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