Transcript 5 - Hacettepe University Department of Mechanical Engineering

Fourth Edition
CHAPTER
5
MECHANICS OF
MATERIALS
Ferdinand P. Beer
E. Russell Johnston, Jr.
John T. DeWolf
Lecture Notes:
J. Walt Oler
Texas Tech University
Analysis and Design
of Beams for Bending
© 2006 The McGraw-Hill Companies, Inc. All rights reserved.
Fourth
Edition
MECHANICS OF MATERIALS
Beer • Johnston • DeWolf
Analysis and Design of Beams for Bending
Introduction
Shear and Bending Moment Diagrams
Sample Problem 5.1
Sample Problem 5.2
Relations Among Load, Shear, and Bending Moment
Sample Problem 5.3
Sample Problem 5.5
Design of Prismatic Beams for Bending
Sample Problem 5.8
© 2006 The McGraw-Hill Companies, Inc. All rights reserved.
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MECHANICS OF MATERIALS
Beer • Johnston • DeWolf
Introduction
• Objective - Analysis and design of beams
• Beams - structural members supporting loads at
various points along the member
• Transverse loadings of beams are classified as
concentrated loads or distributed loads
• Applied loads result in internal forces consisting
of a shear force (from the shear stress
distribution) and a bending couple (from the
normal stress distribution)
• Normal stress is often the critical design criteria

x  
My
I
m 
M c
I

M
S
Requires determination of the location and
magnitude of largest bending moment
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MECHANICS OF MATERIALS
Beer • Johnston • DeWolf
Introduction
Classification of Beam Supports
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MECHANICS OF MATERIALS
Beer • Johnston • DeWolf
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’
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MECHANICS OF MATERIALS
Beer • Johnston • DeWolf
Sample Problem 5.1
SOLUTION:
• Treating the entire beam as a rigid
body, determine the reaction forces
For the timber beam and loading
shown, draw the shear and bendmoment diagrams and determine the
maximum normal stress due to
bending.
• 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
• Identify the maximum shear and
bending-moment from plots of their
distributions.
• Apply the elastic flexure formulas to
determine the corresponding
maximum normal stress.
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MECHANICS OF MATERIALS
Beer • Johnston • DeWolf
Sample Problem 5.1
SOLUTION:
• Treating the entire beam as a rigid body, determine
the reaction forces
from
 F y  0   M B : R B  46 kN
R D  14 kN
• Section the beam and apply equilibrium analyses
on resulting free-bodies
 Fy  0
 20 kN  V1  0
 M1  0
 20 kN 0 m   M 1
 Fy  0
 20 kN  V 2  0
M2  0
 20 kN  2 . 5 m   M 2
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V1   20 kN
0
M1  0
V 2   20 kN
0
M 2   50 kN  m
V 3   26 kN
M 3   50 kN  m
V 4   26 kN
M 4   28 kN  m
V 5   14 kN
M 5   28 kN  m
V 6   14 kN
M6  0
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MECHANICS OF MATERIALS
Beer • Johnston • DeWolf
Sample Problem 5.1
• Identify the maximum shear and bendingmoment from plots of their distributions.
V m  26 kN
M m  M B  50 kN  m
• Apply the elastic flexure formulas to
determine the corresponding
maximum normal stress.
2
2
S  1 b h  1  0 . 080 m  0 . 250 m 
6
6
 833 . 33  10
6
m
3
3
MB
50  10 N  m
m 

6
3
S
833 . 33  10
m
6
 m  60 . 0  10 Pa
© 2006 The McGraw-Hill Companies, Inc. All rights reserved.
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MECHANICS OF MATERIALS
Beer • Johnston • DeWolf
Sample Problem 5.2
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 of a
resulting free-bodies to determine
W10x112 rolled-steel beam. (a) Draw
internal shear forces and bending
the shear and bending-moment diagrams
couples.
for the beam and the given loading. (b)
determine normal stress in sections just
• Apply the elastic flexure formulas to
to the right and left of point D.
determine the maximum normal
stress to the left and right of point D.
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MECHANICS OF MATERIALS
Beer • Johnston • DeWolf
Sample Problem 5.2
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 free-bodies.
From A to C :
 Fy  0
 M1  0
 3x  V  0
3 x 12 x   M
V   3 x kips
0
2
M   1 . 5 x kip  ft
From C to D :
 Fy  0
 24  V  0
V   24 kips
M2  0
24  x  4   M  0
M  96  24 x  kip  ft
From D to B :
V   34 kips
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M   226  34 x  kip  ft
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MECHANICS OF MATERIALS
Beer • Johnston • DeWolf
Sample Problem 5.2
• 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 
M

S
2016 kip  in
126 in
3
 m  16 . 0 ksi
To the right of D :
m 
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M
S

1776 kip  in
126 in
3
 m  14 . 1 ksi
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MECHANICS OF MATERIALS
Beer • Johnston • DeWolf
Relations Among Load, Shear, and Bending Moment
• Relationship between load and shear:
 F y  0 : V  V   V   w  x  0
V   w x
dV
 w
dx
xD
V D  V C    w dx
xC
• Relationship between shear and bending
moment:
 M C  0 :
M
 M   M  V x  wx
x
0
2
2
 M  V  x  1 w  x 
2
dM
V
dx
xD
M
D
 MC 
 V dx
xC
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MECHANICS OF MATERIALS
Beer • Johnston • DeWolf
Sample Problem 5.3
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.
• Apply the relationship between bending
moment and shear to develop the bending
moment diagram.
© 2006 The McGraw-Hill Companies, Inc. All rights reserved.
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MECHANICS OF MATERIALS
Beer • Johnston • DeWolf
Sample Problem 5.3
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  A y  20 kips  12 kips  26 kips  12 kips
A y  18 kips
• Apply the relationship between shear and load to
develop the shear diagram.
dV
 w
dV   w dx
dx
- zero slope between concentrated loads
- linear variation over uniform load segment
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MECHANICS OF MATERIALS
Beer • Johnston • DeWolf
Sample Problem 5.3
• Apply the relationship between bending
moment and shear to develop the bending
moment diagram.
dM
V
dM  V dx
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
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