Structural Analysis: In Theory and Practice

599 Pages·2008·3.44 MB·English

by  Alan Williams

Butterworth-Heinemann is an imprint of Elsevier
30 Corporate Drive, Suite 400, Burlington, MA 01803, USA
Linacre House, Jordan Hill, Oxford OX2 8DP, UK
Copyright © 2009, International Codes Council. All rights reserved.
ALL RIGHTS RESERVED. This Structural Analysis: In Theory and Practice is a
copyrighted work owned by the International Code Council, Inc. Without advance
permission from the copyright owner, no part of this book may be reproduced,
distributed, or transmitted in any form or by any means, including without limitation,
electronic, optical or mechanical means (by way of example and not limitation,
photocopying, or recording by or in an information storage retrieval system). For
information on permission to copy material exceeding fair use, please contact
Publications, 4051 West Flossmoor Road, Country Club hills, IL 60478-5795.
Phone 1-888-ICC-SAFE (422-7233)
Trademarks: “International Code Council,” and the “International Code Council”
logo are trademarks of the International Code Council, Inc.
Library of Congress Cataloging-in-Publication Data
Application submitted
British Library Cataloguing-in-Publication Data
A catalogue record for this book is available from the British Library.
ISBN: 978-1-85617-550-0
For information on all Butterworth–Heinemann publications
visit our Web site at www.elsevierdirect.com
Printed in the United States of America
08 09 10 11 12 10 9 8 7 6 5 4 3 2 1
Foreword
It is with great pleasure that I write this foreword to Structural Analysis: In
Theory and Practice, by Alan Williams. Like many other engineers, I have uti-
lized Dr. Williams ’ numerous publications through the years and have found
them to be extremely useful. This publication is no exception, given the
extensive experience and expertise of Dr. Williams in this area, the credibil-
ity of Elsevier with expertise in technical publications internationally, and the
International Code Council (ICC) with expertise in structural engineering and
building code publications.
Engineers at all levels of their careers will find the determinate and indeter-
minate analysis methods in the book presented in a clear, concise, and practical
manner. I am a strong advocate of all of these attributes, and I am certain that
the book will be successful because of them. Coverage of many other impor-
tant areas of structural analysis, such as Plastic Design, Matrix and Computer
Methods, Elastic-Plastic Analysis, and the numerous worked-out sample prob-
lems and the answers to the supplementary problems greatly enhance and rein-
force the overall learning experience.
One may ask why, in this age of high-powered computer programs, a com-
prehensive book on structural analysis is needed. The software does all of the
work for us, so isn't it sufficient to read the user’s guide to the software or to
have a cursory understanding of structural analysis?
While there is no question that computer programs are invaluable tools that
help us solve complicated problems more efficiently, it is also true that the soft-
ware is only as good as the user’s level of experience and his/her knowledge of
the software. A small error in the input or a misunderstanding of the limita-
tions of the software can result in completely meaningless output, which can
lead to an unsafe design with potentially unacceptable consequences.
That is why this book is so valuable. It teaches the fundamentals of struc-
tural analysis, which I believe are becoming lost in structural engineering.
Having a solid foundation in the fundamentals of analysis enables engineers
to understand the behavior of structures and to recognize when output from a
computer program does not make sense.
Simply put, students will become better students and engineers will become
better engineers as a result of this book. It will not only give you a better
understanding of structural analysis; it will make you more proficient and
efficient in your day-to-day work.
David A. Fanella, Ph.D., S.E., P.E.
Chicago, IL
March 2008
Part One
Analysis of Determinate Structures
1
Principles of statics
Notation
F force
f angle in a triangle opposite sideF
i i
H horizontal force
l length of member
M bending moment
P axial force in a member
R support reaction
V vertical force
W concentrated live load
LL
w distributed dead load
DL
θ angle of inclination
1.1 Introduction
Statics consists of the study of structures that are at rest under equilibrium
conditions. To ensure equilibrium, the forces acting on a structure must bal-
ance, and there must be no net torque acting on the structure. The principles
of statics provide the means to analyze and determine the internal and external
forces acting on a structure.
For planar structures, three equations of equilibrium are available for the
determination of external and internal forces. A statically determinate struc-
ture is one in which all the unknown member forces and external reactions
may be determined by applying the equations of equilibrium.
An indeterminate or redundant structure is one that possesses more
unknown member forces or reactions than the available equations of equilib-
rium. These additional forces or reactions are termed redundants. To determine
the redundants, additional equations must be obtained from conditions of geo-
metrical compatibility. The redundants may be removed from the structure,
and a stable, determinate structure remains, which is known as the cut-back
structure. External redundants are redundants that exist among the external
reactions. Internal redundants are redundants that exist among the member
forces.
4 Structural Analysis: In Theory and Practice
1.2 Representation of forces
A force is an action that tends to maintain or change the position of a struc-
ture. The forces acting on a structure are the applied loads, consisting of both
dead and imposed loads, and support reactions. As shown in F igure 1.1 , the
simply supported beam is loaded with an imposed load W located at point
LL
3 and with its own weight w , which is uniformly distributed over the length
DL
of the beam. The support reactions consist of the two vertical forces located at
the ends of the beam. The lines of action of all forces on the beam are parallel.
W
LL
w
DL
1 2
3
Figure 1.1
In general, a force may be represented by a vector quantity having a magni-
tude, location, sense, and direction corresponding to the force. A vector repre-
sents a force to scale, such as a line segment with the same line of action as the
force and with an arrowhead to indicate direction.
The point of application of a force along its line of action does not affect the
equilibrium of a structure. However, as shown in the three-hinged portal frame
in F igure 1.2 , changing the point of application may alter the internal forces in
the individual members of the structure.
100 kips 100 kips
75 kips 25 kips 25 kips 75 kips
50 kips 50 kips 50 kips 50 kips
(i) (ii)
Figure 1.2
Collinear forces are forces acting along the same line of action. The two hor-
izontal forces acting on the portal frame shown in F igure 1.3 (i) are collinear
and may be added to give the single resultant force shown in (ii).
Principles of statics 5
50 kips 20 kips 30 kips
2 3 2 3
1 4 1 4
(i) (ii)
Figure 1.3
Forces acting in one plane are coplanar forces. Space structures are three-
dimensional structures and, as shown in F igure 1.4 , may be acted on by non-
coplanar forces.
Figure 1.4
In a concurrent force system, the line of action of all forces has a common
point of intersection. As shown in F igure 1.5 for equilibrium of the two-hinged
arch, the two reactions and the applied load are concurrent.
Figure 1.5
6 Structural Analysis: In Theory and Practice
I t is often convenient to resolve a force into two concurrent components. The
original force then represents the resultant of the two components. The direc-
tions adopted for the resolved forces are typically thex- andy-components in a
rectangular coordinate system. As shown in Figure 1.6 , the applied force F on
the arch is resolved into the two rectangular components:
H (cid:2) Fcosθ
V (cid:2) Fsinθ

F
V
θ
H
Figure 1.6
The moment acting at a given point in a structure is given by the product
of the applied force and the perpendicular distance of the line of action of the
force from the given point. As shown in F igure 1.7 , the force F at the free end
of the cantilever produces a bending moment, which increases linearly along
the length of the cantilever, reaching a maximum value at the fixed end of:
M (cid:2) Fl
The force system shown at (i) may also be replaced by either of the force
systems shown at (ii) and (iii). The support reactions are omitted from the fig-
ures for clarity.
l
F F F F
(cid:2) (cid:2)
Forces
1 2 1 2 1 2
F M(cid:2) Fl
Moment
Fl Fl Fl
(i) (ii) (iii)
Figure 1.7
Principles of statics 7
1.3 Conditions of equilibrium
In order to apply the principles of statics to a structural system, the structure
must be at rest. This is achieved when the sum of the applied loads and sup-
port reactions is zero and there is no resultant couple at any point in the struc-
ture. For this situation, all component parts of the structural system are also in
equilibrium.
A structure is in equilibrium with a system of applied loads when the result-
ant force in any direction and the resultant moment about any point are zero.
For a system of coplanar forces this may be expressed by the three equations
of static equilibrium:
∑H (cid:2)0
∑V (cid:2)0
∑M (cid:2)0

where H and V are the resolved components in the horizontal and vertical
directions of a force andM is the moment of a force about any point.
1.4 Sign convention
For a planar, two-dimensional structure subjected to forces acting in the xy
plane, the sign convention adopted is shown in Figure 1.8 . Using the right-
hand system as indicated, horizontal forces acting to the right are positive and
vertical forces acting upward are positive. The z-axis points out of the plane of
the paper, and the positive direction of a couple is that of a right-hand screw
progressing in the direction of the z-axis. Hence, counterclockwise moments
are positive.
y
x
z
Figure 1.8
8 Structural Analysis: In Theory and Practice
Example 1.1
Determine the support reactions of the pin-jointed truss shown in Figure 1.9 .
End 1 of the truss has a hinged support, and end 2 has a roller support.
V (cid:2)20 kips V (cid:2)20 kips
3 4
H (cid:2) 10 kips
3
3 4
8 ft
H (cid:2) 10 kips
1
1 5 2
8 ft 8 ft
V1(cid:2) 15 kips V2(cid:2) 25 kips
(i) Applied loads (ii) Support reactions
Figure 1.9
Solution
To ensure equilibrium, support 1 provides a horizontal and a vertical reaction,
and support 2 provides a vertical reaction. Adopting the convention that hori-
zontal forces acting to the right are positive, vertical forces acting upward are
positive, and counterclockwise moments are positive, applying the equilibrium
equations gives, resolving horizontally:
H (cid:3)H (cid:2)0
1 3
H (cid:2)(cid:4)H
1 3
(cid:2)(cid:4)10kips … acting to the left

T aking moments about support 1 and assuming that V is upward:
2
16V (cid:4)8H (cid:4)4V (cid:4)12V (cid:2)0
2 3 3 4
V (cid:2)(8(cid:5)10(cid:3)4(cid:5)20(cid:3)12(cid:5)20)/16
2
(cid:2) 25kips … acting upward as assumed

R esolving vertically:
V (cid:3)V (cid:3)V (cid:3)V (cid:2)0
1 2 3 4
V (cid:2)(cid:4)25(cid:3)20(cid:3)20
1
(cid:2)15kips … acting upward

T he support reactions are shown at (ii).