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 forces in the cross section of a 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 respect of.
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 it mechanical properties identical 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 Inner forces and voltage
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. U normal voltage Both indices are the same, so one index is placed.
Force factors in the cross section of the 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...
BASIC CONCEPTS OF MECHANICS
DEFORMABLE SOLID
This chapter provides basic concepts that were previously studied in physics courses, theoretical mechanics and resistance of materials.
1.1. Subject of mechanics of deformable solids
Mechanics of deformable solid is the science of the equilibrium and movement of solid bodies and their individual particles, taking into account changes in the distances between individual points of the body that arise as a result of external influences on the solid body. The mechanics of a deformable solid body is based on the laws of motion discovered by Newton, since the speed of motion of real solid bodies and their individual particles relative to each other is significantly less than the speed of light. In contrast to theoretical mechanics, changes in distances between individual particles of a body are considered here. 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 essential properties of deformable bodies and the materials from which they are made. In this case, to describe the properties of the material, the results of experimental studies are used, which served as the basis for creating models of the material. Depending on the material model, the mechanics of a deformable solid is divided into sections: the theory of elasticity, the theory of plasticity, the theory of creep, and the theory of viscoelasticity. In turn, the mechanics of a deformable solid body is part of a more general part of mechanics - mechanics continuum. 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 mechanics of deformable solids is largely associated with the tasks of creating reliable structures and machines. The reliability of the 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 its integrity under external influences without dividing into parts not previously provided for. If the strength is insufficient, the structure or its individual elements are destroyed by dividing the whole into parts. The rigidity of a structure is determined by the measure of change in the shape and size of the structure and its elements under external influences. If changes in the shape and size of a 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 the structure and its elements to maintain its 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 removing the disturbing forces, it returns to its original form of equilibrium. Otherwise, a loss of stability of the original form of equilibrium occurs, which, as a rule, is accompanied by the destruction of the structure. Endurance refers to the ability of a structure to resist the effects of forces that vary over time. 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 magnitude of forces that vary over time. Besides, low frequencies natural vibrations the structure and its elements should not coincide (or be located close to) the oscillation frequencies of external forces. Otherwise, the structure or its individual elements enter into resonance, which may 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 issues of design of structures and machines and problems of technological processes for processing materials. But the scope of application of the mechanics of a deformable solid is not limited to the technical sciences alone. Her methods are widely used in natural sciences, such as geophysics, solid state physics, geology, biology. Thus, in geophysics, with the help of the mechanics of a deformable solid, the processes of propagation of seismic waves and the processes of formation earth's crust, fundamental questions of the structure of the earth’s crust are studied, etc.
1.2. General properties of solids
All solids are made of real materials that have a huge variety of properties. Of these, only a few are of significant importance for the mechanics of a deformable solid. Therefore, the material is endowed with only those properties that make it possible to study the behavior of solids within the framework of the science in question at the least cost.
Mechanics of deformable solids is a science that studies the laws of equilibrium and motion of 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. Correct solution of these tasks 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 refers to the ability of a structure, structure and its individual elements to operate safely, 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 pattern of operation 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 appearance 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 largely relied upon in the disciplines studied by undergraduate students, such as structural mechanics, building structures, structural testing, 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, and to develop methods for solving special tasks, such as, for example, the problem of stress concentration near holes, as well as 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. Scientists and engineers during this period developed enough simple methods calculation of structural elements and laid the foundations further development strength science.
The theory of elasticity began to develop in early XIX centuries like mathematical science, not having an applied nature. Plasticity theory and creep theory both independent sections mechanics of deformable solids were formed in the 20th century.
The mechanics of a deformable solid body is constantly developing science. New methods are being developed for determining the stressed and deformed states of bodies. Various numerical methods for solving problems have become widely used, which is associated with the introduction and use of computers in almost all areas of science and engineering practice.
