combined stress
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Combined StressTRANSCRIPT
11
Classes #13, 14, 15 Classes #13, 14, 15 Civil Engineering Materials – CIVE 2110Civil Engineering Materials – CIVE 2110
Combined Stress Combined Stress
Fall 2010Fall 2010
Dr. GuptaDr. Gupta
Dr. 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
DirectionYinfaceYonStressNormal
DirectionXinfaceXonStressNormal
yx
xy
yxxy
yyy
xxx
+ = 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
- 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
CosASin
CosACos
A
F
y
xy
xy
x
x
x
'
'
0
0
Equations Plane Stress Transformation
-
2222
2
2
22
2
2
2
2
21
2
22
2
21
2
'
'
'
22'
SinCos
CosSin
Cos
CosSinCos
SinCosSinCos
Aoutfactor
xyyxyx
x
yyxy
xxx
yxyxx
yxyxx
Equations Plane Stress Transformation
-
CosASin
CosACos
SinASin
SinACos
A
F
y
xy
xy
x
yx
y
''
'
0
0
Equations Plane Stress Transformation
-
22
2
2
2
2
2121
2
2
2
2
2
21
2
21
''
''
''
22''
SinCos
SinCosCos
SinSinCosCos
SinCosSinCosSinCos
Aoutfactor
yxxyyx
xyxyyx
xyxyxyyx
xxyyxyyx
Equations Plane Stress Transformation
-
2222
2222
218029022:
218029022:
90
'
'
'
SinCoslyconsequent
SinCospreviously
SinSinSinSinnote
CosCosCosCosnote
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