Structural mechanics

The displacement of any cross-section is usually denoted with a symbol (delta) with two indices, the first of which indicates the direction of displacement, and the second one indicates the reason that caused the displacement. So, for example, 1F is denoted the displacement of the

cross-section in the 1st direction, caused by an external load. The sense of the notation 2F and 3F is revealed with the help of Figure 7.1.

Then, it will be necessary to determine the displacements in the direction of several concentrated forces F1, F 2 , , Fn action. Then iF should

be read as follows: this is the displacement of the application point of the force Fi in its direction caused by the load F.

The displacement in the i-th direction caused by the temperature effect is denoted as it , the displacement in the i-th direction caused by

the displacement of the supports is denoted as ic.

Determination of displacements in linearly deformable systems is based on general theorems on elastic systems.

7.2. Work of External Statically Applied Forces

The load on any structure causes the movement of the structure from the initial state to a new, deformed one. We will consider such a load that is applied to the structure so slowly, smoothly, that the resulting accelerations of its elements, and therefore, the inertial forces of their masses can be neglected. The loading process is called static, and the corresponding load is called static.

Let a rod made of a nonlinear elastic material undergo tensile force

F(Figure 7.2).

The stress-strain diagram of this material is shown in Figure 7.3.

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The area of the diagram , as is known from the course “Strength of materials”, is equal to the specific potential energy u0 (in other words,

the energy density is the energy referred to the unit of the initial volume of the element) under a linear stress state.

If we change the scale of the diagram ordinates by introducing the dependencies N A and l l, then we can get the dependence

“load-displacement” that is often used in the practice of calculations (Figure 7.4).

Figure 7.4

In this figure symbol z denotes some intermediate absolute elongation of the rod caused by force F(z), and symbol denotes the displacement

corresponding to the final (maximum) value of the force F.

The work performed by force with infinitely small increase in displacement by dz is determined by the expression: dW F(z)dz.

Summing up the elementary work over the entire range of displacements change we obtain a formula for determining the work performed by a statically applied external force F:

W F(z) dz.

0

For a linear-elastic rod, the ratio between force and displacement is linear (Figure 7.5). Therefore, F(z) k z, where k is the stiffness coef-

ficient of the rod.

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For example, with a static action on the beam of concentrated forces F1, F2 and of concentrated moment M (Figure 7.6) the actual work of

external forces is equal to:

W F1 1 F2 2 M . 2 2 2

Figure 7.6

The minus sign in the last term of the expression is accepted because the direction of the angle of rotation of the cross-section of the beam

and the direction of the moment M are opposite.

7.3. Work of the Internal Forces

in a Plane Linear-Elastic Bars System

Under the static action of external forces on a deformable system, internal forces arise in its cross-sections. To determine the work of these forces, we cut out an element of length dx (Figure 7.7, a) with the help of infinitely close located cross-sections (Figure 7.7, b).

Figure 7.7

With respect to this element, the forces N, M and Q, which replaces the action of the discarded parts of the system on the selected element,

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are external. Internal forces are equal to them, but opposite in direction. Internal forces are resist element deformations. Therefore, the work of internal forces is always negative.

Note. In the formulas of Section 7.3 and below, the following notation will be used:

A – is an area of the bars cross-section;

J – is an axial moment of inertia of a cross-section; the denote of the

forces and the corresponding deformations are opposite). Consequently,

dAN dWN N 2 dx . 2EA

Figure 7.8

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At the angular displacement d of the cross-sections caused by the action of the bending moment M (Figure 7.8, b) its work will be

equal to 12 M d.

the axis of the bar, the expression of the angle of mutual rotation of the

dAM M 2 dx . 2EJ

The tangential stresses in the cross-section, determined by the Zhuravsky`s formula:

cause a mutual shear of the cross-sections Z dx G dx (Figure 7.9).

Figure 7.9

To determine their work, we select the corresponding strips with an area dA at the ends of the element dx. Given the static nature of the load, we find that:

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ing on the shape of the cross-sectional area.

For a rectangular cross-section = 1,2; or round cross-section = 1,18;

for rolling I-beams approximately is equal to the ratio of the area of the I-beam to the area of its wall.

We obtain the full actual work of the internal forces of a plane bars system by integrating the expressions for elementary work along the length of each part of the bar and summing over all parts of the system.

of internal forces is always negative.

The relationship between loads and displacements (forces) is linear in linearly deformable systems. The relationship between the load and work, as follows from formula (7.2), is non-linear. The actual work of a group of simultaneously acting external forces is not equal to the sum of the actual works caused by each of the forces individually. The superposition principle of the action of forces in calculating the actual work is not applicable.

7.4. Application of Virtual Displacements Principle to Elastic Systems

We expand the concepts presented in section 2.4.

An elastic system loaded by a given external action takes a definite deformed position. The displacements of the system points counted from

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the initial (undeformed) state of the system till their corresponding positions in the deformed state are actual displacements.

We set the virtual displacements for the considered system. Since the position of the elastic system in a deformed state is characterized by an infinitely large number of parameters, such a system is a system with an infinitely large number of degrees of freedom. The number of virtual displacements will also be infinitely large.

As noted in section 2.4, while "passing" system from the deformed state to a new, which takes into account the virtual displacements, external actions and internal forces do not change. Therefore, the work of external and internal forces on virtual displacements must be determined by the expressions:

W virt Fi i ,

where Fi – generalized forces;

i – corresponding generalized displacement;

Aintvirt Siei ,

where Si – generalized internal forces;

ei – corresponding generalized deformations. The work of internal forces is always negative.

The formal notation of the principle of virtual displacements is the same as in section 2.4:

W (virt) Аint(virt) 0.

It is assumed that the constraints are ideal in an elastic system, and for virtual displacements, no work is required to overcome friction or to generate and release heat, etc. This is taken into account in inelastic systems.

In practical applications, virtual displacements are the small displacements that can be caused by force actions or other ones. For example, for the beam state shown in Figure 7.10 (state “ i ”), as virtual displacements one can take the displacements of the same beam loaded with

another group of forces (state “ k ”).Then the virtual work of the external

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forces of the state “ i ” at the displacements of the state “ k ” is written in the form:

W (virt) F1 1k F2 2k .

Figure 7.10

The virtual work of the internal forces of the state “ i ”on the beam deformations in the state “ k ” will be equal to:

The principle of virtual displacements is one of the basic principles of mechanics. It allows one to find equilibrium conditions, which are very important, without determining unknown links reactions.

If actual displacements are taken for virtual displacements, then the virtual work of external and internal forces will be determined by the expressions:

i

W (virt) Fi i ,

where W (virt) is virtual work of external forces; Aint(virt) is virtual work of internal forces.

Note that the concept of the virtual displacement (indicated by a symbol ) was introduced by Lagrange. In the classical treatise "Analytical

Mechanics" (1788; Russian transl., Vols. 1–2, 2 ed., 1950), he consid-

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ered the “general formula”, which is the principle of virtual displacements, as the basis of all statics, and the “general formula”, which is a combination of the principle of virtual displacements with the D'Alembert principle, he considered as the basis of all dynamics.

7.5. Theorems of Reciprocity Works and Displacements

Suppose that a linearly deformable system (Figure 7.11, a) is sequentially loaded first with force Fi , and then with force Fk .

Figure 7.11

When the beam proceeds from position 1 to position 2, then the actual work of the force Fi on the displacement ii is equal to Wii 12 Fi ii .

When the beam proceeds from position 2 to position 3, then the actual work of the force Fk is equal to Wkk 12 Fk kk , and the force Fi ,

remaining unchanged at this time, does the virtual work Wik Fi ik on the displacement ik . The total work of two forces will be equal to:

W1 Wii Wkk Wik .

If the beam is loaded in the reverse sequence (first by force Fk , and then by force Fi (Figure 7.11, b)), then we obtain:

W2 Wkk Wii Wki .

Since the value of the work of external forces is equal to the potential energy of the system and, regardless of the loading sequence, in both

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