Timothy A. Philpot - Mechanics of materials _ an integrated learning system-John Wiley (2017)

500
STRESS TRANSFORMATIONS
principal planes
For a given state of plane stress, the stress components σ x , σ y , and τ xy are constants. The dependent
variables σ n and τ nt are actually functions of only one independent variable, θ. Therefore,
the value of θ for which the normal stress σ n is a maximum or a minimum can be determined
by differentiating Equation (12.5) with respect to θ and setting the derivative equal to zero:
dσ
n
σx − σ y
=− (2sin2 θ) + 2τxy
cos2θ
= 0
(12.10)
dθ
2
The solution of this equation gives the orientation θ = θ p of a plane where either a maximum
or a minimum normal stress occurs:
tan2θ
p
=
σ
2τ
x
xy
− σ
y
(12.11)
–
σ x σ y
2
2
+
2θ
p + 180°
FIGURE 12.12
2
τ xy
σ – σ
x y
2
θ
2 p
τ
xy
For a given set of stress components σ x , σ y , and τ xy , Equation (12.11) can be satisfied by
two values of 2θ p , and these two values will be separated by 180°. Accordingly, the values
of θ p will differ by 90°. From this result, we can conclude that
(a) there will be only two planes where either a maximum or a minimum normal stress
occurs, and
(b) these two planes will be 90° apart (i.e., orthogonal to each other).
Notice the similarity between the expressions for dσ n /dθ in Equation (12.10) and τ nt in
Equation (12.6). Setting the derivative of σ n equal to zero is equivalent to setting τ nt equal
to zero; therefore, the values of θ p that are solutions of Equation (12.11) produce values of
τ nt = 0 in Equation (12.6). This property leads us to another important conclusion:
Shear stress vanishes on planes where maximum
and minimum normal stresses occur.
Planes free of shear stress are termed principal planes. The normal stresses acting on these
planes—the maximum and minimum normal stresses—are called principal stresses.
The two values of θ p that satisfy Equation (12.11) are called the principal angles.
When tan 2θ p is positive, θ p is positive and the principal plane defined by θ p is rotated in a
counterclockwise sense from the reference x axis. When tan 2θ p is negative, the rotation is
clockwise. Observe that one value of θ p will always be between positive and negative 45°
(inclusive), and the second value will differ by 90°.
magnitude of principal Stresses
As mentioned, the normal stresses acting on the principal planes at a point in a stressed
body are called principal stresses. The maximum normal stress (i.e., the most positive
value algebraically) acting at a point is denoted as σ p1 , and the minimum normal stress (i.e.,
the most negative value algebraically) is denoted as σ p2 . There are two methods for computing
the magnitudes of the normal stresses acting on the principal planes.
Method one. The first method is simply to substitute each of the θ p values into either
Equation (12.3) or Equation (12.5) and compute the corresponding normal stress. In addition
to giving the value of the principal stress, this method has the advantage that it directly
associates a principal stress magnitude with each of the principal angles.
Method two. A general equation can be derived to give values for both σ p1 and σ p2 .
To derive this general equation, values of 2θ p must be substituted into Equation (12.5).
Equation (12.11) can be represented geometrically by the triangles shown in Figure 12.12.