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Combined Stresses
Objectives
Learn the difference between plain stress and plain strain.
Review generalized Hooke’s Law
Construct Mohr’s circle for 2D and 3D state of stresses.
Generalized Stress State
At a point, the state of stress is dependent on nine stress components, namely
x xy xz
xy y yz
xz yz z
xy yx
xz zx
yz zy
s
Plane Stress
Also known as bi-axial stress.
The stresses exist in a single plane.
There will be no out-of-the-plane stresses.
There will be out-of-the-plane strains. 0
x xy
xy y
z xz yz
s
Given x, y, and xy in the xy coordinate system, what are the components of these stresses in t-n coordinate system?
Stress Transformation
Transformation Equations
Mohr’s Circle for 2-D Stress State
Establish coordinate axes with as abscissa and as ordinate.
Locate points A and B based on the stresses in x and y faces.
Joint points A and B, and identify the center O.
With O as the center and AO as the radius establish a circle.
The circle defines all stress states.
Mohr’s Circle – 2D State of Stress
Point A
Point B
Point O
Determining Principle Stresses & Axes
Key Stresses
Is the maximum shear stress value correct? Correct if we consider the stresses in the plane. Wrong if we consider all the three principle stresses.
2
21,2
2
2max
2 2
2
x y x yxy
x yxy
Transformation Equations
The maximum and minimum stresses are called Principal Stresses.
The principal stresses occur in planes that satisfy
Transformation Equations
Principle directions define- maximum normal stress - minimum normal stress.
Mohr’s Circle for 3-D State of Stress
Determining the Principle Stresses for 3D State of Stress Find the three values of from the following cubic
equation:
3 2
2 2 2
2 2 22 0
x y z
x y x z y z xy yz zx
x y z xy yz zx x yz y xz z xy
Plane Strain The strains exist in a single plane.
There will be no out-of-the-plane strains.
There will be out-of-the-plane stresses.
This results in 3-D state of stress.
Generalized Hooke’s Law
x y z
x
y x z
y
z x y
z
xyxy
yzyz
zxzx
E
E
E
G
G
G