Combined Stresses
(8.1-8.2,8.4)
MAE 314 – Solid Mechanics
Yun Jing
Combined Stresses
1
Introduction
In Chapter 1 and 2, you learned how to determine the normal stress due
to centric loads.
In Chapter 3, you analyzed the distribution of shearing stresses in a
circular member due to a twisting couple.
In Chapter 4, you determined the normal stresses caused by bending
couples.
In Chapters 5 and 6, you evaluated the shearing stresses due to
transverse loads.
In Chapter 7, you learned how the components of stress are transformed
by a rotation of the coordinate axes and how to determine the
principal planes, principal stresses, and maximum shearing stress
at a point.
3- 2
In Chapter 8, you will learn how to determine the stress in a structural
member or machine element due to a combination of loads and how to
find the corresponding principal stresses and maximum shearing
stress.
Principle Stresses in a Beam
Prismatic beam subjected to transverse loading
My
Mc
m 
I
I
VQ
VQ
 xy  
m 
It
It
x  
Principal stresses determined from methods of
Chapter 7
Can the maximum normal stress within the
cross-section be larger than
m 
3- 3
Mc
I
Principle Stresses in a Beam
3- 4
Principle Stresses in a Beam
Cross-section shape results in large values of xy near
the surface where x is also large.
max may be greater than m
3- 5
Sample Problem 8.1
SOLUTION:
Determine shear and bending
moment in Section A-A’
A 160-kN force is applied at the end
of a W200x52 rolled-steel beam.
Neglecting the effects of fillets and of
stress concentrations, determine
whether the normal stresses satisfy a
design specification that they be
equal to or less than 150 MPa at
section A-A’.
3- 6
Calculate the normal stress at top
surface and at flange-web junction.
Evaluate the shear stress at flangeweb junction.
Calculate the principal stress at
flange-web junction
Sample Problem 8.1
SOLUTION:
Determine shear and bending moment in
Section A-A’
M A  160 kN 0.375 m   60 kN - m
VA  160 kN
Calculate the normal stress at top surface
and at flange-web junction.
MA
60 kN  m

S
511106 m3
 117.4 MPa
y
90.4 mm
σ b   a b  117.4 MPa 
c
103mm
 103MPa
a 
3- 7
Sample Problem 8.1
Evaluate shear stress at flange-web junction.
Q   206  12.6  96.7  251 103 mm 3
 251 106 m3
160 kN   251106 m3 
VAQ
b 

It
 52.7 106 m 4   0.0079 m 
 96.5 MPa
Calculate the principal stress at
flange-web junction
 max  12  b 
 12  b 
2
  b2
2
103
2
 103 

 

96.5

 
2
 2 
 160.9 MPa   150 MPa 
3- 8
Design specification is not satisfied.
Stresses Due to Combined Loads

When more than one type of
load acts on a beam, the combined
stress can be found by the superposition of several stress states.
Combined Stresses
9
Example Problem
Two forces are applied to the small
post BD as shown. Knowing that the
vertical portion of the post has a cross
section of 1.5x2.4 inches, determine the
principal stress, principal planes, and
maximum shearing stress at point H.
Combined Stresses
10
Stresses Due to Combined Loads
y
y
H
H
z
x
Combined Stresses
z
x
11
Example Problem
A force of 150N acts at point B on an Lshaped lug wrench as shown. The force
acts vertically downward, perpendicular
to the plane of the wrench. The handle of
the lug wrench is a steel rod with a
diameter of 12.5 mm, and its planform is
shown, as well. Determine the principle
stress at point A, which is on top of the
wrench.
Combined Stresses
12
Example Problem
A 6-kip force is applied to the machine element AB as shown. Knowing that
the uniform thickness of the element Is 0.8 in., determine the normal and
shearing stresses at (a) point a, (b) point b, (c ) point c.
Combined Stresses
13
Example Problem
Member AB has a uniform rectangular cross section of 10*24 mm. For the
loading shown, determine the normal and shearing stresses at (a) point H,
(b) point K.
Combined Stresses
14