Structural mechanics

442

THEME 15. VARIATIONAL PRINCIPLES AND VARIATIONAL

METHODS OF STRUCTURAL MECHANICS.

FINITE ELEMENT METHOD

15.1. Potential Field of Force. Potential Eenergy

A field of force is a space in which a certain force acts on a material point placed there.

This concept is a general one. Examples of force fields are the gravitational fields of planets, the magnetic field of an object, an electrostatic field, etc. A special place among them is taken by potential force fields that have two important physical properties: 1) the force of this field is

positional force, that is F F x, y, z ; 2) the work of the field force

does not depend on the trajectory along which the force applied to a certain point moves, but depends only on the positions of the start and end points; it can be calculated through the integral sum of the corresponding elementary works:

Forces acting in a potential force field are called potential.

If the expression (15.1) under the sign of the integral is the full differ-

443

The work of the potential force is equal to the difference in the values of the force function at the final and initial points of the path of force motion. From relation (15.2) it follows that the force function is found from the equality:

U (Fx dx Fy dy Fz dz) C .

The constant C can have any value. As can be seen from equality (15.3), the work of force does not depend on C.

Gravity forces, as well as the elastic forces of an elastic body, are both potential in an adiabatic process (i.e., a process that takes place without heat exchange with the environment) and isothermal processes (i.e., processes that occur in a physical system at a constant temperature). For these forces there are force-functions.

So, for gravity force F , directed along the axis z (the axis z is directed vertically upwards), we have Fz F and dA F dz . Taking

U 0 when z 0 we receive:

U F z .

The elastic force in the centrally stretched rod (Figure 15.1) is directed in the direction opposite to the external force F .

Figure 15.1

Therefore Fx F r x ( r – is the rigidity coefficient of the elastic rod). The elementary work of this force is equal to dA r x dx . Counting U 0 at x 0 , we find:

U 12 r x2 .

444

In a potential force field, the projections of forces are equal to the partial derivatives of the force function relative to the corresponding coordinates. Indeed, it follows from equality (15.2) that:

Determining the mixed derivatives for U , we find:

These relations are necessary and sufficient conditions for the potentiality of the force field.

The potential energy at the given point M in the field is the amount of work that the force field would have done when moving a material point from a given position to one in which the potential energy is conventionally assumed to be zero (pointO ):

П A(MO) .

Since the functions П(x, y, z) and U (x, y, z) have the same null values (follows from the definitions), from (15.3) when U0 0 we

obtain:

A(MO) U0 U U ,

where U is the value of the force-function at the point M . Thus, we obtain:

П(x, y, z) U (x, y, z) .

445

The work of potential force can be calculated not by expression (15.3), but by the formula:

A(M1М2 ) П1 П2 ,

that is, it is equal to the difference in the values of potential energy in the initial and final positions of the point.

Work and energy, of course, are measured in the same units. Remember that in the SI system the basic units are: meter (m) – unit of length, kilogram (kg) – unit of mass, second (s) – unit of time. The unit of work and energy is the joule (J). 1 J is equal to work that is accomplished by force of 1 N in the path of 1 m.

The technique often uses the МKGFS system. The unit of work is 1 kilogram–force–meter (1 kgf·m). It is the work that is performed with a force of 1 kgf on a distance of 1 m.

Relations between units: 1 kGf·m = 9.81 J; 1 J = 0.102 kGf m.

15.2. Potential Energy of Elastic System Deformation

A special case of the general definition of potential energy given in Section 15.1 is the determination of the potential energy of an elastic deformed body, that is, the field of elasticity forces.

The potential deformation energy U of an elastic system is the amount of work that internal forces would have done in transferring the system from a deformed state to an undeformed one; it is the energy of elasticity forces. It is equal in absolute value, but opposite in sign to the actual work of internal forces, i.e.:

U Aint .

In particular, for a linearly elastic rod under tension-compression:

U (N ) 1 l N 2dx .

2 0 EA

And in pure bending:

U (M ) 1 l M 2dx .

2 0 EJ

446

In general, for a plane bars system:

In these expressions, U is written through efforts (internal forces).

One can represent U through functions expressing the displacements of the points (cross-sections) of the bars. For example, using differential dependencies:

In some cases, the deformation energy of the bar is conveniently expressed not through the functions u x or y x , but through the dis-

placements of individual cross-sections.

For the tensioned bar (Figure 15.2), the horizontal movement of the end of the bar is determined by the parameter l . Then, calculating the potential energy through the work of external forces, we obtain:

Figure 15.2

447

For a bended bar (Figure 15.3), the force F, causing the displacement, is equal to

F 3lEJ3 .

Consequently,

Figure 15.3

So, the energy of elastic deformation can be expressed through efforts, through the functions of displacements, or through discrete parameters of displacements.

Note: In some questions of mechanics, the concept of specific potential energy U0 (in other words, energy density) is used. It is equal to the

area bounded by the curve, the axis and the vertical corresponding to the final value of the relative deformation (Figure 15.4).

Figure 15.4

448

The potential deformation energy of the body is calculated through the specific energy by the expression:

U U0 dx dy dz .

V

By U0add in this figure, additional potential energy (additional work) is denoted. For a linearly elastic bar

UU add .

15.3.Generalized Displacements and Forces. Derivatives of Potential Energy Expressions

The deformation energy U, equal to the work of external forces, is determined by the equality:

It is obteined matrix representation of the quadratic form of n variables F1, F2 , , Fn , where the matrix of the quadratic form is denoted

with A:

449

If in the formula (15.5) the result of the multiplication is represented in the scalar form of the record, then we obtain:

The potential energy of the system is always positive. Therefore, the recorded quadratic form cannot become negative at any value

F1, F2 , , Fn . Such quadratic forms are called positive definite. Expression (15.4) can be represented as:

The vector F can be expressed through the external rigidity matrix:

F = R , R A 1 .

With this in mind, the deformation energy can be written as:

A matrix record of a quadratic form is obtained through generalized displacements. In the formula (15.8), the matrix of the external rigidity of the system

is a quadratic matrix.

450