Rice. 1.5. Beam deformation
At any points of the beam under consideration there is an identical state of stress and, therefore, the linear deformations for all its points are the same. Therefore, the value e can be defined as the ratio of the absolute elongation to the original length of the beam, i.e.
Bars from various materials lengthen differently. For cases where the stresses in the beam do not exceed the proportionality limit, the following relationship has been established by experience:
Where N- longitudinal force in cross sections of timber; F- square cross section timber; E- coefficient depending on physical properties material.
Considering that the normal stress in the cross section of the beam σ = N/F we get ε = σ/E. Where from σ = εE.
The absolute elongation of a beam is expressed by the formula
The following formulation of Hooke's law is more general: the relative longitudinal strain is directly proportional to the normal stress. In this formulation, Hooke's law is used not only in the study of tension and compression of beams, but also in other sections of the course.
Magnitude E called the elastic modulus of the first kind. This is a physical constant of a material that characterizes its rigidity. How more value E, the less, other things being equal, the longitudinal deformation. The modulus of elasticity is expressed in the same units as stress, i.e. in pascals (Pa) (steel E=2* 10 5 MPa, copper E= 1 * 10 5 MPa).
Work E.F. is called the stiffness of the cross section of the beam in tension and compression.
In addition to longitudinal deformation, when a compressive or tensile force is applied to the beam, transverse deformation is also observed. When a beam is compressed, its transverse dimensions increase, and when stretched, they decrease. If the transverse size of the beam before applying compressive forces to it R designate IN, and after applying these forces B - ∆B, then the value ∆V will indicate the absolute transverse deformation of the beam.
The ratio is the relative transverse strain.
Experience shows that at stresses not exceeding the elastic limit, the relative transverse deformation is directly proportional to the relative longitudinal deformation, but has the opposite sign:
The proportionality coefficient q depends on the material of the timber. It is called the transverse strain coefficient (or Poisson's ratio ) and is the ratio of relative transverse to longitudinal deformation, taken in absolute value, i.e. Poisson's ratio along with elastic modulus E characterizes the elastic properties of the material.
Poisson's ratio is determined experimentally. For various materials it has values from zero (for cork) to a value close to 0.50 (for rubber and paraffin). For steel, Poisson's ratio is 0.25...0.30; for a number of other metals (cast iron, zinc, bronze, copper) it
Rice. 1.6. Beam of variable cross section
Determination of the cross-sectional value of the rod is carried out based on the strength condition
where [σ] is the permissible stress.
Let's define the longitudinal displacement δ a points A axis of a beam stretched by force R( rice. 1.6).
It is equal to the absolute deformation of part of the beam ad enclosed between the embedment and the section drawn through the point d, those. longitudinal deformation of the beam is determined by the formula
This formula is applicable only when, within the entire length of the section, the longitudinal forces N and stiffness E.F. the cross sections of the beam are constant. In the case under consideration, on the site ab longitudinal force N is equal to zero (we do not take into account the dead weight of the timber), and in the area bd it is equal R, in addition, the cross-sectional area of the timber in the area ac differs from the cross-sectional area on the site cd. Therefore, the longitudinal deformation of the area ad should be determined as the sum of longitudinal deformations of three sections ab, bc And CD, for each of which the values N And E.F. constant along its entire length:
Longitudinal forces on the considered sections of the beam
Hence,
Similarly, you can determine the displacements δ of any points on the beam axis, and use their values to construct a diagram longitudinal movements (epureδ), i.e. a graph depicting the change in these movements along the length of the axis of the beam.
4.2.3. Conditions of strength. Calculation of rigidity.
When checking cross-sectional area stresses F and longitudinal forces are known and the calculation consists of calculating the calculated (actual) stresses σ in the characteristic sections of the elements. The maximum voltage obtained is then compared with the permissible one:
When selecting sections determine the required areas [F] cross sections of the element (based on known longitudinal forces N and permissible stress [σ]). Accepted cross-sectional areas F must satisfy the strength condition expressed in the following form:
When determining the load capacity by known values F and permissible stress [σ], the permissible values [N] of longitudinal forces are calculated:
Based on the obtained values [N], the permissible values of external loads are then determined [ P].
For this case, the strength condition has the form
The values of standard safety factors are established by standards. They depend on the class of the structure (capital, temporary, etc.), its intended service life, load (static, cyclic, etc.), possible heterogeneity in the manufacture of materials (for example, concrete), and the type of deformation (tension, compression , bending, etc.) and other factors. In some cases, it is necessary to reduce the safety factor in order to reduce the weight of the structure, and sometimes to increase the safety factor - if necessary, take into account the wear of the rubbing parts of machines, corrosion and decay of the material.
The values of standard safety factors for various materials, structures and loads in most cases have the following values: - 2.5...5 and - 1.5...2.5.
By checking the rigidity of a structural element in a state of pure tension-compression, we mean searching for an answer to the question: are the values of the element’s rigidity characteristics (modulus of elasticity of the material) sufficient? E and cross-sectional area F), so that the maximum of all values of displacement of element points caused by external forces, u max, does not exceed a certain specified limit value [u]. It is believed that if the inequality u max< [u] конструкция переходит в предельное состояние.
Let's consider a straight beam of constant cross-section with a length embedded at one end and loaded at the other end with a tensile force P (Fig. 8.2, a). Under the influence of force P, the beam elongates by a certain amount, which is called complete, or absolute, elongation (absolute longitudinal deformation).
At any points of the beam under consideration there is an identical state of stress and, therefore, linear deformations (see § 5.1) for all its points are the same. Therefore, the value can be defined as the ratio of the absolute elongation to the initial length of the beam I, i.e. Linear deformation during tension or compression of beams is usually called relative elongation, or relative longitudinal deformation, and is designated.
Hence,
Relative longitudinal strain is measured in abstract units. Let us agree to consider the elongation strain to be positive (Fig. 8.2, a), and the compression strain to be negative (Fig. 8.2, b).
The greater the magnitude of the force stretching the beam, the greater, other things being equal, the elongation of the beam; how larger area cross-section of the beam, the less elongation of the beam. Bars made from different materials elongate differently. For cases where the stresses in the beam do not exceed the proportionality limit (see § 6.1, paragraph 4), the following relationship has been established by experience:
Here N is the longitudinal force in the cross sections of the beam; - cross-sectional area of the beam; E is a coefficient depending on the physical properties of the material.
Considering that the normal stress in the cross section of the beam we obtain
The absolute elongation of a beam is expressed by the formula
that is, the absolute longitudinal deformation is directly proportional to the longitudinal force.
For the first time, the law of direct proportionality between forces and deformations was formulated (in 1660). Formulas (10.2)-(13.2) are mathematical expressions of Hooke’s law for tension and compression of a beam.
The following formulation of Hooke's law is more general [see. formulas (11.2) and (12.2)]: the relative longitudinal strain is directly proportional to the normal stress. In this formulation, Hooke's law is used not only in the study of tension and compression of beams, but also in other sections of the course.
The quantity E included in formulas (10.2)-(13.2) is called the modulus of elasticity of the first kind (abbreviated as modulus of elasticity). This quantity is a physical constant of the material, characterizing its rigidity. The greater the value of E, the less, other things being equal, the longitudinal deformation.
We will call the product the stiffness of the cross section of the beam under tension and compression.
Appendix I shows the values of elastic modulus E for various materials.
Formula (13.2) can be used to calculate the absolute longitudinal deformation of a section of a beam of length only under the condition that the section of the beam within this section is constant and the longitudinal force N is the same in all cross sections.
In addition to longitudinal deformation, when a compressive or tensile force is applied to the beam, transverse deformation is also observed. When a beam is compressed, its transverse dimensions increase, and when stretched, they decrease. If the transverse size of the beam before applying compressive forces P to it is designated b, and after the application of these forces (Fig. 9.2), then the value will indicate the absolute transverse deformation of the beam.
The ratio is the relative transverse strain.
Experience shows that at stresses not exceeding the elastic limit (see § 6.1, paragraph 3), the relative transverse deformation is directly proportional to the relative longitudinal deformation, but has the opposite sign:
The proportionality coefficient in formula (14.2) depends on the material of the beam. It is called the transverse deformation ratio, or Poisson's ratio, and is the ratio of the relative transverse deformation to the longitudinal deformation, taken in absolute value, i.e.
Poisson's ratio, along with the elastic modulus E, characterizes the elastic properties of the material.
The value of Poisson's ratio is determined experimentally. For various materials it has values from zero (for cork) to a value close to 0.50 (for rubber and paraffin). For steel, Poisson's ratio is 0.25-0.30; for a number of other metals (cast iron, zinc, bronze, copper) it has values from 0.23 to 0.36. Approximate values of Poisson's ratio for various materials are given in Appendix I.
Lecture No. 5
Subject: " Tension and compression»
Questions:
1. Normal stresses in tension and compression
2. Determination of longitudinal and transverse deformation. Hooke's law
4. Temperature stress
5. Mounting stresses
1. Normal stresses in tension and compression
If you apply a grid of lines parallel and perpendicular to the axis of the rod on the surface of a prismatic rod and apply a tensile force to it, you can make sure that the grid lines will remain mutually perpendicular even after deformation (see Fig. 1).
Rice. 1
All horizontal lines, such as cd, will move down while remaining horizontal and straight. It can also be assumed that there will be the same picture inside the rod, i.e. “cross sections of a rod that are flat and normal to its axis before deformation will remain flat and normal to its axis after deformation.” This important hypothesis is called the hypothesis of plane sections or Bernoulli's hypothesis. The formulas obtained on the basis of this hypothesis are confirmed by experimental results.
This picture of deformations gives reason to believe that only normal stresses act in cross sections, identical at all points of the section, and tangential stresses are equal to zero. If tangential stresses occurred, then angular deformation would be observed, and the angles between the longitudinal and transverse lines would no longer be straight. If the normal stresses were not the same at all points of the section, then where the stresses are higher, there would be greater deformation, and therefore the cross sections would not be plane and parallel. By accepting the hypothesis of plane sections we establish that 
.
Since the longitudinal force is the resultant of the internal forces 
, arising on infinitely small areas (see Figure 3.2), it can be represented as:
Rice. 2
Constant quantities can be taken out of the integral sign:
where A is the cross-sectional area.
We obtain a formula for finding normal stresses during tension or compression:
(1)
This is one of the most important formulas in the strength of materials, so we will highlight it in a frame and will do the same in the future.
When stretched 
positive, when compressed - negative.
If only one acts on the beam external force F, That
N= F,
and voltages can be determined by the formula:
2. Determination of longitudinal and transverse deformation
In the elastic stage of operation of most structural materials, stress and strain are related by a direct relationship called Hooke's law:
(2)
where E is the modulus of longitudinal elasticity or Young’s modulus, measured in MPa, characterizing the stiffness of the material, i.e. ability to resist deformation, its values are given in the tables of the reference book;
relative longitudinal deformation, a dimensionless value, since:
; (3)
absolute elongation of the rod, m;
l initial length, m.
The higher the value of the longitudinal elastic modulus E, the less the deformation. For example, for steel E = 2.110 5 MPa, and for cast iron E = (0.75...1.6)10 5 MPa, therefore, a structural element made of cast iron, under the same other conditions, will receive greater deformation than from steel. This should not be confused with the fact that a steel rod brought to the point of rupture will have a significantly greater deformation than a cast iron rod. It's about not about the limiting deformation, but about the deformation in the elastic stage, i.e. without the occurrence of plastic deformations, and under the same load.
Let's transform Hooke's law by replacing from equation (3.3):
Let's substitute the value 
from formula (1):
(4)
We have obtained a formula for the absolute elongation (shortening) of the rod. When stretched 
positive, during compression – negative. Work EA called the stiffness of the beam.
When stretched, the rod becomes thinner, and when compressed, it becomes thicker. The change in cross-sectional dimensions is called transverse deformation. For example, at rectangular section before loading were width b and section height h, and after loading b 1 And h 1 . Relative transverse deformation for section width:
for section height:
Isotropic materials have the same properties in all directions. That's why:
In tension, the transverse strain is negative; in compression, it is positive.
The ratio of transverse to longitudinal strain is called the transverse strain ratio or Poisson's ratio:
(5)
It has been experimentally established that in the elastic stage of operation of any material the value 
and constantly. It lies within 0 
0.5 and for construction materials is given in the tables of the reference book.
From dependence (5) we can obtain the following formula:
(6)
During tension (compression), the cross sections of the beam move in the longitudinal direction. Displacement is a consequence of deformation, but these two concepts must be clearly distinguished. For the rod (see Fig. 3), we determine the magnitude of the deformation and construct a displacement diagram.
Rice. 3
As can be seen from the figure, the segment of the rod AB does not stretch, but will receive movement, since the segment CB will lengthen. Its elongation is:
We denote the displacements of the cross sections by 
. In section C the displacement is zero. From section C to section B, the displacement is equal to the elongation, i.e. increases proportionally to 
in section B. For sections from B to A, the displacements are the same and equal 
, since this section of the rod is not deformed.
3. Statically indeterminate problems
Systems in which the forces cannot be determined using only static equations are considered to be statically indeterminate. All statically indeterminate systems have “extra” connections in the form of additional fastenings, rods and other elements. Such connections are called “superfluous” because they are not necessary from the point of view of ensuring the equilibrium of the system or its geometric immutability, and their arrangement pursues constructive or operational purposes.
The difference between the number of unknowns and the number of independent equilibrium equations that can be constructed for a given system characterizes the number of extra unknowns or the degree of static indetermination.
Statically indeterminate systems are solved by drawing up equations for the displacement of certain points, the number of which must be equal to the degree of indetermination of the system.
Let a force act on a rod rigidly fixed at both ends F(see Fig. 4). Let us determine the reactions of the supports.
Rice. 4
We will direct the reaction of the supports to the left, since the force F acts to the right. Since the weight of the force acts along one line, only one equation of static equilibrium can be drawn up:
-B+F-C=0;
So, two unknown reactions of supports B and C and one equation of static equilibrium. The system is once statically indeterminate. Therefore, to solve it, you need to create one additional equation based on the movements of point C. Let’s mentally discard the right support. Due to force F, the left side of the VD rod will be stretched and section C will shift to the right by the amount of this deformation:
From the support reaction C, the rod will compress and the section will move to the left by the amount of deformation of the entire rod:
The support does not allow section C to move either to the left or to the right, therefore the sum of the displacements from the forces F and C must be equal to zero:
|
Substituting the value of C into the static equilibrium equation, we determine the second reaction of the support:
4. Temperature stress
In statically indeterminate systems, stresses may arise when temperature changes. Let the rod, rigidly sealed at both ends, be heated to a temperature 
hail (see Fig. 5).
Rice. 5
When heated, the bodies expand, and the rod will tend to lengthen by the amount:
Where 
coefficient of linear expansion,
l- original length.
The supports do not allow the rod to lengthen, so the rod is compressed by the amount:
According to formula (4):
=
;
because the:
(7)
As can be seen from formula (7), temperature stresses do not depend on the length of the rod, but depend only on the coefficient of linear expansion, the modulus of longitudinal elasticity and temperature changes.
Temperature stresses can reach high values. To reduce them, special temperature gaps (for example, gaps in rail joints) or compensation devices (for example, elbows in pipelines) are provided in structures.
5. Mounting stresses
Structural elements may have dimensional deviations during manufacture (for example, due to welding). During assembly, dimensions do not match (e.g. bolt holes) and force is applied to assemble the units. As a result, internal forces arise in structural elements without applying an external load.
Let a rod be inserted between two rigid seals, the length of which is equal to A greater than the distance between supports (see Fig. 6). The rod will experience compression. Let's determine the voltage using formula (4):
(8)
Rice. 6
As can be seen from formula (8), installation stresses are directly proportional to the dimensional error A. Therefore it is advisable to have a=0, especially for short rods, since 
inversely proportional to length.
However, in statically indeterminate systems, mounting stresses are specifically resorted to in order to increase bearing capacity designs.
R. Hooke's and S. Poisson's laws
Let us consider the deformations of the rod shown in Fig. 2.2.
Rice. 2.2 Longitudinal and transverse tensile deformations
Let us denote by the absolute elongation of the rod. When stretched, this is a positive value. Through – absolute transverse deformation. When stretched, this is a negative value. The signs of and change accordingly during compression.
Relationship
(epsilon) or 
, (2.2)
called relative elongation. It is positive under tension.
Relationship
Or 
, (2.3)
called relative transverse strain. It is negative when stretched.
R. Hooke in 1660 discovered a law that said: “What is the elongation, such is the force.” In modern writing, R. Hooke's law is written as follows:
that is, the stress is proportional to the relative strain. Here, E. Young’s modulus of elasticity of the first kind is a physical constant within the limits of R. Hooke’s law. It is different for different materials. For example, for steel it is equal to 2 10 6 kgf/cm 2 (2 10 5 MPa), for wood – 1 10 5 kgf/cm 2 (1 10 4 MPa), for rubber – 100 kgf/cm 2 ( 10 MPa), etc.
Considering that , a , we get
where is the longitudinal force at the force section;
– length of the power section;
– rigidity in tension and compression.
That is, the absolute deformation is proportional to the longitudinal force acting on the force section, the length of this section and is inversely proportional to the tensile-compression stiffness.
When calculating based on the action of external loads
where is the external longitudinal force;
– the length of the section of the rod on which it acts. In this case, the principle of independence of the action of forces is applied*).
S. Poisson proved that the ratio is a constant value, different for different materials, that is
or , (2.7)
where is S. Poisson's ratio. This is, generally speaking, a negative value. In reference books, its value is given “modulo”. For example, for steel it is 0.25...0.33, for cast iron - 0.23...0.27, for rubber - 0.5, for cork - 0, that is. However, for wood it can be more than 0.5.
Experimental study deformation processes and
Fracture of tensile and compressed rods
Russian scientist V.V. Kirpichev proved that the deformations of geometrically similar samples are similar if the forces acting on them are placed similarly, and that based on the results of testing a small sample, one can judge the mechanical characteristics of the material. In this case, of course, the scale factor is taken into account, for which a scale factor is introduced, determined experimentally.
Tensile Chart of Mild Steel
Tests are carried out on tensile machines with simultaneous recording of the fracture diagram in the coordinates – force, – absolute deformation (Fig. 2.3, a). Then the experiment is recalculated in order to construct a conditional diagram in coordinates (Fig. 2.3, b).
From the diagram (Fig. 2.3, a) the following can be seen:
– Hooke’s law is valid up to the point;
– from point to point, the deformations remain elastic, but Hooke’s law is no longer valid;
– from point to point, deformations increase without increasing the load. Here the cement frame of ferrite grains of the metal is destroyed, and the load is transferred to these grains. Chernov–Luders shear lines appear (at an angle of 45° to the sample axis);
– from point to point – the stage of secondary hardening of the metal. At the point the load reaches a maximum, and then a narrowing appears in the weakened section of the sample - the “neck”;
– at the point – the sample is destroyed.
Rice. 2.3 Diagrams of steel fracture under tension and compression
The diagrams allow you to get the following basic mechanical characteristics become:
– limit of proportionality – the highest stress up to which Hooke’s law is valid (2100...2200 kgf/cm 2 or 210...220 MPa);
– elastic limit – the highest stress at which deformations still remain elastic (2300 kgf/cm 2 or 230 MPa);
– yield strength – stress at which deformations increase without increasing the load (2400 kgf/cm 2 or 240 MPa);
- tensile strength 
– stress corresponding to the greatest load withstood by the sample during the experiment (3800...4700 kgf/cm 2 or 380...470 MPa);
Let us consider a straight rod of constant cross-section, rigidly fixed at the top. Let the rod have a length and be loaded with a tensile force F . The action of this force increases the length of the rod by a certain amount Δ (Fig. 9.7, a).
When the rod is compressed with the same force F the length of the rod will be reduced by the same amount Δ (Fig. 9.7, b).
Magnitude Δ , equal to the difference between the lengths of the rod after deformation and before deformation, is called the absolute linear deformation (elongation or shortening) of the rod when it is stretched or compressed.
Absolute linear strain ratio Δ to the original length of the rod is called relative linear deformation and is denoted by the letter ε or ε x ( where is the index x indicates the direction of deformation). When the rod is stretched or compressed, the amount ε is simply called the relative longitudinal deformation of the rod. It is determined by the formula:
Repeated studies of the process of deformation of a stretched or compressed rod in the elastic stage have confirmed the existence of a direct proportional relationship between normal stress and relative longitudinal deformation. This relationship is called Hooke's law and has the form:
Magnitude E called the modulus of longitudinal elasticity or the modulus of the first kind. It is a physical constant (constant) for each type of rod material and characterizes its rigidity. The larger the value E , the less will be the longitudinal deformation of the rod. Magnitude E measured in the same units as voltage, that is, in Pa , MPa , etc. The elastic modulus values are contained in the tables of reference and educational literature. For example, the value of the modulus of longitudinal elasticity of steel is taken equal to E = 2∙10 5 MPa , and wood
E = 0.8∙10 5 MPa.
When calculating rods in tension or compression, there is often a need to determine the value of absolute longitudinal deformation if the magnitude of the longitudinal force, cross-sectional area and material of the rod are known. From formula (9.8) we find: . Let us replace in this expression ε its value from formula (9.9). As a result we get = . If we use the normal stress formula , then we obtain the final formula for determining the absolute longitudinal deformation:
The product of the modulus of longitudinal elasticity and the cross-sectional area of the rod is called its rigidity when stretched or compressed.
Analyzing formula (9.10), we can draw a significant conclusion: the absolute longitudinal deformation of a rod during tension (compression) is directly proportional to the product of the longitudinal force and the length of the rod and inversely proportional to its rigidity.
Note that formula (9.10) can be used in the case when the cross section of the rod and the longitudinal force have constant values along its entire length. In the general case, when a rod has a stepwise variable stiffness and is loaded along its length with several forces, it is necessary to divide it into sections and determine the absolute deformations of each of them using formula (9.10).
The algebraic sum of the absolute deformations of each section will be equal to the absolute deformation of the entire rod, that is:
The longitudinal deformation of the rod from the action of a uniformly distributed load along its axis (for example, from the action of its own weight) is determined by the following formula, which we present without proof:
In the case of tension or compression of a rod, in addition to longitudinal deformations, transverse deformations also occur, both absolute and relative. Let us denote by b cross-sectional size of the rod before deformation. When the rod is stretched by force F this size will decrease by Δb , which is the absolute transverse deformation of the rod. This value has a negative sign. During compression, on the contrary, the absolute transverse strain will have a positive sign (Fig. 9.8).
