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Staged Procedure To Get Principal Stresses
1. Compute
Meaning: σ is the average normal stress (recall that σ + σ is an
invariant and so is σ ), whereas R is the radius of Mohr's circle
described later. This R also represents the maximum in-plane shear
value, as discussed in the Lecture notes.
2. The principal stresses are
σ = σ + R , σ = σ − R
3. The above procedure bypasses the computation of principal angles.
Should these be required to find principal directions, use equation (*) of
the Principal Angles slide.
2
2
σav = , R = + + τ xy
1 av^2 av
av av
σ xx + σ
xx
yy
yy
σ xx − σ yy
ASEN 3112 Lecture 6 – Slide 13
Additional Properties
1. The in-plane shear stresses on the principal planes vanish
2. The maximum and minimum in-plane shears are + R and − R,
respectively
3. The max/min in-plane shears act on planes located at +45 and -
from the principal planes. These are the principal shear planes
4. A principal stress element (used in some textbooks) is obtained
by drawing a triangle with two sides parallel to the principal planes
and one side parallel to a principal shear plane
For further details, see Lecture notes. Some of these properties can be
visualized more easily using the Mohr's circle , which provides a
graphical solution to the plane stress transformation problem
ASEN 3112 Lecture 6 – Slide 14
Graphical Solution of Example Using Mohr's Circle
x
y
P σ xx^ =100 psi
τ xy = τ (^) yx =30 psi
σ (^) yy = 20 psi (a)
σ = normal stress
τ = shear stress
0 20 40 60 80 100
50 40 30 20 10 0 − − − − −
H
V
C
σ 1 = 110
σ 2 = 10
coordinates of blue points are H: (20,30), V:(100,-30), C:(60,0)
τ (^) min= −
τ (^) max= 50
Radius R = 50
2 θ 1 = 36.
2 θ 2 = 36.88 +180 = 216.
(b) Mohr's circle
(a) Point in plane stress
ASEN 3112 Lecture 6 – Slide 16
What Happens in 3D?
This topic be briefly covered in class if time allows,
using the following slides.
If not enough time, ask students to read Lecture notes
(Sec 7.3), with particular emphasis on the computation
of the overall maximum shear
ASEN 3112 Lecture 6 – Slide 17
Principal Stresses in 3D (2)
The σ turn out to be the eigenvalues of the stress matrix.
They are the roots of a cubic polynomial (the so-called
characteristic polynomial)
The principal directions are given by the eigenvectors
of the stress matrix.
Both eigenvalues and eigenvectors can be numerically
computed by the Matlab function eig(.)
σ −σ
σ −σ
σ −σ
τ τ
τ
τ τ
τ
xx xy^ xz
yy
zz
yz
zx zy
C(σ) = det yx
i
= −σ + I (^) 1 σ − I (^) 2 σ + I = 0
ASEN 3112 Lecture 6 – Slide 19
3D Mohr Circles
(Yes, There Is More Than One)
All possible stress states
at the material point lie
on the grey shaded
area between the outer
and inner circles
σ = normal
stress
τ = shear
stress
Principal stress σ 1
Outer Mohr's
circle
Inner Mohr's
circles
Principal stress σ 3 2
Overall
+ max shear
The overall maximum shear, which is the radius of the
outer Mohr's circle, is important for assessing strength
safety of ductile materials
ASEN 3112 Lecture 6 – Slide 20
σ 1 , σ 2 , σ =
where σ and σ are the inplane principal stresses obtained
as described earlier in Lecture 6.
Consider plane stress but now account for the third dimension.
One of the principal stresses, call it for the moment σ 3 , is zero:
Plane Stress in 3D: The 3rd Principal Stress
The zero principal stress σ is aligned with the z axis (the
thickness direction) while σ 1 and σ 2 act in the x,y plane:
x
y
z
σ 1
σ 2
Thin plate σ^3
ASEN 3112 Lecture 6 – Slide 22
Plane Stress in 3D: Overall Max Shear
Let us now (re)order the principal stresses by algebraic value as
(A) Inplane principal stresses have opposite signs. Then the
zero stress is the intermediate one: σ 2 , and
To compute the overall maximum shear 2 cases are considered:
σ 1 − σ 3
2
σ
2
(B) Inplane principal stresses have the same sign. Then
If σ σ 0 and σ = 0, τ =
overall max
σ
2
If σ σ 0 and σ = 0, τ = −
overall 3 1 max
τ = τ =
overall inplane max max
σ 1 σ 2 σ 3
ASEN 3112 Lecture 6 – Slide 23
