STEEL CONSTRUCTION: APPLIED STABILITY___________ ________969 STEEL CONSTRUCTION: APPL IED STABILIT Y Lecture 6.2: General Criteria for Elastic Stability OBJECTIVE/SCOPE To establish general criteria for elastic stability and neutral equilibrium as preparation for the use of energy methods in the assessment of critical loads in Lecture 6.4. PREREQUISITES Lecture 6.1: Concepts of Stable and Unstable Elastic Equilibrium RELATED LECTURES Lecture 6.3: Elastic Instability Modes Lecture 6.4: General Methods for Assessing Critical Loads Lecture 6.5: Iterative Met hods for Solving Stability Problems SUMMARY Structural design requires that the equilibrium configuration for the structure, under the prescribed loading, is det ermined and that this can be confirmed as stable; the anal ysis of stability problems is generally done using energy criteria. In this lecture, the Principle of Virtual Work and the Principle of Stationary Total Potential Energy are presented. The general energy criteria for elastic stability derived from these are established and the determination of critical loading corresponding to neutral equilibrium is explained. Only fully conservative systems are considered. The established criteria are illustrated by two basic examples of rod and spring systems. 1. INTRODUCTION The design of structures requires determination of the internal equilibrium forces (moments, shears, etc.) in the structure, under given loadings, and confirmation that the structure is stable under these conditions. It is of fundamental importance to be sure that a structure, slightly disturbed from an equilibrium position by forces, shocks, vibrations, imperfections, residual stresses, etc., will tend to return to it when the disturbance is removed; this required characteristic of elastic stability has become more and more critical
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
To establish general criteria for elastic stability and neutral equilibrium as preparation for
the use of energy methods in the assessment of critical loads in Lecture 6.4.
PREREQUISITES
Lecture 6.1: Concepts of Stable and Unstable Elastic Equilibrium
RELATED LECTURES
Lecture 6.3: Elastic Instability ModesLecture 6.4: General Methods for Assessing Critical Loads
Lecture 6.5: Iterative Methods for Solving Stability Problems
SUMMARY
Structural design requires that the equilibrium configuration for the structure, under the
prescribed loading, is determined and that this can be confirmed as stable; the analysis of
stability problems is generally done using energy criteria. In this lecture, the Principle of
Virtual Work and the Principle of Stationary Total Potential Energy are presented. The
general energy criteria for elastic stability derived from these are established and thedetermination of critical loading corresponding to neutral equilibrium is explained. Only
fully conservative systems are considered. The established criteria are illustrated by two
basic examples of rod and spring systems.
1. INTRODUCTION
The design of structures requires determination of the internal equilibrium forces
(moments, shears, etc.) in the structure, under given loadings, and confirmation that the
structure is stable under these conditions. It is of fundamental importance to be sure that a
structure, slightly disturbed from an equilibrium position by forces, shocks, vibrations,
imperfections, residual stresses, etc., will tend to return to it when the disturbance is
removed; this required characteristic of elastic stability has become more and more critical
nowadays with the increasing use of high strength steels resulting in lighter and slenderer
structures.
The theory of elastic stability (buckling) gives methods for determining the following:• the stability of an equilibrium configuration.
• the critical value of the loading under which the instability occurs.
Most of these methods are derived from general energy criteria which come from energy
principles of mechanics. Therefore, the purpose of this lecture is to briefly present to the
student and the practising engineer the principles of mechanics required to understand the
general criteria of elastic stability, thereby giving a better understanding of the methods
used in buckling investigations, especially the energy methods discussed in Lecture 6.4.
The scope of this lecture is restricted to:
•
conservative loadings and adiabatic elastic systems (fully conservative systems).
• systems whose configurations can be expressed as functions of a finite number of
displacement parameters.
It should be noted that only the static aspect of stability is considered.
2. GENERAL
In this lecture, changes in the configuration of a system from an initial configuration are
considered; any change in the configuration is to be regarded as a displacement. A
configuration can be specified by means of a finite number of independent real variables,called generalised coordinates, denoted here as q 1, q 2, ... q n or more generally q i. A single-
span beam may, of course, possess an infinite set of generalised coordinates, such as the
coefficients q i of a Fourier series, that represent its deflection:
y = q i sin iπx/L
This series, however, can be approximated by a finite number of terms with a finite
number of generalised coordinates which denote the degrees of freedom of the system.
Considering the beam in Figure 1, the generalised coordinates could be the degrees of
freedom of the nodes i and j at the ends of the beam: two translations u and v and onerotation θ per node (all in plane). It is assumed here that the entire elastic deformed shape
of the beam can be defined by using, for example, interpolation functions. The
displacement vector of the beam can be denoted D = (u i, vi, θi, u j, v j, θ j).
Suppose that the system is in a configuration specified by the generalised coordinates q 1,
q 2, ... q n, which is to be tested for equilibrium.
Suppose the system experiences some arbitrarily small displacements from thisconfiguration, merely required to satisfy the boundary conditions, but with the actual loads
acting at their fixed prescribed values. The small displacements considered here are not
necessarily realised; they are imagined to occur purely for comparison purposes, and so
they are called virtual displacements; these virtual displacements are independent of the
loading and are denoted here δqi. Consequently, all work or energy calculations carried out
on the system will lead to virtual work or energy.
For a rigid system, Equations (5) and (8) yield:
δWext = 0 (9)
where δWext is the virtual work of external forces going through the virtual
displacements; the Principle of Virtual Work may be expressed as follows:
" A rigid system is in its equilibrium configuration if the virtual work of all the external
forces acting on it is zero in any virtual displacement which satisfies the boundary
conditions."
For a deformable system, Equation (7) yields:
δWext = δU (10)
where δU is the variation of strain energy in the virtual displacement, and the Principle of
Virtual Work may be expressed as follows:
" A deformable system is in its equilibrium configuration if the virtual work of all the
external forces acting on it is equal to the variation of strain energy in any virtual
displacement which satisfies the boundary conditions."
This is the form of the principle frequently quoted in structural analysis; it is equivalent to
the condition, using Equation (8):
δW = δWint + δWext = 0 (11)
True Equilibrium Configuration
For a system with a finite number of generalised coordinates (q 1, q 2, ...q n), the virtual work
δW corresponding to a virtual displacement from a configuration (q 1, q 2, ...q n) to a
neighbouring configuration (q 1 + δq1, ...q n + δqn) may be represented by a linear form in
where Q1, Q2, ...Qn - are certain functions of generalised coordinates q i, and of internal
(for deformable systems) and external forces.
By analogy to the work performed by a force, the functions Q1, Q2, ... Qn are calledcomponents of generalised forces. The terms Qi do not necessarily have the dimension of
force and they frequently do not all have the same dimension; their dimensions are
determined by the fact that Qi δq i has the dimension of work. Equations (11) and (12)
yield:
Qi δq i = 0 i=1,2,...,n (13)
As δq i are arbitrary, independent of variations in q i, Equation (13) implies that:
Qi = 0 i=1,2,...,n (14)
Solution of these n simultaneous equations of equilibrium yields the values of the q's
corresponding to the true equilibrium configuration.
4. PRINCIPLE OF STATIONARY TOTAL
POTENTIAL ENERGY
The internal and external forces are both conservative (fully conversative system). The
internal forces derive from the single scalar function of the generalised coordinates U(q 1,q 2, ...q n) whose value U is the strain energy which is expressed by Equation (4). Similarly,
the external forces derive from the function Ω(q 1, q 2, ...q n) whose value Ω is the potential
energy of these forces. It yields the result that all forces derive from the single scalarfunction V (q 1, q 2, ...q n) which is called the total potential function and whose value is the
total potential energy of the system. This total potential energy may be expressed as:
V = U + Ω (15)
The total amount of potential energy is generally indeterminate. Only changes of potential
energy are measurable and can be investigated.
Because the system is assumed to be fully conservative,
δW = - δV (16)
where δV is the variation of total potential energy in the virtual displacement, and (11) and(16) yield:
δV = 0 (17)
Equation (17) is an analytical statement of the Principle of Stationary Total PotentialEnergy which states:
Only the load multiplier α is unknown and the condition for neutral equilibrium requires
the solution of the eigenvalue problem:
det [a(α)] = 0 (32)
Solving Equation (32) leads to a set of solutions α, denoted αcr , whose number is equal to
the number of generalised coordinates of the system. The eigenvectors represent the
deformed configuration associated with each solution α. Most of these mathematical
solutions do not correspond to actual behaviour of the structural system; generally, the
designer is only interested in the values of loads above which the system, stable when
unloaded, becomes unstable. These loads are normally obtained with the smallest positive
value α°cr of αcr and so, the critical loads are determined by:
Scr = α°cr S1 (33)
7. ILLUSTRATION ON BASIC EXAMPLES
Example 1
It is interesting to illustrate the stability criterion with the basic example of a pin-ended
compression element shown in Figure 3; however, in order to perform very simple strain
energy calculations, it is assumed that the whole flexibility of the element is concentrated
in a single rotational elastic spring at mid-span, as shown in Figure 4. The two rods, each
of length L/2, are rigid so that their strain energy is zero. The value K of the spring,constant at B, will be discussed later. Sideways movement of the pins A and C are fully
restrained. The load P acts vertically downwards at C, and the external force F, present
from the beginning of loading, acts horizontally leftwards at B.
The value of P at the limit is its critical value Pcr at which elastic buckling occurs. It is
worth noting that this critical value is independent of the external lateral force F acting on
the system. In particular, this critical load is valid for the particular case F = 0, denoting
the classical column buckling problem under axial load only.
A value may be given to K so that the flexibility is the same as the continuous element of
Figure 3. It is defined, therefore, as the value that gives the same lateral displacement δ at
B due to F as the continuous element assuming P is zero.
For the continuous element, simple beam theory gives:
δ = FL3 / (48 EI) (43)
where I is the second moment of area of the element section.
E is Young's modulus.
For the rod and spring system, expressing the moment at B with θ = 4δ/L, gives:
δ = FL2 / (16 K) (44)
Equations (43) and (44) yield the equivalent spring constant: K = 3 EI/L, and the
critical value of P is equal to:
Pcr = 12 EI/L2 (45)
This value is to be compared to the well-known exact value π2 EI/L2; the accuracy of the
result depends, in fact, on the assumptions adopted for the determination of the equivalent
spring constant K.
Example 2
Consider now the rod and spring system shown in Figure 6. The two rods AB and BC,
each of length L, are rigid (no strain energy) and are pinned and linked together at B.
Sideways movement of the pins B and C is restrained by linearly elastic springs, effectivein both tension and compression, of stiffness K 1 and K 2 respectively. The load P acts
vertically downwards at C, and the external forces F1 and F2 act horizontally leftwards at
The analysis of stability problems uses general energy criteria derived from thePrinciple of Virtual Work and from the Principle of Stationary Total Potential
Energy; the first of these principles is the same as the second for fully conservative
systems.
• Any configuration of a system may generally be specified by a set of generalised
coordinates q i. Denoting V as the total potential energy of the system, an
equilibrium configuration satisfies δ2V=0 and the condition for stability of this
equilibrium is δ2V > 0; the first and second variations of V are evaluated for any
virtual displacement δqi satisfying the boundary conditions.
• Critical loadings are derived from the condition for neutral equilibrium given by
δ2V = 0 = minimum.
9. ADDITIONAL READING
1. Mason J.,"Variational, Incremental and Energy Methods in Solid Mechanics and
Shell Theory", Elsevier Scientific Publishing Company, Amsterdam, Oxford, New
York, 1980.
2.
Richards T.H., "Energy Methods in Stress Analysis", Rainbow-Bridge Book
Company, 1977.
3.
Langhaar H.L., "Energy Methods in Applied Mechanics", John Wiley and Sons,
New York, London, 1962.
4.
Massonnet C., "Résistance des matériaux", Volume 2, Dunod, Paris, 1963.5. Timoshenko S.P., "Theory of Elastic Stability", McGraw Hill Book Company,