BASIC CONCEPTS OF MECHANICS
DEFORMABLE SOLID BODY
This chapter presents the basic concepts that were previously studied in the courses of physics, theoretical mechanics and strength of materials.
1.1. The subject of solid mechanics
Solid mechanics is the science of balance and motion solids and their individual particles, taking into account changes in the distances between individual points of the body, which arise as a result of external influences on a solid body. The mechanics of a deformable solid body is based on the laws of motion discovered by Newton, since the speeds of motion of real solid bodies and their individual particles relative to each other are significantly less than the speed of light. In contrast to theoretical mechanics, here we consider changes in the distances between individual particles of the body. The latter circumstance imposes certain restrictions on the principles of theoretical mechanics. In particular, in the mechanics of a deformable solid body, the transfer of points of application of external forces and moments is unacceptable.
Analysis of the behavior of deformable solids under the influence of external forces is carried out on the basis of mathematical models that reflect the most significant properties of deformable bodies and materials from which they are made. In this case, to describe the properties of the material, the results are used experimental studies, which served as the basis for creating material models. Depending on the material model, the mechanics of a deformable solid body is divided into sections: the theory of elasticity, the theory of plasticity, the theory of creep, the theory of viscoelasticity. In turn, the mechanics of a deformable solid body is part of a more general part of mechanics - mechanics of continuous media. Continuum mechanics, being a branch of theoretical physics, studies the laws of motion of solid, liquid and gaseous media, as well as plasma and continuous physical fields.
The development of the mechanics of a deformable solid body is largely associated with the tasks of creating reliable structures and machines. The reliability of a structure and machine, as well as the reliability of all their elements, is ensured by strength, rigidity, stability and endurance throughout the entire service life. Strength is understood as the ability of a structure (machine) and all its (its) elements to maintain their integrity when external influences without division into predetermined parts. With insufficient strength, the structure or its individual elements are destroyed by dividing a single whole into parts. The rigidity of a structure is determined by the measure of the change in the shape and dimensions of the structure and its elements under external influences. If the changes in the shape and dimensions of the structure and its elements are not large and do not interfere with normal operation, then such a structure is considered sufficiently rigid. Otherwise, the rigidity is considered insufficient. The stability of a structure is characterized by the ability of a structure and its elements to maintain their form of equilibrium under the action of random forces not provided for by the operating conditions (disturbing forces). A structure is in a stable state if, after the removal of disturbing forces, it returns to its original form of equilibrium. Otherwise, there is a loss of stability of the original form of equilibrium, which, as a rule, is accompanied by the destruction of the structure. Endurance is understood as the ability of a structure to resist the influence of time-varying forces. Variable forces cause the growth of microscopic cracks inside the material of the structure, which can lead to the destruction of structural elements and the structure as a whole. Therefore, to prevent destruction, it is necessary to limit the magnitudes of the forces that are variable in time. In addition, the lowest frequencies of natural oscillations of the structure and its elements should not coincide (or be close to) the frequencies of oscillations of external forces. Otherwise, the structure or its individual elements enter into resonance, which can cause destruction and failure of the structure.
The vast majority of research in the field of solid mechanics is aimed at creating reliable structures and machines. This includes the design of structures and machines and the problems of technological processes for processing materials. But the scope of application of the mechanics of a deformable solid body is not limited to the technical sciences alone. Its methods are widely used in natural sciences such as geophysics, solid state physics, geology, biology. So in geophysics, with the help of the mechanics of a deformable solid body, the processes of propagation of seismic waves and the processes of formation of earth's crust, fundamental questions of the structure of the earth's crust are being studied, etc.
1.2. General properties of solids
All solids are made up of real materials with a huge variety of properties. Of these, only a few are of significant importance for the mechanics of a deformable solid body. Therefore, the material is endowed with only those properties that make it possible to study the behavior of solids at the lowest cost within the framework of the science under consideration.
Definition 1
Rigid body mechanics is an extensive branch of physics that studies the motion of a rigid body under the influence of external factors and forces.
Figure 1. Solid mechanics. Author24 - online exchange of student papers
This scientific direction covers a very wide range of issues in physics - it studies various objects, as well as the smallest elementary particles of matter. In these limiting cases, the conclusions of mechanics are of purely theoretical interest, the subject of which is also the design of many physical models and programs.
To date, there are 5 types of motion of a rigid body:
- progressive movement;
- plane-parallel movement;
- rotational movement around a fixed axis;
- rotational around a fixed point;
- free uniform movement.
Any complex movement material substance can be ultimately reduced to a combination of rotational and translational motions. The mechanics of motion of a rigid body, which involves a mathematical description of probable changes in the environment, and dynamics, which considers the motion of elements under the action of given forces, is of fundamental and important importance for all this subject matter.
Features of rigid body mechanics
A rigid body that systematically assumes various orientations in any space can be considered as consisting of a huge number of material points. This is just a mathematical method that helps to expand the applicability of theories of particle motion, but has nothing to do with the theory atomic structure real substance. Because the material points of the investigated body will be directed in different directions with different velocities, it is necessary to apply the summation procedure.
In this case, it is not difficult to determine the kinetic energy of the cylinder if the parameter rotating around a fixed vector with an angular velocity is known in advance. The moment of inertia can be calculated by integration, and for a homogeneous object, the balance of all forces is possible if the plate did not move, therefore, the components of the medium satisfy the condition of vector stability. As a result, the relation derived at the initial design stage is fulfilled. Both of these principles form the basis of the theory of structural mechanics and are necessary in the construction of bridges and buildings.
The foregoing can be generalized to the case when there are no fixed lines and the physical body freely rotates in any space. In such a process, there are three moments of inertia related to the "key axes". Conducted postulates in mechanics solid are simplified by using the existing notation mathematical analysis, in which the passage to the limit $(t → t0)$ is assumed, so that there is no need to think all the time how to solve this problem.
Interestingly, Newton was the first to apply the principles of integral and differential calculus in solving complex physical problems, and the subsequent formation of mechanics as integrated science was a matter of such eminent mathematicians, like J. Lagrange, L. Euler, P. Laplace and K. Jacobi. Each of these researchers found in Newton's teachings a source of inspiration for their universal mathematical research.
Moment of inertia
When studying the rotation of a rigid body, physicists often use the concept of moment of inertia.
Definition 2
The moment of inertia of the system (material body) about the axis of rotation is called physical quantity, which is equal to the sum of the products of the indicators of the points of the system and the squares of their distances to the considered vector.
The summation is made over all moving elementary masses into which the physical body is divided. If the moment of inertia of the object under study is initially known relative to the axis passing through its center of mass, then the whole process relative to any other parallel line is determined by the Steiner theorem.
Steiner's theorem states: the moment of inertia of a substance about the rotation vector is equal to the moment of its change about a parallel axis that passes through the center of mass of the system, obtained by multiplying the masses of the body by the square of the distance between the lines.
When an absolutely rigid body rotates around a fixed vector, each individual point moves along a circle of constant radius with a certain speed and the internal momentum is perpendicular to this radius.
Solid body deformation
Figure 2. Solid body deformation. Author24 - online exchange of student papers
Considering the mechanics of a rigid body, the concept of an absolutely rigid body is often used. However, such substances do not exist in nature, since all real objects under the influence of external forces change their size and shape, that is, they are deformed.
Definition 3
The deformation is called constant and elastic if, after the cessation of the influence of extraneous factors, the body assumes its original parameters.
Deformations that remain in the substance after the termination of the interaction of forces are called residual or plastic.
Deformations of an absolute real body in mechanics are always plastic, since they never completely disappear after the termination of the additional influence. However, if the residual changes are small, then they can be neglected and more elastic deformations can be investigated. All types of deformation (compression or tension, bending, torsion) can eventually be reduced to simultaneous transformations.
If the force moves strictly along the normal to a flat surface, the stress is called normal, but if it moves tangentially to the medium, it is called tangential.
A quantitative measure that characterizes the characterizing deformation experienced by a material body is its relative change.
Beyond the elastic limit, residual deformations appear in the solid, and the graph describing in detail the return of the substance to its original state after the final cessation of the force is depicted not on the curve, but parallel to it. The stress diagram for real physical bodies directly depends on various factors. One and the same object can, under short-term exposure to forces, manifest itself as completely fragile, and under long-term exposure - permanent and fluid.
The tasks of science
This is the science of strength and flexibility (rigidity) of engineering structure elements. Methods of mechanics of a deformable body are used for practical calculations and reliable (strong, stable) dimensions of machine parts and various building structures are determined. The introductory, initial part of the mechanics of a deformable body is a course called strength of materials. The basic provisions of the strength of materials are based on the laws of general mechanics of a solid body and, above all, on the laws of statics, the knowledge of which is absolutely necessary for studying the mechanics of a deformable body. The mechanics of deformable bodies also includes other sections, such as the theory of elasticity, the theory of plasticity, the theory of creep, where the same issues are considered as in the resistance of materials, but in a more complete and rigorous formulation.
The resistance of materials, on the other hand, sets as its task the creation of practically acceptable and simple methods for calculating the strength and stiffness of typical, most frequently encountered structural elements. In this case, various approximate methods are widely used. The need to bring the solution of each practical problem to a numerical result makes it necessary in some cases to resort to simplifying hypotheses-assumptions, which are justified in the future by comparing the calculated data with the experiment.
General Approach
It is convenient to consider many physical phenomena using the diagram shown in Figure 13:
Through X here some influence (control) applied to the input of the system is indicated A(machine, test sample of material, etc.), and through Y- reaction (response) of the system to this impact. We will assume that the reactions Y removed from the system output A.
Under managed system A Let us agree to understand any object capable of deterministically responding to some influence. This means that all copies of the system A under the same conditions, i.e. with the same impact x(t), behave in exactly the same way, i.e. issue the same y(t). Such an approach, of course, is only an approximation, since it is practically impossible to obtain either two completely identical systems, or two identical effects. Therefore, strictly speaking, one should consider not deterministic, but probabilistic systems. Nevertheless, for a number of phenomena it is convenient to ignore this obvious fact and consider the system to be deterministic, understanding all the quantitative relationships between the quantities under consideration in the sense of the relationships between their mathematical expectations.
The behavior of any deterministic controlled system can be determined by some relation connecting the output with the input, i.e. X With at. This relation will be called the equation states systems. Symbolically it is written as
where is the letter A, used earlier to denote the system, can be interpreted as some operator that allows you to determine y(t), if given x(t).
The introduced concept of a deterministic system with input and output is very general. Here are some examples of such systems: an ideal gas, the characteristics of which are related by the Mendeleev-Clapeyron equation, an electrical circuit that obeys one or another differential equation, a blade of a steam or gas turbine that deforms in time, forces acting on it, etc. the type most suitable for modeling the behavior of a body deformed under load.
The analysis of any controlled system can in principle be carried out in two ways. The first one microscopic, is based on a detailed study of the structure of the system and the functioning of all its constituent elements. If all this can be done, then it becomes possible to write the equation of state of the entire system, since the behavior of each of its elements and the ways of their interaction are known. So, for example, the kinetic theory of gases allows us to write the Mendeleev-Clapeyron equation; knowledge of the structure of an electrical circuit and all its characteristics makes it possible to write its equations based on the laws of electrical engineering (Ohm's law, Kirchhoff's, etc.). Thus, the microscopic approach to the analysis of a controlled system is based on the consideration of the elementary processes that make up a given phenomenon, and, in principle, is capable of giving a direct, exhaustive description of the system under consideration.
However, the micro-approach cannot always be implemented due to the complex or not yet explored structure of the system. For example, at present it is not possible to write the equation of state of a deformable body, no matter how carefully it is studied. The same applies to more complex phenomena occurring in a living organism. In such cases, the so-called macroscopic phenomenological (functional) approach, in which they are not interested in the detailed structure of the system (for example, the microscopic structure of a deformable body) and its elements, but study the functioning of the system as a whole, which is considered as a connection between input and output. Generally speaking, this relationship can be arbitrary. However, for each specific class of systems, restrictions are imposed on this connection. general, and carrying out a certain minimum of experiments may be sufficient to clarify this connection with the necessary details.
The use of the macroscopic approach is, as already noted, forced in many cases. Nevertheless, even the creation of a consistent microtheory of a phenomenon cannot completely devalue the corresponding macrotheory, since the latter is based on experiment and is therefore more reliable. Microtheory, on the other hand, when constructing a model of a system, is always forced to make some simplifying assumptions that lead to various kinds of inaccuracies. For example, all "microscopic" equations of state of an ideal gas (Mendeleev-Clapeyron, Van der Waals, etc.) have irreparable discrepancies with experimental data on real gases. The corresponding "macroscopic" equations, based on these experimental data, can describe the behavior of a real gas as accurately as desired. Moreover, the micro-approach is such only at a certain level - the level of the system under consideration. At the level of the elementary parts of the system, however, it is still a macro approach, so that the microanalysis of the system can be considered as a synthesis of its constituent parts, analyzed macroscopically.
Since at present the micro-approach is not yet able to lead to an equation of state for a deformable body, it is natural to solve this problem macroscopically. We will adhere to this point of view in the future.
Displacements and deformations
A real rigid body, deprived of all degrees of freedom (the ability to move in space) and under the influence of external forces, deformed. By deformation we mean a change in the shape and size of the body, associated with the movement of individual points and elements of the body. Only such displacements are considered in the resistance of materials.
There are linear and angular displacements of individual points and elements of the body. These displacements correspond to linear and angular deformations (relative elongation and relative shear).
Deformations are divided into elastic, disappearing after the load is removed, and residual.
Hypotheses about the deformable body. Elastic deformations are usually (at least in structural materials such as metals, concrete, wood, etc.) insignificant, so the following simplifying provisions are accepted:
1. The principle of initial dimensions. In accordance with it, it is assumed that the equilibrium equations for a deformable body can be compiled without taking into account changes in the shape and size of the body, i.e. as for a perfectly rigid body.
2. The principle of independence of the action of forces. In accordance with it, if a system of forces (several forces) is applied to the body, then the action of each of them can be considered independently of the action of other forces.
Voltage
Under the action of external forces, internal forces arise in the body, which are distributed over the sections of the body. To determine the measure of internal forces at each point, the concept is introduced voltage. Stress is defined as an internal force per unit sectional area of a body. Let an elastically deformed body be in a state of equilibrium under the action of some system of external forces (Fig. 1). Through a dot (for example, k), in which we want to determine the stress, an arbitrary section is mentally drawn and part of the body is discarded (II). In order for the remaining part of the body to be in balance, internal forces must be applied instead of the discarded part. The interaction of two parts of the body occurs at all points of the section, and therefore the internal forces act over the entire section area. In the vicinity of the point under study, we select the area dA. We denote the resultant of internal forces on this site dF. Then the stress in the vicinity of the point will be (by definition)
N/m 2.
Voltage has the dimension of force divided by area, N/m 2 .
At a given point of the body, the stress has many values, depending on the direction of the sections, which can be drawn through a point through a set. Therefore, speaking of stress, it is necessary to indicate the cross section.
In the general case, the stress is directed at some angle to the section. This total voltage can be decomposed into two components:
1. Perpendicular to the section plane - normal voltage s.
2. Lying in the plane of the section - shear stress t.
Determination of stresses. The problem is solved in three stages.
1. Through the point under consideration, a section is drawn in which they want to determine the stress. One part of the body is discarded and its action is replaced by internal forces. If the whole body is in balance, then the rest must also be in balance. Therefore, for the forces acting on the part of the body under consideration, it is possible to compose equilibrium equations. These equations will include both external and unknown internal forces (stresses). Therefore, we write them in the form
The first terms are the sums of the projections and the sums of the moments of all external forces acting on the part of the body remaining after the section, and the second terms are the sums of the projections and moments of all the internal forces acting in the section. As already noted, these equations include unknown internal forces (stresses). However, for their definition of the equations of statics not enough, since otherwise the difference between an absolutely rigid and deformable body disappears. Thus, the task of determining stresses is statically indeterminate.
2. To compile additional equations, the displacements and deformations of the body are considered, as a result of which the law of stress distribution over the section is obtained.
3. Solving jointly the equations of statics and the equations of deformations, it is possible to determine the stresses.
Power factors. We agree to call the sums of projections and the sums of moments of external or internal forces force factors. Consequently, the force factors in the considered section are defined as the sums of projections and the sums of the moments of all external forces located on one side of this section. In the same way, force factors can also be determined from the internal forces acting in the section under consideration. Force factors determined by external and internal forces are equal in magnitude and opposite in sign. Usually, external forces are known in problems, through which force factors are determined, and stresses are already determined from them.
Model of a deformable body
In the strength of materials, a model of a deformable body is considered. It is assumed that the body is deformable, solid and isotropic. In the strength of materials, bodies are considered mainly in the form of rods (sometimes plates and shells). This is due to the fact that in many practical tasks the design scheme is reduced to a straight rod or to a system of such rods (trusses, frames).
The main types of the deformed state of the rods. Rod (beam) - a body in which two dimensions are small compared to the third (Fig. 15).
Consider a rod that is in equilibrium under the action of forces applied to it, arbitrarily located in space (Fig. 16).
We draw a section 1-1 and discard one part of the rod. Consider the balance of the remaining part. We use a rectangular coordinate system, for the beginning of which we take the center of gravity of the cross section. Axis X direct along the rod in the direction of the outer normal to the section, the axis Y And Z are the main central axes of the section. Using the equations of statics, we find the force factors
three powers
three moments or three pairs of forces
Thus, in the general case, six force factors arise in the cross section of the rod. Depending on the nature of the external forces acting on the rod, it is possible different kinds rod deformation. The main types of rod deformations are stretching, compression, shift, torsion, bend. Accordingly, the simplest loading schemes are as follows.
Stretch-compression. Forces are applied along the axis of the rod. Discarding the right part of the rod, we select the force factors by the left external forces (Fig. 17)
We have one non-zero factor - the longitudinal force F.
We build a diagram of force factors (epure).
Rod torsion. In the planes of the end sections of the rod, two equal and opposite pairs of forces are applied with a moment M kr =T, called torque (Fig. 18).
As can be seen, only one force factor acts in the cross section of the twisted rod - the moment T = F h.
Cross bend. It is caused by forces (concentrated and distributed) perpendicular to the axis of the beam and located in a plane passing through the axis of the beam, as well as pairs of forces acting in one of the main planes of the bar.
The beams have supports, i.e. are non-free bodies, a typical support is an articulated support (Fig. 19).
Sometimes a beam with one embedded and the other free end is used - a cantilever beam (Fig. 20).
Consider the definition of force factors on the example of Fig.21a. First you need to find the support reactions R A and .
Lecture #1
Strength of materials as a scientific discipline.
Schematization of structural elements and external loads.
Assumptions about the properties of the material of structural elements.
Internal forces and stresses
Section method
displacements and deformations.
The principle of superposition.
Basic concepts.
Strength of materials as a scientific discipline: strength, stiffness, stability. Calculation scheme, physical and mathematical model of the operation of an element or part of a structure.
Schematization of structural elements and external loads: timber, rod, beam, plate, shell, massive body.
External forces: volumetric, surface, distributed, concentrated; static and dynamic.
Assumptions about the properties of the material of structural elements: the material is solid, homogeneous, isotropic. Body deformation: elastic, residual. Material: linear elastic, non-linear elastic, elastic-plastic.
Internal forces and stresses: internal forces, normal and shear stresses, stress tensor. Expression of internal forces in the cross section of the rod in terms of stresses I.
Section method: determination of the components of internal forces in the section of the rod from the equilibrium equations of the separated part.
Displacements and deformations: displacement of a point and its components; linear and angular strains, strain tensor.
Superposition principle: geometrically linear and geometrically nonlinear systems.
Strength of materials as a scientific discipline.
Disciplines of the strength cycle: strength of materials, theory of elasticity, structural mechanics are united by the common name " Mechanics of a solid deformable body».
Strength of materials is the science of strength, rigidity and stability elements engineering structures.
by design It is customary to call a mechanical system of geometrically invariable elements, relative movement of points which is possible only as a result of its deformation.
Under the strength of structures understand their ability to resist destruction - separation into parts, as well as an irreversible change in shape under the action of external loads .
Deformation is a change relative position of body particles associated with their movement.
Rigidity is the ability of a body or structure to resist the occurrence of deformation.
Stability of an elastic system called its property to return to a state of equilibrium after small deviations from this state .
Elasticity - this is the property of the material to completely restore the geometric shape and dimensions of the body after removing the external load.
Plastic - this is the property of solids to change their shape and size under the action of external loads and retain it after the removal of these loads. Moreover, the change in the shape of the body (deformation) depends only on the applied external load and does not happen on its own over time.
Creep - this is the property of solids to deform under the influence of a constant load (deformations increase with time).
Building mechanics call science about calculation methods structures for strength, rigidity and stability .
1.2 Schematization of structural elements and external loads.
Design model It is customary to call an auxiliary object that replaces the real construction, presented in the most general form.
The strength of materials uses design schemes.
Design scheme - this is a simplified image of a real structure, which is freed from its non-essential, secondary features and which accepted for mathematical description and calculation.
The main types of elements into which the whole structure is subdivided in the design scheme are: beam, rod, plate, shell, massive body.
Rice. 1.1 Main types of structural elements
bar is a rigid body obtained by moving a flat figure along a guide so that its length is much greater than the other two dimensions.
rod called straight beam, which works in tension/compression (significantly exceeds the characteristic dimensions of the cross section h,b).
The locus of points that are the centers of gravity of cross sections will be called rod axis .
plate - a body whose thickness is much less than its dimensions a And b in respect of.
A naturally curved plate (curve before loading) is called shell .
massive body characteristic in that all its dimensions a ,b, And c have the same order.
Rice. 1.2 Examples of bar structures.
beam is called a bar that experiences bending as the main mode of loading.
Farm called a set of rods connected hingedly .
Frame – is a set of beams rigidly connected to each other.
External loads are divided on focused And distributed .
Fig 1.3 Schematization of the operation of the crane beam.
force or moment, which are conventionally considered to be attached at a point, are called concentrated .
Figure 1.4 Volumetric, surface and distributed loads.
A load that is constant or very slowly changing in time, when the speeds and accelerations of the resulting movement can be neglected, called static.
A rapidly changing load is called dynamic , calculation taking into account the resulting oscillatory motion - dynamic calculation.
Assumptions about the properties of the material of structural elements.
In the resistance of materials, a conditional material is used, endowed with certain idealized properties.
On fig. 1.5 shows three characteristic strain diagrams relating force values F and deformations at loading And unloading.
Rice. 1.5 Characteristic diagrams of material deformation
Total deformation consists of two components, elastic and plastic.
The part of the total deformation that disappears after the load is removed is called elastic .
The deformation remaining after unloading is called residual or plastic .
Elastic - plastic material is a material exhibiting elastic and plastic properties.
A material in which only elastic deformations occur is called perfectly elastic .
If the deformation diagram is expressed by a non-linear relationship, then the material is called nonlinear elastic, if linear dependence , then linearly elastic .
The material of structural elements will be further considered continuous, homogeneous, isotropic and linearly elastic.
Property continuity means that the material continuously fills the entire volume of the structural element.
Property homogeneity means that the entire volume of the material has the same mechanical properties.
The material is called isotropic if its mechanical properties are the same in all directions (otherwise anisotropic ).
The correspondence of the conditional material to real materials is achieved by the fact that experimentally obtained averaged quantitative characteristics of the mechanical properties of materials are introduced into the calculation of structural elements.
1.4 Internal forces and stresses
internal forces – increment of the forces of interaction between the particles of the body, arising when it is loaded .
Rice. 1.6 Normal and shear stresses at a point
The body is cut by a plane (Fig. 1.6 a) and in this section at the point under consideration M a small area is selected, its orientation in space is determined by the normal n. The resultant force on the site will be denoted by . middle the intensity on the site is determined by the formula . The intensity of internal forces at a point is defined as the limit
(1.1) The intensity of internal forces transmitted at a point through a selected area is called voltage at this site .
Voltage dimension .
The vector determines the total stress on a given site. We decompose it into components (Fig. 1.6 b) so that , where and - respectively normal And tangent stress on the site with the normal n.
When analyzing stresses in the vicinity of the considered point M(Fig. 1.6 c) select an infinitesimal element in the form of a parallelepiped with sides dx, dy, dz (carry out 6 sections). The total stresses acting on its faces are decomposed into normal and two tangential stresses. The set of stresses acting on the faces is presented in the form of a matrix (table), which is called stress tensor
The first index of the voltage, for example , shows that it acts on a site with a normal parallel to the x-axis, and the second shows that the stress vector is parallel to the y-axis. For normal stress, both indices are the same, therefore one index is put.
Force factors in the cross section of the rod and their expression in terms of stresses.
Consider the cross section of the loaded rod rod (rice 1.7, a). We reduce the internal forces distributed over the section to the main vector R, applied at the center of gravity of the section, and the main moment M. Next, we decompose them into six components: three forces N, Qy, Qz and three moments Mx, My, Mz, called internal forces in the cross section.
Rice. 1.7 Internal forces and stresses in the cross section of the rod.
The components of the main vector and the main moment of internal forces distributed over the section are called internal forces in the section ( N- longitudinal force ; Qy, Qz- transverse forces ,Mz,My- bending moments , Mx- torque) .
Let us express the internal forces in terms of the stresses acting in the cross section, assuming they are known at every point(Fig. 1.7, c)
Expression of internal forces through stresses I.
(1.3)
1.5 Section method
When external forces act on a body, it deforms. Consequently, the relative position of the particles of the body changes; as a result of this, additional forces of interaction between particles arise. These interaction forces in a deformed body are domestic efforts. Must be able to identify meanings and directions of internal efforts through external forces acting on the body. For this, it is used section method.
Rice. 1.8 Determination of internal forces by the method of sections.
Equilibrium equations for the rest of the rod.
From the equilibrium equations, we determine the internal forces in the section a-a.
1.6 Displacements and deformations.
Under the action of external forces, the body is deformed, i.e. changes its size and shape (Fig. 1.9). Some arbitrary point M moves to a new position M 1 . The total displacement MM 1 will be
decompose into components u, v, w parallel to the coordinate axes.
Fig 1.9 Full displacement of a point and its components.
But the displacement of a given point does not yet characterize the degree of deformation of the material element at this point ( example of beam bending with cantilever) .
We introduce the concept deformations at a point as a quantitative measure of material deformation in its vicinity . Let's single out an elementary parallelepiped in the vicinity of t.M (Fig. 1.10). Due to the deformation of the length of its ribs, they will receive an elongation.
Fig 1.10 Linear and angular deformation of a material element.
Linear relative deformations at a point defined like this():
In addition to linear deformations, there are angular deformations or shear angles, representing small changes in the original right angles of the parallelepiped(for example, in the xy plane it will be ). Shear angles are very small and are of the order of .
We reduce the introduced relative deformations at a point into the matrix
. (1.6)
Quantities (1.6) quantitatively determine the deformation of the material in the vicinity of the point and constitute the deformation tensor.
The principle of superposition.
A system in which internal forces, stresses, strains and displacements are directly proportional to the acting load is called linearly deformable (the material works as linearly elastic).
Bounded by two curved surfaces, the distance...
