Page 239 - Mechanical Behavior of Materials

240 Chapter 6 Review of Complex and Principal States of Stress and Strain

y '
25 MPa 14.9 o
x '
20 MPa 19.7 60
y 60
95 MPa ' 100.3
x y 30.1 o
x '
40.3

(a) (b) (c)

Figure E6.1 Example of a state of stress (a) and its equivalent representations that contain
the principal normal stresses (b) and the principal shear stress (c).

This representation of the state of stress is shown as Fig. E6.1(c). The uncertainty as to the
sign of the shear stress can be resolved by noting that the positive shear diagonal (dashed line)
must be aligned with the larger of σ 1 and σ 2 . Alternatively, a more rigorous procedure is to use
◦
θ = θ s =−30.1 in Eq. 6.5, which gives τ 3 = 40.3 MPa. The positive sign indicates that the

shear stress is positive in the new x -y coordinate system.

As for any case of plane stress, the largest of σ 1 and σ 2 is the maximum normal stress that
occurs on any plane at this point in the material, so that σ max = 100.3MPa (Ans.). However, we
cannot determine τ max from what has been presented so far. Note that the principal shear stress
τ 3 is merely the largest shear stress for any rotation in the x-y plane, and the true maximum shear
stress may lie on planes that we have not yet considered. (See Section 6.3.2 and Ex. 6.4.)

6.2.3 Mohr’s Circle
A convenient graphical representation of the transformation equations for plane stress was
developed by Otto Mohr in the 1880s. On σ versus τ coordinates, these equations can be shown
to represent a circle, called Mohr’s circle, which is developed as follows.
Although it may not be immediately apparent, Eqs. 6.4 and 6.5 do represent a circle on a σ-τ
plot in parametric from, where 2θ is the parameter. This can be shown by combining these two
equations to eliminate 2θ. First, isolate all terms containing 2θ on one side of Eq. 6.4. Then square
both sides of Eqs. 6.4 and 6.5, sum the result, and invoke simple trigonometric identities to eliminate
2θ, obtaining

2 2

σ x + σ y 2 σ x − σ y 2
σ − + τ = + τ xy (6.13)
2 2
This equation is of the form

2
2
(σ − a) + (τ − b) = r 2 (6.14)