Structural mechanics

help of certified software systems for many unit loads and compiling an appropriate influence matrix.

By definition, Sik is the force in the i -th section of the structure caused by a unit force applied in section k. Elements of the i -th row of the matrix LS give the ordinates of the influence line for effort Si . In

order to find the elements of the k-th column of the influence matrix, it is necessary to calculate the given structure for loading it by force Fk 1.

The number of such unit loads is n.

The following example will explain the features of calculating and plotting the influence line for efforts using the matrix of the influence of efforts.

313

The design diagram of the frame is shown in Figure 9.39, a. The shape of any influence line in the area between adjacent nodes can be represented by the values of the ordinates of efforts in three equally spaced sections. Therefore, we will calculate the frame by loading it with forces F 1 applied sequentially in each characteristic section between the nodes and compose, for example, the matrix of the influence of bending moments LM . This matrix has the form:

According to the second and fifth rows of matrix LM , Inf. Line M2 (Figure 9.39, b) and Inf. Line M5 (Figure 9.39, c) are constructed. Since

the stiffness of the rods in tension-compression in this calculation is assumed to be infinity, i.e. their longitudinal deformations are neglected, then at the points corresponding to nodes A and B, the ordinates of the influence lines are zero.

Of course, it is advisable to perform calculations of such and more complex systems for many loads with the help of software systems available in design organizations.

In the same way it would be possible to compose matrices of the influence of transverse or longitudinal forces and construct the necessary influence line for these efforts. Figure 9.39, d shows the influence line for Q5.

The calculation results also allow one to obtain influence matrices of displacements. Let us pay attention to the line of influence of vertical displacement of section 2 (Figure 9.39, f) and influence line for the angle of rotation of node B (Figure 9.39, e). Their shape is consistent with the recommendations of section 7.11: to obtain the first one should use force F = 1 applicated in section 2; in the second case, at point B, moment М = 1 is applied (directed counterclockwise). The numerical values of

the ordinates correspond to EJ = 13.5 МН м2.

314

9.12. Calculating Frames Taking into Account Longitudinal Deformation of the Bars

As noted in Section 9.1, each rigid node of a plane frame has three degrees of freedom, each hinged node – two. For this reason, the degree of kinematic indeterminacy of the frames, in the calculation of which the longitudinal deformations of the bars are taken into account, is much higher than the frames, in the calculation of which the longitudinal deformations are neglected.

In order to form the primary system of the displacement method, three additional links are superimposed on each rigid node of a given system, and two linear links on each hinged one. To determine the coefficients and free terms of the canonical equations, it is necessary to construct diagrams of bending moments and longitudinal forces in the primary system, caused by unit displacements of additional links and given exposures.

The values of the coefficients and free terms are found by the static method from the equilibrium equations of the nodes of the main system.

The same values of the coefficients of the canonical equations of the displacement method can be found with kinematic method by the formula:

rik Mi Mk dx Ni Nk dx . EJ EA

The second term in this expression is calculated in the same ways as the first, in particular, for example, by “multiplying” the corresponding diagrams of longitudinal forces.

The influence of this term on the value of rik, as follows from the above expression, increases with a decrease in the stiffness of the bars on tension-compression.

Some features of calculating the coefficients and distribution of bending moments will be shown on the example of the frame shown in Figure 9.40 a.

The primary system and the positive directions of the primary unknowns are shown in Figure 9.40, b. We restrict ourselves to considering

diagrams M1, N1 and M F (Figures 9.40, c–e).

From the equilibrium condition of node 3 (Figure 9.40, f)

316

Naturally, the same value r11 will be obtained by the kinematic method:

To determine the free terms of the canonical equations, one should use the distribution of forces at the nodes shown in Figures 9.40, g, h, or use the kinematic method. In the latter case:

where Mi , Ni – functions of bending moments and longitudinal forces

from unit displacements of nodes in the primary system of the displacement method;

M F0 , NF0 – functions of bending moments and longitudinal forces

from the given load in the primary system of the force method.

The influence of longitudinal deformations on the distribution of internal forces in the frame can be judged by the final bending moments diagrams constructed at the ratio

EA h2 10,

EJ

where h = 1 m (Figure 9.40, i), and at EA (there are no longitudinal deformations) (Figure 9.40, j). At

EA h2 10,

EJ

the calculation results will be even more different from those that correspond to the calculation option with EA .

An increase in the flexibility of the bars under tension-compression leads to an increase in the displacements of the nodes, and therefore, the calculation of such frames according to an undeformed scheme should be considered as approximate.

318

THEME 10. SIMULTANEOUS APPLICATION OF THE FORCE AND DISPLACEMENT METHODS. MIXED METHOD

10.1. Force and Displacement Methods in Comparison

Both the force method and the displacement method have its advantages and disadvantages. Each of them, taking into account the assumptions used in the calculation, is accurate. In both methods, it is also possible to take into account the influence of longitudinal and shear deformations in addition to bending deformations. Which one should be used for calculation?

With manual calculations, the search for the best method for calculating of the given system is reduced, in most cases, to the search for a calculation variant with the least laboriousness. Most often the choice of one or another method depends on the number of unknowns.

As a rule the best way to do the calculation of frames, the nodes of which do not have linear mobility is by the displacement method. Diagrams of effort are easy to construct, have local character and, because of this, the system of canonical equations is rarefied. However, when longitudinal deformations are taken into account the number of displacement method unknowns increases significantly.

The choice of a rational primary system of the method of forces and the construction of diagrams are associated with a more complex logic of understanding the structure of the system. The operation to calculate the coefficients and free terms of the canonical equations is also quite timeconsuming. In the displacement method, this part of the calculations, carried out, for example, in a static way, is less time-consuming. As an advantage of the force method, we note that the degree of static indeterminacy of a given system does not depend on whether or not the influence of longitudinal deformations is taken into account in the calculation.

The above remarks on the methods discussed are a qualitative characteristic of them. Note, in addition, that in each particular case, the engineer has the right to choose any of them, guided by their own level of knowledge of these calculation methods.

The decision to choose a method for automated computing is associated not so much with the computational procedures for each of them, but with the features of the primary system. The logic of automating the process of choosing the primary system of the displacement method is

319

simpler. The ideas of the displacement method are widely used in the development of software systems for calculating and designing building structures.

10.2. Force and Displacement Methods.

Simultaneous Application

To choose a rational calculation method an engineer must deep understand the main principles of the force and displacement methods. In particular, joint application of these methods is possible for the calculation of both symmetric and asymmetric systems.

Let us first consider the features of the calculation of symmetric systems. The load applied to such a system can always be decomposed into symmetrical and inverse-symmetric (otherwise, skew-symmetric) components. As a rule, it turns out that it is convenient to calculate a frame under the symmetric load component by one method, for example, by the displacement method, and it is convenient to calculate the same frame under the inverse-symmetric load component by the other method, for example, by the force method. The final result of calculating the frame for a given load is obtained by summing the results of its calculation on both load components. Now let us consider an example.

The frame (Figure 10.1, a) has four unknowns by the displacement method (the primary system and the primary unknowns are shown in Figure 10.1, b) and four unknowns by the force method (Figure 10.1, c). The symmetric and inverse-symmetric components of the given load are presented in Figures 10.1, d, e. The calculation of the frame due to the action of the symmetrical load is performed by the displacement method, since in this case the only unknown, not equal to zero, will be unknown Z1. To calculate the frame on the inverse-symmetric load, we use the

force method, because in this case only X 3 will be non-zero.

The bending moment diagrams corresponding to the action on the frame of the symmetrical and inverse-symmetric loads are shown in Figures 10.1, f, g. And the final diagram of bending moments is shown in Figure 10.1, h.

The presented version of the frames calculation in the educational literature is sometimes called the combined method of calculation.

320