CHAPTER ONE
Introduction
1.1. CLASSIFICATIONS AND SCOPE
STRUCTURAL DESIGN
The design of a structure is an art in which the experience of past successful and unsuccessful construction, the laws of physics and mathematics, and
the results of research are utilized to provide structures which can function
efficiently and safely, which are economical to build and maintain, and which
are aesthetically pleasing. This definition of structural design is a grossly
abbreviated definition of an operation which, for a major project, may involve
the cooperation among, and the pooling of the knowledge of, hundreds of
experts from a variety of disciplines. One could not even attempt to place
only the major phases of structural design within the covers of one book,
and it would be impossible to find a person who would be an expert in all
the fields of knowledge involved. The purpose in this book is to deal with
1
2
INlRODUCTION
SEC.
1.1
only one of the many facets of the design process, namely, the analysis of
the strength of metal frames and their structural components.
One step in the design of structural frames is determining the geometrical
configuration of the members comprising the load-carrying skeleton of the
frame. This step would be taken after the overall geometry of the frame has
been defined and the load-combinations acting on it have been specified.
The design of the members is usually not a direct process. From a preliminary analysis, tempered by judgment and experience, one determines the
sizes of the members which could approximately fulfill the requirements of
safety and economy. Unless one is experienced or the structure simple, one is
not at first sure whether the requirements have been met. Thus we must
analyze the structure to ascertain whether it behaves satisfactorily. If it does
not seem to, and this will usually be the case in our first analysis, we make
adjustments and analyze again until we are satisfied. One phase of the design
process is, therefore, to perform individual analyses in which the response of
a given structure to a given set of loads is determined.
The analysis of a given structure for the specified loads acting on it is
performed in two steps: first, the force distribution in the structural frame is
determined by an analysis based on either the elastic or the plastic theory,
and second, the members and connections are checked to ascertain whether
they are able to support the forces acting on them without exceeding an
allowable stress or an allowable moment, for example, or without fracturing or
becoming unstable. This latter part of the analysis also involves an examination of the overall stability of the frame. The first part of the problem, the
analysis of forces, is covered in texts on indeterminate structural analysis
(see, for example, Refs. 1.1 through 1.5) and in texts of plastic analysis and
design.(1.6,1.7,1.8) For these problems the computer has become an invaluable
and increasingly necessary tool. (U, 1.10, 1.11) The major emphasis in this book
is on the second part of the problem: what are the limiting requirements to
which the structure and its components must conform to be safe? Structural
analysis will be discussed only as it applies to the second problem, and no
attempt will be made to formulate fully an efficient means of analysis. An
understanding of the limits of structural usefulness entails an understanding
of the behavior of the structure. The primary purpose of this book is to deal
with the behavior of structural elements and structural frames under load,
especially near the failure point of the structure. In many cases problems of
elastic and inelastic instability are involved. For metal structures such instability problems must always be considered, although they are sometimes
not the primary cause of failure. Metal structures may also fail by brittle
fracture or by elastic or plastic fatigue.(1.12) Fracture and fatigue will not be
considered here, even though they are very important. The principal limits of
usefulness discussed here are those due to instability. For a large number of
cases they become of an overriding importance.
This book will concern itself, therefore, with the behavior of metal frames
SEC.
1.1
INTRODUCTION
3
and structural components which are subjected to static nonrepetitive loads
and which ultimately fail because of some form of instability when loaded
into the inelastic range. Most civil engineering steel structures as well as many
aluminum structures fall into this category.
CLASSIFICATION OF LOADS
The determination of the loads acting on a structure is a separate and
important study. We shall not be concerned with this topic further except to
define the types of loads acting on the structures to be analyzed.
Loads may be either static or dynamic. The weight of the structure, called
the dead load, and certain specific fixed loads which do not change during
the life of the structure are the only true static loads. In usual practice, however, the live loads due to occupancy and, in many instances, the wind loads,
also, are treated as static loads. For static loads we can neglect the effects
of inertia arising from the acceleration of the mass of the structure and the
effects of rapid load changes on the material properties. For dynamic loading
these effects may not be neglected; they may well playa predominant role.
Dynamic loading arises from the acceleration caused by wind, earthquake,
blast, or impact. In the past such loads have usually been considered as quasistatic, and in the analysis of the structure no distinction was made between
their effects and those of true static loads. With the development of methods
of dynamic analysis and the use of the computer, it is now possible to make
an analysis for dynamic effects. Because this is a separate and important
field of study of structural behavior, it will not be dealt with in this
book. (\. 13, 1. 14, 1.15,1.16)
With the exception of the weight of the structure, which usually remains
constant during the life of the structure, the loads fluctuate. These load repetitions may lead to design considerations involving the fatigue cracking of
the material and to failure due to successively larger deflections after each
load repetition. We shall restrict ourselves here to nonrepetitive loads.
Thus the loads on the structures to be analyzed will be static and nonrepetitive. We shall further specify that some or all of the loads are related
by a constant factor of proportionality (proportional loading) and that the
loads will retain the same direction throughout the whole loading history.
The reason for these latter restrictions is that in the inelastic range the response of the structure is dependent on the sequence in which the various
loads acting on the structure are applied.
CLASSIFICATION OF STRUCTURES
Structures can be classified in many ways. (1.17,1.18,1.19) For our purposes
the subdivisions into shell and frame structures, as given in Ref. 1.19, is
adequate. In shell structures, the load-carrying element also serves the functional requirements of enclosing space. The structural frame, or skeleton,
4
INTRODUCTION
SEC.
1.2
usually serves only to support the loads transmitted from the functional
elements of the structure. We shall deal with frame structures only. Some
examples of such structures are simple and continuous beams, rigid frames,
trusses, and plate girders.
1.2. THE RESPONSE OF STRUCTURES TO LOADS
THE LOAD-DEFORMATION BEHAVIOR
The behavior of a frame under loads is best visualized from a curve which
relates the load to the deflection of any characteristic point on the structure.
For example, if the two-story rigid frame in Fig. 1.1 were sUbjected to the
vertical loads P and the horizontal loads aP (where a is a constant factor of
proportionality) and loading were started at P = 0, an experimenter would
obtain a curve like the one shown in this figure for the relationship between
P and the horizontal deflection v of the top of the structure.
The load-deflection relationship in Fig. 1.1 is typical of the response of
'00",,'0,
Load P
'"' 'OOT
D"",,'o, 00'''00
~---=,,-Maximum
lood
Working lood
I
I
1
I
I
~/
/
I
/
.
I1_______.
Unloading and reloading
Deflection v
Fig. 1.1. The load-deformation relationship of a frame
SEC.
1.2
INTRODUCTION
5
frame structures to static proportional loading. As P increases from zero, the
structure behaves elastically until the elastic limit is reached. For any load
below this limit the structure is elastic, that is, it will return to its original
undeformed position upon complete removal of the load.
Beyond the elastic limit some portions of the frame begin to yield. As a
result, the frame members become less stiff, and increasingly larger deflections result from equal increments ofload until finally a peak is reached on the
curve. This is the maximum load which can be supported. Under some conditions the load may drop very sharply after the peak of the curve is reached,
and in some instances this drop is very gradual, resulting in a flat plateau.
With further deformation the load must decrease if static equilibrium is to be
maintained. If the loads are removed anywhere in the inelastic region, then
the structure will not return along its path of loading and a permanent deflection results when P is zero (see dashed line in Fig. 1.1). Subsequent reloading
will follow approximately the unloading curve. It should be noted that even
in the elastic region the deflection is not necessarily a linear function of the
loading. This nonlinearity is introduced by the changes of the geometry ofthe
deformed structure.
LIMITS OF STRUCTURAL USEFULNESS
From a load-deflection curve we can make several observations about the
usefulness of the structure. The most obvious of these is the maximum load.
If the load is due to dead weight, the structure will collapse when this load is
reached. In design we must be certain that the working load (see Fig. 1.1),
which is to be supported under service conditions, is substantially less than the
maximum load. The ratio of the maximum load to the working load is called
the loadJactor, and it is usually prescribed in structural specifications. For the
type of structure shown in Fig. 1.1, for instance, the load factor prescribed by
the 1963 AISC specification is 1.40 if the horizontal loads are due to wind. (1.20)
Under certain conditions the use of the structure dictates deflection limitations under working loads. A load-deflection curve can also serve as a
check on this condition. In fact, the load-deflection curve is a record of the
history of the structure. If we have such a curve for our structure, we can
check for various criteria of structural usefulness. When one such limit is
reached, we have arrived at what we call the Jailure of the structure. Under
static nonrepetitive loads we can have three important criteria of failure: (1)
limiting deflection, (2) maximum load, and (3) the start of unstable behavior.
Of these, the first criterion is often dictated by rather hard-to-define factors
(such as plaster cracking), but the other two are real and definite limits to
usefulness. Thus failure will generally mean that either the maximum load
has been reached or that the load-deformation path has arrived at a point
at which instability sets in.
6
INTRODUCTION
SEC.
1.3
1.3. INSTABILITY
INELASTIC INSTABILITY
The load-deflection curve in Fig. 1.1 represents the locus of points for
which the structure is in equilibrium with the applied loads. This equilibrium
may be either stable or unstable. The state of the equilibrium is of vital importance because we cannot tolerate excursions into the unstable range; we are
particularly interested in the point at which it goes from the stable into the
unstable condition, as this represents a real limit to the usefulness of the
structure.
A structure is stable if it tends to return toward its original position after
a small disturbance is applied to it and then removed. On the other hand, a
structure is unstable if a small disturbance produces a further increase of
deflection. In the first instance an addition of energy is required to produce the
disturbance, and in the second instance energy is released.(1.21,1.22,1.23)
In the mathematical treatment of stability problems the disturbance is
usually virtual, that is, it does not change the existing force system.(1.21) In
an actual structure these disturbances are of course real, and their effect is
reflected not only on the structure but also on the loading system. Thus we
must consider the response of both the structure and the loading device for a
test of stability.". 23, 1.24)
Let us first consider the stability of a structure subjected to dead, or gravity,
loads (Fig. 1.2). The addition of weight to the structure causes an increase
in potential energy, and the load-deformation characteristics of the load system can be represented by a series of straight lines parallel to the deflection
axis, as shown by the dashed lines in Fig. 1.2. Each line corresponds to a
different weight or energy level defined by the intercept with the load
axis. (I. 24)
The intersections between the load characteristics and the structure loaddeflection curve correspond to equilibrium points. For example, the points
A and B on the load characteristic CD in Fig. 1.2 are equilibrium situations.
In order to check for stability we disturb the structure a small amount, displacing A to A' and B to B'. For point A this disturbance requires an increase
of energy, that is, the load characteristic tends toward a higher energy level
A ". The unbalanced force, representing the difference between the two characteristics, is directed toward the point A. An increase of energy is required to
make this disturbance, and thus point A is stable. For point B the disturbance
tends toward a lower energy level B", and the unbalanced force is directed
away from B. The energy is released, and B is therefore unstable.
Similar tests on all points on the ascending portion of the curve will show
that these are stable; on the other hand, the descending portion is unstable.
SEC.
1.3
7
INTRODUCTION
Limit of stability
Load
Stable region
---t----..
Unstable region
Stable
C-
-- -
---""')~=-'
-~------------
-----
------
-------
Load characteristics
Structure characteristic
Deformation
Fig. 1.2. State of the equilibrium under dead loading
The boundary between the two states of equilibrium is at the peak of the
load-deflection curve. This point, being neither stable nor unstable, is in neutral
equilibrium. It represents the point at which the structure will collapse under
dead loads.
Not all load characteristics are like those shown for dead loads in Fig.
1.2. Another type of loading, commonly encountered in screw-type testing
machines and in loads transmitted from adjacent elastic structures and
representing the elastic response of the load system, is shown as a series of
parallel lines in Fig. 1.3. (1.24,1.25) Applying the same test for stability as for
the loading in Fig. 1.2, we find that points A, E, C, F, and G are stable and
that point E is unstable. Neutral equilibrium exists at point D, where the
gradient of the load characteristic is equal to the gradient of the structure
curve. It should be noted that point E, which is at the peak of the curve,
and point C, which is already beyond it, are both stable.
Because disturbances are naturally present in any test, the structure curve
will not follow its path from the start of instability at D through E and F to
G, where it is again stable, unless it is externally restrained to do so, but it
8
SEC.
INTRODUCTION
1.3
Load
Testing machine characteristics
Stable
Unstable
Stable
Deformation
Fig. 1.3. Equilibrium of frames in a testing machine
will rapidly pass from D to G. This phenomenon is called the dynamic jump,
and it usually involves large, if not catastrophic, changes in geometry and
can in most cases not be tolerated for satisfactory structure performance. 1
According to our previous discussion then, a structure is stable if the
gradient of the load-deflection curve of the structure gs is larger than the
gradient of the load characteristic gL, or
gs >gL
stable equilibrium
gs
neutral equilibrium
=
gL
unstable equilibrium
g" < gL
In metal frame structures of the type discussed here, the point of neutral
1 Stable points beyond the peak of the load deflection curve have been frequently
encountered in tests on structures (for example, see Ref. 1.26). The phenomenon of the
dynamic jump is well known for shells and rigid bar systems, (1. 23) where it is called oil
canning, or snap-through, and it has been observed for frame structures during tests. (1.27)
A theoretical treatment of the dynamic jump and a review of the literature is contained
in Ref. 1.28.
SEC.
1.3
9
INTRODUCTION
equilibrium will occur at the peak of the load-deflection curve or on the descending portion of it. Since these parts of the curve are already in the inelastic
region, we shall call this form of instability inelastic instability. In other than
laboratory tests we do not know the load characteristic very precisely. For
our purposes we shall conservatively use the peak of the curve (gs = 0) as
the point of neutral equilibrium, and so this failure load of the structure will
also be its maximum load. 2
BIFURCATION OF THE EQUILIBRIUM
Inelastic instability may often be preceded by a phenomenon called variously buckling or bifurcation of the equilibrium (Fig. 1.4). As the load is
increased from zero, the structure begins to deform in a pattern characteristic
of the type of structure and loading. Basing our supposition on this initial
deformation pattern, we can expect the structure to deform according to curve
OAB. However, under some conditions it is possible that at a certain critical
load the deformation configuration suddenly changes into a different pattern.
The equilibrium is said to bifurcate (there is a fork at point A in Fig. 1.4), or
the structure buckles, into branch AC.
The actual load-deformation curve then consists of two stable branches:
curve OA, the prebuckling branch, and curve AC, the postbuckling branch.
Branch AB is unstable and will not
be followed by the structure. Lood
Buckling does not mean that the
Unstoble branch
structure has necessarily failed; the
post buckling branch is still stable
"'
by our previous definition and
.... ....
Critical--.... /. /'
'\
failure will be due to inelastic inload
rtl
C
\
stability, as discussed before. HowStable branch
B
(postbuckling curve)
ever, in many cases, and for reasons
which will become evident in later
Point of bifurcation
chapters, the initiation of buckling
(that is, the point of bifurcation) is
considered the limit of usefulness of
the structure. Buckling can occur
anywhere along the original curve
oAB: in the elastic region as well as
beyond the peak. After its advent the
course of the original curve usually
o
Deformation
is changed downward, resulting in
Fig. 1.4. Bifurcation of the equilibrium
a weakening effect.
----\
"
2 This applies if we consider the whole structure. We shall utilize the descending stable
portions of the curve for individual members in later studies.
10
INTRODUCTION
SEC.
1.4
Examples of buckling are column buckling, lateral-torsional buckling
of beams and beam-columns, local buckling, and frame instability under
symmetric axial loads. Because of the importance of buckling we shall discuss
it in considerable detail in later portions of this book. 3
1.4. FRAMES AND THEIR COMPONENTS
EQUILIBRIUM AND COMPATIBILITY
The preceding comments concerning the behavior of complete structures
apply equally well to the components of the frame. In fact, when we set about
to analyze a frame, we start with the analysis of components, building up the
complete structure by maintaining equilibrium at the juncture of the components and observing the proper boundary conditions. In rigidly jointed
structures we assume that the deformations at the joint are compatible with
each other. We can thus synthesize the behavior of the complete frame from
the knowledge of the behavior of its components.
FRAME COMPONENTS
We shall subdivide frames into the following components: members and
connections. Members are elements which are much longer than they are deep,
and connections are the devices used to connect two or more members so
that the forces occurring at the ends of the members are transmitted and that,
in rigidly jointed structures, continuity is maintained.
TYPES OF MEMBERS
We shall employ two kinds of classifications for members. One of these
will be prescribed by the kind of loading, and the other will depend on the
nature of the cross section.
We can classify members as beams, if bending predominates and the
effect of axial load can be ignored; as beam-columns, if significant amounts of
both bending and axial force are present; and as columns, if compressive
axial forces predominate to the extent that bending can be neglected.
Members are also classified according to the geometry of the cross section
into solid (or thick-walled) members and into thin-walled members. For the
3 Our discussion on instability and buckling was kept purposely clear of mathematical
formulations to illustrate the phenomena rather than their mathematical properties. The
student should consult the following references for a sampling of the more formal treatments of stability: Chap. 3 in Ref. 1.21, Chaps. I and 6 in Ref. 1.22, Chap. 6 in Ref. 1.23,
Chap. 2 in Ref. 1.29, Chap. 2 in Ref. 1.30, and Refs. 1.31, and 1.32. We shall discuss
some of the mathematical properties of buckling problems in Chapter 3.
SEC.
1.4
INTRODUCTION
11
former all the dimensions of the cross section are of the same order ofmagnitude, whereas for the latter the thickness of the plate elements from which the
cross section is made up is of a smaller order of magnitude than the depth
and width of the member. Examples of solid members are reinforced concrete
beams or heavy steel columns. Most shapes used in metal construction, on
the other hand, are thin-walled. Because of their practical importance and
the predominant role of instability we shall concentrate on thin-walled members in this book.
MATERIAL AND
CROSS-SECTIONAL PROPERTIES
The behavior of members depends on their geometry, on the loads, and on
the properties of the cross section. The cross-sectional properties in turn
depend on their geometry, on the material properties, and on the residual
strains present as a result of the fabrication process.
The material properties which will be of importance are obtained from a
tension or compression test on a small coupon of the material from which the
member is made. Such a test furnishes stress-strain curves, two of which are
shown in Fig. 1.5. Only the initial portions of the curves are shown here, as
we are not going to utilize the full curves up to rupture. The curve in Fig.
1.5(a) is typical of structural carbon steels; the curve in Fig. 1.5(b) is characteristic of structural aluminum.
For both types of curves the stress is proportional to the strain up to the
yield stress U y or the proportional limit Up. In this elastic range U = Ee,
where E is the modulus of elasticity, U is the average stress PIA, and € is the
corresponding strain. In structural carbon steel the curve exhibits a pronounced plastic plateau, where strain increases but the stress remains equal
to Uy. Strain hardening commences at a strain €ST; the slope of the curve upon
further straining is EST, the strain-hardening modulus.
The solid curve in Fig. 1.5(a) represents a static condition. In reality it is
not possible to test a coupon in this manner. If the coupon is strained in a
testing machine at a slow and constant strain rate, then we obtain the dashed
curve. This curve generally exhibits an upper yield point Uru and a lower yield
point Un. The plastic range is a somewhat wavy line, and the static yield
stress (or static yield level) is obtained by stopping the straining (see dip in
the dashed curve). The dynamic yield stress Urn is always higher than u r . (1.33)
Since we have adopted static loading as the basis for our work here, we shall
use the static yield level as the termination of elastic behavior.
If no pronounced yield level exists [Fig. 1.5(b)], then a yield strength UYO
is defined by the per cent offset method (see dashed line). For such a material
we shall use this value as our limit of elastic behavior, calling it for convenience also u r .
12
SEC.
INTRODUCTION
1.4
Stress (]'
~
~_~/~("~.(: ~:_~:YU:_=(]',: =~'5l.: _:~ _=_=~: YD=_:
:
~_:.: _:-:-::-::-:'-:--::-1-;-~---------
__=_.:__
\/
!
~T
I
Unloading
path
-/
IE
/
'E
I
Strain E:
(al
Stress (]'
(]'YOI---~/~
I
I
I
I
I
I
E
I
I
I
I
!.-+
Strain E:
0.002 in.! in.
(bl
Fig. 1.5. Stress-strain curves for steel and aluminum
If the loads are removed from the test coupon any time after yielding has
commenced, the stress-strain curve will not follow back along its original path
but will unload elastically. Upon the complete removal of the load the specimen will have a permanent set [see Fig. 1.5(a)).
Knowledge of the geometry of the cross section and the stress-strain curve
is not sufficient to fully define the response of the cross section to forces
acting on it. Residual stresses exist in the member as a result of the manufacturing process, and these must be added to the stresses caused by the loads
to obtain the complete stress distribution.
SEC.
1.4
INTRODUCTION
13
Residual stresses are caused by a variety of factors, and in some cases they
may be as large as the yield stress. (1.34,1.35,1.36) They are a result of plastic
deformation during the manufacturing process. For example, uneven cooling
of steel shapes after hot rolling or welding, and cold straightening by gagging
or rotarizing, all result in residual stresses. For hot-rolled wide-flange shapes
the flange tips cool faster than the metal at the flange-web juncture. As the
slower-cooling portions finally cool and contract, they induce compressive
residual stresses on the parts which are already cold. Thus the flange tips have
compressive residual stresses, whereas the flange centers are in tension.
Residual stresses can be greatly reduced by stress-relieving the member,
but this is usually done only in exceptional cases because of the great expense
involved. We must therefore assume that steel members will have substantial
residual stresses. The distribution and magnitude of residual stresses is a
function of a variety of factors. In Refs. 1.34 through 1.36 there are shown a
great number of diagrams with the measured residual stresses, caused by hot
rolling, welding, and flame cutting, on steel cross sections of various shapes
and sizes. The residual stresses due to cold straightening are usually localized
and can in most cases be neglected. (1.37)
The diagram in Fig. 1.6 shows an idealization of the residual stress distribution in a hot-rolled wide-flange shape. This pattern has been used in a
number of determinations of the inelastic strength of beam-columns, (1.37, I. 38)
bu,.c
d
w
b
Fig. 1.6. Residual stress distribution in a steel wide-flange cross section
14
INTRODUCTION
SEC.
1.4
and we shall adopt it throughout this book to show the effect of residual
stresses in various inelastic studies. The pattern closely approximates the
residual stresses measured on ASTM-A7 steel 8WF31 sections. (I. 35. 1.36) In
Fig. 1.6 the web has tensile stresses Urt which are constant, and the stresses in
the flanges are assumed to vary linearly from Urt at the flange center to a
compressive stress U re at the four flange tips. The stresses are also assumed
to be constant across the thickness of the plates.
Since no external forces exist, equilibrium requires that the sum of the
stresses over the whole cross section must be zero. Thus, from Fig. 1.6,
LUr dA = 0 = urt(d -
2t)w
+ 4t (ur) (~
-
2ureb~e2uJ
-
4t(Ure ) (
T
bU re
)
2u re + 2u rt
where U r is the residual stress, dA is an area element, and integration is performed over the whole area A. After some algebraic manipulation we find
that
(1.1)
We shall now show how residual stresses can affect the load-deformation
behavior of the cross section. In Chapter 4 we shaH find that the relationship
between a compressive axial load and the resulting axial strain is of importance for the determination of the critical load of columns. In the short wideflange column of Fig. 1.7 the axial stress U is shown superimposed on the
residual stresses. Since U is a compressive stress, the maximum combined
stress in the member will be at the flange tips, where the compressive residual
stress is largest. When U = Uy - Ure, yielding will start at the four flange
tips; the plastic zones will penetrate toward the center of the flanges until the
cross section is completely plastified.
The stress condition in the flanges of the wide-flange member of Fig. 1.7
is shown more clearly in Fig. 1.8. Equilibrium over the whole cross section
requires that
p=
LUdA = UA -
4t ( ; )
(~
(1.2)
- db)
where d is a coefficient defining the extent of yield penetration in the flanges.
From Fig. 1.8 we see that ii = U + Ure -'Uy ; it can be also proved from the
geometry of similar triangles that
U = (1 - 2d)(u re
+ Urt) + Uy -
Ure
(1.3)
With ii and U substituted into Eq. (1.2), we obtain the foHowing formula for
the average stress U A:
UA =
i
= (1 - 2d)(urc
+ Urt) + Uy -
Urc -
~
(1 - 2d)2(U rc
+ Urt)
(1.4)
SEC.
1.4
15
INTRODUCTION
Fig. 1.7. Stresses in a short column under axial load
" l;Jrr
O'r-~c
"
iXb
....I
(]'
f~V/7f
Yielded portion of flonge
I.
b/2
Fig. 1.8. Applied and residual stresses in the flange of the wide-flange
column
16
INTRODUCTION
SEC.
1.4
The axial shortening of the member is determined by the strains in the
elastic core of the column, or
(1.5)
If we define a yield strain at the hypothetical start of yielding in the absence of
residual stresses as
ay
(1.6)
€y=/J
and substitute a from Eq. (l.3) into Eq. (1.5), we find that
IX
=
art
+ ay 2(a re
(e/ey)ay
(1.7)
+ art)
If we now substitute IX into Eq. (1.4), we obtain the relationship
aA. _.!.- _ [
btal'
] (~_ I
ey
A (arc
art) ey
a1' -
+
+ a re )2
(1.8)
ay
Equation (1.8) represents the relationship between the average stress
PIA and the resulting axial strain e. Since the member is very short, this is
also a property of each cross section. It is valid in the range between initiation
of yielding and full plastification, that is, I - arc/a}" =< e/ey =< I
arday.
In the elastic range aA/ay = e/e!" and after full yielding a A/ar = 1.0, regardless of the value of e, provided we neglect the effect of strain hardening
[Fig. 1.5(a)].
The curve showing the relations discussed above for an 8WF31 rolled
shape with arc = 0.3 ar is given in Fig. 1.9.' The dashed lines give the situation in which no residual stresses are present. The effect of the residual stresses
is to bring about a rounding of the curve due to earlier yielding which results
in a reduced stiffness; the residual stresses do not, however, inhibit the column
from reaching its fully yielded condition when aA = ar.
The preceding discussion pertained to a very simple case of loading. Similar
relationships for beams and beam-columns will be discussed in later chapters.
For these members the significant relationship is between the moment acting
at a cross section and the resulting curvature. The conclusions reached from
this simple example, however, remain valid: Residual stresses will cause
yielding earlier than is expected if they are neglected, and they cause a reduction in the stiffness of the member. The stiffness of the cross section is here
defined as the slope of the relevant force-deformation relationship. Now from
Fig. 1.9 one could superficially conclude that the effect of the residual stresses
is not very great. However, we should realize that the behavior of the whole
member is obtained by the integration of the characteristics of each cross
section along the whole length of the member, and we shall see that the inelastic
+
, The dimensions of an 8WF31 shape are given in Ref. 1.39.
SEC.
1.5
17
INTRODUCTION
E=1.I89Ey
Partial yielding af
1-_ _ _E_a_st_ic_r_a_ng"-e_ _ _f+----.:..:f=a:.:.'ng~e:::.s-_.l_F~I~~~!!.c~t~ _
eTA
WU
811F31
o;.c = 0.3eTy
0.5
O
ttttt
OA=PIA
0.5
1.0
Fig. 1.9. The average stress-strain curve of a wide-flange cross section
behavior of these members cannot be properly predicted unless residual
stresses are accounted for. We shall also see that buckling is sensitive to
member stiffness and that therefore residual stresses playa significant role
in a buckling analysis.
1.5. ORGANIZATION OF THE BOOK
The material in this book deals with that step in the structural design
process when the assumed structure has been analyzed and its internal force
distribution is known and it is desired to determine whether the forces will
exceed the limits of structural usefulness. These limits of usefulness will be
those of maximum capacity or instability, and they will be defined in terms
of the load-deformation relationship of the individual members or of the
whole frame. The emphasis will be on structural behavior.
This book is organized in the following manner: Chapter 2 will introduce
certain topics on the elastic behavior of prismatic members of a thin-walled
open cross section. In this chapter we shall study the problem of torsion, and
we shall set up the differential equations of combined bending and torsion.
These elastic equations will serve as the starting point for each of the following three chapters: Beams (Chapter 3), Columns (Chapter 4), and BeamColumns (Chapter 5). In each of these chapters, the general behavior of the
18
INTRODUCTION
SEC.
1.5
member is described first. This is followed by a study of elastic buckling, and
the differential equations developed at the end of Chapter 2 will serve as the
starting point. The elastic studies will be followed by a consideration of
inelastic behavior and inelastic instability. The final portions of each of these
chapters will deal with structural specifications and their relationships to
the actual limits of structural usefulness. The final chapter (Chapter 6) will
be concerned with the overall behavior of frames. The assumptions underlying the various structural analyses will be examined, and elastic as well as
inelastic frame behavior will be discussed. Finally the provisions of the
relevant structural specifications will be presented and examined in the light
of what we have learned about the behavior offrames.
Before entering into these studies, it will be well to consider briefly the
sources of our knowledge on structural behavior. These sources are: (1)
successful construction experience, (2) unsuccessful experience (that is,
failures during fabrication, erection, or in service), and (3) structural research. The benefits of the first two sources have been available to our builders
from the beginning of history, and they form an important basis for the
design of all future structures. Research has been a relatively recent source
of knowledge, starting with Galileo's famous experiments on beams, (1.40)
and assuming a really vital role since the second half of the 19th century.
Research now no longer follows new developments in construction, as was
the case in the latter part of the 19th century (witness, for example, Tetmaier's
experiments on columns, (1.40) which were motivated by the failure of the
compression chords of railroad trusses), but it precedes construction. A
classic example of this is the acceptance of plastic design by the steel construction industry in the 1950's. This method was accepted, in terms of specifications and buildings designed, only after research showed it to be both as
safe as, and more economical than, the previously used elastic design
methods. (1. 41, 1. (2)
In the 1960's it is becoming evident .that one of the important design
considerations is the maximum strength of the structure. Structural research
in the 1945-1965 period has led to a thorough understanding of the behavior
of structures under known loads. This knowledge must yet be integrated with
parallel efforts which consider the probabilistic aspects of the design problem
to provide a truly realistic means of designing structures.
REFERENCES
1.1. C. H. Norris and J. B. Wilbur, Elementary Structural Analysis (New York:
McGraw-Hill Book Company, 1960).
1.2. A. S. Hall and R. W. Woodhead, Frame Analysis (New York: John Wiley &
Sons, Inc., 1961).
CHAP.
1
INTRODUCTION
19
1.3. S. O. Asplund, Structural Mechanics: Classical and Matrix Methods (Englewood Cliffs, N.J.: Prentice-Hall, Inc., 1966).
1.4. J. M. Gere and W. Weaver, Jr., Analysis of Framed Structures (Princeton,
N.J.: D. Van Nostrand Company, Inc., 1965).
1.5. R. K. Livesley, Matrix Methods of Structural Analysis (New York: Pergamon
Press, 1964).
1.6. L. S. Beedle, Plastic Design of Steel Frames (New York: John Wiley & Sons,
Inc., 1958).
1.7. B. G. Neal, The Plastic Methods of Structural Analysis (New York: John
Wiley & Sons, Inc., 1956).
1.8. P. G. Hodge, Jr., Plastic Analysis of Structures (New York: McGraw-Hill
Book Company, 1959).
1.9. P. C. Wang, Numerical and Matrix Methods in Structural Mechanics (New
York: John Wiley & Sons, Inc., 1966).
1.10. M. F. Rubinstein, Matrix Computer Analysis of Structures (Englewood Cliffs,
N.J.: Prentice-Hall, Inc., 1966).
1.11. C. K. Wang, Matrix Methods of Structural Analysis (Scranton, Pa.: International Textbook Company, 1966).
1.12. H. S. Reemsnyder, "Fatigue" in Structural Steel Design, Lambert Tall, ed.
(New York: The Ronald Press Company, 1964).
1.13. G. L. Rogers, Dynamics of Framed Structures (New York: John Wiley &
Sons, Inc., 1959).
1.14. C. H. Norris et aI., Structural Designfor Dynamic Loads (New York: McGrawHill Book Company, 1959).
1.15. G. B. Warburton, The Dynamical Behavior of Structures (New York:
Pergamon Press, 1964).
1.16. W. C. Hurty and M. F. Rubinstein, Dynamics of Structures (Englewood
Cliffs, N.J.: Prentice-Hall, Inc., 1964).
1.17. L. S. Beedle, "Introduction " in Structural Steel Design, Lambert Tall, ed.
(New York: The Ronald Press Company, 1964).
1.18. V. Z. Vlasov, Thin Walled Elastic Beams, Y. Schechtman, trans. (Moscow,
1959; Israel Program for Scientific Translation, Jerusalem, 1961), p. 3.
1.19. B. Bresler and T. Y. Lin, Design of Steel Structures (New York: John Wiley
& Sons, Inc., 1960), p. 2.
1.20. American Institute of Steel Construction, Specification for the Design, Fabrication and Erection of Structural Steel for Buildings (adopted April 17, 1963).
1.21. N. J. Hoff, The Analysis of Structures (New York: John Wiley & Sons, Inc.,
1956), Sec. 3.1.
1.22. H. L. Langhaar, Energy Methods in Applied Mechanics (New York: John
Wiley & Sons, Inc., 1962), Sec. 1.11.
20
INTRODUCTION
CHAP.
1
1.23. K. Marguerre, "Knick- und Beulvorgange" in Neuere Festigkeitsprobleme
des Ingenieurs, K. Marguerre, ed. (Berlin: Springer Verlag, 1950).
1.24. M. G. Lay, "The Static Load-Deformation Behavior of Planar Steel Structures" (Ph.D. dissertation, Lehigh University, 1964).
1.25. J. M. T. Thompson, "Stability of Elastic Structures and Their Loading
Devices," Journal of Mechanical Engineering Science, p. 153, June 1961.
1.26. M. G. Lay and T. V. Galambos, "The Experimental Behavior of Beam and
Column Subassemblages," Welding Research Council Bulletin No. 110,
November 1965.
1.27. A. J. Francis and L. K. Stevens, "Struts in Triangulated Frames," Civil
Engineering Transactions, Institution of Engineers, Australia, March 1961.
1.28. G. Davies and B. G. Neal, "The Dynamic Behavior of a Strut in a Truss
Framework," Proc. of the Royal Society, Series A, Vol. 253, p. 542, 1959.
1.29. S. P. Timoshenko and J. M. Gere, Theory of Elastic Stability (New York:
McGraw-Hill Book Company, 1961).
1.30. F. Bleich, The Buckling Strength of Metal Structures (New York: McGrawHill Book Company, 1952).
1.31. S. J. Britvec and A. H. Chilver, "Elastic.Buckling of Rigidly Jointed Braced
Frames," Proceedings of the ASCE, Vol. 89, No. EM6, p. 217, December 1963.
1.32. G. Augusti, "Some Problems in Structural Instability, with Particular Reference to Beam-Columns of I-Section" (Ph.D. dissertation, Cambridge University, 1964).
1.33. S. J. Errera, "Materials," in Structural Steel Design, Lambert Tall, ed. (New
York: The Ronald Press Company, 1964).
1.34. L. Tall, "Compression Members," in Structural Steel Design, Lambert Tall,
ed. (New York: The Ronald Press Company, 1964).
1.35. A. W. Huber and L. S. Beedle, "Residual Stress and the Compressive Strength
of Steel," Welding Journal, Vol. 33, p. 589, December 1954.
1.36. L. S. Beedle and L. Tall, "Basic Column Strength," Proceedings of the ASCE,
Vol. 86, No. ST7, p. 139, July 1960.
1.37. R. L. Ketter, E. L. Kaminsky, and L. S. Beedle, "Plastic Deformation of
Wide-Flange Beam-Columns," Transactions of the ASCE, Vol. 120, p. 1028,
1955.
1.38. R. L. Ketter and T. V. Galambos, "Columns Under Combined Bending and
Thrust," Transactions of the ASCE, Vol. 126(1), p. 1, 1961.
1.39. AISC, Steel Construction Manual, 6th ed. (New York: American Institute of
Steel Construction, 1963).
1.40. S. P. Timoshenko, History of the Strength of Materials (New York: McGrawHill Book Company, 1953).
1.41. ASCE-WRC Committee, Commentary on Plastic Design in Steel, ASCE
Manual No. 41, 1961.
CHAP.
1
INTRODUCTION
21
1.42. M. G. Lay, "The Experimental Bases of Plastic Design," Welding Research
Council Bulletin, No. 99, September 1964.
PROBLEM
1.1. Develop expressions for the f/fy versus U A/Uy relationships for an ideal
wide-flange cross section with residual stress (Fig. 1.6) and for which the axial force
is in tension. Plot the curve and compare it with the curve for axial compression
(Fig. 1.9).