11

Classes #13, 14, 15 Classes #13, 14, 15 Civil Engineering Materials – CIVE 2110Civil Engineering Materials – CIVE 2110

Combined Stress Combined Stress

Fall 2010Fall 2010Dr. GuptaDr. GuptaDr. PickettDr. 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 StressesProcedure:- 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.

Combined Stress-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:

zyyz

zxxz

yxxy

yzxzxyzyxstressesSix ,,,, ,

Stress TransformationGeneral State of Stress:- 3 dimensional

Plane Stress- 2 dimensional

Remember:

zyyz

zxxz

yxxy

yzxzxyzyxstressesSix ,,,, ,

xyyxstressesThree ,,

Stress Transformation

Plane Stress 2 dimensional

Stress Components are:

DirectionXinfaceYonStressShear

DirectionYinfaceXonStressShear

DirectionYinfaceYonStressNormalDirectionXinfaceXonStressNormal

yx

xy

yxxy

yyy

xxx

+ = CCW, upward on right face

Plane Stress Transformation

State of Plane Stress at a POINTMay need to be determinedIn 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 - direction

xyyx

yxyx

''''

+ = 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.

''' yxx and

Ay = (ΔA)SinΔAx = (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:

'y

'''' xyyx

Ay = (ΔA)CosΔ

Ax = (ΔA)SinΔ

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

'''' xyyx

+ = 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

'''' xyyx

+ = CCW, upward on right face

Equations Plane Stress Transformation

-

SinASin

SinACos

CosASinCosACos

AF

y

xy

xy

x

x

x

'

'

00

Equations Plane Stress Transformation

-

2222

22

22

22

2

221

222

221

2

'

'

'

22'

SinCos

CosSinCos

CosSinCos

SinCosSinCos

Aoutfactor

xyyxyx

x

yyxy

xxx

yxyxx

yxyxx

Equations Plane Stress Transformation

-

CosASin

CosACos

SinASinSinACos

A

F

y

xy

xy

x

yx

y

''

'

0

0

Equations Plane Stress Transformation

-

22

2

22

22121

22

22

221

221

''

''

''

22''

SinCos

SinCosCos

SinSinCosCos

SinCosSinCosSinCos

Aoutfactor

yxxyyx

xyxyyx

xyxyxyyx

xxyyxyyx

Equations Plane Stress Transformation

-

2222

2222

218029022:218029022:

90

'

'

'

SinCoslyconsequent

SinCospreviously

SinSinSinSinnoteCosCosCosCosnote

setfor

xyyxyx

y

xyyxyx

x

y

Equations of Plane Stress Transformation

The equations for the transformation of Plane Stress are:

22

2

2222

2222

''

'

'

SinCos

SinCos

SinCos

yxxyyx

xyyxyx

y

xyyxyx

x

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