Post on 23-Oct-2014
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