Consider two various states(in the order of loading) of the same elastic system: state 1 under the action of a group of forces and state 2 under the action of a group of forces on the example of a beam in Fig. 33, a. Let us define and compare the work of external forces under the following assumptions. First, the system is gradually loaded by the forces of state 1, and then, when the forces reach the final value, the system will be gradually loaded by the forces of state 2. In the second variant, the sequence of application of forces changes. First, the system is loaded by the forces of state 2, and then by the forces of state 1. Let's assume that, first, the load of the first state gradually began to act on the system, and then the second. The total work of external forces will be expressed by the algebraic sum .

Consider now the application of the load in reverse order, when the load of the second, and then the first state is applied first. In this case, the total work of external forces is expressed by the following algebraic sum: , where is the work of external forces of the second state on the displacements caused by the action of the forces of the first state.

According to expression (63), the total work W external forces is equal in absolute value to the work A internal forces, taken with the opposite sign, or the potential energy of deformation U.

It is known that in a linearly deformable system, the potential energy of deformation does not depend on the sequence of application of external forces, but depends only on the initial and final states of the system. Since the initial and final states of the system are the same in both cases of loading, the total work of external forces will be equal, i.e. or from where

The resulting analytical dependence expresses a theorem on the reciprocity of work and is formed as follows: in a linearly deformable body, the possible work of external or internal forces of the first state on the displacements of the points of their application, caused by the action of the forces of the second state, is equal to the possible work of external or internal forces of the second state, displacements caused by the action forces of the first state. This is the so-called Betty-Rayleigh theorem.

The displacement reciprocity theorem can be represented as a special case of the work reciprocity theorem. Let only one unit force act on the beam in the first state, and also one unit force in the second state (Fig. 34, a, b). The force is applied at point 1, and the force is applied at point 2. Based on the theorem on the reciprocity of work, we equate the possible work of external forces of the first state on displacements of the second state to the work of forces of the second state on displacements of the first state:

This is an analytical expression for the theorem of reciprocity of displacements, which is formulated as follows: the displacement of the point of application of the first unit force in the direction, caused by the action of the second unit force, is equal to the displacement in the direction of the second unit force, caused by the action of the first unit force, this is the so-called Maxwell's theorem, which is of fundamental importance in structural mechanics.

Figure 34 - Determining the reciprocity of movements

Literature:

Main: 6[bit.3: from 29-31; section 5: from 36-47].

Control questions:

1 Why is it necessary to reduce the size of the panels and for what purpose are additional two-support truss trusses introduced, and how many and what categories are distinguished in truss trusses, and how are the forces in the elements of the main and additional trusses determined?

2 What functions express deformations (displacements) in elastic systems and how can this be written analytically, and under what assumptions, name them, displacements and deformations of the considered elastic systems obey the law of independence of the action of forces?

3 Why analyze the work of external and internal forces of an elastic body and what concepts are used in structural mechanics, and also by what dependence is the work of deformation of the elements of a structure determined under the static application of external forces, define Claiperon's theorem?

4 By what dependence is the work of all external forces acting on the beam determined and through what forces can the work of the internal forces of an elastic rod system be expressed?

5 By what dependence is the total work of internal forces determined and why is the work of external and internal forces called possible?

6 What analytical dependence expresses the work reciprocity theorem and how is it formulated (Betty-Rayleigh theorem)?

5 Single state o.S.

The horizontal displacement of t. B, caused by the action of forces, is equal to zero.

According to Maxwell's theorem on the reciprocity of displacements: , i.e. mutual vertical displacement of sections C 1 and C 2 caused by the action of force , equals zero.

To determine the displacement ∆ 3 F , it is necessary to build and multiply diagrams of the forces and.

1 1 P

D

3 Single state o.S.

Cargo condition O.S.

Solving the system of canonical equations, they find extra unknowns x i. The final diagram of bending moments is built using the principle of superposition.

To check the correctness of the diagram M, static and kinematic checks are used. Static consists in checking the balance of all frame nodes isolated from the structure and under the action of bending moments in converging rods and external moments applied in the nodes. For example, for a plot M F we get for the node D:

D - node D is in equilibrium.

The kinematic check consists in the absence of total displacements in a given system in the direction of the discarded bonds:

those. it is necessary to multiply each of the single diagrams by the final diagram M. If zero is not obtained, then at least one calculation error has been made (there are several such errors during the first calculation). In order to avoid uncertainty in finding an error, a system of step-by-step intermediate checks has been developed.

The diagram of the transverse forces Q for simple frames is built using the section method. For similar frames, the Q diagram is built by cutting out individual frame bars and then considering their equilibrium under the action of external loads and internal forces at the ends of the bars. Since the loads are known, the bending moments can be taken from the final diagram M, and the longitudinal forces do not participate in the formulation of the equilibrium equations, it is possible to calculate the transverse forces at the ends of the rods.

Consider a rod ij:

y P q M

M ij i j M ij

N ij

N ij

Q ij Q ij

The plot of longitudinal forces N for a simple frame can be plotted using the section method. For a similar frame, the plot N is built by considering the balance of the cut out nodes under the action of active loads of transverse forces taken from the plot Q, and longitudinal forces. It is necessary to sequentially consider nodes in which no more than two longitudinal forces are unknown. For example, for node k we get:

P 1 N kj j

P 2 K Q kj

Q ki

N ki

For a static check of the entire frame as a whole, it is necessary to apply all the support reactions and draw up three equilibrium equations that must be identically fulfilled ():

Movement method

Let us consider an alternative method to the method of forces for revealing the static indeterminacy of rod systems, called the method of displacements. In the method of forces, reactions and (or) internal forces in extra bonds are taken as unknowns, which are found from the equality to zero of displacements in the direction of discarded bonds. In the displacement method, the displacements of the moving nodes of the structure are taken as unknowns, which are found from the equality to zero of the reactions in imaginary support links that prevent the nodes from moving: in the force method, some of the links are discarded, and in the displacement method, on the contrary, a certain number of new links are introduced. At first glance, it seems that we complicate the task by introducing additional connections, but thanks to the original approach, this is not so. The fact is that by introducing a number of virtual connections into a real structure, we get a set of basic cases of loading beams used in the calculation of a large variety of bar systems. This approach lends itself easily to EMW programming.

Consider a simple U-shaped frame and imagine a possible scheme for its deformation under the influence of external loads, taking into account the following simplifying assumptions:

The rods bend when bent, but do not change their length;

Rigid nodes are rotated so that the angles between adjacent bars do not change.

Rigid nodes D, E, F, G will rotate through some angles θ 1 -θ 4 and move horizontally by ∆ 1 and ∆ 2 . Because rods are not extensible, then DD 1 =EE 1 =∆ 1 and FF 1 =GG 1 =∆ 2. Thus, the total number of unknowns is equal to the degree of kinematic indeterminacy n k =n y +n l =4+2=6.

The number of corner unknowns n y is equal to the number of rigid frame nodes. The number of linear unknowns n l is equal to the number of degrees of freedom of the hinged model. n l \u003d W sh.m. \u003d 3D-2U w -C \u003d 3 * 6-2 * 8-0 \u003d 2.

Choose main system displacement method, introducing virtual (imaginary) terminations in rigid nodes that prevent rotation, and linear connections at nodes E, G, preventing horizontal movement.

If we now rotate the virtual terminations at angles θ 1 -θ 4 and shift the linear connections by ∆ 1 and ∆ 2 , and in addition apply external loads, then we get equivalent system , fully adequate to the given system both in the kinematic sense (the corresponding displacements are equal) and in the static sense (the corresponding reactions in real and virtual links are equal). We denote the unknowns by the letters Z i .

Equivalent system

D D 1 P 1 E E 1 Z 1 P 1 Z 5 Z 2

Target system

Articulated model

Main system

Let us calculate the reactions in virtual bonds caused by angular and linear displacements Z i, as well as external given loads , using the principle of superposition. Contact i we get in the equivalent system:

where is the reaction in connection i, caused by the action of a single displacement j communication, - reaction due i from the action of an external load.

Since there are no virtual connections in the given system, then for it .

Based on the adequacy of the equivalent and given systems, we obtain , i.e.

Revealing to all i, we obtain the system of canonical equations of the displacement method:

Reactions in the main system from various influences can be found by the force method. There are two main cases of supporting beams:

Blind seals on both sides;

One blind seal and one hinged support.

As an example, consider certain reactions that occur when the embed A is rotated through an angle.

The beam is statically indeterminate 2 times n=R-U=4-2=2.

X 1

e.s. X 2

1 E.S. , 1 E. Ep.

2 E.S. , 2 E. Ep

1 l*1

We select the main system of the force method, discard the connections in support B, and show the equivalent system. We write down the system of canonical equations:

We consider 1 and 2 single and cargo state of the main system. The angle of rotation of the left support acts as an external load. Calculate compliance and displacement ∆ iF .

Substitute in the system:

or

From the equilibrium equations we find:

Based on the data obtained, a plot of bending moments in a given beam is constructed from a single angle of rotation.

The diagram is built on stretched fibers.

Consider the action of a force R on the beam.

Target system

Main system

Equivalent system

pl/4

We have received two elements of the library of basic load cases. Similarly, solutions were found for other cases, which are used as "bricks" in the calculation of frames.

Consider the I unit and load states of the main system.

P 2

P 1

M F

r 11

For the frame shown above, for example, you can write:

where h and l- lengths of posts and crossbars converging at node D; Y c and Y p are the moments of inertia of the posts and crossbars. Similarly, we find:

After finding "single" r ij and cargo R iF reactions, a system of equations is solved for the displacements of nodes Z i. Then the final diagram of the bending moments is built.

where - the diagram of bending moments in the main system of the displacement method from a single displacement - the same from an external load.

Similar to the force method, the displacement method has a number of intermediate and final checks on the correctness of the solution of the problem.

The beginning of possible displacements, being a general principle of mechanics, is of great importance for the theory of elastic systems. As applied to them, this principle can be formulated as follows: if the system is in equilibrium under the action of an applied load, then the sum of the work of external and internal forces on possible infinitesimal displacements of the system is equal to zero.

where - external forces;
- possible movements of these forces;
- the work of internal forces.

Note that in the process of making a possible displacement by the system, the magnitude and direction of external and internal forces remain unchanged. Therefore, when calculating work, one should take half, and the full value of the product of the corresponding forces and displacements.

Consider two states of a system in equilibrium (Fig. 2.2.9). Capable of the system is deformed by the generalized force (Fig. 2.2.9, a), in the state - force (Fig. 2.2.9, b).

The work of the forces of the state on state transitions , as well as the work of state forces on state transitions , will be possible.

(2.2.14)

Let us now calculate the possible work of the internal forces of the state on displacements caused by state load . To do this, consider an arbitrary element of the rod with length
in both cases. For flat bending, the action of the remote parts on the element is expressed by the system of forces ,,
(Fig. 2.2.10, a). Internal forces have directions opposite to external ones (shown by dashed lines). On fig. 2.2.10, b shows external forces ,,
acting on the element
capable of . Let us determine the deformations caused by these forces.

Obviously elongation of the element
caused by forces

.

Work of internal axial forces on this possible move

. (2.2.15)

Mutual angle of rotation of element faces caused by pairs
,

.

Work of internal bending moments
on this move

. (2.2.16)

Similarly, we determine the work of transverse forces on movements caused by forces

. (2.2.17)

Summing up the work obtained, we obtain the possible work of the internal forces applied to the element
rod, on displacements caused by another, completely arbitrary load, marked with the index

Summing up the elementary work within the rod, we obtain the total value of the possible work of internal forces:

(2.2.19)

Let us apply the beginning of possible displacements, summing up the work of internal and external forces on the possible displacements of the system, and obtain a general expression for the beginning of possible displacements for a flat elastic rod system:

(2.2.20)

That is, if the elastic system is in equilibrium, then the work of external and internal forces is in the state on possible displacements caused by another, completely arbitrary load, marked with the index , equals zero.

Theorems on the reciprocity of work and displacement

Let us write down the expressions for the beginning of possible displacements for the beam shown in Fig. 2.2.9 by accepting for the state as possible displacements caused by the state , and for the state - movements caused by the state .

(2.2.21)

(2.2.22)

Since the expressions for the work of internal forces are the same, it is obvious that

(2.2.23)

The resulting expression is called the work reciprocity theorem (the Betti theorem). It is formulated as follows: the possible work of external (or internal) forces of the state on state transitions is equal to the possible work of external (or internal) forces of the state on state transitions .

Let us apply the work reciprocity theorem to a special case of loading, when in both states of the system one single generalized force is applied
and
.

Rice. 2.2.11

Based on the work reciprocity theorem, we obtain the equality

, (2.2.24)

which is called the theorem on the reciprocity of displacements (Maxwell's theorem). It is formulated as follows: the movement of the point of application of the first force in its direction, caused by the action of the second unit force, is equal to the movement of the point of application of the second force in its direction, caused by the action of the first unit force.

Theorems on the reciprocity of work and displacements greatly simplify the solution of many problems in determining displacements.

Using the work reciprocity theorem, we determine the deflection
beams in the middle of the span under the action on the moment support
(Fig. 2.2.12, a).

We use the second state of the beam - the action at point 2 of the concentrated force . Angle of rotation of the reference section
we determine from the condition of fixing the beam at point B:

Rice. 2.2.12

According to the reciprocity work theorem

,

Consider two states of an elastic system in equilibrium. In each of these states, a certain static load acts on the system (Fig. 4a). Let's denote the displacements in the directions of forces F1 and F2 through, where the index "i" shows the direction of displacement, and the index "j" - the cause that caused it.

We denote the work of the load of the first state (force F1) on the displacements of the first state through A11, and the work of the force F2 on the displacements caused by it - A22:

Using (1.9), works A11 and A22 can be expressed in terms of internal force factors:

Let us consider the case of static loading of the same system (Fig. 5a) in the following sequence. First, a statically increasing force F1 is applied to the system (Fig. 23b); when the process of its static increase is over, the deformation of the system and the internal forces acting in it become the same as in the first state (Fig. 23a). The work of the force F1 will be:

Then a statically growing force F2 begins to act on the system (Fig. 5b). As a result, the system receives additional deformations and additional internal forces arise in it, the same as in the second state (Fig. 5a). In the process of increasing the force F2 from zero to its final value, the force F1 , remaining unchanged, moves down by the amount of additional deflection and, therefore, performs additional work:

The force F2 does the work:

The total work A under sequential loading of the system by forces F1, F2 is equal to:

On the other hand, in accordance with (1.4), the total work can be defined as:

Equating expressions (1.11) and (1.12) to each other, we obtain:

A12=A21 (1.14)

Equality (1.14) is called the reciprocity of work theorem, or Betti's theorem: the work of the forces of the first state on displacements in their directions, caused by the forces of the second state, is equal to the work of the forces of the second state on displacements in their directions, caused by the forces of the first state. Omitting intermediate calculations, we express the work of A12 in terms of bending moments, longitudinal and transverse forces arising in the first and second states:

Each integrand on the right-hand side of this equality can be viewed as a product inner effort, arising in the section of the rod from the forces of the first state, to the deformation of the element dz, caused by the forces of the second state.

Proof of the work reciprocity theorem

Let's mark two points 1 and 2 on the beam (Fig. 15.4, a).

We apply statically at point 1 the force . It will cause a deflection at this point, and at point 2 -.

We use two indices to designate displacements. The first index means the place of movement, and the second - the reason causing this movement. That is, almost like on a letter envelope, where we indicate: where and from whom.

So, for example, means the deflection of the beam at point 2 from the load.

After the growth of strength is completed. we apply at point 2 to the deformed state of the beam the static force (15.4, b). The beam will receive additional deflections: at point 1 and at point 2.

Let's make an expression for the work that these forces do on their respective displacements: .

Here the first and third terms are the elastic work of forces and . According to Clapeyron's theorem, they have a coefficient . The second term does not have this coefficient, since the force does not change its value and does the possible work on the displacement caused by another force.