1. Complex Static Stresses and Torsion
UAA School of Engineering
CE Properties of Materials
Lecture # 17 1

2. Quiz 1: The State of StressP A B C D E
Stresses at point A:
Stresses at point B:
Stresses at point C:
Stresses at point D:
Stresses at point E: 1

3. Quiz 2: Stresses Acting on Other Directions
250 psi
600 psi
Welding Plan
 =37o
=?
=?
600 psi
250 psi 3

4. Complex Static Stresses
Complex Stress States:
 and  in three directions
Typical complex stress states:
Biaxial stresses are stresses  applied
along two axes of the element and  = 0.
Triaxial stresses are stresses  applied along three axes of the element and  = 0. 4

5. Biaxial Stresses
Consider a small plane element with plane stresses. Cut an angle  through the element and derive the equations for the stress normal to the new plane and the shear stress.
 = 0
  0 5

6. Stress Normal to Plane 6

7. Shear Stress on Plane 7

8. Mohr’s Circle for Biaxial Stress8

9. Triaxial Stress Normal stresses are in three dimensions.
Shear stress in each of these three dimensions is zero.
The following element is in triaxial stress state. 9

10. Mohr’s Circle for Triaxial Stress10

11. Triaxial Strain Stresses usually act in all three directions
It is necessary to modify Hooke’s Law regarding stress and strain in the elastic range in order to include Poisson’s effects in each direction, caused by the strains in the other two directions.
The generalized equations for Hooke’s Law are given below: 11

12. Purposes of Torsional Tests
The purposes of torsion tests usually parallel those for tension tests:
Determine the proportional limit and the ultimate strength in shear
Determine the modulus of rigidity (G)
Plot stress-strain curve for material behavior in shear 12

13. Torsion Procedure and Set up
Torsion specimens: round bars
Torsion apparatus: similar to tension
The ends of a specimen should be securely gripped
Apply torque
Measure the torque and the angle of twist 13

14. Torsional Specimens
The Torsion test is a test which places the specimen in pure shear. 14

15. Torsional Quantities 15

16. Elastic Shear Strength16
(unit = radian/length)

17. Torsional Modulus of Resilience
(unit = radian/length) 17

18. Torsional Plastic Strength
Yielding begins at outer surface.
Stress variation is no longer linear.
Yielding continues until the cross section is fully yielded. 18

19. Torsional Plastic Shear Strengthu 19

20. Torsional Fracture Ductile materials fracture in shear.
Brittle materials fracture in tension. 20

21. Review Mechanics 21