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STABILITY OF STRUCTURES
Principles and Applications
By CHAI H. YOO SUNG C. LEE
Butterworth-Heinemann
Copyright © 2011 Elsevier Inc.
All right reserved.
ISBN: 978-0-12-385123-9
Chapter One
Buckling of Columns Contents
1.1. Introduction 1 1.2. Neutral Equilibrium 3 1.3. Euler Load 4 1.4. Differential Equations of Beam-Columns 8 1.5. Effects of Boundary Conditions on the Column Strength 15 1.6. Introduction to Calculus of Variations 18 1.7. Derivation of Beam-Column GDE Using Finite Strain 24 1.8. Galerkin Method 27 1.9. Continuous Beam-Columns Resting on Elastic Supports 29 1.9.1. One Span 29 1.9.2. Two Span 30 1.9.3. Three Span 31 1.9.4. Four Span 34 1.10. Elastic Buckling of Columns Subjected to Distributed Axial Loads 38 1.11. Large Deflection Theory (The Elastica) 44 1.12. Eccentrically Loaded Columns—Secant Formula 52 1.13. Inelastic Buckling of Straight Column 56 1.13.1. Double-Modulus (Reduced Modulus) Theory 57 1.13.2. Tangent-Modulus Theory 60 1.14. Metric System of Units 66 General References 67 References 68 Problems 69
1.1. INTRODUCTION
A physical phenomenon of a reasonably straight, slender member (or body) bending laterally (usually abruptly) from its longitudinal position due to compression is referred to as buckling. The term buckling is used by engineers as well as laypeople without thinking too deeply. A careful examination reveals that there are two kinds of buckling: (1) bifurcation-type buckling; and (2) deflection-amplification-type buckling. In fact, most, if not all, buckling phenomena in the real-life situation are the deflection-amplification type. A bifurcation-type buckling is a purely conceptual one that occurs in a perfectly straight (geometry) homogeneous (material) member subjected to a compressive loading of which the resultant must pass though the centroidal axis of the member (concentric loading). It is highly unlikely that any ordinary column will meet these three conditions perfectly. Hence, it is highly unlikely that anyone has ever witnessed a bifurcation-type buckling phenomenon. Although, in a laboratory setting, one could demonstrate setting a deflection-amplification-type buckling action that is extremely close to the bifurcation-type buckling. Simulating those three conditions perfectly even in a laboratory environment is not probable.
Structural members resisting tension, shear, torsion, or even short stocky columns fail when the stress in the member reaches a certain limiting strength of the material. Therefore, once the limiting strength of material is known, it is a relatively simple matter to determine the load-carrying capacity of the member. Buckling, both the bifurcation and the deflection-amplification type, does not take place as a result of the resisting stress reaching a limiting strength of the material. The stress at which buckling occurs depends on a variety of factors ranging from the dimensions of the member to the boundary conditions to the properties of the material of the member. Determining the buckling stress is a fairly complex undertaking.
If buckling does not take place because certain strength of the material is exceeded, then, why, one may ask, does a compression member buckle? Chajes (1974) gives credit to Salvadori and Heller (1963) for clearly elucidating the phenomenon of buckling, a question not so easily and directly explainable, by quoting the following from Structure in Architecture:
A slender column shortens when compressed by a weight applied to its top, and, in so doing, lowers the weight's position. The tendency of all weights to lower their position is a basic law of nature. It is another basic law of nature that, whenever there is a choice between different paths, a physical phenomenon will follow the easiest path. Confronted with the choice of bending out or shortening, the column finds it easier to shorten for relatively small loads and to bend out for relatively large loads. In other words, when the load reaches its buckling value the column finds it easier to lower the load by bending than by shortening.
Although these remarks will seem excellent to most laypeople, they do contain nontechnical terms such as choice, easier, and easiest, flavoring the subjective nature. It will be proved later that buckling is a phenomenon that can be explained with fundamental natural principles.
If bifurcation-type buckling does not take place because the aforementioned three conditions are not likely to be simulated, then why, one may ask, has so much research effort been devoted to study of this phenomenon? The bifurcation-type buckling load, the critical load, gives the upper-bound solution for practical columns that hardly satisfies any one of the three conditions. This will be shown later by examining the behavior of an eccentrically loaded cantilever column.
1.2. NEUTRAL EQUILIBRIUM
The concept of the stability of various forms of equilibrium of a compressed bar is frequently explained by considering the equilibrium of a ball (rigidbody) in various positions, as shown in Fig. 1-1 (Timoshenko and Gere 1961; Hoff 1956).
Although the ball is in equilibrium in each position shown, a close examination reveals that there are important differences among the three cases. If the ball in part (a) is displaced slightly from its original position of equilibrium, it will return to that position upon the removal of the disturbing force. A body that behaves in this manner is said to be in a state of stable equilibrium. In part (a), any slight displacement of the ball from its position of equilibrium will raise the center of gravity. A certain amount of work is required to produce such a displacement. The ball in part (b), if it is disturbed slightly from its position of equilibrium, does not return but continues to move down from the original equilibrium position. The equilibrium of the ball in part (b) is called unstable equilibrium. In part (b), any slight displacement from the position of equilibrium will lower the center of gravity of the ball and consequently will decrease the potential energy of the ball. Thus in the case of stable equilibrium, the energy of the system is a minimum (local), and in the case of unstable equilibrium it is a maximum (local). The ball in part (c), after being displaced slightly, neither returns to its original equilibrium position nor continues to move away upon removal of the disturbing force. This type of equilibrium is called neutral equilibrium. If the equilibrium is neutral, there is no change in energy during a displacement in the conservative force system. The response of the column is very similar to that of the ball in Fig. 1-1. The straight configuration of the column is stable at small loads, but it is unstable at large loads. It is assumed that a state of neutral equilibrium exists at the transition from stable to unstable equilibrium in the column. Then the load at which the straight configuration of the column ceases to be stable is the load at which neutral equilibrium is possible. This load is usually referred to as the critical load.
To determine the critical load, eigenvalue, of a column, one must find the load under which the member can be in equilibrium, both in the straight and in a slightly bent configuration. How slightly? The magnitude of the slightly bent configuration is indeterminate. It is conceptual. This is why the free body of a column must be drawn in a slightly bent configuration. The method that bases this slightly bent configuration for evaluating the critical loads is called the method of neutral equilibrium (neighboring equilibrium, or adjacent equilibrium).
At critical loads, the primary equilibrium path (stable equilibrium, vertical) reaches a bifurcation point and branches into neutral equilibrium paths (horizontal). This type of behavior is called the buckling of bifurcation type.
1.3. EULER LOAD
It is informative to begin the formulation of the column equation with a much idealized model, the Euler column. The axially loaded member shown in Fig. 1-2 is assumed to be prismatic (constant cross-sectional area) and to be made of homogeneous material. In addition, the following further assumptions are made:
1. The member's ends are pinned. The lower end is attached to an immovable hinge, and the upper end is supported in such a way that it can rotate freely and move vertically, but not horizontally.
2. The member is perfectly straight, and the load P, considered positive when it causes compression, is concentric.
3. The material obeys Hooke's law.
4. The deformations of the member are small so that the term (y')2 is negligible compared to unity in the expression for the curvature, y"/[1 + (y')2]3/2. Therefore, the curvature can be approximated by y".
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.3.1)
From the free body, part (b) in Fig. 1-2, the following becomes immediately obvious:
EIy" -M(x) = -Py or EIy" + Py = 0
Equation (1.3.2) is a second-order linear differential equation with constant coefficients. Its boundary conditions are
y = 0 at x = 0 and x = l (1.3.3)
Equations (1.3.2) and (1.3.3) define a linear eigenvalue problem. The solution of Eq. (1.3.2) will now be obtained. Let k2 = P/EI, then y" + k2y = 0. Assume the solution to be of a form y = αemx for which y' = αmemx and y" = αm2emx. Substituting these into Eq. (1.3.2) yields (m2 + k2) αemx = 0.
Since αemx cannot be equal to zero for a nontrivial solution, m2 + k2 = 0, m = [±ki. Substituting gives
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
A and B are integral constants, and they can be determined by boundary conditions.
y = 0 at x = 0 -> A = 0
y = 0 at x = l -> B sin kl = 0
As B [not equal to] 0 (if B = 0, then it is called a trivial solution; 0 = 0), sin kl = 0 -> kl = nπ
where n = 1, 2, 3, ... but n [not equal to] 0. Hence, k2 = P/EI = n2π2/l2, from which it follows immediately
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.3.5)
The eigenvalues Pcr, called critical loads, denote the values of load P for which a nonzero deflection of the perfect column is possible. The deflection shapes at critical loads, representing the eigenmodes or eigenvectors, are given by
y = B sin nπx/l (1.3.5)
Note that B is undetermined, including its sign; that is, the column may buckle in any direction. Hence, the magnitude of the buckling mode shape cannot be determined, which is said to be immaterial.
The smallest buckling load for a pinned prismatic column corresponding to n = 1 is
PE = π2EI/l2 (1.3.6)
If a pinned prismatic column of length l is going to buckle, it will buckle at n = 1 unless external bracings are provided in between the two ends.
A curve of the applied load versus the deflection at a point in a structure such as that shown in part (a) of Fig. 1-3 is called the equilibrium path. Points along the primary (initial) path (vertical) represent configurations of the column in the compressed but straight shape; those along the secondary path (horizontal) represent bent configurations. Equation (1.3.4) determines a periodic bifurcation point, and Eq. (1.3.5) represents a secondary (adjacent or neighboring) equilibrium path for each value of n. On the basis of Eq. (1.3.5), the secondary path extends indefinitely in the horizontal direction. In reality, however, the deflection cannot be so large and yet satisfies the assumption of rotations to be negligibly small. As P in Eq. (1.3.4) is not a function of y, the secondary path is horizontal. A finite displacement formulation to be discussed later shows that the secondary equilibrium path for the column curves upward and has a horizontal tangent at the critical load.
(Continues...)
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