notice that for the principal coordinate system, all shearing stresses vanish and thus the state includes only normal stresses. These issues should be compared to the equivalent comments made for the strain tensor at the end of Section 2.4.
y
sy tyz tzy
2
1
txy
sx x
z (General Coordinate System)
FIGURE 3-5
s1
tyx
tzx t xz sz
s2
s3
3 (Principal Coordinate System)
Comparison of general and principal stress states.
Stress and Equilibrium
55