A short note on Mohr circle equations in Stress and Strength of Materials.

1. Course Topics
1. Stress-strain relationships for general dimensional cases and
special cases.
2. Stress transformation equations.
3. Graphical determination of stresses and strains using Mohr’s
circle.
4. Theories of failure.
5. Stress concentration and relief of stress concentration.
6. Moments of inertia and area.
7. Products of inertia and area.
8. Unsymmetrical bending and shear centers in curved beams.
9. Torsion and stress concentration.
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2. Content
- Why transform stresses ?
- Applications of 2d stress transformation analysis
- Assumptions
- Derivation of transformation equations (plane stress).
- Examples
- Practice Questions
- Assignment
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3. Calculating Stresses in a Solid Body
One of the key concepts taught in the Mechanics of Materials
course concerns the nature of stress in a solid body.
Internal forces in a body produce stresses that:
- tend to elongate or contract the material and
- stresses that tend to warp or skew the material.
Students learn how to calculate stresses that arise in a variety of
structural and mechanical components.
In the typical problem-solving process, internal forces initially
are considered individually.

4. For example, consider a simple component such as the elbow (pipe)
shown in the figure below. Two forces, P and W, act on the end of the
pipe.
Force P:
- pulls on the component, causing
elongation at points A and B.
- also causes the pipe to bend
about the vertical (y) axis.
Force W:
- causes the pipe to twist about its
longitudinal (x) axis.
- also causes the component to bend
about the horizontal (z) axis.
Each of these effects produce stresses in the pipe component.

5. Furthermore, the stresses produced in the pipe depend on which point
we choose to examine.
Stresses produced at point A by loads P and W are different from the
stresses produced at point B.
To make matters worse, the individual stresses produced at either
point A or point B by the two loads are calculated in the x, y, and z
directions.
The combined effect of all stresses acting at a point will produce
stresses in the pipe material that act simultaneously in all directions,
and, in general, the largest stresses will not occur in the x, y, or z
directions.
Consequently, the engineer must be able to consider all possible
combinations of stress acting in any direction.

6. To properly design the pipe component, the engineer must:
- be able to compute each of the stresses acting at any point.
- from this set of stresses in the x, y, and z directions, compute
the most critical stresses acting at any possible orientation.
To assess the combined effect of stresses on a solid body, stress
transformation relationships are used.
Based on the necessity of satisfying equilibrium conditions, a set of
equations can be derived that expresses the variation of two types of
stress, normal stress and shear stress, for any orientation with respect
to the original xyz coordinate system.
Although abstract in nature, stress transformations are an important
tool necessary to design components such as beams and pipes that
are safe and reliable.

7. Applications of 2D Stress Transformation

8. 8
Other Applications …..

9. Practical Applications of 2D stress transformation

10. ASSUMPTIONS
• Material is homogenous and isotropic.
• Material operates in the region of elastic regime.
• Assumption of state of plane stress exists within the material.
Thus, normal and shearing stresses on the z-face of element is
assumed equal to zero.
i.e. σz = τzx = τzy = 0
• By implication no forces are exerted on the z faces of the
element.
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12. Derivations
• x’, y’, x’y’ as a function of x, y, xy and θ.
• Transformation equation Vs Equation of circle.
• Mohr Circle.
• Principal planes.
• Principal stresses.
• Maximum shearing stress planes.
• Maximum shearing stress.
• Important Notes.
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13. STRAIN ROSSETE

14. The material stress may only be
calculated from equation:
σ = E ε,
if the elongation in the force direction
has been measured and the stress state
is single-axis.

16. Class Practice 1
For the initial stress element shown, employing Mohr circle approach,
determine
a. the principal stresses.
b. the maximum shearing stress and corresponding normal stress.
c. principal planes,
d. maximum shearing plane
e. show these values on properly oriented stress elements.
y
x
300 MPa
400 MPa
200 MPa
Initial stress element
Y
X
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17. Class Practice 2
For the state of plane stress shown in the figure below, using Mohr’s
circle determine the principal planes, maximum shearing stress
plane, principal stresses and the maximum shearing stress. Show
these values on properly oriented stress elements.
y
x
10 MPa
50 MPa
Initial stress element
B
A
40 MPa
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18. Example 1
For the given state of stress shown on the right, determine the
normal and shearing stresses after the element shown has been
rotated thru:
(a) 25o clockwise (+ve)
(b) 10o counter clockwise (-ve).
(c) Using MS Excel determine
the normal and shear stresses
for 0o to 90o element rotation
using a step of 1O.
(d) Subsequently determine the
i. maximum normal stress.
ii. minimum normal stress.
iii. maximum shearing stress.
(e) What are the corresponding
faces of the stresses determined
in (d).
60 MPa
20 MPa
40 MPa

19. Assignment
• Read up slide 6 – Strain and Deformation
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20. Assignment 2
Pick any question of your
choice and implement the
scene on the right.
Remember to color code
the output plots as
suggested in the figure on
the right.