Structural mechanics

In the scalar form of the record, we obtain:

This expression is a record of Castiliano's first theorem (1875): the derivative of the potential strain energy relative to force is equal to the displacement of the point of application of this force in its direction.

Differentiating expression (15.9) relative to the variable 1 and tak-

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This expression is a record of the second Lagrange theorem: in the equilibrium position, the derivative of the potential strain energy relative to displacement is equal to the corresponding force.

15.4. Total Energy of the Deformable System

From the energy view point, the phenomenon of the body deformation is a process of energies exchange of two systems of forces (force fields): internal and external.

Therefore, for a complete energy characteristic of a body in a deformed state, it is not enough to consider only the deformation energy U, since it represents a part of the energy of the interacting force fields.

We will consider only conservative external forces. Their work depends only on the initial and final state and does not depend on the way of transition from one position to another. Conservative forces include, for example, gravity forces.

If we take the energy of the system in the initial (undeformed) state equal to zero, then the potential П of external forces in the deformed state will be measured by the amount of work that these forces can perform when the system is transferred from given state to the initial one.

The total energy of the loaded body is taken equal to:

where U is the potential energy of deformation (elastic potential or, otherwise, the energy of elastic forces, the potential of internal forces);

P is the energy of external forces (potential of external forces). External forces are gravity forces. With a relatively small change in

the distance between bodies in near-Earth space, gravitational forces practically do not change. Therefore, gravity forces form a homogeneous force field, that is, a field in which the value of each force is constant, independent of the displacements of their application points. Their work is calculated as the work of unchanging forces when moving the system from a given position to the initial one.

For a centrally tensioned rod (Figure 15.5)

P F l ,

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and for a bended bar loaded with a distributed load (Figure 15.6):

l

P q(x) y(x) dx .

0

Thus, the total energy of the system can be expressed either through the functions of displacements or through discrete parameters.

For the last example:

is a functional (function of function) E E( y) .

For a discrete linear-elastic system, the potential of internal forces is (see formula (15.9)):

1 n n

U 2 rij i j .

i 1 j 1

Replacing the notation of the generalized displacement by Z , we obtain:

1 n n

U 2 rijZi Z j .

i 1 j 1

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The potential of external forces is:

where RiF are reactions in additional constraints of the primary system

of the displacement method.

Then the expression for the total energy of the system is represented as:

15.5. Displacements Variation Principle

This principle expresses the equilibrium condition of a deformable system, recorded through its displacements, using the introduced concept of total energy E.

For a tensioned rod (Figure 15.5) u x is a function that determines the longitudinal displacements of the cross-sections; u x are true dis-

placements for which the balance between external and internal forces is established.

In a deformed state, the total energy of the rod is equal to the work of internal and external forces on displacements u :

E(u) U P ( Aint Wext )

Let us give the points of the system additional infinitesimal displacements u u(x) ; u is an arbitrary function with infinitesimal ordi-

nates. It is called a variation of function u(x) . In a state u u , the energy will be equal to:

E(u u) ( Aint Aint Wext Wext ) .

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Subtracting the expression E u from the last equality, we obtain an

infinitesimal change in energy (the first variation of energy) caused by the variation of the function u :

E E(u u) E(u) ( Aint Wext ) .

For a system that is in equilibrium, when the displacements u(x) oc-

curs, the right-hand side in the last equality is equal to zero, since, in accordance with the principle of virtual displacements (see Section 7.4), the work of all the forces of the system on virtual displacements u

must be equal to zero:

A Aint Wext 0 ,

therefore,

This is a formal notation of the displacements variation principle (the Lagrange principle): of all the displacements allowed by the constraints of the system, the true displacements u(x) have the property that the

total energy of the system has a stationary value when these displacements occure. Such a property of energy will be observed when it has an extreme value for the true displacements in comparison with all nearest ones.

Consider the scheme shown in Figure 15.7.

Figure 15.7

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or

E U y P 12 EJ y 2dx F 12 F F .

The terms in these expressions are converted on the basis of the numerical equality of the potential energy of elastic deformation and the actual work of external forces.

We investigate the change in the total energy of the system depending on the change (variation) of the deformed beam axis. For example, we increase the ordinates of the deflections of the beam axis by a factor of k. We get:

Е

Figure 15.8

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When k 1, that is, in the real state of equilibrium, min E y takes

place.

In this example, it would be possible to vary not the equation of the bent beam axis, but the corresponding function of bending moments, i.e., the stress state.

The result of the calculations would naturally be the same.

Let us consider a second example. In a discrete linearly deformable system under one-parameter loading, all generalized parameters are interconnected linearly. Therefore, using the parameter of a generalized displacement Z , we can write the total energy in the form:

displacements, corresponding to the unit parameter of the generalized load F.

Since there is equality for the system:

then the expression for energy can be represented in this form of notation:

E Z 2 Z ,2

where

n n

rij Zi Z j . i 1 j 1

Function E Z has a minimum at a point (1.0; –0.5).

We increase the displacement Z by a factor k. Then, given that for the final load value the parameters F and Z are both fixed, expressing through the actual work of external forces, we obtain:

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leads to the conclusion that of all possible deformed states of the system, the true state occurs at k 1. The total energy of the system in this state is minimal.

Investigation of the behavior of functionals E at stationary points us-

ing the second variation 2E gives reason to judge the quality of the system equilibrium. P.G.L. Dirichlet (German mathematician, 1805– 1859) proved that:

– if E 0 and 2E 0 , then E Emin (stable equilibrium);

– if E 0 and 2E 0 , then E Emax (unstable equilibrium);

– if E 0 and 2E 0 , then E const (indifferent equilibrium).

A thorough study of the equilibrium states of mechanical systems will be carried out in the section “Stability of structures”.

In the problems of the structural statics, methods for calculating stable systems are studied. Therefore, the stationarity condition E 0 for them is identified with the condition of minimum total energy.

15.6. Ways to Solve Variational Problems

The functions y x , that realize the extremum of the functional E y , can be found in two ways:

1. By solving differential equations obtained from condition E 0

(15.14).

2. Using the so-called direct methods of variations calculus.

The problem of finding y x by solving the differential equation is addressed in those cases when, for the element (object) under study, the

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energy can be written as a function depending on the displacements and their derivatives of the first, second, and higher order. A necessary condition for the minimum of a definite integral

b

E Ф y, y , y , , y(к) dx ,

a

that is, the stationarity condition

As an example, we show the solution to the problem of bending the cantilever beam on an elastic Winkler base (Figure 15.9).

Figure 15.9

We determine the total energy of the interacting forces:

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In this case:

The differential equation corresponding to condition E 0 , will have the form:

E J y k y q .

For k 0 we get the usual differential equation of transverse bending:

E J y q .

The general solution of the equation will contain four arbitrary constants. To obtain a particular solution, four additional conditions must be set.

Direct methods of variations calculus allow us to reduce the problem of finding the functional minimum to the problem of finding the minimum of a function of many variables by solving a system of linear algebraic equations. These include the Rayleigh – Ritz, Bubnov – Galerkin methods, the callocation method, etc. Let us show the essence of direct methods using the example of the Rayleigh – Ritz method.

From an infinite system of functions 1(x), 2 (x), ... , r (x), ..., sat-

isfying the boundary conditions of the problem, we select the first r functions i (x) and form a new function fr of the following form, using a linear combination:

r

fr (x) a1 1(x) a2 2 (x) ... ar r (x) ai i (x) ,

i 1

where ai are the arbitrary coefficients. Functions i (x) are called coor-

dinate or basic functions.

The functional E(Ф(х)) after replacing Ф(х) by fr (x) turns into a function E(a1 ,a2 ,..., ar ) of r independent variables. The necessary

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