Structural mechanics

Comparing with (14.3), we see that the transformation matrix is transposed with respect to the equilibrium matrix. Hence we can write:

These equations are called geometric equations. They are the equations of continuity of deformations of the bars system.

The matrix AT is called the deformation matrix. With its help, deformations of system elements are calculated through displacements of nodes.

So that the reader does not have an opinion that the deformation matrix for the considered example turned out to coincide accidentally with the transposed equilibrium matrix, we study the question of the relationship of these matrices in more detail.

14.6. Duality Principle

The equilibrium equations were compiled for the undeformed state of the system, that is, under the assumption of small deformations of its elements, causing small displacements of nodes.

Due to this assumption, the equilibrium equations and geometric equations turned out to be linear. Systems to which this assumption applies are called geometrically linear.

An important property of equations is that the matrices of equilibrium equations and geometric equations are mutually transposed. This relationship can be shown in general terms. Let, for example, between vec-

tors z and there is dependence in the form:

A1 z .

In accordance with the virtual displacement principle for a system in equilibrium, the sum of the virtual works of external and internal forces is zero. Actual displacements can be considered as a special case of virtual. In this case:

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Substituting

FT ST AT and A1 z, into the equations (14.5), we will have

ST AT z ST A1 z 0,

which implies the equality

A1 AT .

The obtained dependence is common for linear systems and expresses a static-geometric analogy of the calculated relations.

In the case of large displacements, the problem of determining the stress-strain state becomes nonlinear. Systems in which large displacements and small deformations take place, together with the corresponding problems are called geometrically nonlinear. An example of geometrically nonlinear systems can be some cable-stayed systems. For these systems, equilibrium equations are compiled for their deformed state taking into account nodal displacements. The matrix A of equilibrium

equations will depend on the displacements z, the matrix A1 of geometric equations will also be dependent on z, but

The static-geometric analogy for geometrically non-linear problems appears in a more complex form. Its consideration is beyond the scope of this tutorial.

In the following presentation of the theory of calculating bars systems, geometrically linear systems are considered.

14.7. Physical Equations

The relationship between the strain vector and the force vector for an individual bar is established linear:

i Di Si .

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Recall that:

To determine the component Bi , one should consider loading the bar with the end moments M Bi and M Ei (load state) and loading with a

unit moment at the beginning of the bar (Figure 14.4). “Multiplying" the diagrams, we get:

Similar reasoning will allow us to write the expression:

Figure 14.4

Given that the compliance of the bar from the longitudinal force is equal to li / (EA)i , the matrix of internal compliance of the bar, taking

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into account tensile-compression and bending deformations, will be as follows:

The frame (Figure 14.3) consists of two bars, so the matrix of internal compliance of the system is quasi-diagonal:

and the physical equations are written in the form:

D S.

14.8.Features of the Calculation of the Systems for Temperature Changes, Settlements of Supports

and Inaccuracy in the Manufacture of Bars

These factors are taken into account by appropriate adjustment of

physical equations. The deformations vector caused by the efforts from the load F should be summed with the new deformations vector

from other exposures.

As the temperature changes with respect to a certain initial state, the frame bars become deformed (Figure 14.5). Denote by t1 the tempe-

rature change along the upper face of the bar, through t2 – along the bottom.

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Figure 14.5

Let

t2 t1.

A change in temperature along the axis of the rod

t t1 t2

2

causes its extension

l t l.

The temperature difference

t t2 t1

causes the rotation of the end sections by angles determined by the formula (7.12):

B E t l . h 2

The directions of rotation of the end sections of the bar in Figure 14.5 are shown as positive. In this case, the strain vector for this bar is written as follows:

ti li , Bi , Ei .

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For the entire system a strain vector is formed by joining vectors for individual bars.

When calculating the system for the inaccuracy of manufacturing its elements, the vector B is known by the condition of the problem. The

components of the strain vector from inaccurate production of bars are determined by the difference between the real and design values of the dimensions of the bars.

The vector of deformation of the bars from the settlement of the supports can be obtained as follows. We select from the matrix A the rows associated with the equilibrium conditions of the support nodes in the directions of the support links. Displacements can have all support nodes or only a part of them. The number of such lines, equal to the number of support links, is denoted by r . The corresponding equilibrium conditions for the support nodes of the system are written in the form:

A r S 0.

We will divide the matrix A r into blocks using a vertical partition (Table 14.1) and we will consider it as a complex matrix

links:

Sr R .

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The equilibrium equations for the support nodes can be written as follows:

Anr r Sn r Arr Sr 0 .

Since Arr is the identity matrix, then:

R Sr Anr r Sn r .

For given displacements of the support links, the deformation vector of the rods is determined by the expression:

c Anr r z .

The order of the vector z is r .

To take into account the considered exposures, the physical equations should be written in the form:

14.9. Calculating the Bar Systems. General Equations.

The Mixed Method

Equations of equilibrium (14.3), geometric (14.4) and physical equations (14.6) together form a common system of equations for calculating a linearly deformable bars system. Imagine them in the following form:

The sought-for quantities in (14.7) are the n – dimensional force vector S, the m – dimensional displacement vector z, and the n -dimensio- nal strain vector . Total unknowns – (2n m). The number of equations in the system is also equal to 2n m : equilibrium equations – n,

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geometric equations – m, physical equations – n. Therefore, this common system of linear independent equations has a unique solution. This

means that the exposures F and acting on the structure, according to the solution of the system of equations, cause a single picture of the distribution of forces, displacements and deformations in it. Such a system of determining mathematical relationships is called the mathematical model for calculating the bars system.

The order of the system of equations (14.7) can be reduced. For ex-

ample, if we find the strain vector from the third group of equations and substitute it into the second group of equations, then the system of equations (14.7) is transformed to:

The forces and displacements are unknown in this version of the mathematical model. Therefore, the system of equations of the form (14.8) or (14.9) is called the system of equations of the mixed method.

Solving the equations of the mixed method allows you to find the forces in the bars of the system and the displacements of its nodes.

14.10. Displacement Method

We represent the equations of equilibrium in displacements. If there are no infinitely rigid elements in the bars system, the quasi-diagonal matrix D is a nonsingular matrix; its determinant is nonzero. Therefore,

from the second group of equations (14.8) we can find the vector S:

S D 1 A z K A z ,

where K is the matrix of the internal stiffness of the bars system.

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