Structural mechanics

Table 9.1 (сontinuation)

271

Table 9.1 (ending)

9.3. Canonical Equations

The forces in the elements of the primary system change when it is transfered into a position corresponding to the deformed position of the given system. From the given effect reactions arise in additional links of the primary system. If displacements equal to the displacements in the corresponding directions of the given system are given each additional angular and linear link, then the reactions in the additional links must be equal to zero.

Consequently, reactions in the additional links are functions of nodal displacements and loads, and the condition of static equivalence of the primary and given systems is reduced to equations of the form

272

where Ri is the complete reaction in the i -th additional link caused by

displacements and load.

The number of such equations is naturally equal to the total number of displacement method unknowns.

Based on the principle of independence of the action of forces, functional dependence (9.1) can be represented as

where Rik is the reaction in link i caused by the true value of the displacement of link k k 1,n ; RiF is the reaction due to load.

Substituting (9.3) into equation (9.2) and accepting i 1, 2, , n, we obtain the following system of linear equations:

These equations are called the canonical equations of the displacement method. As follows from the previous reasoning, the physical meaning of i-th equation is that the total reaction in this additional link caused by displacements Z1, Z2 , , Zn and a given external load has

to be zero.

273

Coefficients (reactions) r11, r22 , , rnn located on the main diagonal are called the main reactions; coefficients (reactions) rik (i k) are called

secondary, and free terms R1F , R2F , , RnF are called load reactions.

When the structure is exposed to temperature changes, the free terms of the equations are replaced by R1t , R2t , , Rnt , and when there are

support shifts – by R1c , R2c , , Rnc.

When determining the reaction in i-th additional link, its positive direction should coincide with the positive direction of displacement Zi

adopted in the primary system.

In the matrix notation, the equations (9.4) have the form:

is matrix of coefficients of canonical equations (system rigidity matrix in the directions of additional links);

Z Z1, Z2 , , Zn T

is matrix (vector for one load option) of primary unknowns;

RF R1F , R2F , , RnF T

is matrix (vector for one load option) of the free members of the canonical equations (load reactions).

9.4.Static method for determining the coefficients and free terms of canonical equations

Coefficients and free terms of canonical equations are reactions in additional links. To determine them, it is necessary to know the distribu-

274

tion of forces in the primary system due to the unit displacements of these links and due to the load.

Plotting diagrams of bending moments from these effects are shown on the example of a two-span frame (Figure 9.14, a).

The same figure also shows the possible deformed state of the frame, which allows you to visually determine the number of primary unknowns: one angular and the other linear. We confirm, however, determining the number of unknowns according to the general rules. The frame has one rigid node. As follows from the kinematic analysis of the hinge-rod system (Figure 9.14, b), the degree of linear mobility of its nodes is also equal to one. The possible direction of movement of the nodes in the figure is shown by the arrow . The total number of

unknowns is equal n na nl 1 1 2 . The primary system is shown in Figure 9.14, c.

Figure 9.14

Plotting the bending moment diagrams is performed using the data in table 9.1. Diagram M F (load diagram) and diagrams M1 , M2 (unit

diagrams) are shown in Figures 9.14, d, e, f.

In addition, in Figures 9.14, e, f the dashed line shows the curved axes of the bars, which allow you to set the position of the stretched fibers on each of them and correctly depict the plot of the moments. The same figures show the reactions in the additional links. Their directions are

275

accepted as positive (in the directions of the positive displacement of the links). Recall that in the designation of the reaction rik , the first index

i indicates the number of the link in which the reaction occurs, and the second k indicates the number of the displacement that caused this reaction. In the designation RiF , the second index F means that the

cause of the reaction is load F.

To determine the reactions by a static way equilibrium equations are used. In particular, since only a moment can occur in a floating support, to determine it, one should use the equation of equilibrium of the formM 0. So, to determine R1F , we will show the internal forces acting

on the node in the cut-out state (Figure 9.15, a), and draw up the equation:

R1F q28l2 q18l2 0,

from which we find

R1F q18l2 q28l2 .

Figure 9.15

276

The equilibrium equation for determination of the unit reaction r11 (Figure 9.15, b) can be written as follows:

r11 6EJl 6EJl 4EJh 0 .

Therefore,

r11 12EJ 4EJ . l h

An equation also in the form of the sum of moments relative to the node (Figure 9.15, c)

M r12 6hEJ2 0

is used to determine the unit reactive moment r12:

r12 6hE2J .

Reactions in additional links that impede linear movements of nodes are determined from the equilibrium conditions of a frame fragment. All external and internal forces acting on the fragment, except the reaction to be calculated, must be known.

For the example under consideration, when determining the reactions R2F , r21, r22 , such fragments can be the frame diagrams shown in Fi-

gures 9.15, d, e, f.

Writing down the corresponding conditions of equilibrium of forces shown in each of these figures, we obtain equations for determining unknown reactions

In particular, the load reaction R2F (Figure 9.15, d) is determined by the equation

277

the secondary unit reaction r21 (Figure 9.15, e) by the equation:

and the main unit reaction r22 (Figure 9.15, f) by the equation:

9.5. Kinematic Method for Determining the Coefficients and Free Terms of Canonical Equations

Consider some basic system of the method of displacements in unit states “ k ” and “ m ” (Figure 9.16, a, b). The work of the external forces of state “ k ” on the displacements of state “ m ” is:

Wkm rmk 1.

It is known that the work of external forces Wkm is equal (with a minus sign) to the work of internal forces. Therefore, expressing the work of internal forces through bending moments Mk in state " k " on the cor-

responding deformations Mm dx of the frame in state " m ", we obtain:

EJ

Calculation of integrals of the form

Mk Mm dx EJ

reduces to numerical integration, in the simplest cases – to “multiplication” of the bending moments diagrams.

278

Having determined the work of the external forces of state “ m ” on the displacements of state “ k ” we get:

Wmk rkm 1.

Based on the reciprocity theorem, we get:

Wkm Wmk ,

It is a formal record of the reaction reciprocity theorem (the first theorem of J. Rayleigh (1842–1919)): the reaction in link “ m ” from the unit displacement of link “ k ” in its direction is equal to the reaction in the link “ k ” from the unit displacement of link “ k ” in its direction.

Consider two more frame states. In the first state " k ", unit displace-

Work of the external forces of state “ k ” on the displacements of state “ i ” (there are no movements of the nodes) is zero:

Wki 0 .

Therefore, the work of internal forces is equal to zero too:

Mk Mi dx 0 . EJ

However, the work of the external forces of state “ i ” on the displacements of state “ k ” is:

Wik 1 ik rki 1 .

By the reciprocity theorem we obtain:

Wik Wki 1 ik rki 1 0.

280