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Mechanics of Materials
Goal: Load Deformation
Factors that affect deformation of a structure
P PPP
Stress: intensity of internal force
Normal Stress ()
P
A
Definition: stresses that act in a direction perpendicular to the cut surface
P
n
P
n
tensile stresses (+) compressive stresses (-)
Uniformly distributed stresses:
P: normal force acting at the cut surfaceA: cross sectional area
Non-uniformly distributed stresses:0
limA
P
A
Normal Stress – Example 1Find stresses at cross sections AA and BB. The cross sectional areas of AA and BB are SAA and SBB respectively.
P1 P3
P2
A
A
B
B
Deformation and Normal Strain
oLL
Deformation:
P P
L0
L
oLL
Normal Strain:
Strain at one point:
change in size
change in shape
oL
0limL L
Normal Strain – Example 2Find the total deformation of the structure shown below if the values in the strain gauges are and 6
26
1 10200,10100 mm 100 ,mm 100 21 LL
P1
P2
L1 L2
P3
strain gauge 1strain gauge 2
Stress-Strain Relationship
Review:
load
stress strain
deformation
Constitutive Law - Stress and Strain Relationship
Tensile Test:
- Apply load - Measure strain- Plot stress vs. strain curve
Stress - Stain Diagram - Ductile Material (structural steel)
True Diagram Partially Enlarge Diagram
• Proportional limit
• Yield stress
• Ultimate stress
Stress - Stain Diagram - Ductile Material Proportional Limit: the largest value of stress for which Hooke’slaw may be applied for a given material.
Yield Point ): a critical point, after the yield point, the specimen undergoes a large deformation with a relatively small increase in the applied load.
Plastic Deformation: deformation that remains after the load is applied.
Ultimate Stress (): the maximum stress developed in a materialbefore rupture.
Breaking Stress (): stress at rupture.
Stress - Strain Diagram - Aluminum Alloy
- no noticeable yield point- offset method - yield occurs at offset. %2.0
Stress - Strain Diagram - Brittle Material
- rupture occurs without noticeable any prior change in the rate of elongation.- no difference between and .
Linear Elasticity
lateral strain
axial strain
Hooke’s Law:
Poisson’s ratio: the ratio of the lateral or perpendicular strain to
the longitudinal or axial strain.
E
Stress-Strain Relationship – Example 3
Find the total deformation of the structure shown below. Express the answer in terms of P’s, S’s, L’s and E.
P1
P2
L1 L2
P3
Shear Stress
Shear force: force that acts tangential to the surface.
Average shear stress:
V V
V
A
©2001 Brooks/Cole, a division of Thomson Learning, Inc. Thomson Learning™ is a trademark used herein under license.©2001 Brooks/Cole, a division of Thomson Learning, Inc. Thomson Learning™ is a trademark used herein under license.
Shear Stress - double shear
22
2, is the diameter of the bolt
12
4
V P Pd
A dd
Shear stress:
Bearing stress:
F.B.D. of bolt
, is the thickness of the clevis2
bb
b
F Ph
A d h
©2001 Brooks/Cole, a division of Thomson Learning, Inc. Thomson Learning™ is a trademark used herein under license.©2001 Brooks/Cole, a division of Thomson Learning, Inc. Thomson Learning™ is a trademark used herein under license.
Shear Stress - single shear
F.B.D. of bolt
22
4, is the diameter of the bolt
14
V P Pd
A dd
Shear stress:
Bearing stress:
, is the thickness of the bar or flangebb
b
F Ph
A d h
Shear Strain
2
Shear strain: changes in shape (angle).
V d
L
If deformation is small, i.e., is small, tand
L
Sign Conventions for Shear Stress
Positive faces: outward normal direction is in the positivedirection of a coordinate axis. Negative faces: the opposite faces
Positive shear stress:
on a positive face, acts in the positive direction of one of the coordinate axes.
on a negative face, acts in the negative direction of oneof the coordinate axes.
Negative shear stress:
on a positive face, acts in the negative direction of one of the coordinate axes.
on a negative face, acts in the positive direction of oneof the coordinate axes.
x
y
z
Sign Conventions for Shear Strain
Positive shear strain:
Negative shear stress:
If the angle between two positive faces (or two negativefaces) are reduced.
If the angle between two positive faces (or two negativefaces) are increased.
Shear Stress vs. Strain - Hooke’s Law in Shear
G
G: shear modulus or modulus of rigidity
Shear Stress and Strain – Example 4
A punch for making holes in steel plates is shown in the following figure. Assume that a punch having diameter d = 20 mm. is used to punch a hole in a 8 mm. plate, as shown in the cross-sectional view. If a force P = 110kNis required to create the hole, what is the average shear stress in the plate and the average compressive stress in the punch?