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Overview
Two additional chapters offer an introduction to the text and derive the differential equations governing the elastic deformations of prismatic thin-walled open members. Numerous examples throughout the treatment illustrate theory and applications.
Product Details
| ISBN-13: | 9780486811840 |
|---|---|
| Publisher: | Dover Publications |
| Publication date: | 04/10/2016 |
| Series: | Dover Books on Engineering |
| Sold by: | Barnes & Noble |
| Format: | eBook |
| Pages: | 400 |
| File size: | 32 MB |
| Note: | This product may take a few minutes to download. |
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Read an Excerpt
Structural Members and Frames
By Theodore V. Galambos
Dover Publications, Inc.
Copyright © 1996 Theodore V. GalambosAll rights reserved.
ISBN: 978-0-486-81184-0
CHAPTER 1
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 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. For these problems the computer has become an invaluable and increasingly necessary tool. 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. 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 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 play a 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 quasi-static, 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.
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 non-repetitive. 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. 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, 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 αP (where α 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 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 of load 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 of the 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 load factor, 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.
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 failure 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.
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.
In the mathematical treatment of stability problems the disturbance is usually virtual, that is, it does not change the existing force system. 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.
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.
The intersections between the load characteristics and the structure load-deflection 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. 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. Applying the same test for stability as for the loading in Fig. 1.2, we find that points A, B, 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 B, 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,]IT L where it is again stable, unless it is externally restrained to do so, but it 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.
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 = gL neutral equilibrium
gs< gL unstable equilibrium
In metal frame structures of the type discussed here, the point of neutral 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.
(Continues...)
Excerpted from Structural Members and Frames by Theodore V. Galambos. Copyright © 1996 Theodore V. Galambos. Excerpted by permission of Dover Publications, Inc..
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