Course review for final examination
1. The meaning of each parameter in the formula of your Lecture Notes should be fully
understood;
2. Sign convention must be followed;
3. Understand and independently solve the assignment questions, practice questions and
examples in the lecture notes.
Chapter 1: Sections 1.1-1.5
(a) How to draw FBD.
(b) Method of sections to determine the internal loads (may include normal force (N), shear force
(V), bending moment (M) and torsional moment (T); the bending moment is a moment about an
axis within the cross section, while the torsional moment is a moment about the longitudinal
axis)
(c) Equilibrium equations of mechanical member.
(d) How to calculate average normal and average shear stresses.

N
A

V
A
N: Internal normal force; V: Internal shear force; A: acting area of internal normal force and
shear force
Chapters 2 and 3: Sections 2.1, 2.2, 3.1, 3.2, 3.3, 3.5, 3.6
(a) How to calculate normal strain ɛ and shear strain ? Normal strain is used to describe the
change is length, while the shear strain is associated with the change of angle.
(b) Hook’s law (the stress-strain relation):   E
  G
E: Young’s modulus of the material; G: Shear modulus of the material
Chapter 4: Sections 4.1-4.5
(a) How to calculate the normal stress according to the internal normal force N for an axially
loaded member? The normal stress distribution is uniform on the cross section and the average
normal stress is calculated as,

N
A
(b) How to calculate the longitudinal displacement (or deformation) of an axially loaded
member? Understand how to use the following three equations.
L
L
N ( x)dx
   d  
EA( x)
0
0
NL

EA
n
 
i 1
N i Li
Ei Ai
N (Ni): Internal normal force; E: Young’s modulus of the material; A: cross-sectional area of
the mechanical member; L(Li): segment length of the mechanical member.
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(c) How to solve the statically indeterminate problems? These kinds of problems usually need
one more equation (compatibility condition) in addition to the equilibrium equations to solve the
unknowns. Or superposition technique can also be used to solve this kind of problems.
(d) From the equations in (a) and (b), it is indicated that the normal stress developed in the
axially loaded mechanical member and its deformation can be expressed in terms of the applied
loads (internal normal force can be determined in terms of the applied loads from the method of
sections and equilibrium equations) and the geometric dimensions of the mechanical member.
Therefore, the following problems could be solved from the calculation of stress and
deformation:
(1) If the applied loads, material (material properties and allowable stress are known for
example) and dimensions of the mechanical member are given, we can justify if the mechanical
member could sustain the applied loads. The criterion is that the calculated maximum normal
stress can not exceed the allowable stress, or the criterion could be set for the maximum
deformation;
(2) If the material (material properties and allowable stress are known for example) and
dimensions of the mechanical member are given, we can determine the maximum loads could be
sustained by the mechanical member, which is based on the same criterion as stated in (1).
(3) If the applied loads, material (material properties and allowable stress are known for
example) of the mechanical member are given, we can design the dimension of the mechanical
member. The criterion is the same as in (1).
……
Chapter 5: Sections 5.1, 5.2, 5.4, 5.5.
(a) How to calculate the shear stress distribution of a circular shaft according to the internal
torsional moment (or torque) T?
T
 ( ) 
J
T: Internal torsional moment (or torque); J: Polar moment of inertia of the cross-sectional
area of the shaft; ρ: Radial position of the point interested
When ρ=c (c is the outer radius of the shaft), shear stress reaches the maximum value on the
cross section.
(b) How to determine the twist angle ϕ of a circular shaft? The following equations may be used.
L

0
T ( x)
dx
J ( x)G

TL
JG
 
Ti Li
J i Gi
L (Li): segment length; G: shear modulus of the material.
(c) How to calculate the polar moment inertia J?
(d) How to solve the statically indeterminate problems? These kinds of problems usually need
one more equation (compatibility condition) in addition to the equilibrium equations to solve the
unknowns.
(e) From the equations in (a) and (b), it is indicated that the shear stress developed in the
torsional shaft and its deformation can be expressed in terms of the applied loads (internal torque
can be determined in terms of the applied loads from the method of sections and equilibrium
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equations) and the geometric dimensions of the mechanical member. Therefore, the following
problems could be solved from the calculation of stress and deformation:
(1) If the applied loads, material (material properties and allowable stress are known for
example) and dimensions of the mechanical member are given, we can justify if the mechanical
member could sustain the applied loads. The criterion is that the calculated maximum shear
stress can not exceed the allowable shear stress, or the criterion could be set for the maximum
deformation;
(2) If the material (material properties and allowable stress are known for example) and
dimensions of the mechanical member are given, we can determine the maximum loads could be
sustained by the mechanical member, which is based on the same criterion as in (1).
(3) If the applied loads, material (material properties and allowable stress are known for
example) of the mechanical member are given, we can design the dimension of the mechanical
member. The criterion is the same as in (1).
…
Chapter 6: Sections 6.1, 6.3, 6.4.
(a) How to draw the shear force and moment diagrams? How to determine the shear force and
moments on any particular cross section (or how to determine the shear and moment functions?)?
This can be done either by method of sections or by the integration based on the relation among
distributed load, shear force and bending moment.
(b) How to calculate the bending (or normal) stress according to the internal bending moment
M?
My
 ( y)  
I
M: internal bending moment; y: interested point position measured from the neutral axis,
positive above the neutral axis, while negative below the neutral axis. I: moment of inertia of
the entire x-section about its neutral axis.
(c) How to determine the neutral axis position and calculate the second moment inertia I about
the neutral axis?
Chapter 7: Sections 7.1, 7.2.
(a) How to calculate the shear stress according to the shear force V? The transverse shear stress
on the cross section is always the same direction as the internal shear force. Complimentary
theory of shear stress should be understood.
 ( y) 
VQ
It
′
V: internal shear force; I: moment of inertia of the entire x-section about its neutral axis; t:
width of cross section through the point of interest; A': area above/below the point of interest;
: distance from the neutral axis of A' to the neutral axis of entire cross section.
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Chapter 12: Sections 12.1-12.3, 12.5-12.7, 12.9
(a) How to calculate the slope and the deflection of the beam using integration based on the
following equation? 2
d  M ( x)

dx 2
EI
The integration constants can be determined from the boundary conditions and continuity
conditions for the connecting point between two adjacent sections. The boundary conditions for
different supports should be known. Pay attention to the continuity condition when the
coordinate axes for these two sections are pointing to the opposite directions. Discontinuity
function (Macaulay function) can also be used to determine M, which means that one single
function is sufficient to determine the bending moment for the entire beam. Correspondingly,
one single function will be derived to calculate the slope and the deflection of the beam.
(b) How to solve the statically indeterminate problems for the bending beam? Two methods:
(i) Superposition technique: Using the Table for the slopes and deflections of the bending
beams directly (or by your own calculation). The procedures are summarized as:
(1) Draw the FBD for the entire beam, and write down the equilibrium equations. (Equations are
not sufficient to get all the unknowns, one more equation is needed)
(2) Take any support reaction as a “redundant”. This original problem is decomposed into two
subproblems: subproblem 1 and subproblem 2.
(3) In subproblem 1, this “redundant” is removed temporarily and the beam is subjected to the
applied loads only with other support constraints. Based on this loading condition, the deflection
or the slope at the point with “redundant” can be determined from the results in the Table for the
slopes and deflections of the bending beams (or you can calculate yourself since this Subproblem
1 is a statically determinate problem).
(4) In Subproblem 2, the “redundant” is put back and the beam is subjected to this “redundant”
only (no other applied loads) with the other support constraints. Under this loading condition, the
deflection or slope at this point with the “redundant” can also be determined from the Table or
your own calculation.
(5) After you solve these two subproblems, the deflection or the slope of the beam in the original
problem can be determined as the summation of the deflection or the slope of the beams of the
two subproblems. The compatibility condition for the point with the “redundant” can also be
written. This compatibility condition together with the equilibrium equations are sufficient to
solve the problem.
(ii) Integration method: The procedures are summarized as the following:
(1) Draw the FBD for the entire beam, and write down the equilibrium equations.
(2) Determine the function of internal bending moment for each section of the beam. These
functions may be in terms of the unknown reactions (leave all unknowns as they are)
(3) Using integration to determine the slopes and deflections for all sections. At this stage, the
unknowns are unknown reactions and integration constants.
(4) Apply all boundary conditions at all supports and continuity conditions at all the connecting
points between two adjacent sections. Solve these equations, you will determine everything.
* When using Integration method, the Macaulay function can also be applied. In this way,
only one single function needs to be determined for the shear, moment, slope and deflection.
Then applied all the boundary conditions, you will solve all the unknowns combined with the
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equilibrium equations. When using Macaulay function, the continuity conditions are NOT
necessary since everything is expressed as a single function.
To understand these two methods in solving statically indeterminate problems of bending beams,
you can check the examples in you lecture notes. The solutions with these two methods were
provided.
Chapter 13: Sections 13.1-13.3
(a) How to determine the critical buckling load and critical buckling stress for compressive
columns with various types of supports?
Pcr 
 2 EI
(kL) 2
 cr 
Pcr
A
Both critical buckling load and buckling stress depend on the material and geometric
properties of the column.
E: Young’s modulus of the material; I: moment of inertia of the cross section about its neutral
axis; k: effective-length factor depending on the end supports; A: cross section area of the
column.
(b) Determine whether the compressive column yields (plastically deforms) first or buckle first
by comparing the critical buckling stress and the yielding stress.
Chapter 8: Sections 8.1, 8.2
(a) How to calculate the stresses developed in the cylindrical and spherical vessels under internal
pressure p?
(b) How to determine the stress components of a material point in the mechanical member
according to combined loads?
This is usually realized by calculating the stress corresponding to each individual internal load.
The same kinds of stresses can be superimposed. For example, the normal stresses in the same
direction (x direction for example) can be summed up. However, the normal stresses in the
different directions (x and y directions for example) can not be summed up. The shear and
normal stresses can not be summed up either.
(c) Understand the concept of state of stress for a material point in the mechanical member and
know how to determine the state of stress. This is usually realized by determining the stress
components for this material point on the cross section area. The procedures for determining the
state of stress for any material point are summarized as:
(i) Draw the FBD for the entire mechanical member and determine the reactions;
(ii) Using the method of sections to determine the internal loads on the cross section area through
this point;
(iii) Calculate the stress according to each individual internal load, then the same kinds of
stresses can be summed up;
(iv) Cut a stress element with two surfaces being the cross sections, then the stress components
on this element can be determined. (complimentary theory of shear stress may be used here if
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shear stress develops at the material point along the cross section). Plot the stress components on
this element. Thus, the sate of stress for a material point is determined. Based on the state of
stress, stress transformation can be performed, and principal stresses, maximum shear stress can
be calculated correspondingly as discussed in Chapter 9.
Chapter 9: Sections 9.1, 9.2, 9.3, 9.4
(a) Equations of plane-stress transformation:
 x' 
x  y
2
 x' y'  
 y' 

 x  y
 x  y
2
x  y
2

2
cos 2   xy sin 2
sin 2   xy cos 2
 x  y
2
cos 2   xy sin 2
When do the stress transformation, the state of stress or stress components (  x ,  y ,  xy ) for a
material point must be determined first, which will be realized by the procedures shown in
Chapter 8(c) above. Then the stress components can be plotted on the stress element as described
in Chapter 8(c). A reference coordinate system xy will be established to describe this element.
The corresponding stress components  x  y  xy are defined according to this reference coordinate
system (Sign convention must be followed, check you lecture notes). Based on these stress
transformation equations, you can solve the following problems:
(i) Determine the stress components on any inclined surface (or sustained by any inclined
surface). This inclined surface is described by a coordinate system x’y’, in which you can
assume x’ is normal to the inclined surface, and the construction of y’ follows the right-hand
rule. Thus the orientation  of this inclined surface is measured from x to x’. Using the first two
equations in stress transformation above, you can calculate the normal stress  x ' and shear stress
 x ' y ' sustained by the inclined surface.
(ii) Use all the three equations above, you can calculate the stress components of any oriented
stress element. This stress element is described by the coordinate system x’y’ and the orientation
 is measured from x to x’.
(iii) Determine the principal stresses and maximum shear stress for any material point. These
stresses can be determined by using the following equations in (b) and (c) directly, which are
derived from the stress transformation equations.
(b) How to calculate principal stresses and principal directions (principal planes)
 1,2 
x  y
2
2
  x  y 
2
 
   xy
2 

1   2
6
tan 2 p 
 xy
( x   y ) 2
Using these equations, the principal stresses can be determined directly. The principal planes are
90o apart. After calculating  p for the principal directions, substitute them back into the first
equation in (a) (stress transformation equations), you can determine which angle corresponds to
which normal stress.
(c) How to calculate the maximum in-plane shear stress and the plane with such maximum shear
stress? Use the following equation directly.
2
 max
  x  y 
2
 
   xy
2 

tan 2 s 
( x   y ) 2
 xy
(d) How to construct a Mohr’s circle? How to determine the stress components on any inclined
surface? How to determine the principal stresses and maximum shear stress? You can use the
above equations directly for mathematical calculation or use the Mohr’s circle.
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