Definition 1
Rigid body mechanics is a broad branch of physics that studies the motion of a solid body under the influence of external factors and strength.
Figure 1. Solid mechanics. Avtor24 - online exchange of student works
Given scientific direction covers a very wide range of issues in physics - it studies various objects, as well as the smallest elementary particles substances. 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.
Today, there are 5 types of motion of a rigid body:
- forward motion;
- plane-parallel motion;
- rotational movement around a fixed axis;
- rotational around a fixed point;
- free uniform movement.
Any complex movement of material matter can ultimately be reduced to a combination of rotational and translational movements. Fundamental and important for this entire topic is the mechanics of rigid body motion, which involves a mathematical description of probable changes in the environment and dynamics, which considers the movement of elements under the influence of given forces.
Features of solid mechanics
A solid body that systematically takes on a variety of orientations in any space can be considered to consist of a huge number of material points. It's simple mathematical method, which helps expand the applicability of theories of particle motion, but has nothing to do with the theory atomic structure real substance. Because material points Since the body under study will be directed in different directions with different speeds, 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 rotating around a stationary vector with angular velocity parameter. The moment of inertia can be calculated by integration, and for a homogeneous object, equilibrium 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 relationship 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 above can be generalized to the case when there are no fixed lines and the physical body rotates freely in any space. In such a process, there are three moments of inertia related to the “key axes”. Postulates in mechanics solid are simplified if we use existing notations mathematical analysis, in which the passage to the limit $(t → t0)$ is assumed, so there is no need to constantly think about how to solve this issue.
Interestingly, Newton was the first to apply the principles of integral and differential calculus to solve complex problems. physical problems, and the subsequent formation of mechanics as a complex science was the work of such outstanding mathematicians as J. Lagrange, L. Euler, P. Laplace and K. Jacobi. Each of these researchers found in Newton's teaching 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 a system (material body) relative to the axis of rotation is called physical quantity, which is equal to the sum of the products of the indicators of the system points and the squares of their distances to the vector in question.
The summation is carried out over all moving elementary masses into which the physical body is divided. If the moment of inertia of the object under study relative to the axis passing through its center of mass is initially known, then the entire process relative to any other parallel line is determined by Steiner’s theorem.
Steiner's theorem states: the moment of inertia of a substance relative to the rotation vector is equal to the moment of its change relative to a parallel axis that passes through the center of mass of the system, obtained by multiplying the mass 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. Deformation of a solid body. Avtor24 - online exchange of student works
When considering rigid body mechanics, 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
Deformation is called permanent and elastic if, after the cessation of the influence of extraneous factors, the body returns to its original parameters.
Deformations that remain in a substance after the cessation of 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 cessation of additional influence. However, if the residual changes are small, then they can be ignored and more elastic deformations can be studied. All types of deformation (compression or tension, bending, torsion) can ultimately be reduced to transformations occurring simultaneously.
If the force moves strictly 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 characteristic deformation experienced by a material body is its relative change.
Beyond the elastic limit, residual deformations appear in a solid and a graph detailing 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. Voltage diagram for real physical bodies directly depends on various factors. The same object can, under short-term exposure to forces, manifest itself as completely fragile, but under long-term influence, it can become permanent and fluid.
Mechanics of a deformable solid is a science that studies the laws of equilibrium and motion solids under conditions of their deformation under various influences. The deformation of a solid body means that its size and shape change. An engineer constantly encounters this property of solids as elements of structures, structures and machines in his practical activities. For example, a rod elongates under the action of tensile forces, a beam loaded with a transverse load bends, etc.
Under the action of loads, as well as thermal influences, internal forces arise in solid bodies, which characterize the resistance of the body to deformation. Internal forces per unit area are called stresses.
The study of the stressed and deformed states of solids under various influences is the main task of the mechanics of a deformable solid.
Strength of materials, theory of elasticity, theory of plasticity, theory of creep are sections of the mechanics of deformable solids. In technical, in particular construction, universities, these sections are of an applied nature and serve to develop and substantiate methods for calculating engineering structures and structures on strength, rigidity And sustainability. The correct solution of these problems is the basis for the calculation and design of structures, machines, mechanisms, etc., since it ensures their reliability throughout the entire period of operation.
Under strength usually understood as ability safe work structures, structures and their individual elements, which would exclude the possibility of their destruction. The loss (exhaustion) of strength is shown in Fig. 1.1 using the example of beam destruction under the action of force R.
The process of exhaustion of strength without changing the operating pattern of a structure or the form of its equilibrium is usually accompanied by an increase in characteristic phenomena, such as the appearance and development of cracks.
Stability of the structure - this is its ability to maintain the original form of balance until destruction. For example, for the rod in Fig. 1.2, A up to a certain value of the compressive force, the initial rectilinear form of equilibrium will be stable. If the force exceeds a certain critical value, then the curved state of the rod will be stable (Fig. 1.2, b). In this case, the rod will work not only in compression, but also in bending, which can lead to its rapid destruction due to loss of stability or to the occurrence of unacceptably large deformations.
Buckling is very dangerous for structures and structures as it can occur within a short period of time.
Structural rigidity characterizes its ability to prevent the development of deformations (elongations, deflections, twist angles, etc.). Typically, the rigidity of structures and structures is regulated by design standards. For example, the maximum deflections of beams (Fig. 1.3) used in construction should be within /= (1/200 + 1/1000)/, the twist angles of the shafts usually do not exceed 2° per 1 meter of shaft length, etc.
Solving problems of structural reliability is accompanied by a search for the most optimal options in terms of operational efficiency or operation of structures, material consumption, manufacturability of construction or manufacturing, aesthetics of perception, etc.
Material resistance in technical universities is essentially the first engineering discipline in the learning process in the field of design and calculation of structures and machines. The course on strength of materials mainly outlines methods for calculating the simplest structural elements - rods (beams, beams). At the same time, various simplifying hypotheses are introduced, with the help of which simple calculation formulas are derived.
In the field of strength of materials, methods of theoretical mechanics and higher mathematics, as well as experimental data, are widely used. The strength of materials as a basic discipline is heavily relied upon by undergraduate students, such as structural mechanics, building structures, testing of structures, dynamics and strength of machines, etc.
The theory of elasticity, the theory of creep, and the theory of plasticity are the most general sections of the mechanics of a deformable solid. The hypotheses introduced in these sections are of a general nature and mainly concern the behavior of the body material during its deformation under the influence of load.
In the theories of elasticity, plasticity and creep, the most accurate or sufficiently rigorous methods of analytical problem solving are used, which requires the involvement of special branches of mathematics. The results obtained here make it possible to provide methods for calculating more complex structural elements, such as plates and shells, to develop methods for solving special problems, such as the problem of stress concentration near holes, and to establish areas of use for solutions to the strength of materials.
In cases where the mechanics of a deformable solid cannot provide methods for calculating structures that are simple enough and accessible to engineering practice, various experimental methods are used to determine stresses and strains in real structures or in their models (for example, the strain gauge method, the polarization optical method, the holography, etc.).
The formation of strength of materials as a science can be dated back to the middle of the last century, which was associated with the intensive development of industry and the construction of railways.
Requests from engineering practice gave impetus to research in the field of strength and reliability of structures, structures and machines. During this period, scientists and engineers developed fairly simple methods for calculating structural elements and laid the foundations for the further development of the science of strength.
The theory of elasticity began to develop in early XIX centuries like mathematical science, not having an applied nature. The theory of plasticity and the theory of creep as independent sections of the mechanics of deformable solids were formed in the 20th century.
Mechanics of deformable solids is a constantly developing science in all its branches. New methods are being developed for determining the stressed and deformed states of bodies. Various numerical methods problem solving, which is associated with the introduction and use of computers in almost all areas of science and engineering practice.
Lecture No. 1
Strength of materials as a scientific discipline.
Schematics of structural elements and external loads.
Assumptions about the material properties of structural elements.
Internal forces and stresses
Section method
Movements and deformations.
Superposition principle.
Basic concepts.
Strength of materials as a scientific discipline: strength, rigidity, stability. Calculation diagram, physical and mathematical model of the operation of an element or part of a structure.
Schematics 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 material properties of structural elements: the material is continuous, homogeneous, isotropic. Body deformation: elastic, residual. Material: linearly elastic, nonlinearly elastic, elastoplastic.
Internal forces and stresses: internal forces, normal and tangential stresses, stress tensor. Expression of internal efforts in cross section rod through stress I.
Method of sections: determination of the components of internal forces in the cross section of a rod from the equilibrium equations of the separated part.
Displacements and deformations: point displacement and its components; linear and angular deformations, strain tensor.
Superposition principle: geometrically linear and geometrically nonlinear systems.
Strength of materials as a scientific discipline.
The disciplines of the strength cycle: strength of materials, theory of elasticity, structural mechanics are united under the common name “ Mechanics of a solid deformable body».
Strength of materials is the science of strength, rigidity and stability elements engineering structures.
Design it is customary to call a mechanical system of geometrically unchangeable 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 irreversible change in shape under the influence 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 deformation.
Stability of the elastic system call its property of returning to a state of equilibrium after small deviations from this state .
Elasticity – this is the property of a material to completely restore the geometric shape and dimensions of a body after removing the external load.
Plastic - this is the property of solids to change their shape and size under the influence of external loads and maintain it after removing these loads. Moreover, the change in body shape (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).
Structural mechanics called science about calculation methods structures for strength, rigidity and stability .
1.2 Schematics of structural elements and external loads.
Design model it is customary to call an auxiliary object that replaces the real structure, presented in the most general form.
Strength of materials uses calculation schemes.
Calculation 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 divided in the design scheme include: beam, rod, plate, shell, massive body.
Rice. 1.1 Main types of structural elements
timber is a rigid body obtained by moving a flat figure along a guide so that its length is significantly greater than the other two dimensions.
The rod called straight beam, which works in tension/compression (significantly exceeds the characteristic cross-sectional dimensions h,b).
The geometric locus of the points that are the centers of gravity of the cross sections will be called rod axis .
Plate - this is a body whose thickness is significantly less than its dimensions a And b in plan.
A naturally curved plate (curve before loading) is called shell .
Massive body characterized by the fact that all its sizes a ,b, And c have the same order.
Rice. 1.2 Examples of rod structures.
Beam called a beam that experiences bending as the main method of loading.
Fermoy called a set of rods connected by hinges .
Frame – This is a set of beams rigidly connected to each other.
External loads are divided on concentrated And distributed .
Fig. 1.3 Schematic diagram of the operation of the crane beam.
Force or moment, which are conventionally considered to be applied at a point, are called focused .
Figure 1.4 Volumetric, surface and distributed loads.
A load that is constant or varies very slowly over time, when we can neglect the speeds and accelerations of the resulting movement, called static.
A rapidly changing load is called dynamic , calculation taking into account the resulting oscillatory motion - dynamic calculation.
Assumptions about the material properties of structural elements.
In resistance of materials, a conditional material is used, endowed with certain idealized properties.
In Fig. 1.5 shows three characteristic deformation diagrams relating force values F and deformation during loading And unloading.
Rice. 1.5 Characteristic diagrams of material deformation
The total deformation consists of two components: elastic and plastic.
The part of the total deformation that disappears after removing the load is called elastic .
The deformation remaining after unloading is called residual or plastic .
Elastic - plastic material - This is a material exhibiting elastic and plastic properties.
A material in which only elastic deformations occur is called ideally elastic .
If the deformation diagram is expressed by a nonlinear relationship, then the material is called nonlinearly elastic, if linear dependence , then linearly elastic .
We will further consider the material of structural elements continuous, homogeneous, isotropic and linearly elastic.
Property continuity means that the material continuously fills the entire volume of the structural element.
Property uniformity means that the entire volume of 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 introducing experimentally obtained averaged quantitative characteristics of the mechanical properties of materials into the calculation of structural elements.
1.4 Internal forces and stresses
Inner forces – increment of interaction forces between particles of a body that arise when it is loaded .
Rice. 1.6 Normal and shear stresses at a point
The body is dissected 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. We denote the resultant force on the site by . Average We will determine the intensity at the site using the formula. We define the intensity of internal forces at a point 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 voltage at a given site. Let us decompose it into components (Fig. 1.6 b) so that , where and – respectively normal And tangent stress on the area with the normal n.
When analyzing stresses in the vicinity of the point under consideration M(Fig. 1.6 c) select an infinitesimal element in the shape of a parallelepiped with sides dx, dy, dz (6 sections are carried out). 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 is voltage, for example , shows that it acts on an area with a normal parallel to the x-axis, and the second shows that the stress vector is parallel to the y-axis. For normal voltage, both indices coincide, so one index is used.
Force factors in the cross section of a rod and their expression through stress.
Let's consider the cross section of the loaded rod (Fig. 1.7a). Let us 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- shear forces , Mz,My- bending moments , Mx- torque) .
Let us express the internal forces in terms of stresses acting in the cross section, assuming they are known at each point(Fig. 1.7, c)
Expression of internal efforts through tension I.
(1.3)
1.5 Section method
When external forces act on a body, it becomes deformed. Consequently, the relative arrangement of the particles of the body changes; As a result, additional interaction forces between particles arise. These interaction forces in a deformed body are internal efforts. It is necessary to be able to determine meaning and direction of internal efforts through external forces acting on the body. For this purpose it is used section method.
Rice. 1.8 Determination of internal forces using the section method.
Equilibrium equations for the remaining part of the rod.
From the equilibrium equations we determine the internal forces in the section a-a.
1.6 Movements and deformations.
Under the influence 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 Complete movement of a point and its components.
But the movement of a given point does not yet characterize the degree of deformation of the material element at this point ( example of bending a beam with a cantilever) .
Let's introduce the concept deformations at a point as a quantitative measure of material deformation in its vicinity . Let us select 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 elongation.
Figure 1.10 Linear and angular deformations of a material element.
Linear relative deformations at a point will be defined like this():
In addition to linear deformations, angular deformations or shear angles, representing small changes in the initially right angles of the parallelepiped(for example, in the xy plane it would be ). The shear angles are very small and of the order of magnitude.
We reduce the introduced relative deformations at a point into a matrix
. (1.6)
Values (1.6) quantitatively determine the deformation of the material in the vicinity of a point and constitute the deformation tensor.
Superposition principle.
A system in which internal forces, stresses, deformations and displacements are directly proportional to the acting load is called linearly deformable (the material acts as linearly elastic).
Limited by two curved surfaces, the distance...
Problems of science
This is the science of strength and compliance (rigidity) of elements of engineering structures. Using the methods of mechanics of a deformable body, practical calculations are carried out 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 principles of the resistance of materials are based on the laws of general mechanics of a solid body and, above all, on the laws of statics, 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, and the theory of creep, where the same issues are considered as in the strength of materials, but in a more complete and rigorous formulation.
Strength of materials aims to create 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 forces one to resort in a number of cases to simplifying hypotheses and assumptions, which are further justified by comparing calculated data with experiment.
General approach
Many physical phenomena It is convenient to consider using the diagram shown in Figure 13:
Through X this indicates some influence (control) applied to the system input 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 are removed from the system output A.
Under managed system A Let us agree to understand any object capable of responding deterministically to some influence. This means that all copies of the system A under the same conditions, i.e. under the same influences x(t), behave strictly the same, i.e. give out the same y(t). This 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 probabilistic rather than deterministic systems. However, for a number of phenomena it is convenient to ignore this obvious fact and consider the system to be deterministic, understanding all quantitative relationships between the quantities under consideration in the sense of relationships between their mathematical expectations.
The behavior of any deterministic controlled system can be determined by a certain relationship connecting the output to the input, i.e. X With at. We will call this relation the equation state systems. Symbolically it is written like this
where is the letter A, used earlier to denote the system, can be interpreted as a certain operator that allows us to determine y(t), if specified 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, deformed in time, by the forces acting on it, etc. Our goal is not to study an arbitrary controlled system, and therefore in the process of presentation we will introduce the necessary additional assumptions, which, limiting the generality, will allow us to consider a system of particular 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 accomplished, then it becomes possible to write the equation of state of the entire system, since the behavior of each of its elements and the methods of their interaction are known. 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 law, 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 providing a direct, comprehensive 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 has been 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 one is not interested in the detailed structure of the system (for example, the microscopic structure of a deformable body) and its elements, but studies the functioning of the system as a whole, which is considered as a connection between input and output. Generally speaking, this connection can be arbitrary. However, for each specific class of systems, general restrictions are imposed on this connection, and carrying out a certain minimum of experiments may be sufficient to clarify this connection in the necessary detail.
The use of a macroscopic approach is, as already noted, in many cases forced. However, even the creation of a consistent microtheory of a phenomenon cannot completely invalidate the corresponding macrotheory, since the latter is based on experiment and is therefore more reliable. Microtheory, 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.) equations have irremovable 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, it is still a macro approach, so microanalysis of the system can be considered as a synthesis of its components, 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 solid 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 a body associated with the movement of individual points and elements of the body. In the strength of materials only such movements are considered.
There are linear and angular movements of individual points and elements of the body. These movements correspond to linear and angular deformations (relative elongation and relative shift).
Deformations are divided into elastic, disappearing after the load is removed, and residual.
Hypotheses about a deformable body. Elastic deformations are usually (at least in structural materials such as metals, concrete, wood, etc.) insignificant, therefore the following simplifying provisions are accepted:
1. The principle of initial sizes. In accordance with it, it is accepted that 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 an absolutely 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 a body, then the action of each of them can be considered independently of the action of other forces.
Voltages
Under the influence 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 the internal force per unit cross-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 point (for example, k), in which we want to determine the stress, we mentally draw an arbitrary section and discard part of the body (II). In order for the remaining part of the body to be in equilibrium, internal forces must be applied instead of the discarded part. The interaction of two parts of the body occurs at all points of the cross-section, and therefore internal forces act over the entire cross-sectional area. In the vicinity of the point under study, we select an area dA. Let us denote the resultant of internal forces on this area dF. Then the voltage in the vicinity of the point will be (by definition)
N/m 2.
Stress has the dimension of force divided by area, N/m2.
At a given point of the body, the stress has many values, depending on the direction of the sections, of which many can be drawn through the point. Therefore, when talking about voltage, it is necessary to indicate the cross section.
In general, the stress is directed at a certain angle to the section. This total stress can be decomposed into two components:
1. Perpendicular to the section plane – normal voltage s.
2. Lying in the section plane – shear stress t.
Determination of stresses. The problem is solved in three stages.
1. A section is drawn through the point under consideration, in which they want to determine the stress. One part of the body is thrown away and its action is replaced by internal forces. If the whole body is in balance, then the rest of the body must also be in balance. Therefore, equilibrium equations can be drawn up for the forces acting on the part of the body under consideration. 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 projections and the sums of moments of all external forces acting on the part of the body remaining after the section, and the second ones are the sums of projections and moments of all internal forces acting in the section. As already noted, these equations include unknown internal forces (stresses). However, to determine them the equations of statics not enough, since otherwise the difference between an absolutely solid and a 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. By solving the static equations and the deformation equations together, stresses can be determined.
Power factors. Let us agree to call the sum of projections and the sum of moments of external or internal forces power factors. Consequently, the force factors in the section under consideration are defined as the sum of projections and the sum of moments of all external forces located on one side of this section. In the same way, power factors can be determined by 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, in problems, external forces are known, through which the force factors are determined, and from them the stresses are already determined.
Deformable body model
In strength of materials, the model of a deformable body is considered. It is assumed that the body is deformable, continuous and isotropic. In the strength of materials, mainly bodies in the form of rods (sometimes plates and shells) are considered. This is explained by the fact that in many practical problems the design diagram is reduced to a straight rod or to a system of such rods (trusses, frames).
Main types of deformed state of rods. A rod (beam) is a body in which two dimensions are small compared to the third (Fig. 15).
Let us consider a rod that is in equilibrium under the action of forces applied to it, arbitrarily located in space (Fig. 16).
We draw a 1-1 section and discard one part of the rod. Let us consider the equilibrium of the remaining part. We will use a rectangular coordinate system, the origin of which will be the center of gravity of the cross section. Axis X direct along the rod towards the outer normal to the section, axis Y And Z– the main central axes of the section. Using static equations we will find the force factors
three forces
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 various types rod deformation. The main types of rod deformations are stretching, compression, shift, torsion, bend. Accordingly, the simplest loading schemes look like this.
Tension-compression. Forces are applied along the axis of the rod. Having discarded the right part of the rod, we highlight the force factors based on the left external forces (Fig. 17)
We have one non-zero factor - longitudinal force F.
We build a diagram of force factors (diagram).
Torsion of the rod. In the planes of the end sections of the rod, two equal and opposite pairs of forces are applied with a moment M cr =T, called torque (Fig. 18).
As you can see, in the cross section of the twisted rod there is only one force factor - the moment T = Fh.
Transverse 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 by pairs of forces acting in one of the main planes of the rod.
Beams have supports, i.e. are non-free bodies, a typical support is a hinged-movable support (Fig. 19).
Sometimes a beam with one embedded end and the other free end is used - a cantilever beam (Fig. 20).
Let's consider the definition of force factors using the example of Fig. 21a. First you need to find the reactions of the supports R A and .
