Stress and Strain
(3.8-3.12, 3.14)
MAE 316 – Strength of Mechanical Components
NC State University Department of Mechanical & Aerospace Engineering
1
Stress and Strain
Introduction

MAE 316 is a continuation of MAE 314 (solid mechanics)

Review topics




Beam theory
Columns
Pressure vessels
Principle stresses
New topics





2
Contact Stress
Press and shrink fits
Fracture mechanics
Fatigue
Stress and Strain
Normal Stress (3.9)
Normal Stress (axial loading)

F

A
Sign Convention



3
σ > 0 Tensile (member is in tension)
σ < 0 Compressive (member is in compression)
Stress and Strain
Shear Stress (3.9)

Shear stress (transverse loading)

“Single” shear
average shear stress
 ave 
P F

A A
“Double” shear

 ave
4
F
P
F
2
 

A
A 2A
Stress and Strain
Stress and Strain Review (3.3)
Bearing stress

P P
b 

Ab td
Single shear case
Bearing stress is a normal stress

5
Stress and Strain
Strain (3.8)
Normal strain (axial loading)



L
Hooke’s Law

  E
Where E = Modulus of Elasticity (Young’s modulus)
6
Stress and Strain
Stress and Strain Review (3.12)
Torsion (circular shaft)


Shear strain


T
L

Shear stress
T

J

Angle of twist
TL

GJ
Where G = Shear modulus (Modulus of rigidity)
and J = Polar moment of inertia of shaft cross-section
7
Stress and Strain
Thin-Walled Pressure Vessels (3.14)
Thin-walled pressure vessels


Cylindrical
 t  1 

Spherical
pri
t
Circumferential “hoop”
stress
1   2 
8
Stress and Strain
l  2 
pri
2t
Longitudinal stress
pri
2t
Beams in Bending (3.10)
Beams in pure bending
Strain


x  
y

y
 y  z 

Where ν = Poisson’s Ratio and
ρ = radius of curvature
Stress

My
x  
I
9
Where I = 2nd moment of inertia of the
cross-section
Stress and Strain
Beam Shear and Bending (3.10-3.11)


Beams (non-uniform bending)
Shear and bending moment
dM
V
dx
dV
 w
dx

Shear stress
 avg 

VQ
It
Where Q = 1st moment of the cross-section
Design of beams for bending
 max 
10
M
max
I
c

M
max
Factor of Safety 
S
Stress and Strain
allowable
applied
Example
Draw the shear and bending-moment diagrams for the beam and loading
shown.
11
Stress and Strain
Example Problem
An extruded aluminum beam has the cross section shown. Knowing that
the vertical shear in the beam is 150 kN, determine the shearing stress
a (a) point a, and (b) point b.
12
Shear Stress in Beams
Combined Stress in Beams

In MAE 314, we calculated stress and strain for each type
of load separately (axial, centric, transverse, etc.).

When more than one type of load acts on a beam, the
combined stress can be found by the superposition of
several stress states.
13
Stress and Strain
Combined Stress in Beams

Determine the normal and shearing stress at point K.
The radius of the bar is 20 mm.
14
Stress and Strain
2D and 3D Stress
(3.6-3.7)
MAE 316 – Strength of Mechanical Components
NC State University Department of Mechanical & Aerospace Engineering
15
Stress and Strain
Plane (2D) Stress (3.6)

Consider a state of plane stress: σz = τxz = τyz = 0.
φ
φ
φ
φ
φ
φ
φ
Slice cube at an angle φ to the x-axis
(new coordinates x’, y’)
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Stress and Strain
φ
φ
Plane (2D) Stress (3.6)

Sum forces in x’ direction and y’ direction and use trig
identities to formulate equations for transformed stress.
 x' 
x  y
2

 x  y
2
cos  2    xy sin  2 
φ
 y' 
x  y
2
 x' y'  
17

 x  y
2
 x  y
2
cos  2    xy sin  2 
φ
φ
φ
φ
sin  2    xy cos  2 
Stress and Strain
Plane (2D) Stress (3.6)

Plotting a Mohr’s Circle, we can also develop equations
for principle stress, maximum shearing stress, and the
orientations at which they occur.
 max,min 
 x  y
2
  x  y 
   xy2
 
 2 
2
 max   ave  R
 min   ave  R
 max  R
 min   R
  x  y 
   xy2
  
 2 
2
 max,min
tan  2P  
2 xy
 x  y
 x  y
tan  2S   
2 xy
18
Stress and Strain
Plane (2D) Strain

Mathematically, the transformation of strain is the same as
stress transformation with the following substitutions.
   and    2
 x' 
 y' 
x  y
2
x  y
2


x y
2
x y
2
cos  2  
cos  2  
 xy
2
 xy
2
sin  2 
 max,min 
sin  2 
 x ' y '     x   y  sin  2    xy cos  2 
19
x y
tan  2P  
2
  x   y    xy 
  

 
 2   2 
 xy
x  y
Stress and Strain
2
2
3D Stress (3.7)

Now, there are three possible principal stresses.

Also, recall the stress tensor can be expressed in matrix
form.
 x  xy  xz 


 yx  y  yz 
 zx  zy  z 


20
Stress and Strain
3D Stress (3.7)

We can solve for the principle stresses (σ1, σ2, σ3) using a
stress cubic equation.
 i  I1 i  I 2 i  I3  0
3

2
Where i = 1,2,3 and the three constant I1, I2, and I3 are
expressed as follows.
I1   x   y   z
I 2   x y   x z   y z   xy   yz   xz
2
2
2
I 3   x y z  2 xy yz xz   x yz   y xz   z xy
2
21
Stress and Strain
2
2
3D Stress
Let nx, ny and nz be the direction cosines of the normal vector to surface ABC
with respect to x, y, and z directions respectively.
F
F
F
22
x
0

( nx ) A   x (nx A )   xy (n y A)   xz (nz A)  0
y
0

( n y ) A   xy (nx A)   y (n y A )   yz (nz A)  0
z
0

( nz ) A   xz (nx A)   yz (ny A)   z (nz A )  0
Stress and Strain
3D Stress (3.7)



How do we find the maximum shearing stress?
The most visual method is to observe a 3D Mohr's
Circle.
Rank principle stresses largest to smallest: σ1 > σ2 > σ3
 max 
σ2
σ3
σ1
23
Stress and Strain
1   3
2
3D Stress

(5.5)
A plane that makes equal angles with the principal planes
is called an octahedral plane.
1
3
1

( 1   2 ) 2  ( 2   3 ) 2  ( 3   1 ) 2
3
 oct  ( 1   2   3 )
 oct
24
Stress and Strain
3D Stress (3.7)

For the stress state shown below, find the principle
stresses and maximum shear stress.
Stress tensor
9 4 0 
4  6 0


0 0 0 
25
Stress and Strain
3D Stress (3.7)

Draw Mohr’s Circle for the stress state shown below.
Stress tensor
0 
60 30
30 35 0 


 0 0  25
26
Stress and Strain
Curved Beams
(3.18)
MAE 316 – Strength of Mechanical Components
NC State University Department of Mechanical & Aerospace Engineering
27
Stress and Strain
Curved Beams (3.18)


Thus far, we have only analyzed stress in straight beams.
However, there many situations where curved beams are
used.
Hooks
Chain links
Curved structural beams
28
Stress and Strain
Curved Beams (3.18)

Assumptions




Pure bending (no shear and axial forces present – will add
these later)
Bending occurs in a single plane
The cross-section has at least one axis of symmetry
What does this mean?



29
σ = -My/I no longer applies
Neutral axis and axis of symmetry (centroid) are no longer the
same
Stress distribution is not linear
Stress and Strain
Curved Beams (3.18)
Flexure formula for tangential stress:

My
Ae(rn  y)
Where
M = bending moment about centroidal axis (positive M puts inner surface in
tension)
y = distance from neutral axis to point of interest
A = cross-section area
e = distance from centroidal axis to neutral axis
rn = radius of neutral axis
30
Stress and Strain
Curved Beams (3.18)

If there is also an axial force
present, the flexure formula
can be written as follows.


P
My

A Ae(rn  y )
Table 3-4 in the textbook
shows rn formulas for several
common cross-section shapes.
31
Stress and Strain
Example
Plot the distribution of stresses across section A-A of the crane hook shown
below. The cross section is rectangular, with b=0.75 in and h =4 in, and the
load is F = 5000 lbf.
32
Stress and Strain
Curved Beams (3.18)

Calculate the tangential stress at A and B on the curved
hook shown below if the load P = 90 kN.
33
Stress and Strain