Structural mechanics

where Fk 1;

F are the forces acting in the state a (this force is a uniformly distributed load q in Figure 8.19, a);

Fk is the displacement caused by Fk 1 in the direction of force F, (in this example, the area of the diagram of vertical displacements of the horizontal element).

Figure 8.19

Since the diagrams in the states a (Figure 8.19, a) and c (Figure 8.19, c) coincide completely, the expression (8.26) is applicable to the frame states b (Figure 8.19, b) and c. In this case, as F, in Figure 8.19, c, the distributed load and the primary unknowns X1 and X2

are accepted. But the work of the primary unknowns on the displacements of the frame in the state b is equal to zero. Therefore:

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i. e., the right side of the expression (8.27) is the work of external forces applied to the primary system. This work is done on the displacements of a statically indeterminate system in state k.

Note that in the above explanations, there were no restrictions on the choice of the primary system.

Writing the expression (8.27) through the work of bending moments, we obtain:

where M F0 is the bending moments diagram in the primary system (Fi-

gure 8.19, e).

Thus, when determining displacements in statically indeterminate systems, one of the “multiplied” diagrams can be built in a given statically indeterminate system, and the second – in any statically determinate one obtained from a given system.

Let us turn to the calculations. In Figure 8.20, a diagram of bending moments in a statically indeterminate frame caused by a given load is shown, and in Figure 8.20, b – diagram of bending moments in the same

In Figure 8.20, c the diagram of moments in a statically determinate frame (primary system) caused by Fk 1 is shown, and in Figure 8.20, d –

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plot of moments in the primary system caused by a given load. By the formula (8.25) we get:

Figure 8.20

According to the formula (8.28):

It is clear that the calculations of displacements using formulas (8.25) or (8.28) are simpler than using the formula (8.24).

8.10. Calculating Frames Subjected to Change of Temperature and to Settlement of Supports

When calculating the frames subjected to the thermal effect, the canonical equations of the force method are recorded in the form:

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To calculate the free terms of the equations, formula (7.12) is used. In statically determinate systems, there are not the efforts caused by

the action of the temperature. Therefore, the final diagram of bending moments in a given frame is constructed by summing up unit diagrams of moments multiplied by found from the equations values of corresponding unknowns:

Kinematic check comes down to the verification of the frame displacements in the direction of redundant constraints, i. e. checking the condition:

When calculating the frames subjected to the settlements of supports, the canonical equations are written in the form:

The free terms of the equations are calculated, in the general case, by the formula (7.13).

E x a m p l e. Construct diagrams M , Q and N caused by the ac-

tion of temperature change in the frame (Figure 8.21, a). The height of the cross-section of the elements AC and BD equals to h1 0.3 m, the

element CD equals to h2 0.4 m. The coefficient of thermal linear

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expansion of the material equals to 1.2 10 5 1/( C), the bendimg rigidity is EJ 60 МN·m2.

Figures 8.21

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The primary system in the initial and deformed states is shown in Figure 8.21, b. The coefficients of the canonical equations will be determined taking into account the influence of only bending moments. Using

responding segments are shown in Figure 8.21, b) using formula (7.12), we write the used values of the calculating parameters in the Table 8.1.

Table 8.1

Recall that in the calculations by formula (7.12) each term in it is assumed to be positive in the case when the corresponding directions of the elements deformation caused by unit forces and thermal action coincide.

Following the calculation algorithm (section 8.7), we write the system of canonical equations in numerical form:

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The static indeterminacy of the frame disclosed. There is not the diagram of moments caused by the external exposure in the statically determinate primary system subjected to the thermal effect.

Therefore, we construct the final diagram of bending moments by the expression:

M M1X1 M2 X2 .

This diagram is shown in Figure 8.21, h. Bending moments in the frame depend on the values of the rigidity of the elements, i. e., one of the general properties of statically indeterminate systems is confirmed (Section 8.1). In parentheses are the ordinates for the initial data accepted in the example.

We are performing a kinematic verification. The total diagram of unit moments Ms is shown in Figure 8.21, g.

5636.67 6250.0 0.

The condition (8.30) is satisfied. The diagrams Q and N are shown in Figures 8.21, i, k.

E x a m p l e. Construct diagrams M , Q and N in the frame subjected to the settlements of supports indicated in Figure 8.22, a. It is as-

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sumed that the rigidity of the frame elements equal to EJ 60 МN·m2, and the settlements of supports equals to c1 c2 c 0.01 m.

Figure 8.22

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The given frame is twice statically indeterminate. Selecting the primary system of the force method (Figure 8.22, b), we write the canonical equations in the form:

11X1 12 X2 1c 0;21X1 22 X2 2c 0.

We will construct the unit moments diagrams (Figures 8.22, c, d) and calculate the coefficients of canonical equations:

the formula (7.13) we get:

1c Rk1ck 1 c1 2.5 c2 c1 2.5 c2 3.5 c;

2c Rk 2ck 0.5 c2 0.5 c2 0.5 c.

The canonical equations, after simple transformations, get the following form:

Since the displacements of the supports does not cause efforts in a statically determinate system the final diagram of the bending moments is constructed by the expression:

M M1X1 M2 X2.

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It is shown in Figure 8.22, f. In parentheses there are the values of the ordinates of the moments for the accepted source data. Kinematic verification, as when calculating the thermal effect, is reduced to verifying the fulfillment of the condition:

MMs dx ic 0 . EJ

We will check it using the total unit diagram Ms (Figure 8.22, e):

3.5 c 0.5 c 0.

The check is performed. The diagrams Q and N are shown in Figures 8.22, g, h.

8.11. Influence Line for Efforts

To construct the influence line for any effort, it is necessary, first, using the well-known methods of structural mechanics, to obtain the dependence (analytical or numerical) of this effort due to the position of

the force F 1 S f x , and then, using this dependence, determine

the ordinates of the influence line for all characteristic sections.

If the methods of statics are used to determine the dependence S f x then the corresponding method of constructing the influence

line is called static one.

In statically indeterminate systems, the effort in the cross-section of the element is determined by the expression (8.14). If it is used to con-

struct influence lines mind, that the values of the primary unknowns Xi

and the value of the effort in the cross-section k of the primary system change due to the moving load F 1. Therefore, the expression (8.14) for constructing the influence line for the effort in the cross-section k should be rewritten in the form:

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