1. Civil Engineering Materials – CIVE 2110
Classes #13, 14, 15
Civil Engineering Materials – CIVE 2110
Combined Stress
Fall 2010
Dr. Gupta
Dr. Pickett

2. Combined Stresses Assume: Linear Stress-Strain relationship
Elastic Stress-Strain relationship
Homogeneous material
Isotropic material
Small deformations
Stress determined far away from points of stress concentrations (Saint-Venant principle)

3. Combined Stresses Procedure: Draw free body diagram.
Obtain external reactions.
Cut a cross section, draw free body diagram.
Draw force components acting through centroid.
Compute Moment loads about centroidal axis.
Compute Normal stresses associated with each load.
Compute resultant Normal Force.
Compute resultant Shear Force.
Compute resultant Bending Moments.
Compute resultant Torsional Moments.
Combine resultants (Normal, Shear, Moments) from all loads.

4. Example: # 8.6 Pg. 451-452 Hibbeler, 7th edition
Combined Stress
Example: # 8.6 Pg
Hibbeler, 7th edition

5. Example: # 8.6 Pg. 451-452 Hibbeler, 7th edition
Combined Stress
Example: # 8.6 Pg
Hibbeler, 7th edition

6. Combined Stress
Problem: # 8-43, 8-44
Pg. 458
Hibbeler, 7th edition
Remember:
for Shear Stress

7. Areas and Centroids, Mechanics of Materials, 2nd ed, Timoshenko, p. 727

8. Stress Transformation
General State of Stress:
- 3 dimensional
Remember:

9. Stress Transformation
General State of Stress:
- 3 dimensional
Plane Stress
- 2 dimensional
Remember:

10. Stress Transformation
Plane Stress
2 dimensional
Stress Components are:
+ = CCW, upward on right face

11. Plane Stress Transformation
State of Plane Stress
at a POINT
May need to be determined
In various
ORIENTATIONS, .
+ = CCW, upward on right face

12. Plane Stress Transformation
Must determine:
To represent the same stress as:
Must transform:
Stress – magnitude
- direction
Area – magnitude
+ = CCW, upward on right face

13. Steps for Plane Stress Transformation
To determine
acting on X’ face, :
- Draw free body diagram
at orientation .
- Apply equilibrium equations:
ΣFx’=0 and ΣFy’=0
by multiplying stresses
on each face
by the area of each face.
Ay = (ΔA)SinΔ
Ax = (ΔA)CosΔ

14. Steps for Plane Stress Transformation
To determine
acting on Y’ face, :
- Draw free body diagram
at orientation .
- Apply equilibrium equations:
ΣFx’=0 and ΣFy’=0
by multiplying stresses
on each face
by the area of each face.
- Remember:
Ax = (ΔA)SinΔ
Ay = (ΔA)CosΔ

15. Plane Stress Transformation
Problem: # 9-6, 9-9, 9-60
Pg. 484
Hibbeler, 7th edition

16. Equations Plane Stress Transformation
A simpler method,
General Equations:
- Draw free body diagram
at orientation .
- Apply equilibrium equations:
ΣFx’=0 and ΣFy’=0
by multiplying stresses
on each face
by the area of each face.
- Sign Convention:
+ = Normal Stress = Tension
+ = Shear Stress = CCW, Upward on right face
+ =  = CCW from + X axis
+ = CCW, upward on right face

17. Equations Plane Stress Transformation
- Draw free body diagram
at orientation .
- Apply equilibrium equations:
ΣFx’=0 and ΣFy’=0
by multiplying stresses
on each face
by the area of each face.
- Sign Convention:
+ = Normal Stress = Tension
+ = Shear Stress =
+ = CCW, Upward on right face,
+ =  = CCW from + X axis
+ = CCW, upward on right face

18. Equations Plane Stress Transformation-

19. Equations Plane Stress Transformation-

20. Equations Plane Stress Transformation-

21. Equations Plane Stress Transformation-

22. Equations Plane Stress Transformation-

23. Equations of Plane Stress Transformation
The equations for the transformation of
Plane Stress are:

24. Plane Stress Transformation
Problem: # 9-6, 9-9, 9-60
Pg. 484
Hibbeler, 7th edition