External forces in strength of materials are divided into active And reactive(connection reactions). Loads are active external forces.
Loads by application method
By application method loads there are voluminous(own weight, inertial forces) acting on each infinitesimal element of volume, and surface ones. Surface Loads are divided into concentrated loads And distributed loads.
Distributed Loads characterized by pressure - the ratio of the force acting on a surface element normal to it to the area of this element and is expressed in the International System of Units (SI) in pascals, megapascals (1 PA = 1 N/m2; 1 MPa = 106 Pa), etc. etc., and in the technical system - in kilograms of force per square millimeter, etc. (kgf/mm2, kgf/cm2).
In compromising materials, they are often considered surface loads, distributed along the length of the structural element. Such loads are characterized by intensity, usually denoted q and expressed in newtons per meter (N/m, kN/m) or in kilograms of force per meter (kgf/m, kgf/cm), etc.
Loads according to the nature of changes over time
Based on the nature of changes over time, they are distinguished static loads- increasing slowly from zero to its final value and then not changing; And dynamic loads causing large inertial forces.
28. Dynamic, cyclic loading, the concept of endurance limit.
Dynamic load is a load that is accompanied by the acceleration of particles of the body in question or parts in contact with it. Dynamic loading occurs when rapidly increasing forces are applied or in the case of accelerated movement of the body under study. In all these cases, it is necessary to take into account the forces of inertia and the resulting movement of the masses of the system. In addition, dynamic loads can be divided into impact and repeated-variable loads.
Impact load (impact) is a loading in which the acceleration of body particles sharply changes its value in a very short period of time (sudden application of load). Note that, although impact is a dynamic type of loading, in a number of cases, when calculating impact, inertial forces are neglected.
Repeatedly variable (cyclic) loading – loads that change in magnitude (and possibly in sign) over time.
Cyclic loading is a change in the mechanical and physical properties of a material under the long-term action of stresses and strains that cyclically change over time.
Endurance limit(Also limit fatigue) - in the sciences of strength: one of the strength characteristics of a material that characterizes it endurance, that is, the ability to absorb loads that cause cyclic stresses in the material.
29. The concept of fatigue of materials, factors influencing resistance to fatigue failure.
Material fatigue- in materials science - the process of gradual accumulation of damage under the influence of variable (often cyclic) stresses, leading to a change in its properties, the formation of cracks, their development and destruction material for the specified time.
Effect of stress concentration
In places where there is a sharp change in the transverse dimensions of the part, holes, grooves, grooves, threads, etc., as shown in paragraph 2.7.1, a local increase in stress occurs, significantly reducing the endurance limit compared to that for smooth cylindrical samples. This reduction is taken into account by introducing into the calculations effective stress concentration factor, representing the ratio of the endurance limit of a smooth sample under a symmetrical cycle to the endurance limit of a sample of the same dimensions, but having one or another stress concentrator:
.
2.8.3.2. Influence of part dimensions
It has been experimentally established that as the size of the test sample increases, its endurance limit decreases ( scale effect). This is explained by the fact that with increasing size, the probability of heterogeneity in the structure of materials and its internal defects (cavities, gas inclusions) increases, and also by the fact that when producing small-sized samples, hardening (hardening) of the surface layer takes place to a relatively greater depth than that of the samples large sizes.
The influence of the dimensions of parts on the value of the endurance limit is taken into account by the coefficient ( scale factor), which is the ratio of the endurance limit of a part of given dimensions to the endurance limit of a laboratory sample of a similar configuration having small dimensions:
.
2.8.3.3. Influence of surface condition
Traces of a cutting tool, sharp marks, scratches are the source of fatigue microcracks, which leads to a decrease in the endurance limit of the material.
The influence of the surface condition on the endurance limit in a symmetrical cycle is characterized by coefficient surface quality, which is the ratio of the endurance limit of a part with a given surface treatment to the endurance limit of a thoroughly polished sample:
.
2.8.3.4. Effect of surface hardening
Various methods of surface hardening (mechanical hardening, chemical-thermal and heat treatment) can significantly increase the value of the surface quality coefficient (up to 1.5 ... 2.0 or more times instead of 0.6 ... 0.8 times for parts without hardening). This is taken into account in calculations by introducing the coefficient.
2.8.3.5. Impact of cycle asymmetry
The cause of fatigue failure of a part is long-acting alternating stresses. But, as experiments have shown, with an increase in the strength properties of a material, their sensitivity to cycle asymmetry increases, i.e. the constant component of the cycle “contributes” to the reduction in fatigue strength. This factor is taken into account by the coefficient.
Constant loads.(q) Depending on the duration of action, loads are divided into permanent and temporary. Constant loads are the weight of load-bearing and enclosing structures of buildings and structures, the weight and pressure of soils, the effect of prestressing reinforced concrete structures.
Temporary loads. Long-term loads(P) . These include: weight stationary equipment on the floors of machines, apparatus, engines, containers, etc.; pressure of gases, liquids, granular bodies in containers; the weight of specific contents in warehouses, refrigerators, archives, libraries and similar buildings and structures; the part of the live load established by the standards in residential buildings, in office and domestic premises; long-term temperature technological effects from stationary equipment; loads from one suspended or one overhead crane, multiplied by coefficients: 0.5, 0.6...depending on the type of crane
Short-term loads.(S) These include: the weight of people, parts, materials in equipment maintenance and repair areas - aisles and other areas free from equipment; part of the load on the floors of residential and public buildings; loads arising during the manufacture, transportation and installation of structural elements; loads from overhead and overhead cranes used in the construction or operation of buildings and structures; snow and wind loads; temperature climatic influences.
Special loads. These include: seismic and explosive effects; loads caused by malfunction or breakdown of equipment and sudden disruption technological process(for example, with a sharp increase or decrease in temperature, etc.); the effects of uneven deformations of the base, accompanied by a radical change in the structure of the soil (for example, deformation of subsidence soils during soaking or permafrost soils during thawing), etc.
Standard loads. They are established by standards or nominal values. Standard permanent loads are taken based on the design values of the geometric and structural parameters and on the average density values. Standard temporary technological and installation loads are established according to highest values intended for normal use; snow and wind - according to the average of annual unfavorable values or according to unfavorable values corresponding to a certain average period of their repetitions.
Design loads. Their values when calculating structures for strength and stability are determined by multiplying the standard load by the load safety factor γf, usually greater than unity. Reliability factor under the action of the weight of concrete and reinforced concrete structures γ f -1>1. The reliability coefficient under the influence of the weight of structures, used in calculating the stability of the position against floating, capsizing and sliding, as well as in other cases when a decrease in mass worsens the operating conditions of the structure, is accepted γ f=0.9. When calculating structures at the construction stage, the calculated short-term loads are multiplied by a factor of 0.8. When calculating structures based on deformations and displacements (according to the second group of limit states), the design loads are taken equal to the standard values with a coefficient γt = 1.
Combination of loads. Structures must be designed for various combinations loads or the forces corresponding to them, if the calculation is carried out according to the inelastic state scheme. Depending on the composition of the loads taken into account, the following are distinguished: basic combinations, including constant, long-term and short-term loads or forces from them; special combinations, including constant, long-term, possible short-term and one of the special loads or efforts from them.
In the main combinations, when taking into account at least two temporary loads, their calculated values (or the corresponding efforts) are multiplied by combination coefficients equal to: for long-term loads f1 = 0.95; for short-term f2=0.9. When taking into account one temporary load, f1 = f2 = l. When taking into account three or more short-term loads, the standards allow their calculated values to be multiplied by combination coefficients: f 2 =l- for the first most important short-term load; f 2 = 0.8 - for the second; f2 = 0.6 - for the rest.
In special combinations for long-term loads f1 = 0.95, for short-term loads f 2 = 0.8, except for cases specified in the design standards for buildings and structures in seismic areas.
Classification of loads.
Statistical load (Fig. 18.2 A) do not change over time or change very slowly. When subject to statistical loads, strength calculations are carried out.
Re-variables loads (Fig. 18.26) repeatedly change value or value and sign. The action of such loads causes metal fatigue.
Dynamic loads (Fig. 18.2c) change their value in a short period of time, they cause large accelerations and inertia forces and can lead to sudden destruction of the structure.
It is known from theoretical mechanics that, depending on the method of applying loads, there can be focused or distributed on the surface.
In reality, the transfer of load between parts occurs not at a point, but at a certain area, i.e. the load is distributed.
However, if the contact area is negligibly small compared to the dimensions of the part, the force is considered concentrated.
When calculating real deformable bodies in the resistance of materials, it is not necessary to replace the distributed load with a concentrated one.
The axioms of theoretical mechanics in the strength of materials are used to a limited extent.
You cannot transfer a pair of forces to another point on a part, move a concentrated force along the line of action, and you cannot replace a system of forces with a resultant when determining displacements. All of the above changes the distribution of internal forces in the structure.
Shapes of structural elements
All the variety of forms is reduced to three types based on one characteristic.
1. Beam- any body whose length is significantly greater than other dimensions.
Depending on the shape of the longitudinal axis and cross sections, several types of beams are distinguished:
Straight permanent beam cross section(Fig. 18.3a);
Straight stepped beam (Fig. 18.35);
Curved beam (Fig. 18.Sv).
2. Plate- any body whose thickness is significantly less than other dimensions (Fig. 18.4).
3. Array- a body that has three sizes of the same order.
Control questions and tasks
1. What is called strength, rigidity, stability?
2. By what principle are loads classified in the resistance of materials? What type of damage does repeated variable loads lead to?
4. What body is called a beam? Draw any beam and indicate the axis of the beam and its cross section. What bodies are called plates?
5. What is deformation? What deformations are called elastic?
6. At what deformations is Hooke’s law satisfied? Formulate Hooke's law.
7. What is the principle of initial sizes?
8. What is the assumption of the continuous structure of materials? Explain the assumption of homogeneity and isotropy of materials.
LECTURE 19
Topic 2.1. Basic provisions. External and internal loads, section method
Know the method of sections, internal force factors, stress components.
Be able to determine types of loads and internal force factors in cross sections.
During operation, structural elements experience external influence, which is assessed by the magnitude of the external force. External forces include active forces and reactions of supports.
Under the influence of external forces, internal elastic forces arise in the part, striving to return the body to its original shape and size.
External forces must be determined by methods of theoretical mechanics, and internal forces must be determined by the main method of strength of materials - the method of sections.
In the resistance of materials, bodies are considered in equilibrium. To solve problems, equilibrium equations obtained in theoretical mechanics for a body in space are used.
The coordinate system associated with the body is used. More often, the longitudinal axis of a part is designated z, the origin of coordinates is aligned with the left edge and placed at the center of gravity of the section.
Section method
The method of sections consists of mentally dissecting a body with a plane and considering the equilibrium of any of the cut off parts.
If the whole body is in balance, then each part of it is in balance under the influence of external and internal forces. Internal forces are determined from equilibrium equations compiled for the body part in question.
We dissect the body across the plane (Fig. 19.1). Let's look at the right side. External forces act on it F 4; F 5 ; F 6 and internal elastic forces q to, distributed over the section. The system of distributed forces can be replaced by the main vector Ro , placed at the center of gravity of the section, and the total moment of forces.
The main moment is also usually represented in the form of moments of pairs of forces in three projection planes:
M x- torque relative to Oh;M y - torque relative to O y, M z - torque relative to Oz.
The resulting components of elastic forces are called internal power factors. Each of the internal force factors causes a certain deformation of the part. Internal force factors balance the external forces applied to this element of the part. Using six equilibrium equations, we can obtain the magnitude of the internal force factors:
From the above equations it follows that:
N z - longitudinal force, Oz external forces acting on the cut-off part of the beam; causes tension or compression;
Q x - shear force, equal to the algebraic sum of projections onto the axis Oh
Q y - shear force, equal to the algebraic sum of projections onto the axis OU external forces acting on the cut-off part;
forces Q x and Q y cause a shear of the section;
M z - torque, equal to the algebraic sum of the moments of external forces relative to the longitudinal axis Oz-, causes the beam to twist;
M x - bending moment, equal to the algebraic sum of the moments of external forces relative to the Coolant axis;
M y - bending moment, equal to the algebraic sum of the moments of external forces relative to the Oy axis.
The moments M x and M y cause the beam to bend in the corresponding plane.
Voltages
Section method allows you to determine the value of the internal force factor in the section, but does not make it possible to establish the law of distribution of internal forces over the section. To assess strength, it is necessary to determine the magnitude of the force at any point in the cross section.
The intensity of internal forces at a cross-section point is called mechanical stress. Stress characterizes the amount of internal force per unit cross-sectional area.
Consider a beam to which an external load is applied (Fig. 19.2). By using section method let's cut the beam with a transverse plane, discard the left part and consider the equilibrium of the remaining right part. Select a small area on the cutting plane ΔA. The resultant internal elastic forces act on this area.
Voltage direction p avg coincides with the direction of the internal force in this section.
Vector p avg called full tension. It is customary to decompose it into two vectors (Fig. 19.3): τ - lying in the section area and σ - directed perpendicular to the site.
If the vector ρ - spatial, then it is divided into three components:
Classification of external loads acting on structural elements.
General classification structural elements.
Technical objects and structures consist of individual parts and elements that vary widely in shape, size, and other parameters and characteristics. From the standpoint of engineering calculations, it is customary to distinguish four main groups of structural elements: rods, plates, shells, and arrays.
Rods– these are straight or curved structural elements in which one dimension (length) significantly exceeds two other dimensions (in a spatial orthogonal coordinate system), see Figure 20. Examples of structural elements such as rods: legs of a chair or table, column of a building structure, lifting rope cars, car gearbox shift lever, etc.
Z Curved rod
Straight rod
Figure 20. Diagrams of structural elements of the rod type
t (plate thickness)
Figure 21. Diagram of a plate type design element
Figure 22. Diagram of a shell type structural element (cylindrical)
Rice. 23. Diagram of an array type structural element
Plates- these are flat structural elements in which one size (thickness) is significantly smaller than the other two. Examples of plates: table top; walls and ceilings of buildings, etc., see Figure 21, from which it is clear that the thickness of the plate is significantly less than its two dimensions in plan.
Shells– these are not flat thin-walled elements structures in which one size (wall thickness) is significantly smaller than other sizes. Examples of casings: pipelines for transporting liquid and gaseous products (cylindrical casings); cylindrical, spherical or combined containers for liquids; conical bins for bulk materials; non-flat coatings of various structures, etc., see Figure 22, which shows a cylindrical shell (thin-walled cylindrical pipe), in which the wall thickness is significantly less than its diameter and length.
Arrays- these are structural elements in which all three sizes are comparable. Examples of arrays: foundation blocks machines, machines and building structures; massive bridge supports, etc., see Figure 23.
In the courses “Engineering Mechanics” and “Strength of Materials” greatest attention is devoted to the fundamental study of structural elements such as rods. Plates, shells, and arrays are studied in advanced Strength of Materials and specialty courses.
Concentrated Forces- these are forces applied to a structural element on its surface area, the dimensions of which, compared to the dimensions of the entire surface of the structural element, can be neglected. As a rule, concentrated forces are the result of the influence of another body (in particular, another structural element) on a given body (structural element). In many practically important cases, concentrated
forces can be considered applied to a structural element at a point without noticeable damage to the accuracy of engineering calculations. Units of measurement of concentrated forces N (Newton), kN (kilonewton), etc.
Volume forces- these are forces applied throughout the entire volume of a structural element, for example distributed gravity forces. Units of measurement of distributed volumetric forces N/m 3, kN/m 3, etc. The total force of gravity (N, kN) of a structural element is often conventionally taken into account in calculations as a concentrated force applied at a point called its center of gravity.
Distributed forces (loads)- these are forces applied to a part of the area (or length) of a deformable body, commensurate with the dimensions of the entire body. There are superficially distributed forces (loads), the units of measurement of which are N/m 2, kN/m 2, etc. (for example, distributed snow loads on building roofs), as well as linearly distributed loads (along the length of structural elements), the units of measurement of which are N/m, kN/m, etc. (for example, distributed pressure forces of slabs supported on beams of building structures).
Static forces (loads)– these are forces (loads) that do not change (or insignificantly change) their value, position and direction of action during the operation of the structure.
Dynamic forces (loads)– these are forces (loads) that significantly change their value, position and/or direction in short periods of time and cause vibrations of the structure.
Rated loads– these are the normal maximum loads that arise during the operation of the structure.
Control questions:
1) What is studied in the Strength of Materials course? What is its significance for highly skilled technical professionals?
2) What are external loads and internal forces?
3) Explain the concepts of deformation, strength, stiffness and stability.
4) Explain the concepts of homogeneity, continuity, isotropy and anisotropy.
5) Give a classification of structural elements.
6) Give a classification of external loads acting on structural elements.
1. Alexandrov A.V. and others. Strength of materials. Textbook for universities - M.: Higher. school, 2001. – 560 p. (p. 5...20).
2. Stepin P.A. Strength of materials. – M.: Higher. school, 1983. – 303 p. (p. 5...20).
3. Handbook on strength of materials/Pisarenko G.S. and others - Kyiv: Naukova Dumka, 1988. - 737 p. (p. 5...9).
Test tasks for SRS– with the help of educational literature, expand information on the following issues:
1) what are elastic forces?
2) what is the essence of the principle of the absence of initial internal efforts in the body (p. 9-10)?
3) what are the principles for schematizing external loads acting on structural elements used in engineering calculations (p. 8-11)?
4) explain the principle of independence of the action of forces (, pp. 18-20; , p. 10)?
5) explain the principle of Saint-Venant (, pp. 10-11);
6) what is the difference between deformation and displacement (, pp. 17-18; , pp. 13-14)?;
7) general concept about the method of sections (, pp. 13-16;, pp. 14-17);
8) the general concept of stresses in a deformable body, designations of normal and tangential stresses (, pp. 13-15;, pp. 17-20).
9) classification of external loads acting on structural elements (see clause 5.3).
Lecture 6. Topic 6. “Central tension-compression of straight rigid rods”
Purpose of the lecture– outline introductory provisions on the topic, the essence and application of the section method for determining internal forces in rods under central tension-compression; give basic concepts about diagrams of internal efforts.
External forces in strength of materials are divided into active And reactive(connection reactions). Loads are active external forces.
Loads by application method
According to the method of application, loads can be volumetric (own weight, inertial forces), acting on each infinitesimal element of volume, and surface. Surface Loads are divided into concentrated loads And distributed loads.
Distributed Loads characterized by pressure - the ratio of the force acting on a surface element normal to it to the area of this element and is expressed in the International System of Units (SI) in pascals, megapascals (1 PA = 1 N/m2; 1 MPa = 106 Pa), etc. d., and in technical system– in kilograms of force per square millimeter etc. (kgf/mm2, kgf/cm2).
In compromising materials, they are often considered surface loads, distributed along the length of the structural element. Such loads are characterized by intensity, usually denoted q and expressed in newtons per meter (N/m, kN/m) or in kilograms of force per meter (kgf/m, kgf/cm), etc.
Loads according to the nature of changes over time
Based on the nature of changes over time, they are distinguished static loads- increasing slowly from zero to its final value and then not changing; And dynamic loads causing big