MME2202 CourseReview

8/3/2019 MME2202 CourseReview

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Course review: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 and examples in the

lecture notes

Chapter 1: Sections 1.1, 1.2

(a) How to draw the FBD?

(b) Method of sections to determine the internal loads (may include normal force, shear

force, bending moment and torsional moment; the bending moment is a moment about an

axis within the cross section, while the torsional moment is a moment about thelongitudinal axis)

(c) Equilibrium equations of mechanical member.

(d) How to calculate average normal and shear stresses

Chapters 2 and 3: Sections 2.1, 2.2, 3.2, 3.4, 3.7

(a) How to calculate normal and shear strain

(b) Hooks law (the stress-strain relation)

Chapter 4: Sections 4.1-4.5

(a) How to calculate the normal stress according to the normal force N? which is the

average normal stress

(b) How to calculate the longitudinal displacement? Understand how to use the following

three equations.

(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.

Chapter 5: Sections 5.1, 5.2, 5.4, 5.5.

(a) How to calculate the shear stress distribution according to the internal torsionalmoment (or torque) T?

(b) How to determine the twist angle of the shaft? The following equations may be used.

N

A

=

L L

0 0

( )d

( )

P x dx

EA x = =

PL

EA =

1

ni

i

i i i

P L

E A

=

=

( )T

J

=

TL

JG =

0

( )

( )

LT x

dx J x G

= TL

JG =


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(c) How to calculate the polar moment inertia J?

(d) How to solve the statically indeterminate problem?

Chapter 6: Sections 6.1, 6.3, 6.4.

(a) How to draw the shear force and moment diagrams? How to determine the shear forceand moments on any particular cross section? This can be done either by method ofsections 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?

(c) How to calculate the second moment inertia Iabout the neutral axis?

Chapter 7: Sections 7.1, 7.2.

(a) How to calculate the shear stress according to the shear force V? Complimentary theory

of shear stress should be understood.

(b) How to calculate Q?

* For Chapters 5, 6 and 7, how to calculate the geometric parameters should also be

known. Such as the position of the neutral axis, the polar moment of inertia, the (2 nd)moment of inertia about the neutral axis, the 1st moment of inertia of an area about the

neutral axis.

Chapter 8: Sections 8.1, 8.2

(a) How to calculate the stresses developed in the cylindrical and spherical vessels underinternal pressure? (check the formula for stresses in your lecture notes or textbook)

(b) How to determine the stress components 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 samedirection (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 fordetermining 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 areathrough this point;

(iii) Calculate the stress according to each individual internal load, then the same kinds of

stresses can be summed up;

( )My

yI

=

( ')VQ

yIt

=


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(iv) Cut a stress element with two surfaces being the cross sections, then the stresscomponents on this element can be determined. (complimentary theory of shear stress may

be used here if shear stress develops at the material point). Plot the stress components on

this element. Thus, the sate of stress for a material point is determined. Based on the stateof stress, stress transformation and principal stresses, maximum shear stress can be

calculated correspondingly as discussed in Chapter 9.

Chapter 9: Sections 9.1-9.4

(a) Equations of plane-stress transformation:

When do the stress transformation, the state of stress for a material point must be

determined first, which will be realized by the procedures shown in Chapter 8(c) above.

For the stress element determined above, 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 xy, in which x isnormal 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 stress

element. This stress element is described by the coordinate system xy 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)

' cos 2 sin 22 2

x y x y

x xy

+ = + +

' 'sin 2 cos 2

2

x y

x y xy

= +

' cos 2 sin 22 2

x y x y

y xy

+ =

2

2

1,22 2 xy

x y x y

+

= +

1 2


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Using these equations, the principal stresses can be determined directly. The principal

planes are 90

o

apart. After calculating p for the principal directions, substitute them backinto 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.

(d) How to construct a Mohrs circle? How to determine the stress components on anyinclined surface? How to determine the principal stresses and maximum shear stress? You

can use the above equations directly for mathematical calculation or use the Mohrs circle.

Chapter 12: Sections 12.1-12.3, 12.5-12.7, 12.9

(a) How to calculate the slope the deflection of the beam using integration based on thefollowing equation?

The integration constants can be determined from the boundary conditions and continuity

conditions for the connecting point between two adjacent sections. Pay attention to thecontinuity condition when the coordinate axes for these two sections are pointing to the

opposite directions. Discontinuity function (Macaulay function) can also be used todetermine 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 slope and 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 bendingbeams 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. The original problem becomes a problemthat the beam is subjected to the applied loads and this redundant reaction. 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 tothe 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 (chart) for the slopes and deflections of the bending beams (or you can calculateyourself since this Subproblem 1 is a determinate problem).

( ) 2tan2

x y

s

xy

=

2

2

max2 xy

x y

= +

tan2( ) 2

xy

p

x y

=

2

2

d M EI

dx

=


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MME2202 CourseReview