Structural mechanics

From the first of them it follows that if det A 0 , then:

S A 1 F .

The condition that the determinant of the matrix A is equal to zero is a sign that the calculated system is partially geometrically variable or instantly variable.

The second group of equations allows you to calculate the displacement vector z:

z A 1 T D S .

In the absence of external load, we obtain:

These relations confirm the well-known property of statically determinate systems: a change in temperature, displacements of supports or inaccuracy in the manufacture of elements in statically determinate systems do not cause internal forces, but cause only displacements.

14.13. General Equations for a Bar

Consider a frame loaded with a nodal load (Figure 14.16, a), and a fragment of its discrete scheme (Figure 14.16, b). The directions of the nodal forces and the forces of interaction in the sections shown in the figure correspond to the directions of the axes of the general coordinate system.

We establish the relationship of the load in the nodes i, j and the ef-

forts in the sections adjacent to the nodes. This dependence is easier to obtain first in the local coordinate system (for the bar i j – the system

), and then, using the rules of linear transformations, in the general system XY.

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Figure 14.16

The directions of efforts in the end sections of the bar and nodal forces oriented along the axes of the local coordinate system are shown in Figures 14.17, a, b, c.

Figure 14.17

In general, the efforts vectors at the beginning of the bar

SB NB , QB , MB T

and at the end of it

SE NE , QE , M E T

contain three components each. In relation to the bar, these forces are external and dependent; they are connected by three equations of equilibrium:

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Through a* , the bar equilibrium matrix is denoted in the local coordinate system:

Upon transition to the general coordinate system, the equilibrium equations of the bar (14.15) are transformed.

Consider the problem of transforming the coordinates of the vector of nodal forces in the transition from a local coordinate system to a common one.

From the equations of projections of linear forces in the i-th node on the axis of the general coordinate system (Figure 14.18) it follows that:

Fix Fi cos Fi sin ,

Fi y Fi sin Fi cos .

Figure 14.18

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Given that the moment mi remains unchanged when the coordinate

system is rotated, we present the expression for the transformation of the forces of the i-th node in the form:

where CT is matrix of the rotation operator when the vector is rotated through an angle clockwise.

Through C it is customary to denote the matrix of the operator of rotation of the vector counterclockwise.

Similar relations hold for forces in the j-th node:

Fj CT Fj* .

Then, for the load vector in the nodes connected by the bar, the rotation transformation will be performed using the expression:

So, if equality (14.15) is multiplied on the left by the matrix V T , then in the general coordinate system the vector of nodal forces F will be expressed through the vector of efforts S in the following form:

where a is a bar equilibrium matrix in the general coordinate system, i.e.:

In this form, the equilibrium matrix is written for the bar with both pinched ends. As follows from (14.18), equilibrium matrices for bars with other conditions of supporting the ends can be obtained from this one by deleting rows and columns corresponding to zero forces in the bar.

In particular, if the left end of the bar has a hinge support ( M B 0 ),

and the other end is pinched, the equilibrium matrix is obtained from the original by deleting the second column and the third row. For bars with different options of support fastenings, the equilibrium matrices in the general coordinate system are written in the Table 14.2.

Let us determine the relationship between the deformations of the bar and the displacements of its ends. We write the displacement vector in the general coordinate system for the rod rigidly fixed at the ends in the form:

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z zBx , zBy , B , zEx , zEy , E, T ,

where, as before, the indices “ B ” and “ E ” denote the beginning and ending of the bar. The displacements of the bar ends are the displacements of the nodes that it connects.

Figure 14.19 shows the initial and deformed positions of the bar in the local coordinate system.

Figure 14.19

Table 14.2

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