Let us consider a straight beam of constant cross-section with length l, embedded at one end and loaded at the other end with a tensile force P (Fig. 2.9, a). Under the influence of force P, the beam elongates by a certain amount?l, 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, the linear deformations for all its points are the same. Therefore, the value can be defined as the ratio of the absolute elongation?l to the initial length of the beam l, 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. 2.9, a), and the compression strain to be negative (Fig. 2.9, b).
The greater the magnitude of the force stretching the beam, the greater, with other equal conditions, beam extension; the larger the area cross section beam, the less elongation of the beam. 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:
Here N is the longitudinal force in the cross sections of the beam;
F - cross-sectional area of the beam;
E - coefficient depending on physical properties 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
those. 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 by R. Hooke (in 1660).
The following formulation is more general Hooke's law 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 value E included in the formulas is called the longitudinal elastic modulus (abbreviated as the elastic modulus). This quantity is physical material constant, characterizing its rigidity. The greater the value of E, the less, other things being equal, the longitudinal deformation.
The product EF is called the cross-sectional stiffness of the beam in tension and compression.
If the transverse size of the beam before applying compressive forces P to it is denoted by b, and after the application of these forces b +?b (Fig. 9.2), then the value?b 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 strain is directly proportional to the relative longitudinal strain e, but has the opposite sign:
The proportionality coefficient in formula (2.16) depends on the material of the beam. It is called the transverse strain ratio, or Poisson's ratio, and is the ratio of transverse strain to longitudinal strain, taken according to 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.
Table 2.1 Elastic modulus values.
Table 2.2 Transverse strain coefficient values (Poisson's ratio)
9. Absolute and relative strain in tension (compression). Poisson's ratio.
If, under the influence of a force, a beam of length changes its longitudinal value by , then this value is called absolute longitudinal deformation (absolute elongation or shortening). In this case, transverse absolute deformation is also observed.
The ratio is called relative longitudinal strain, and the ratio is called relative transverse strain.
The ratio is called Poisson's ratio, which characterizes the elastic properties of the material.
Poisson's ratio is significant. (for steel it is equal to )
10. Formulate Hooke's law in tension (compression).
I form. In the cross sections of a beam under central tension (compression), the normal stresses are equal to the ratio of the longitudinal force to the cross-sectional area:
II form. The relative longitudinal strain is directly proportional to the normal stress, whence.
11. How are stresses determined in the transverse and inclined sections of a beam?
– force equal to the product of stress and the area of the inclined section:
12. What formula can be used to determine the absolute elongation (shortening) of a beam?
The absolute elongation (shortening) of a beam (rod) is expressed by the formula:
, i.e.
Considering that the value represents the stiffness of the cross section of a beam with a length, we can conclude: the absolute longitudinal deformation is directly proportional to the longitudinal force and inversely proportional to the stiffness of the cross section. This law was first formulated by Hooke in 1660.
13. How are temperature deformations and stresses determined?
As the temperature increases, the mechanical strength characteristics of most materials decrease, and as the temperature decreases, they increase. For example, for steel grade St3 at and ;
at and , i.e. .
The elongation of a rod when heated is determined by the formula , where is the coefficient of linear expansion of the material of the rod, and is the length of the rod.
Normal stress arising in the cross section. As the temperature decreases, the rod shortens and compressive stresses arise.
14. Characterize the tension (compression) diagram.
Mechanical characteristics materials are determined by testing samples and constructing corresponding graphs and diagrams. The most common is the static tensile (compression) test.
Limit of proportionality (up to this limit Hooke’s law is valid);
Material yield strength;
Strength limit of the material;
Breaking (conditional) stress;
Point 5 corresponds to the true breaking stress.
1-2 material flow area;
2-3 zone of material hardening;
and - the magnitude of plastic and elastic deformation.
Modulus of elasticity in tension (compression), defined as: , i.e. .
15. What parameters characterize the degree of plasticity of a material?
The degree of plasticity of a material can be characterized by the following values:
Residual relative elongation - as the ratio of the residual deformation of the sample to its original length:
where is the length of the sample after rupture. Value for various brands steel ranges from 8 to 28%;
Residual relative narrowing - as the ratio of the cross-sectional area of the sample at the point of rupture to the original area:
where is the cross-sectional area of the torn sample at the thinnest point of the neck. The value ranges from a few percent for brittle high-carbon steel to 60% for low-carbon steel.
16. Problems solved when calculating tensile (compressive) strength.
Lecture outline
1. Deformations, Hooke’s law during central tension-compression of rods.
2. Mechanical characteristics of materials under central tension and compression.
Let's consider a structural rod element in two states (see Figure 25):
External longitudinal force F absent, the initial length of the rod and its transverse size are equal, respectively l And b, cross-sectional area A the same along the entire length l(the outer contour of the rod is shown by solid lines);
The external longitudinal tensile force directed along the central axis is equal to F, the length of the rod received an increment Δ l, while its transverse size decreased by the amount Δ b(the outer contour of the rod in the deformed position is shown by dotted lines).
l Δ l
Figure 25. Longitudinal-transverse deformation of the rod during its central tension.
Incremental rod length Δ l is called its absolute longitudinal deformation, the value Δ b– absolute transverse deformation. Value Δ l can be interpreted as longitudinal movement (along the z axis) of the end cross section of the rod. Units of measurement Δ l and Δ b same as initial dimensions l And b(m, mm, cm). In engineering calculations it is used next rule signs for Δ l: when a section of the rod is stretched, its length and value Δ increase l positive; if on a section of a rod with an initial length l internal compressive force occurs N, then the value Δ l negative, because there is a negative increment in the length of the section.
If absolute deformations Δ l and Δ b refer to initial sizes l And b, then we obtain relative deformations:
– relative longitudinal deformation;
– relative transverse deformation.
Relative deformations are dimensionless (as a rule,
very small) quantities, they are usually called e.o. d. – units of relative deformations (for example, ε = 5.24·10 -5 e.o. d.).
The absolute value of the ratio of the relative longitudinal strain to the relative transverse strain is a very important material constant called the transverse strain ratio or Poisson's ratio(after the name of the French scientist)
As you can see, Poisson's ratio quantitatively characterizes the relationship between the values of relative transverse deformation and relative longitudinal deformation of the rod material when applying external forces along one axis. The values of Poisson's ratio are determined experimentally and are given in reference books for various materials. For all isotropic materials, the values range from 0 to 0.5 (for cork close to 0, for rubber and rubber close to 0.5). In particular, for rolled steels and aluminum alloys in engineering calculations it is usually taken for concrete.
Knowing the value of longitudinal deformation ε (for example, as a result of measurements during experiments) and Poisson's ratio for a specific material (which can be taken from a reference book), you can calculate the value of the relative transverse strain
where the minus sign indicates that longitudinal and transverse deformations always have opposite algebraic signs (if the rod is extended by an amount Δ l tensile force, then the longitudinal deformation is positive, since the length of the rod receives a positive increment, but at the same time the transverse dimension b decreases, i.e. receives a negative increment Δ b and the transverse strain is negative; if the rod is compressed by force F, then, on the contrary, the longitudinal deformation will become negative, and the transverse deformation will become positive).
Internal forces and deformations arising in structural elements under the influence of external loads, represent a single process in which all factors are interconnected. First of all, we are interested in the relationship between internal forces and deformations, in particular, during central tension-compression of structural rod elements. In this case, as above, we will be guided Saint-Venant's principle: the distribution of internal forces significantly depends on the method of applying external forces to the rod only near the point of loading (in particular, when forces are applied to the rod through a small area), and in parts quite remote from the places
application of forces, the distribution of internal forces depends only on the static equivalent of these forces, i.e., under the action of tensile or compressive concentrated forces, we will assume that in most of the volume of the rod the distribution internal forces will be uniform(this is confirmed by numerous experiments and experience in operating structures).
Back in the 17th century, the English scientist Robert Hooke established a direct proportional (linear) relationship (Hooke's law) of the absolute longitudinal deformation Δ l from tensile (or compressive) force F. In the 19th century, the English scientist Thomas Young formulated the idea that for each material there is a constant value (which he called the elastic modulus of the material), characterizing its ability to resist deformation under the action of external forces. At the same time, Jung was the first to point out that linear Hooke's law is true only in a certain region of material deformation, namely – during its elastic deformations.
In the modern concept, in relation to uniaxial central tension-compression of rods, Hooke’s law is used in two forms.
1) Normal stress in the cross section of a rod under central tension is directly proportional to its relative longitudinal deformation
, (1st type of Hooke's law),
Where E– the modulus of elasticity of the material under longitudinal deformations, the values of which for various materials are determined experimentally and are listed in reference books that technicians use when carrying out various engineering calculations; yes, for rental carbon steels, widely used in construction and mechanical engineering; for aluminum alloys; for copper; for other materials value E can always be found in reference books (see, for example, “Handbook on Strength of Materials” by G.S. Pisarenko et al.). Units of elastic modulus E same as units of measurement normal stress, i.e. Pa, MPa, N/mm 2 and etc.
2) If in the 1st form of Hooke’s law written above, the normal stress in the section σ express in terms of internal longitudinal force N and cross-sectional area of the rod A, i.e. , and the relative longitudinal deformation – through the initial length of the rod l and absolute longitudinal deformation Δ l, i.e., then after simple transformations we get the formula for practical calculations(longitudinal deformation is directly proportional to the internal longitudinal force)
(2nd type of Hooke's law). (18)
From this formula it follows that with increasing value of the elastic modulus of the material E absolute longitudinal deformation of the rod Δ l decreases. Thus, the resistance of structural elements to deformation (their rigidity) can be increased by using materials with higher elastic modulus values. E. Among the structural materials widely used in construction and mechanical engineering high value elastic modulus E have steel. Value range E For different brands small steels: (1.92÷2.12) 10 5 MPa. For aluminum alloys, for example, the value E approximately three times less than that of steels. Therefore for
structures with increased rigidity requirements, preferred materials are steel.
The product is called the rigidity parameter (or simply rigidity) of the cross section of the rod during its longitudinal deformations (the units of measurement of the longitudinal stiffness of the section are N, kN, MN). Magnitude c = E A/l is called the longitudinal stiffness of the rod length l(units of measurement of the longitudinal stiffness of the rod With – N/m, kN/m).
If the rod has several sections ( n) with variable longitudinal stiffness and complex longitudinal load (a function of the internal longitudinal force on the z coordinate of the cross section of the rod), then the total absolute longitudinal deformation of the rod will be determined by more general formula
where integration is carried out within each section of the rod of length , and discrete summation is carried out over all sections of the rod from i = 1 before i = n.
Hooke's law is widely used in engineering calculations of structures, since most structural materials during operation can withstand very significant stresses without collapsing within the limits of elastic deformations.
For inelastic (plastic or elastic-plastic) deformations of the rod material direct application Hooke's law is invalid and, therefore, the above formulas cannot be used. In these cases, other calculated dependencies should be applied, which are discussed in special sections of the courses “Strength of Materials”, “Structural Mechanics”, “Mechanics of Solid Deformable Body”, as well as in the course “Theory of Plasticity”.
Have an idea of longitudinal and transverse deformations and their relationship.
Know Hooke's law, dependencies and formulas for calculating stresses and displacements.
Be able to carry out calculations of the strength and stiffness of statically determined beams in tension and compression.
Tensile and compressive strains
Let us consider the deformation of a beam under the action of a longitudinal force F(Fig. 4.13).
Initial sizes timber: - initial length, - initial width. The beam is lengthened by an amount Δl; Δ1- absolute elongation. When stretched, the transverse dimensions decrease, Δ A- absolute narrowing; Δ1 > 0; Δ A<0.
During compression, the following relation is fulfilled: Δl< 0; Δ a> 0.
In the strength of materials, it is customary to calculate deformations in relative units: Fig.4.13
Relative extension;
Relative narrowing.
There is a relationship between longitudinal and transverse deformations ε′=με, where μ is the transverse deformation coefficient, or Poisson’s ratio, a characteristic of the plasticity of the material.
End of work -
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Theoretical mechanics
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All topics in this section:
Axioms of statics
The conditions under which a body can be in equilibrium are derived from several basic provisions, applied without proof, but confirmed by experience and called axioms of statics.
Connections and reactions of connections
All laws and theorems of statics are valid for a free rigid body. All bodies are divided into free and bound. A body that is not tested is called free.
Determination of the resultant geometrically
Know the geometric method of determining the resultant system of forces, the equilibrium conditions of a plane system of converging forces.
Resultant of converging forces
The resultant of two intersecting forces can be determined using a parallelogram or triangle of forces (4th axiom) (Fig. 1.13).
Projection of force on the axis
The projection of force onto the axis is determined by the segment of the axis, cut off by perpendiculars lowered onto the axis from the beginning and end of the vector (Fig. 1.15).
Determination of the resultant system of forces by an analytical method
The magnitude of the resultant is equal to the vector (geometric) sum of the vectors of the system of forces. We determine the resultant geometrically. Let's choose a coordinate system, determine the projections of all tasks
Equilibrium conditions for a plane system of converging forces in analytical form
Based on the fact that the resultant is zero, we obtain: FΣ
Methodology for solving problems
The solution to each problem can be divided into three stages. First stage: We discard the external connections of the system of bodies whose equilibrium is being considered, and replace their actions with reactions. Necessary
Couple of forces and moment of force about a point
Know the designation, module and definition of the moments of a force pair and a force relative to a point, the equilibrium conditions of a system of force pairs. Be able to determine the moments of force pairs and the relative moment of force
Equivalence of pairs
Two pairs of forces are considered equivalent if, after replacing one pair with another pair, the mechanical state of the body does not change, that is, the movement of the body does not change or is not disrupted
Supports and support reactions of beams
Rule for determining the direction of bond reactions (Fig. 1.22). The articulated movable support allows rotation around the hinge axis and linear movement parallel to the supporting plane.
Bringing force to a point
An arbitrary plane system of forces is a system of forces whose lines of action are located in the plane in any way (Fig. 1.23). Let's take the strength
Bringing a plane system of forces to a given point
The method of bringing one force to a given point can be applied to any number of forces. Let's say h
Influence of reference point
The reference point is chosen arbitrarily. An arbitrary plane system of forces is a system of forces whose line of action is located in the plane in any way. When changing by
Theorem on the moment of the resultant (Varignon’s theorem)
In the general case, an arbitrary plane system of forces is reduced to the main vector F"gl and to the main moment Mgl relative to the selected center of reduction, and gl
Equilibrium condition for an arbitrarily flat system of forces
1) At equilibrium, the main vector of the system is zero (=0).
Beam systems. Determination of support reactions and pinching moments
Have an idea of the types of supports and the reactions that occur in the supports. Know the three forms of equilibrium equations and be able to use them to determine reactions in the supports of beam systems.
Types of loads
According to the method of application, loads are divided into concentrated and distributed. If the actual load transfer occurs on a negligibly small area (at a point), the load is called concentrated
Moment of force about a point
The moment of a force about an axis is characterized by the rotational effect created by a force tending to rotate a body around a given axis. Let a force be applied to a body at an arbitrary point K
Vector in space
In space, the force vector is projected onto three mutually perpendicular coordinate axes. Vector projections form edges rectangular parallelepiped, the force vector coincides with the diagonal (Fig. 1.3
Bringing an arbitrary spatial system of forces to the center O
A spatial system of forces is given (Fig. 7.5a). Let's bring it to the center O. The forces must be moved in parallel, and a system of pairs of forces is formed. The moment of each of these pairs is equal
Some definitions of the theory of mechanisms and machines
With further study of the subject of theoretical mechanics, especially when solving problems, we will encounter new concepts related to the science called the theory of mechanisms and machines.
Point acceleration
Vector quantity characterizing the rate of change in speed in magnitude and direction
Acceleration of a point during curvilinear motion
When a point moves along a curved path, the speed changes its direction. Let us imagine a point M, which, during a time Δt, moving along a curvilinear trajectory, has moved
Uniform movement
Uniform motion is motion at a constant speed: v = const. For straight uniform motion(Fig. 2.9, a)
Uneven movement
With uneven movement, the numerical values of speed and acceleration change. The equation uneven movement V general view is the equation of the third S = f
The simplest motions of a rigid body
Have an idea about forward movement, its features and parameters, about the rotational motion of the body and its parameters. Know the formulas for determining parameters progressively
Rotational movement
Movement in which at least the points of a rigid body or an unchanging system remain motionless, called rotational; a straight line connecting these two points,
Special cases of rotational motion
Uniform rotation (angular velocity is constant): ω = const. Equation (law) of uniform rotation in in this case has the form: `
Velocities and accelerations of points of a rotating body
The body rotates around point O. Let us determine the parameters of motion of point A, located at a distance r a from the axis of rotation (Fig. 11.6, 11.7).
Rotational motion conversion
Conversion rotational movement carried out by various mechanisms called transmissions. The most common are gear and friction transmissions, as well as
Basic definitions
A complex movement is a movement that can be broken down into several simple ones. Simple movements are considered to be translational and rotational. To consider the complex motion of points
Plane-parallel motion of a rigid body
Plane-parallel, or flat, motion of a rigid body is called such that all points of the body move parallel to some fixed one in the reference system under consideration.
Method for determining the instantaneous velocity center
The speed of any point on the body can be determined using the instantaneous center of velocities. Wherein complex movement represented as a chain of rotations around different centers. Task
Friction concept
Absolutely smooth and absolutely solid bodies do not exist in nature, and therefore, when one body moves over the surface of another, resistance arises, which is called friction.
Sliding friction
Sliding friction is the friction of motion in which the velocities of bodies at the point of contact are different in value and (or) direction. Sliding friction, like static friction, is determined by
Free and non-free points
A material point whose movement in space is not limited by any connections is called free. Problems are solved using the basic law of dynamics. Material then
The principle of kinetostatics (D'Alembert's principle)
The principle of kinetostatics is used to simplify the solution of a number of technical problems. In reality, inertial forces are applied to bodies connected to the accelerating body (to connections). d'Alembert proposal
Work done by a constant force on a straight path
The work of force in the general case is numerically equal to the product of the modulus of force by the length of the distance traveled mm and by the cosine of the angle between the direction of force and the direction of movement (Fig. 3.8): W
Work done by a constant force on a curved path
Let point M move along a circular arc and force F makes a certain angle a
Power
To characterize the performance and speed of work, the concept of power was introduced.
Efficiency
The ability of a body to do work when transitioning from one state to another is called energy. There is energy general measure various forms mother's movements and interactions
Law of change of momentum
The quantity of motion of a material point is a vector quantity equal to the product of the mass of the point and its speed
Potential and kinetic energy
There are two main forms of mechanical energy: potential energy, or positional energy, and kinetic energy, or motional energy. Most often they have to
Law of change of kinetic energy
Let a constant force act on a material point of mass m. In this case, point
Fundamentals of the dynamics of a system of material points
Totality material points, interconnected by interaction forces, is called a mechanical system. Any material body in mechanics is considered as a mechanical
Basic equation for the dynamics of a rotating body
Let a rigid body, under the action of external forces, rotate around the Oz axis with angular velocity
Moments of inertia of some bodies
Moment of inertia of a solid cylinder (Fig. 3.19) Moment of inertia of a hollow thin-walled cylinder
Strength of materials
Have an idea of the types of calculations in the strength of materials, the classification of loads, internal force factors and resulting deformations, and mechanical stresses. Zn
Basic provisions. Hypotheses and assumptions
Practice shows that all parts of structures are deformed under the influence of loads, that is, they change their shape and size, and in some cases the structure is destroyed.
External forces
In the resistance of materials, external influences mean not only force interaction, but also thermal interaction, which arises due to uneven changes in temperature
Deformations are linear and angular. Elasticity of materials
Unlike theoretical mechanics, where the interaction of absolutely rigid (non-deformable) bodies was studied, in the strength of materials the behavior of structures whose material is capable of deformation is studied.
Assumptions and limitations accepted in strength of materials
Real Construction Materials, from which various buildings and structures are erected, are quite complex and heterogeneous solids, having various properties. Take this into account
Types of loads and main deformations
During the operation of machines and structures, their components and parts perceive and transmit to each other various loads, i.e. force influences that cause changes in internal forces and
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
Section method. Voltage
Know the method of sections, internal force factors, stress components. Be able to determine types of loads and internal force factors in cross sections. For ra
Tension and compression
Tension or compression is a type of loading in which only one internal force factor appears in the cross section of the beam - longitudinal force. Longitudinal forces m
Central tension of a straight beam. Voltages
Central tension or compression is a type of deformation in which only longitudinal (normal) force N occurs in any cross section of the beam, and all other internal
Tensile and compressive stresses
During tension and compression, only normal stress acts in the section. Stresses in cross sections can be considered as forces per unit area. So
Hooke's law in tension and compression
Stresses and strains during tension and compression are interconnected by a relationship called Hooke's law, named after the English physicist Robert Hooke (1635 - 1703) who established this law.
Formulas for calculating the displacements of beam cross sections under tension and compression
We use well-known formulas. Hooke's law σ=Eε. Where.
Mechanical tests. Static tensile and compression tests
These are standard tests: equipment - a standard tensile testing machine, a standard sample (round or flat), a standard calculation method. In Fig. 4.15 shows the diagram
Mechanical characteristics
Mechanical characteristics of materials, i.e. quantities characterizing their strength, ductility, elasticity, hardness, as well as elastic constants E and υ, necessary for the designer to
Have an idea of longitudinal and transverse deformations and their relationship.
Know Hooke's law, dependencies and formulas for calculating stresses and displacements.
Be able to carry out calculations of the strength and stiffness of statically determined beams in tension and compression.
Tensile and compressive strains
Let us consider the deformation of a beam under the action of a longitudinal force F (Fig. 21.1).
In the strength of materials, it is customary to calculate deformations in relative units:
There is a relationship between longitudinal and transverse deformations
Where μ - coefficient of transverse deformation, or Poisson's ratio, - characteristic of the plasticity of the material.
Hooke's law
Within the limits of elastic deformations, deformations are directly proportional to the load:
- coefficient. IN modern form:
Let's get a dependency
Where E- modulus of elasticity, characterizes the rigidity of the material.
Within elastic limits, normal stresses are proportional to elongation.
Meaning E for steels within (2 – 2.1) 10 5 MPa. All other things being equal, the stiffer the material, the less it deforms:
Formulas for calculating the displacements of beam cross sections under tension and compression
We use well-known formulas.
Relative extension
As a result, we obtain the relationship between the load, the dimensions of the beam and the resulting deformation:
Δl- absolute elongation, mm;
σ - normal stress, MPa;
l- initial length, mm;
E - elastic modulus of the material, MPa;
N- longitudinal force, N;
A - cross-sectional area, mm 2;
Work AE called section rigidity.
conclusions
1. The absolute elongation of a beam is directly proportional to the magnitude of the longitudinal force in the section, the length of the beam and inversely proportional to the cross-sectional area and elastic modulus.
2. The relationship between longitudinal and transverse deformations depends on the properties of the material, the relationship is determined Poisson's ratio, called transverse deformation coefficient.
Poisson's ratio: steel μ from 0.25 to 0.3; at the traffic jam μ = 0; near rubber μ = 0,5.
3. Transverse deformations are less than longitudinal ones and rarely affect the performance of the part; if necessary, the transverse deformation is calculated using the longitudinal one.
Where Δa- transverse narrowing, mm;
and about- initial transverse size, mm.
4. 
Hooke's law is satisfied in the elastic deformation zone, which is determined during tensile tests using a tensile diagram (Fig. 21.2).
When working plastic deformations should not occur, elastic deformations are small compared to the geometric dimensions of the body. The main calculations in the strength of materials are carried out in the zone of elastic deformations, where Hooke's law operates.
In the diagram (Fig. 21.2), Hooke’s law operates from the point 0 to the point 1 .
5. Determining the deformation of a beam under load and comparing it with the permissible one (which does not impair the performance of the beam) is called rigidity calculation.
Examples of problem solving
Example 1. The loading diagram and dimensions of the beam before deformation are given (Fig. 21.3). The beam is pinched, determine the movement of the free end.
Solution
1. 
The timber is stepped, so diagrams should be drawn longitudinal forces and normal stresses.
We divide the beam into loading areas, determine the longitudinal forces, and build a diagram of the longitudinal forces.
2. We determine the values of normal stresses along sections, taking into account changes in the cross-sectional area.
We build a diagram of normal stresses.
3. At each section we determine the absolute elongation. We summarize the results algebraically.
Note. Beam pinched occurs in the patch unknown reaction in the support, so we start the calculation with free end (right).
1. Two loading sections:
section 1:
stretched;
section 2:
 |
Three voltage sections:
 |
Example 2. For a given stepped beam (Fig. 2.9, A) construct diagrams of longitudinal forces and normal stresses along its length, and also determine the displacements of the free end and section WITH, where the force is applied R 2. Modulus of longitudinal elasticity of the material E= 2.1 10 5 N/"mm 3.
Solution
1. The given beam has five sections /, //, III, IV, V(Fig. 2.9, A). The diagram of longitudinal forces is shown in Fig. 2.9, b.
2. Let's calculate the stresses in the cross sections of each section:
for the first
for the second
for the third
for the fourth
for the fifth
The normal stress diagram is shown in Fig. 2.9, V.
3. Let's move on to determining the displacements of cross sections. The movement of the free end of the beam is defined as the algebraic sum of the lengthening (shortening) of all its sections:
Substituting numerical values, we get
4. The displacement of section C, in which the force P 2 is applied, is defined as the algebraic sum of the lengthening (shortening) of sections ///, IV, V:
Substituting the values from the previous calculation, we get
Thus, the free right end of the beam moves to the right, and the section where the force is applied R 2, - to the left.
5. The displacement values calculated above can be obtained in another way, using the principle of independence of the action of forces, i.e., determining the displacements from the action of each force P 1; R 2; R 3 separately and summing up the results. We recommend that the student do this independently.
Example 3. Determine what stress occurs in a steel rod of length l= 200 mm, if after applying tensile forces to it its length becomes l 1 = 200.2 mm. E = 2.1*10 6 N/mm 2.
Solution
Absolute elongation of the rod
Longitudinal deformation of the rod
According to Hooke's law
Example 4. Wall bracket (Fig. 2.10, A) consists of a steel rod AB and a wooden strut BC. Rod cross-sectional area F 1 = 1 cm 2, cross-sectional area of the strut F 2 = 25 cm 2. Determine the horizontal and vertical displacements of point B if a load is suspended in it Q= 20 kN. Modules of longitudinal elasticity of steel E st = 2.1*10 5 N/mm 2, wood E d = 1.0*10 4 N/mm 2.
Solution
1. To determine the longitudinal forces in the rods AB and BC, we cut out node B. Assuming that the rods AB and BC are stretched, we direct the forces N 1 and N 2 arising in them from the node (Fig. 2.10, 6 ). We compose the equilibrium equations:
Effort N 2 turned out with a minus sign. This indicates that the initial assumption about the direction of the force is incorrect - in fact, this rod is compressed.
2. Calculate the elongation of the steel rod Δl 1 and shortening the strut Δl 2:
Traction AB lengthens by Δl 1= 2.2 mm; strut Sun shortened by Δl 1= 7.4 mm.
3. To determine the movement of a point IN Let's mentally separate the rods in this hinge and mark their new lengths. New point position IN will be determined if the deformed rods AB 1 And B 2 C bring them together by rotating them around the points A And WITH(Fig. 2.10, V). Points IN 1 And AT 2 in this case they will move along arcs, which, due to their smallness, can be replaced by straight segments V 1 V" And V 2 V", respectively perpendicular to AB 1 And SV 2. The intersection of these perpendiculars (point IN") gives the new position of point (hinge) B.
4. In Fig. 2.10, G the displacement diagram of point B is shown on a larger scale.
5. Horizontal movement of a point IN
Vertical
where the component segments are determined from Fig. 2.10, g;
Substituting numerical values, we finally get
When calculating displacements, the absolute values of the lengthening (shortening) of the rods are substituted into the formulas.
Test questions and assignments
1. A steel rod 1.5 m long is stretched by 3 mm under load. What is the relative elongation? What is relative contraction? ( μ = 0,25.)
2. What characterizes the transverse deformation coefficient?
3. State Hooke's law in modern form for tension and compression.
4. What characterizes the elastic modulus of a material? What is the unit of elastic modulus?
5. Write down the formulas for determining the elongation of the beam. What characterizes the work AE and what is it called?
6. How is the absolute elongation of a stepped beam loaded with several forces determined?
7. Answer the test questions.