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

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)

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.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Equations Plane Stress Transformation -

Equations Plane Stress Transformation -

Equations Plane Stress Transformation -

Equations Plane Stress Transformation -

Equations Plane Stress Transformation -

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

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