Structural mechanics

cement of supports or other exposures, the required total displacement is determined by summing the components from each exposure separately.

The features of determining displacements in statically indeterminate systems will be described below.

7.10. Matrix Form of the Displacements Determination

Сonsider this question in relation to the plane trusses. In practical problems of trusses calculating, it is important to be able to determine the displacements of each node in horizontal and vertical directions. The total number of unknown displacements with this approach will be equal to the number of degrees of freedom of the nodes m 2N L (there are no displacements of nodes in the directions of the support links). In Figure 7.34, a unknown displacements of nodes are shown by arrows.

Figure 7.34

To determine the displacement i we take the auxiliary state as shown in Figure 7.34, b: load Fi 1 is applied in the direction of the required displacement. In this figure, a designation of the force Nki ari-

sing in the rods is shown near each rod of the truss, where the index k corresponds to the number of the rod. The index n corresponds to the number of the last truss member.

From formula (7.6) it follows that

where Nki is the force in the k -th rod caused by Fi 1;lk is absolute deformation of the k -th truss rod.

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An expanded record of the last expression with respect to all calculated displacements will appear as the following equations:

where is the vector of nodal displacements;

LTN is the matrix transposed with respect to the influence matrix LN ;

l is the vector of absolute deformations of the rods.

For statically determinate truss m 2N L B, that is m n and in this case the matrix LN will be square.

So, in order to find the displacements of the truss nodes, it is necessary to know the deformations l of the rods, determined in accordance with the action set on the system.

When the temperature changes:

lk tk lk ,

where is the coefficient of linear thermal expansion; tk is the temperature change of the k -th rod.

If there are displacements due to inaccuracy in the manufacture of the rods, lk is determined as the differences between the real and design

values of the lengths of the rods.

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When calculating a physically nonlinear system under the action of a load F, it is possible, using a nonlinear tensile (compression) diagram,

to determine the corresponding elongation (shortening) lk by a known

Then for the vector of deformations caused by a given load, there is a dependence:

Substituting expression (7.15) into formula (7.14), we obtain a matrix notation of the formula for determining the nodal displacements of the truss due to the load F:

To determine the displacements of bended systems due to the load F, we will use the Simpson's formula. At the k-th section of the bar with

variable bending rigidity, the Mohr's integral is written in the form:

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where the superscripts B, M and E indicate the values Mi , M F , ...

and EJ at the beginning, middle and end of the integration section. We represent this expression in matrix form:

and then, at EJ const, the computations in the section are reduced to:

Summing up the results of calculations for all sections, we obtain:

Using the sequential docking of the bending moment vectors in all n parts of the system and introducing the matrix of pliability D for the entire system into the calculation, the displacements calculation can be represented as follows:

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If it is necessary to determine the displacements of several points of the system, the row-vector LTi should be replaced by a matrix LT , in

each row of which values of bending moments caused by the i-th auxiliary state are recorded.

If the problem is to determine the displacements caused by different loadings, it is necessary to replace the vector M F with a matrix, in each

column of which values of efforts corresponded to a certain load are recorded.

With these remarks, the expression for determining the displacements of a bended system in the general case can be written as:

In this expression, the index m corresponds to the number of determined displacements for one loading, the index t corresponds to the number of independent loadings.

If M L, the matrix will be a matrix of external pliability A of the flexible bars system:

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The same remark applies to formula (7.16). Replacing the vector NF with the matrix N LN , as a result of the calculations we obtain the truss pliability matrix:

7.11. Influence Lines for Displacements

The theorem of reciprocal displacements is used to solve various problems in mechanics. In particular, the influence lines for displacements are relatively easy to obtain. Suppose, for example, it is necessary to construct the influence line for the rotation angle k (Figure 7.35, a).

Each new position of the unit force (Figure 7.35, b) corresponds to a certain value of the rotation angle ( k1, k 2 , ...). At the same time, on the

basis of the reciprocity theorem, these displacements can be determined each time by uploading the beam with a fixed generalized force Mk 1

(Figure 7.35, c). Consequently, the shape of the influence lines for k

coincides with the diagram of the vertical displacements of the beam axis caused by force Mk 1. The equation corresponding to this load for the

bent axis of the beam is written in Section 7.5.

Figure 7.35

An analysis of the results of the last example (Figure 7.35) shows that the practical task of constructing influence lines for displacements of a

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linearly deformable system, on the one hand, can be associated with its calculation on the set of unit loads in characteristic sections, and then with the determination of the required displacement for each of them. On the other hand, this task may be connected with the calculation of the system for one load and the determination of the corresponding displacements in those cross sections in which the unknown shape of the influence line can be represented by the found displacements. The second solution is generally preferred.

We illustrate it with the example of a multi-span statically determinate beam (Figure 7.36), for which we will construct the influence line for 3. From the calculation of the loading beam by force F1 1 we can

find only one ordinate 31 of the influence line for 3 (Figure 7.36, b), from the calculation at the action of the force F2 1 we can find the ordinate 32 and so on. A simpler technique is to construct an influence line 3 as a diagram of vertical displacements of the axis of the beam from the action of the force F3 1 (Figure 7.36, c). In Figure 7.36, d it is shown the view of Inf. line for 3 taking into account generally accepted

construction rules: positive ordinates are located above the axis of the beam, negative ones are below.

Figure 7.36

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7.12. Influence Matrix for Displacements

The vertical displacement, due to the given load, of the cross-section i, for which the influence line for displacement is constructed, can be calculated by the formula:

iF i1 F1 i2 F2 in Fn ,

where F1, F2 , ..., Fn – are concentrated vertical forces applied in char-

acteristic sections.

With the value of the index i = 3 we get the expression for calculation3F using the influence line (Figure 7.36, d).

Applying the expression for iF to each characteristic cross-section and using the matrix form for recording the transformations, we obtain the value of the displacement vector F :

is the influence matrix for displacements.

The components of the k-th column are the ordinate values of the displacements diagrams constructed due to Fk 1, which corresponds to the

general definition of the influence matrices. Since the conditions ik ki

are fulfilled, the matrix A is a symmetric matrix and, therefore the influence lines for i can be constructed from the elements of the i-th column

or i-th row.

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In the case of systems of arbitrary outline, not necessarily the beams, displacements ik may have different orientations in space. They deter-

mine the pliability of the system at some point i in a given direction (i- th) caused by the unit force applied at a point к. Therefore, the matrix A is called the pliability matrix of the system. To calculate it, one can use formulas (7.20) and (7.21).

E x a m p l e . Calculate the matrix A of the external pliability of the frame in the given directions (Figure 7.37).

Figure 7.37

Diagrams of bending moments caused by the action of unit forces in given directions are shown in Figure 7.38.

Figure 7.38

When compiling the influence matrix LM , we will consider the ordinates of the diagrams M , located inside the frame contour as positive.

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The pliability matrix A is calculated as follows:

4

6EJ

4 4

6EJ

4

6EJ

5

6 2EJ

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