NAZARIN B. NORDIN
nazarin@icam.edu.my
Introduction
Classification of Beam Supports
Shear and Bending Moment Diagrams
• Determination of maximum normal and
shearing stresses requires identification of
maximum internal shear force and bending
couple.
• Shear force and bending couple at a point are
determined by passing a section through the
beam and applying an equilibrium analysis on
the beam portions on either side of the
section.
• Sign conventions for shear forces V and V’
and bending couples M and M’
Sample Problem 8.1
SOLUTION:
• Treating the entire beam as a rigid
body, determine the reaction forces
• Section the beam at points near
supports and load application points.
Apply equilibrium analyses on
resulting free-bodies to determine
internal shear forces and bending
couples
For the timber beam and loading
shown, draw the shear and bendmoment diagrams and determine
the maximum normal stress due to • Identify the maximum shear and
bending-moment from plots of their
bending.
distributions.
• Apply the elastic flexure formulas to
determine the corresponding
maximum normal stress.
Sample Problem 8.1
SOLUTION:
• Treating the entire beam as a rigid body, determine
the reaction forces
from
 Fy  0   M B : RB  46 kN RD  14 kN
• Section the beam and apply equilibrium analyses
on resulting free-bodies
 Fy  0
 20 kN  V1  0
V1  20 kN
 M1  0
20 kN0 m   M1  0
M1  0
 Fy  0
 20 kN  V2  0
V2  20 kN
 M2  0
20 kN2.5 m   M 2  0
M 2  50 kN  m
V3  26 kN
M 3  50 kN  m
V4  26 kN M 4  28 kN  m
V5  14 kN M 5  28 kN  m
V6  14 kN M 6  0
Sample Problem 8.1
• Identify the maximum shear and bendingmoment from plots of their distributions.
Vm  26 kN M m  M B  50 kN  m
• Apply the elastic flexure formulas to
determine the corresponding maximum
normal stress.
S  16 b h 2  16 0.080 m 0.250 m 2
 833 .33  10  6 m3
MB
50  103 N  m
m 

S
833 .33  10  6 m3
 m  60.0  10 6 Pa
Sample Problem 8.2
SOLUTION:
• Replace the 10 kip load with an
equivalent force-couple system at D.
Find the reactions at B by considering
the beam as a rigid body.
• Section the beam at points near the
support and load application points.
Apply equilibrium analyses on
The structure shown is constructed
resulting free-bodies to determine
of a W10x112 rolled-steel beam. (a)
internal shear forces and bending
Draw the shear and bendingcouples.
moment diagrams for the beam and
the given loading. (b) determine
normal stress in sections just to the • Apply the elastic flexure formulas to
determine the maximum normal
right and left of point D.
stress to the left and right of point D.
Sample Problem 8.2
SOLUTION:
• Replace the 10 kip load with equivalent forcecouple system at D. Find reactions at B.
• Section the beam and apply
equilibrium analyses on resulting freebodies.
From A to C :
 Fy  0  3x  V  0
 M1  0
3x 12 x  M
From C to D :
 Fy  0  24  V  0
V  3x kips
 0 M  1.5 x 2 kip  ft
V  24 kips
 M 2  0 24 x  4   M  0 M  96  24 x  kip  ft
From D to B :
V  34 kips
M  226  34 x  kip  ft
Sample Problem 8.2
• Apply the elastic flexure formulas to
determine the maximum normal stress to
the left and right of point D.
From Appendix C for a W10x112 rolled
steel shape, S = 126 in3 about the X-X
axis.
To the left of D :
M 2016 kip  in

S
126 in 3
To the right of D :
 m  16.0 ksi
M 1776 kip  in

S
126 in 3
 m  14.1 ksi
m 
m 
Sample Problem 8.3
SOLUTION:
• Taking the entire beam as a free body,
determine the reactions at A and D.
• Apply the relationship between shear and
load to develop the shear diagram.
Draw the shear and bending
moment diagrams for the beam
and loading shown.
5 - 11
• Apply the relationship between bending
moment and shear to develop the bending
moment diagram.
Sample Problem 8.3
SOLUTION:
• Taking the entire beam as a free body, determine the
reactions at A and D.
MA  0
0  D24 ft   20 kips6 ft   12 kips14 ft   12 kips28 ft 
D  26 kips
 Fy  0
0  Ay  20 kips  12 kips  26 kips  12 kips
Ay  18 kips
• Apply the relationship between shear and load to
develop the shear diagram.
dV
 w
dx
dV   w dx
- zero slope between concentrated loads
- linear variation over uniform load segment
5 - 12
Sample Problem 8.3
• Apply the relationship between bending moment
and shear to develop the bending moment
diagram.
dM
V
dx
dM  V dx
- bending moment at A and E is zero
- bending moment variation between A, B,
C and D is linear
- bending moment variation between D
and E is quadratic
- net change in bending moment is equal to
areas under shear distribution segments
- total of all bending moment changes across
the beam should be zero
Sample Problem 8.5
SOLUTION:
• Taking the entire beam as a free body,
determine the reactions at C.
• Apply the relationship between shear
and load to develop the shear diagram.
Draw the shear and bending
moment diagrams for the beam
and loading shown.
• Apply the relationship between
bending moment and shear to develop
the bending moment diagram.
Sample Problem 8.5
SOLUTION:
• Taking the entire beam as a free body,
determine the reactions at C.
 Fy  0   12 w0 a  RC
a

 M C  0  12 w0 a L    M C
3

RC  12 w0 a
a

M C   12 w0 a L  
3

Results from integration of the load and shear
distributions should be equivalent.
• Apply the relationship between shear and load
to develop the shear diagram.
a
2 
 
x
 x
VB  V A    w0 1   dx    w0  x  

a
2a 
 
0 
0
a
VB   12 w0 a    area under load curve
- No change in shear between B and C.
- Compatible with free body analysis
Sample Problem 8.5
• Apply the relationship between bending moment
and shear to develop the bending moment
diagram.
a
a
2 


 x 2 x3 
x


M B  M A    w0  x   dx   w0   

 2 6a 

2a  

0


 0
M B   13 w0a 2
L


M B  M C    12 w0 a dx   12 w0 aL  a 
a
a w0 
a
M C   16 w0 a3L  a  
L 
2 
3
Results at C are compatible with freebody analysis
Sample Problem 8.8
SOLUTION:
• Considering the entire beam as a freebody, determine the reactions at A and
D.
A simply supported steel beam is
to carry the distributed and
concentrated loads shown.
Knowing that the allowable normal
stress for the grade of steel to be
used is 160 MPa, select the wideflange shape that should be used.
5 - 17
• Develop the shear diagram for the
beam and load distribution. From the
diagram, determine the maximum
bending moment.
• Determine the minimum acceptable
beam section modulus. Choose the
best standard section which meets this
criteria.
Sample Problem 8.8
• Considering the entire beam as a free-body,
determine the reactions at A and D.
 M A  0  D5 m   60 kN1.5 m   50 kN4 m 
D  58.0 kN
 Fy  0  Ay  58.0 kN  60 kN  50 kN
Ay  52.0 kN
• Develop the shear diagram and determine the
maximum bending moment.
V A  Ay  52.0 kN
VB  VA  area under load curve  60 kN
VB  8 kN
• Maximum bending moment occurs at
V = 0 or x = 2.6 m.
M max  area under shear curve, A to E 
 67.6 kN
5 - 18
Sample Problem 8.8
• Determine the minimum acceptable beam
section modulus.
M max 67.6 kN  m
S min 

 all
160 MPa
 422.5 10  6 m3  422.5 103 mm3
Shape
S  10 mm
W410  38.8
637
W360  32.9
474
W310  38.7
549
W250  44.8
535
W200  46.1
448
3
3
• Choose the best standard section which
meets this criteria.
W 360 32.9
•
Shear force diagram (SFD): Graph of shear force V vs x
•
Bending moment diagram (BMD): Graph of bending moment
M vs x
Example 8.1: Draw the shear force and bending moment
dagrams for the beam shown in Fig. 1.
SOLUTIONS
QUESTION 1
If the beam carries loads at the positions shown in
figure, what are the reactive forces at the supports?
The weight of the beam may be neglected.
QUESTION 2
If the beam carries loads at the positions shown in
figure, what are the reactive forces at the beam? The
weight of the beam may be neglected.
QUESTION 3
Determine the shear force and bending moment at points 3.5m
and 8.0m from the right-hand end of the beam. (neglect the
weight of the beam)
QUESTION 4
A beam of length 5.0m and neglect the weight rests on supports at
each end and a concentrated load of 255N is applied at its
midpoint. Determine the shear force and bending moment at
distances from the right-hand end of the beam of
a) 1.5m
b) 2.4m
c) Draw the shear force and bending moment diagram.
QUESTION 5
A cantilever has a length of 2m and a concentrated load of 8kN is
applied to its free end. Determine the shear force and bending
moment at distances of
a) 0.5m
b) 1.0m
c) Draw the shear force and bending moment diagram.
(neglect the weight of the beam.
QUESTION 6
A beam of length 5.5m supports at each end and a concentrated
load of 135N is applied at 2.5m from the left hand end. Determine
the shear force and bending moment at distances of;
a) 0.8m
b) 1.2m
c) Draw the shear force and bending moment diagram.
(neglect the weight of the beam)
QUIZ
A beam of length 1m supports at each end and a concentrated load
of 1.5N is applied at the centre. Determine the shear force and
bending moment at distances of;
a) 0.25m
b) 0.65m
c) Draw the shear force and bending moment diagram.
(neglect the weight of the beam)