PR INC IPAL ES PAT RONES D E D ISEÑO UT IL IZ ADOS

Introduction

This text is primarily concerned with the presentation of methods of analysis of structures and structural components, such as beams, that have a linear relationship between the levels of the applied loads and the e€ects of these loads with regards to the strains and deforma-tions of the structure and all its components. It is common experience to expect that if a beam is loaded by a force normal to its axis the deformation of the beam will be in the direction of the load. If we double the magnitude of the force we would expect that the level of the deformation would increase by a factor of two. Essentially the common experience forms the basis for linear behaviour of structures.

We should be aware, however, that structures do not always obey this linear correspondence between load and e€ect. In chapter 12 we con-sider the situation of beam and frame buckling where the application of a load aligned with the axis of the beam can result in deformations normal to the axis. The level of the deformations is in such a case non-linearly related to the level of the load. In other words, if we increase the level of the load the magnitude of the deformations are increased disproportionally. Another source of non-linear relation between load-ing and consequent deformation would result from non-linear material properties, either a non-linear stress±strain path or yielding of the material at a constant stress. This source of non-linearity is not included in the presentations of this text.

The linear behaviour of structures and its analysis constitutes a central feature of structural engineering and this text considers a wide range of types of structures and components that exhibit this form of behaviour. There are some general principles that form the fundamental basis of this linear behaviour and of the analysis methods developed to explain and predict the response of structure to applied loading. This chapter presents these principles of superposition and

reciprocity, in a general manner. Although the principles are illustrated with regard to simple beams, it should be remembered that the princi-ples are completely general and apply to all forms of structures, such as trusses, frames and plates.

5.1 The principle of superposition of the e€ects in the linear theory of elasticity (existence and uniqueness of the solution)

So far we have introduced the ®rst examples of elastic structures, i.e.

bars and beams, and dealt with their treatment using energy principles.

It is now timely to give some attention to the framework in which we are operating, i.e. the linear theory of elasticity. In order to make com-pletely clear the foundations of this theory, let us recall what we have already determined in the previous chapter, namely

. equations (4.7) and (4.19) describing the elastic behaviour of Bernoulli's model of beam deformations are both linear. This means that e€ects resulting from axial forces or bending moments are proportional to the intensity of these actions on the element . operating in the theory of in®nitesimal (very small) displace-ments allows the displacement and strain ®elds corresponding to di€erent load conditions simply to be added.

These are the two pillars on which the linear theory of elasticity is founded, as indicated in Fig. 5.1.

The ®rst of the points above, which involves the linearity of the con-stitutive relationships of the model, is clearly a physical assumption,

Linear theory of elasticity

Linearised constitutive laws Theory of infinitesimal displacements

Fig. 5.1

while the second, related to the linearisation of the displacement and strain ®elds, is essentially a mathematical axiom. As may be inferred from its description, in this theory all the equations are linear, with comprehensible and acceptable simpli®cations in the treatment of a vast class of structural problems.

First of all it is clear that in this framework the principle of superposition of e€ects holds true. In other words we are allowed to add and subtract the e€ects due to di€erent separate loading conditions. In its simplest form this means that if we, say, were to double the magnitude of the external forces, i.e. by adding to a certain set of loading a second set of loading equal to the ®rst one, we would expect the consequent deforma-tions also to increase by a factor of two. As a further example, let us consider the beam shown in Fig. 5.2, loaded by a uniformly distributed load q and by a concentrated load P at the middle of the span.

With respect to any e€ect, we can analyse the beam model shown in Fig. 5.3 and add the resulting solution to the one already obtained for the beam shown in Fig. 4.15 and analysed in chapter 4.

The ®eld of lateral displacements, i.e. normal to the longitudinal axis of the beam, for the beam in Fig. 5.3, subject to the uniformly dis-tributed load q is given by the solution of Euler's equation of beam deformation, (4.43), under the following boundary conditions

uy…0† ˆ 0 and …0† ˆ 0 at A

u_y…l† ˆ 0 and M…l† ˆ 0 at B …5:1†

l /2 l /2

B A

P

q

Fig. 5.2

B A

q

Fig. 5.3

From equation (4.44) we have uy…z† ˆ qz²

48EIx…2z²ÿ 5 l z ‡ 3l²† …5:2†

The solution for the beam in Fig. 5.2, subject to both the evenly distrib-uted load q and the concentrated load P at the mid-point of the span, is thus given by the simple addition of equation (5.2) to equations (4.80) and (4.81), i.e.

u1y…z† ˆ z²

96EI_x‰ÿ9Pl ‡ 6ql²‡ …11P ÿ 10ql†z ‡ 4qz²Š …5:3†

for z 2 ‰0; l=2Š and u2y…z† ˆ…l ÿ z†

96EIx‰2Pl²ÿ 10Plz ‡ …5P ‡ 6ql†z²ÿ 4qz³Š …5:4†

for z 2 ‰l=2; lŠ.

Of course we would have obtained the same expressions by writing the condition of equilibrium for the beam subjected to both these loads simultaneously and developing all the calculations accordingly.

However, as will become clearer in the following, the principle of superposition of the e€ects is a very powerful tool not only from a practical, but also from a theoretical point of view.

For example, let us ®x our attention on two apparently very simple questions

. Does the general condition of equilibrium (1.28) always lead to a solution of the problem?

. Is this hypothetical solution unique?

In the linear theory of elasticity the answer to both these questions is yes.With regards to the ®rst question, the demonstration of such a state-ment is quite a complicated matter from a general point of view and it has been fully established only in relatively recent times. Nevertheless, it is very easy to reach the correct conclusion in many speci®c problems, given the linearity of the equations that describe the models. This has been the case, for example, of Euler's equation of beam deformation, (4.43).

With regards to the second question, the answer was provided by Kirchho€¹about the middle of the 19th century and its demonstration

1Kirchho€, Gustav Robert (KoÈnigsberg, 1824 ± Berlin, 1887), German mathematician and physicist.

constitutes an instructive application of the principle of superposition of e€ects. In order to demonstrate the assertion of uniqueness, let us start our reasoning from the most general possible standpoint.

Let V be a generic linearly elastic body with certain geometrical con-straints on the boundary and subject to a generic system of forces Pⁱ, as shown in Fig. 5.4.

Let us initially assume the existence of two distinct solutions of this problem, say A and B. Also we assume that these solutions are de®ned by the ®elds of displacements u^A…x; y; z† and u^B…x; y; z†. As we know, the variation of the strain energy of the body is dependent on its state of deformation only, that is

U ˆ …

V

f …"ij† dV …5:5†

Moreover, in chapter 3 we noted that in the linear theory of elasticity this strain energy is a quadratic function of the strain components "_ij, see equation (3.107),

U ˆ E

2…1 ‡ †…1 ÿ 2†

…

V



…1 ÿ †…"²xx‡ "²yy‡ "²zz†

‡ 2…"_xx"_yy‡ "_xx"_zz‡ "_yy"_zz†

‡2…1 ÿ 2†

…1 ‡ † …"²xy‡ "²xz‡ "²yz†



dV …5:6†

Since these components are functions of the displacement ®eld u…x; y; z†, we can derive a di€erent strain ®eld for each of the two

V

Pⁱ

u = 0

Fig. 5.4

hypothetical solutions of the elastic problem at hand

Finally, as the principle of superposition of e€ects holds true, we can write the principle of conservation of energy with reference to the dif-ference between the solutions A and B of the problem. This di€erence is characterised by the strain ®eld

"^_ij ˆ "^A_ij ÿ "^B_ij …5:9†

and by the total zero sum of external actions, that is

Pⁱÿ Pⁱ ˆ 0 …5:10†

The principle of conservation of energy, equation (1.9), can be there-fore written as

E ˆ
U ˆ W ˆ 0 …5:11†

and requires
U ˆ 0, because of the zero value of the external work, W ˆ 0.

But, on account of equation (5.6), we have

U ˆ E

where
U is zero only if "^A_ij ˆ "^B_ij. With reference to the relationships in equations (5.7) and (5.8), this means that the displacement ®elds u^A…x; y; z† and u^B…x; y; z†, which represent two di€erent solutions of the problem of the elastic equilibrium, must di€er at most by a con-stant, corresponding to a rigid displacement of the whole body. There-fore, if, as is the usual condition, the geometrical constraints are such as to prevent any rigid displacements, the two solutions A and B must coincide and the uniqueness of the solution is demonstrated.

5.2 Reciprocal theorems in the linear theory of elasticity

Apart from the self-consistency of the theory, from a technical point of view the success of the mathematical theory of linear elasticity lies undoubtedly on the extensive availability of powerful and practical tools for the analysis of engineering structures. Among these tools a very important place is occupied by the so-called reciprocal theorems.

In order to introduce these theorems, let us consider the simple sup-ported beam shown in Fig. 5.5, which we consider to be loaded in sequence ®rst by the group of forces P¹ and then by the group of forces P².

The work done by all these forces is

W ˆ W ₁₁‡ W₂₂‡ W₁₂ …5:13†

where W₁₁is the component work from the group P¹, W₂₂is the com-ponent work from the group P²(remember that for both these compo-nents of work Clapeyron's theorem holds true) and W₁₂is the so-called displacement work of P¹, i.e. the work of P¹ due to the displacement

A B

P¹

A B

P¹ P²

Fig. 5.5

®eld produced by P². Sometimes the component work, i.e. the work done by a force along its own deformation, is also called eigenwork (the word eigen is a German word and means `own').

If we reverse the order of loading by ®rst applying P² and then P¹, we have, as shown in Fig. 5.6,

W ˆ W ₂₂‡ W₁₁‡ W₂₁ …5:14†

where W21 is the displacement work of P².

Of course, since the principle of superposition of the e€ects holds true, the ®nal con®guration of the beam will be identical in both cases and we have from the principle of conservation of energy

U ˆ W ˆ W …5:15†

By comparison of equations (5.13) and (5.14) we are led to the conclu-sion that

W12ˆ W21 …5:16†

This is called Betti's theorem, which can be stated as follows:

Given two forces or two groups of forces acting on a linearly elastic structure, the work done by the ®rst force, or group of forces, on account of the displacements due to the second force is equal to the work done by the second force, or group of forces, on account of the displacements due to the ®rst force.

A second theorem, due to Maxwell, can be straightforwardly derived by Betti's theorem. It states that for the same load applied at di€erent points A and B we must have the condition

uABˆ uBA …5:17†

A B

A B

P¹ P²

P²

Fig. 5.6

where u_AB is the generalised displacement² at A due to the force at B and u_BA is the generalised displacement at B due to the same force at A. In fact, if P is a generic load applied both at A and B, we have from equation (5.16)

W_ABˆ W_BA ) P u_ABˆ P u_BA …5:18†

and we immediately get the relationship (equation (5.17)).

2Here as generalised displacements we intend the components of displacement enabling the applied forces to perform work. In this sense we can introduce generalised forces as well. Thus, generalised forces and corresponding generalised displacements (or vice versa) are quantities characterised by the fact that the sum of their products always represents a true work.

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