Analysis of Beams in Bending
(5.1-5.3)
MAE 314 – Solid Mechanics
Yun Jing
Analysis of Beams in Bending
1
Bending Moment Along a Beam




In this chapter, we will learn how to find the bending moment M along
the beam.
M is not necessarily constant; sometimes M is a function of x.
We will also solve for the shear force V(x), which will be used in
Chapter 6.
As before, we need a new FBD every time the loading changes.
1
2
1
3
Analysis of Beams in Bending
2
2
Review of Beam Supports


3 equilibrium equations: Σ FY = 0, Σ FX = 0, Σ M = 0
Ignore the horizontal (x-direction) components, because these are
axial loading.
R1
R3
R2
R1
R2
R1
R3
R2
R3
R4
R1 M
1
R2
Analysis of Beams in Bending
R3
R1 M
1
R1 M1
R2
R2
R4
R3 M2
3
Sign Convention

Recall the applied loading results in both a bending moment M and a
shear force V.
Positive shear and bending moment
Analysis of Beams in Bending
4
Analysis of Beams in Bending
5
Analysis of Beams in Bending
6
Analysis of Beams in Bending
7
Example Problem
Draw the shear and bending moment diagrams for the beam and
loading shown, and determine the maximum absolute value (a) of the
shear and (b) of the bending moment.
Analysis of Beams in Bending
8
Relations Between F, V, and M

For beams with more complicated loading, it is helpful to develop a
relationship between load, shear and bending moment.

Sum forces in the vertical direction.
F
y
V  (V  V )  wx  0  V   wx
dV
 w
dx
xC '
or
VC '  VC    wdx
xC
Analysis of Beams in Bending
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Relations Between F, V, and M

Sum moment about C’.
 M C M  M   M  Vx  wx
x
0
2
1
2
M  Vx  wx 
2

Neglect (Δx)2 term since it is much smaller than
Δx term.
dM
V
dx
xC '
or
M C '  M C   V dx
xC
Analysis of Beams in Bending
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Example Problem
Determine (a) the equations of the shear and bending moment curves
for the beam and loading shown, and (b) the maximum absolute value
of the bending moment in the beam.
Analysis of Beams in Bending
11
Example Problem
Draw the shear and bending-moment diagrams for the beam and loading
shown.
Analysis of Beams in Bending
12
Design of Beams in Bending
(5.4)
MAE 314 – Solid Mechanics
Yun Jing
Design of Beams in Bending
13
Design of Beams for Bending

Recall the largest normal stress in the beam subject to bending occurs
at the surface and can be defined as
 max 

M
max
I
c

M
max
where
S
I
S
c
A safe design requires the maximum stress is no more than the
allowable stress (σmax ≤ σall), so
Smin 
M
max
 all
Design of Beams in Bending
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Procedure for Design





Determine the value for σall.
Draw shear and moment diagrams.
From the diagrams, determine the maximum absolute
bending moment.
Determine the minimum allowable value Smin.
Use Smin to determine best cross section dimensions.


Timber beam: Smin = bh2/6
Rolled-steel beam: Use Appendix C in textbook
Design of Beams in Bending
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Example Problem
For the beam and loading shown, design the cross section of the beam,
knowing that the grade of timber used has an allowable normal stress of
12 MPa.
Design of Beams in Bending
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Example Problem
Knowing that the allowable stress for the steel used is 160 MPa, select
the most economical S-shape beam to support the loading shown.
Design of Beams in Bending
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Example Problem

A beam is to be made of steel that has an allowable bending stress of
170MPa. Select an appropriate W shape that will carry the loading
shown in the figure below
Design of Beams in Bending
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