Mechanics of Materials
Goal:
Deformation
Load
Factors that affect
deformation of a
structure
P
P
Stress: intensity of internal force
P
P
Normal Stress (s)
Definition: stresses that act in a direction perpendicular to the cut surface
s
s
n
n
P
P
tensile stresses (+)
Uniformly distributed stresses:
compressive stresses (-)
P
s
A
P: normal force acting at the cut surface
A: cross sectional area
Non-uniformly distributed stresses:
P
s  lim
A0 A
Normal Stress – Example 1
Find stresses at cross sections AA and BB. The cross sectional areas of
AA and BB are SAA and SBB respectively.
A
P1
B
P2
A P3
B
Deformation and Normal Strain
change in size
Deformation:
change in shape

L0
  L  Lo
  L  Lo
P
P
L
Normal Strain:
Strain at one point:


Lo

  lim
L  0 L
Normal Strain – Example 2
Find the total deformation of the structure shown below if the values in the
strain gauges are 1  100106 ,  2  200106 and
L1  100mm, L2  100mm
strain gauge 1
P1
P2
strain gauge 2
P3
L1
L2
Stress-Strain Relationship
Review:
load
stress
deformation
strain
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’s
law may be applied for a given material.
Yield Point s): 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 (s): the maximum stress developed in a material
before rupture.
Breaking Stress (s): stress at rupture.
Stress - Strain Diagram - Aluminum
Alloy
- no noticeable yield point
- offset method
- yield occurs at   0.2% offset.
Stress - Strain Diagram - Brittle Material
- rupture occurs without
noticeable any prior change
in the rate of elongation.
- no difference between
and
.
Linear Elasticity
Hooke’s Law:
s  E 
Poisson’s ratio: the ratio of the lateral or perpendicular strain to
the longitudinal or axial strain.
lateral strain

 

axial strain

    
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.
P2
P1
P3
L1
L2
Shear Stress
V
V
Shear force: force that acts tangential to the surface.
Average shear stress:

V
A
Shear Stress - double shear
F.B.D. of bolt
V
P
2P


, d is the diameter of the bolt
A 2 1d2 d2
4
Fb
P
s


, h is the thickness of the clevis
Bearing stress: b
Ab 2d  h
Shear stress:  
©2001 Brooks/Cole, a division of Thomson Learning, Inc. Thomson Learning™ is a trademark used herein under license.
Shear Stress - single shear
Shear stress:

V
P
4P


, d is the diameter of the bolt
A 1d2 d2
4
Bearing stress:
F.B.D. of bolt
sb 
©2001 Brooks/Cole, a division of Thomson Learning, Inc. Thomson Learning™ is a trademark used herein under license.
Fb
P

,
Ab d  h
h is the thickness of the bar or flange
Shear Strain
d
V
L 

Shear strain: changes in shape (angle).

  
2
d
tan






If deformation is small, i.e.,  is small,
L
Sign Conventions for Shear Stress
Positive faces: outward normal direction is in the positive
direction of a coordinate axis.
Negative faces: the opposite faces
y
on a positive face, acts in the positive direction of one
of the coordinate axes.
x
z
Positive shear stress:
on a negative face, acts in the negative direction of one
of the coordinate axes.
on a positive face, acts in the negative direction of one
of the coordinate axes.
Negative shear stress:
on a negative face, acts in the positive direction of one
of the coordinate axes.
Sign Conventions for Shear Strain
Positive shear strain:
If the angle between two positive faces (or two negative
faces) are reduced.
Negative shear stress:
If the angle between two positive faces (or two negative
faces) 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 = 110kN
is required to create the hole, what is the average shear stress in the plate and
the average compressive stress in the punch?