In general stress in a material can be defined by three normal stresses and three shear stresses. For many cases we can simplify to three stress components. If no stresses act on the outside surfaces we call the state of stress plane stress. In this case we can treat the problem in 2 dimensions.

To determine the maximum normal and shear stresses it is necessary that we can transform stresses from one co-ordinate system to another.

To transform stress from one orientation to another we need to take into account

1) magnitude of each stress

2) orientation of each stress

3) the orientation of each area the stress acts on

So the transformation is similar to that used for force components but is a little more involved.

Take sections Draw the free body diagram

Force equilibrium

The state of plane stress at a point on the surface of the airplane fuselage is represented on the element oriented as shown in the figure. Represent the state of stress at the point on an element that is oriented 30° clockwise from the position.

Step 1 Take sections

A

B Step 2 Draw the two free body diagrams.

A

B Step 3 Force Equilibrium for A

A

B

0 xF

030cos30sin2530sin30sin80

30sin30cos2530cos30cos50

AA

AAAx

MPax 15.4

A

B

0 yF

030sin30sin2530cos30sin80

30cos30cos2530sin30cos50

AA

AAAyx

MPayx 8.68

2sin2cos22 xy

yxyxx

2sin2cos22 xy

yxyxy

2cos2sin2 xy

yxyx

General Equations of Plane-Stress Transformation

B Step 3 Force Equilibrium for A

2sin2cos22 xy

yxyxy

MPa

y

85.25

60sin2560cos2

5080

2

5080

Answer

xxy

xy

2.0

2.0

Graph showing the variation of normal stress and shear stress with angle.

The principal in-plane stresses are the maximum and minimum stresses. These will occur at the maxima and minima of the curve.

yx

xyp

xyyxx

d

d

22tan

02cos22sin1

2

2

2,1 22 xyyxyx

The magnitude and direction of the principal stresses can therefore be defined as a circle.

Mohr’s circle can be used as a graphical technique to transform stress and/or strain.

2sin2cos22 xy

yxyxx

2cos2sin2 xy

yxyx

22

2

2

2

22Rxy

yxyx

yxx

Stress transformations

Combine and add

Circle with radius R

22

2 xyyxR

Christian Otto Mohr (October 8, 1835 - October 2, 1918) was a German civil engineer, one of the most celebrated of the nineteenth century.

Starting in 1855, his early working life was spent in railroad engineering for the Hanover and Oldenburg state railways, designing some famous bridges and making some of the earliest uses of steel trusses.

Even during his early railway years, Mohr's interest had been attracted by the theories of mechanics and the strength of materials, and in 1867, he became professor of mechanics at Stuttgart Polytechnic and, in 1873, at Dresden Polytechnic. Mohr had a direct and unpretentious lecturing style that was popular with his students.

Mohr was an enthusiast for graphical tools and developed the method, for visually representing stress in three dimensions, previously proposed by Carl Culmann. In 1882, he famously developed the graphical method for analysing stress known as Mohr's circle and used it to propose an early theory of strength based on shear stress.

Constructing Mohr’s Circle

1) Co-ordinate system with normal stress +ve to right and shear stress +ve down

2) Using the sign convention shown plot the center of the circle C at the average normal stress on the x axis

3) Plot a reference point at x xy – this represents =0

4) Connect A with C and determine the radius.

5) Draw the circle

Principle stresses are at B and D

p is measured counter-clockwise from AC to BC & DC

s is measured clockwise from AC to EC or FC