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ELEMENTARY THEORY OF STRUCTURES
McGraw-Hill Harmer
Babbitt
Civil Engineering Series
E. Davis, Consulting Editor
Engineering in Public Health
•
Babbitt and Dolaxd
Benjamin
•
Water Supply Engineering
•
Statically Indeterminate Structures
Davis, Troxell, and Wiskocil
The Testing and Inspection
•
of
Engineering Materials
Dunham Foundations of Structures Dunham The Theory and Practice of Reinforced Concrete Dunham and Young Contracts, Specifications, and Law for •
•
•
Engineers
Gaylord and Gaylord Structural Design Hennes and Ekse Fundamentals of Transportation Engineering Henry Design and Construction of Engineering Foundations Krynine and Judd Principles of Engineering Geology and Geotechnics LiNSLEY and Franzini Elements of Hydraulic Engineering LiNSLEY, KoHLER, AND Paulhus Applied Hydrology •
•
•
•
'
*
LiNSLEY, KoHLER, AND Paulhus
Matson, Smith, and Hurd Mead, Mead, and Akerman •
*
Hydrology
for Engineers
Traffic Engineering •
Contracts, Specifications, and
Engineering Relations
NoRRis, Hansen, Holley, Biggs, Namyet, and Minami
•
Structural
Design for Dynamic Loads
Peurifoy
•
Construction Planning, Equipment, and Methods
Peurifoy
•
Estimating Construction Costs
Troxell and Davis Tschebotarioff
•
•
Composition and Properties
of
Concrete
Soil Mechanics, Foundations, and Earth Structures
Urquhart, O'Rourke, and Winter Design of Concrete Structures Wang and Eckel Elementary Theory of Structures •
•
Elementary Theory of Structures
CHU-KIA WANG,
Ph.D.
Professor of Architectural Engineering University of Illinois
CLARENCE LEWIS ECKEL,
C.E,
Professor of Civil Engineering
Dean
of the College of Engineering
University of Colorado
McGRAW-HILL BOOK COMPANY, New York
Toronto 1957
London
INC.
ELEMENTARY THEORY OF STRUCTURES
©
Copyright 1957 by the McGraw-Hill Book Company, Inc. Printed in the United States of America. All rights reserved. This book, or parts thereof, may not be reproduced in any form without permission of the publishers. Library of Congress Catalog Card
Number 56-11058
III
THE MAPLE PRESS COMPANY, YORK,
PA.
PREFACE designed to present the essential principles of structural analysis in a first course for architectural and civil engineering students. The analysis of statically determinate structures is based on the laws of
This text
statics,
is
while that of statically indeterminate structures depends on both
the principles of statics and the geometric conditions of the deformed structure.
These principles are relatively simple; nevertheless experi-
ence shows that, in order to acquire proficiency and
facility,
a student
must expect to work a considerable number of problems involving the appropriate conditions of statics and geometry. Incidental to the presentation of basic principles in this text, special
emphasis has been given to illustrative examples. It is hoped that this feature will relieve the teacher of undue blackboard routine and thereby permit time for lively and fruitful class discussion. 1 through Chapters 11 through 15 are devoted to an introduction to the analyWith the exception of Chaps. sis of statically indeterminate structures. 6, 7, and 10, which deal with applications to the analysis of structures such as roof trusses, building bents, and bridge trusses, the basic concern
Statically determinate structures are discussed in Chaps.
10.
of this text is the
use of general principles and methods of structural
analysis.
In schools where " unified '^ courses in structural analysis and design are offered, this text may be used as a principal source book for the "analysis" portion of
'^ design" assignments. Teachers who prefer to give a ''unified" treatment of statically determinate and statically indeterminate structures will find that Chaps. 3 and 11, Chaps. 4 and 12, and Chaps. 5 and 13 may be conveniently used
in pairs.
Chapters 8 and
moving
loads,
9,
which deal with influence diagrams and
are general in nature.
The
criteria for
topics discussed in these
chapters are essential for an understanding of the structural analysis of bridge trusses or other structures carrying
moving
loads.
Although
students in architecture or architectural engineering are likely to be pri-
marily interested in building structures, they will find these chapters,
PREFACE
VI
and perhaps Chap.
10, of
value in adding to their over-all understanding
of the procedures of structural analysis.
and moment distribution are treated Again a choice is permitted in that the slope-deflection method, and then the moment-distribution method, may be studied; or both methods may be discussed in relation to a given problem at the same time. If the latter choice is made, Chaps. 14 and
The methods
of slope deflection
separately in Chaps. 14 and 15.
15
may
be used together.
Although great care has been taken script, the
in checking calculations
and manu-
authors will appreciate notices of errors and suggestions for
improvement
in future editions.
The authors wish
to
thank Mrs. C. K.
Wang
for her valuable assist-
ance, especially in typing the final manuscript of this text.
C. K.
Wang
C. L.
Eckel
CONTENTS Preface
Chapter 1-1.
v
General Introduction
1.
Theory
tures.
Chapter
1-3.
of Structures Defined.
Loads on Structures.
1
Layout and Classification Methods of Analysis.
1-2. 1-4.
of Struc-
Equilibrium of Coplanar-force Systems
2.
The Free Body.
4
Equilibrium of Coplanar-concurrent-force SysEquilibrium of Coplanar-parallel-force Systems. 2-4. Equilibrium of General Coplanar-force Systems. 2-5. Reactions on a Three-hinged Arch. 2-1.
tems.
Chapter
2-2.
2-3.
3.
Shears and Bending Moments in Beams
26
3-2. Relationship between 3-1. Definition of Shears and Bending Moments. Load, Shear, and Bending Moment. 3-3. Shear and Bending-moment Equa3-4. Shear and Bending-moment Diagrams. 3-5. Bending-moment tions. Diagram by the Graphic Method.
Chapter
4.
Analysis of Statically Determinate Rigid Frames and Composite Structures
Determinate Rigid Frames. Statically Determinate Composite Structures. 4-1. Analysis
Chapter 5-1.
5.
Statically
Chapter
6.
4-2.
5-4.
Method
of
59
Method
5-2.
Moments and
Shears.
of Joints. 5-5.
5-3.
Method
of
The Graphic Method.
Analysis of Roof Trusses
General Description.
Loads as Recommended
6-2.
in the
48
Analysis of
Stresses in Trusses
Stress Analysis of Trusses.
Sections.
6-1.
of
77
Dead, Snow, and Wind Loads. 1940
ASCE
Final Report.
6-4.
6-3.
Wind
Combina-
tions of Loads.
Chapter 7-1.
7.
Analysis of Building Bents
General Description.
Chapter
8.
7-2.
Methods
97 of Analysis.
Influence Diagrams
106
8-3. Influence Diagrams for 8-2. Definition. General Introduction. Reactions on a Beam. 8-4. Influence Diagram for Shear in a Beam. 8-5. 8-6. Influence Diagram Influence Diagram for Bending Moment in a Beam.
8-1.
vii
Vm
CONTENTS
as a Deflection Diagram.
8-7. Influence Diagrams for Simple Trusses. 8-8. Diagram between Panel Points of a Truss. 8-9. Influence Diagrams for Reactions on a Truss. 8-10. Influence Diagram for Shear in a Parallel-chord Truss. 8-11. Influence Diagram for Bending Moment at a Panel Point in the Loaded Chord of a Truss. 8-12. Influence Diagram for 8-13. Influence Pier or Floor-beam Reaction. Diagram for Bending Moment at a Panel Point in the Unloaded Chord of a Truss. 8-14. Influence Diagram for Stress in a Web Member of a Truss with Inclined Chords.
Influence
Chapter
Criteria for Maxima: Moving Loads
9.
138
General Introduction. 9-2. Maximum Reactions and Shears in Simple Beams: Uniform Loads. 9-3. Maximum Reactions and Shears in Simple Beams: Concentrated Loads. 9-4. ]\Iaximum Bending Moment at a Point in a Simple Beam: Uniform Load. 9-5. Maximum Bending Moment at a Point 9-6. Absolute Maximum Bending in a Simple Beam: Concentrated Loads. Moment in a Simple Beam: Concentrated Loads. 9-7. Maximum Reactions on Trusses. 9-8. Maximum Shear in a Panel of a Parallel-chord Truss. 9-9. Maximum Bending Moment at a Panel Point in the Loaded Chord of a Truss. 9-10. Maximum Bending Moment at a Panel Point in the Unloaded Chord 9-1.
of a Truss.
9-11.
Maximum
Stress in a
Web Member
of a
Truss with
Inclined Chords.
Chapter
10.
Analysis of Highway and Railw^ay Bridges
177
10-2. Dead Load. 10-3. Live Load on Highway General Description. 10-4. Live Load on Railway Bridges. 10-5. Impact. 10-6. Use Bridges. 10-7. Analysis of Bridge Portals. of Counters in Trusses.
10-1.
Chapter
11.
Analysis of Statically Indeterminate Beams
230
Indeterminate Beams. 11-2. 1 1-3. The MomentDeflections and Slopes in Statically Determinate Beams. 11-1. Statically
Determinate
area Method.
11-4.
Method.
Law
11-6.
vs.
Statically
The Conjugate-beam Method. of
Reciprocal
Deflections.
11-5.
The Unit-load
11-7. Statically
Inde-
11-8. Statically Indeterminate terminate Beams with One Redundant. Beams with Two Redundants. 11-9. Influence Diagrams for Statically Indeterminate Beams.
Chapter
12.
Analysis of Statically Indeterminate Rigid Frames
.
.
267
Indeterminate Rigid Frames. Determinate vs. 12-2. Deflections of Statically Determinate Rigid Frames: the Moment-area Method. 12-3. Deflections of Statically Determinate Rigid Frames: the Unit-load Method. 12-4. Analysis of Statically Indeterminate Rigid Frames by the Method of Consistent Deformation. 12-5. Influence Diagrams for Statically Indeterminate Rigid Frames. 12-1. Statically
Chapter
13.
Statically
Analysis of Statically Indeterminate Trusses
....
Determinate vs. Statically Indeterminate Trusses. 13-2. 13-3. Deflections of Statically Determinate Trusses: the Unit-load Method. 13-4. Deflections of Statically Determinate Trusses: the Graphical Method. Analysis of Statically Indeterminate Trusses by the Method of Consistent 13-5. Influence Diagrams for Statically Indeterminate Deformation. 13-1. Statically
Trusses.
295
Chapter
14.
CONTENTS
IX
The Slope-deflection Method
318
14-2. Derivation General Description of the Slope-deflection Method. 14-3. Application of the Slope-deflection of the Slope-deflection Equations. Method to the Analysis of Statically Indeterminate Beams. 14-4. Application of the Slope-deflection Method to the Analysis of Statically Indetermi14-5. Application nate Rigid Frames. Case 1. Without Joint Movements. of the Slope-deflection Method to the Analysis of Statically Indeterminate Rigid Frames. Case 2. With Joint Movements. 14-1.
Chapter
15.
The Moment-distribution Method
345
15-2. AppliGeneral Description of the Moment-distribution Method. Method to the Analysis of Statically Inde15-3. Check on Moment Distribution. 15-4. Stiffness terminate Beams. 15-5. Factor at the Near End of a Member When the Far End Is Hinged. Application of the Moment-distribution Method to the Analysis of Statically Indeterminate Rigid Frames. Case 1. Without Joint Movements. 15-6. Application of the Moment-distribution Method to the Analysis of Statically Indeterminate Rigid Frames. Case 2. With Joint Movements. 15-1.
cation of the Moment-distribution
Answers to Problems
377
Index
385
CHAPTER
1
GENERAL INTRODUCTION
1-1.
Theory
of Structures Defined.
Engineers design structures such
machine
parts, as well as various kinds of
as bridges, buildings, ships,
equipment and other structural installations. Incident to design, the engineer must first determine the layout of the structure, its shape, and Then he must estimate or otherwise determine its constituent members. the loads which the structure is to carry. The theory of structures deals with the principles and methods by which the direct stress, the shear and bending moment, and the deflection at any section of each constituent
member design
is
in the structure
may
to proportion the
working stresses
be calculated. The next phase of the in accordance with the allowable
members
of the material
and other requirements
for the proper
A typical roof truss Fig. 1-1
functioning of the structure. texts
on strength
cussed in this text.
may have
This work
is
generally within the scope of
of materials or structural design It
may be well
to be repeated a
and
will
not be dis-
to point out that the process of design
number
of times before a satisfactory final
design can be found. Consider, for example, the design of a typical roof truss such as
shown
in Fig. 1-1.
out of the truss
is
is
The
process of design involves four stages: (1) a layassumed; (2) the loading, which may consist of dead
load (weight of the roofing material and the truss load, or other loading,
is
estimated;
(3)
itself),
snow
load,
the direct stresses in the
wind
members
and (4) the sizes of the members are determined in accordance with the design specifications. This text will concern itself primarily with the third stage, but with occasional reference to the second of the truss are found;
1
ELEMENTARY THEORY OF STRUCTURES
J stage.
stages 1-2.
The eventual is
reconciliation
between the
first
and the fourth
largely a matter of experience.
Layout and Classification of Structures.
The
laj^out of
any
struc-
ture depends largely on the function of the structure, the loading conditions, and the properties of the material to be used. Except in routine situations, the determination of the layout of any structure requires knowledge, judgment, and experience. Usually after two or more lay-
outs for the same structure are carried through the initial design stage, a is made to determine the preferred design. Sometimes the
comparison
preliminary layout has to be modified to meet unanticipated conditions encountered in the later stages of design. Basicalh^ most structures may be classified as beams, rigid frames, or trusses or combinations of these elements. A beam is a structural member subjected to transverse loads onl3\ It is completeh^ analj^zed when the shear and bending-moment values have been found. A rigid frame is
a structure composed of
A
members connected by
rigid joints
(welded
frame is completelj^ analyzed when the variations in direct stress, shear, and bending moment along the lengths A truss is a structure in which all of all members have been found. members are usually assumed to be connected by frictionless hinges. A truss is completeh^ analyzed when the direct stresses in all members have been determined. There are also structural members or machine parts which may be subjected to the action of direct stress, shear and bending moment, and t"v\'isting moment. 1-3. Loads on Structures. Generally, the loads on structures consist of dead load, live load, and the dynamic or impact effects of the live load. joints, for instance).
rigid
Dead load includes the weight of the structure itself; live load is the loading to be carried by the structure; and impact is the dynamic effect Thus, in building design, the weight of the application of the live load. of the flooring,
beams, girders, and columns makes up the dead load;
while the weight of movable partitions, furniture, the wind load are considered as live load.
etc.,
the snow load, and
Often the live load comes on a
structure rather suddenly or as a mo^-ing or rolling load, as, for example,
when a train passes over a bridge. In this case the live load is increased by an estimated percentage to include its dynamic effect. This increase is
called the
impact load.
obvious that most of the dead load, except such items as the roofing on roof trusses, ceiling plaster under floors, and handrails on bridges, cannot be determined until the members have actually been designed; therefore, dead load has to be first assumed and then checked after the Except for unusual strucsizes of the members have been determined. It is
tures the dead-load stress normally constitutes only a relatively small
GENEIL\L INTRODUCTION
3
percentage of the total stress in a member; so that in routine designs a modification of the first design is seldom necessary. In its passage across the structure, the position which the live load assumes in order to cause a maximum direct stress, shear, or bending
member is of great importance and comprehensive treatment in this text. 1-4. Methods of Analysis. In Art. 1-1 it was stated that the theory of structures deals with the principles and methods by which the direct stress, shear, and bending moment at any section of the member may be found under given conditions of loading. Because the forces acting on a structural member may usually be assumed to lie in the same plane and are in equilibrium, fundamental structural analysis involves the use of the three equations of equilibrium for a general coplanar-force system; These three equations, together viz., 2Fx = 0, ZFy = 0, and Z3/ = 0. with a good working knowledge of simple arithmetic, algebra, geometry,
moment
at a particular section in a
will receive
trigonometry, and some calculus, are the necessary prerequisites for
studying the elementary theory of structures.
CHAPTER
2
EQUILIBRIUM OF COPLANAR-FORCE SYSTEMS
2-1. The Free Body. No matter how complicated a structure may be may be assumed it to be cut into various members, parts, or sections, each of which is under the action of a system of coplanar forces. Any one
member,
from the whole structure, is called drawn and complete with the magnitudes and directions (both known and unknown) of all the forces acting on it, is called a free-body diagram. The facility and ease with which the free-body diagrams are chosen and drawn are the key to the subject of part, or section, thus set free
a free body.
A
free body, clearly
structural analysis.
The
free body, being at rest within the structure,
librium under the action of
all
must be
the coplanar forces acting on
it.
in equiIf
the
magnitude, or the direction, or both, of some of these forces are unknown, they can be found by the principles of statics, which are the three equations of equilibrium 2Fx = 0, 2Fy = 0, and 'EM = 0. In this chapter, the methods of solving for these unknown magnitudes or directions will be explained. 2-2. Equilibrium of Coplanar-concurrent-force Systems. If the free body happens to be a point (a pin, for instance), the forces acting on it are concurrent. The resultant of a coplanar-concurrent-force system must be a single force, the x component of which is XF^ of the component forces and and the y component is XFy. Thus the two equations XFx =
EFy
=
are necessary
and
sufficient to ensure that the resultant is zero
or that the coplanar-concurrent-force system
is
in equilibrium.
These
two conditions for equilibrium permit the calculation of two unknowns, which may be the magnitudes of two forces with known directions, or the magnitude of one force with known direction and the direction of another force with known magnitude. It should be noted that the x and y directions are purely arbitrary; thus, in applying the equation XFx = 0, any direction may be considered Also, as long as the concurrent forces are in equilibrium as the X axis.
and have no resultant, the sum of the moments of the component forces about any point in the plane must be zero. Thus in cases where they may be more conveniently applied, the moment equations EM a = and 4
EQUILIBRIUM OF COPLANAR-FORCE SYSTEMS
O
may be substituted for either or both of the resolution equa23/^ = and i:Fy = 0. tions ^F, = In the graphic method of finding the resultant of a coplanar-concurrentforce system, a zigzag line is drawn connecting successively the component vectors taken in any convenient order; the resultant is then given by the vector extending from the starting point of the first component vector Should the resultant be zero, to the end point of the last component. the end point of the last component force must coincide with the starting Thus the graphic condition for the equipoint of the first component. librium of a coplanar-concurrent-force system is that the force polygon must close. For example, if the four coplanar, concurrent forces ah, he, cd, and de as sho^\^l in the space diagram of Fig. 2-la are in equilibrium, the points A and E in the force polygon ABODE of Fig. 2-16 must coincide. Note that the forces in Fig. 2-la may be designated in an irregular order, although they are normally named in alphabetical order around point
(a)
Space diagram
Fig. 2-1
in either the clock^^^se or counterclockwise direction.
It should also be noted that the position and direction, but not the magnitude, of the forces must be plotted accurately in the space diagram; while both magnitude and direction of the forces, and not the position, are represented in the
force polj'gon.
In the examples which follow, both the algebraic and graphic solutions are given.
Example 2-1. If the four coplanar, concurrent forces Fi, Fo, Fz, and 7^4 shown in Fig. 2-2a are in equilibrium, find the magnitude and direction of F4 which is arbitrarily assumed to act in the direction shown. ALGEBRAIC SOLUTION. From ZF^ = 0, 50 cos 15° {Fa).
=
+
100 cos 45°
-(50)(0.966)
-
Therefore (Fa), acts to the right as
assumed
-
80 sin 30°
left as
in Fig. 2-2a.
shown
= = -79.0
1b
and not
to the
-h (F,),
(100)(0.707) -h (80)(0.500) in Fig. 2-26
ELEMENTARY THEORY OF STRUCTURES
A -149^
lb
Ji-lOOIb
-F3-80Ib
From ZFy = (F4),
=
(F4)y acts
0.
50 sin 15° - 100 cos 45° - 80 cos 30° -h (FA)y = -(o0){0.2o9) + (100)(0.707) + (80)(0.866) = +127.0 lb
upward
assumed
as
and as shown
in Fig. 2-2a
in Fig. 2-26.
Referring to Fig. 2-26.
CHECK.
By
'9.0
