Using the section method. Section method
All materials, structural elements and structures, under the influence of external forces, to one degree or another experience displacement (movement relative to the loaded state) and change their shape (deform). The interaction between parts (particles) within a structural element is characterized by internal forces.
Inner forces− forces of interatomic interaction that arise when external loads are applied to a body and tend to counteract deformation.
To calculate structural elements for strength, rigidity and stability, it is necessary to use section method identify emerging internal power factors.
The essence of the section method is that external forces applied to the cut-off part of the body are balanced by internal forces that arise in the section plane and replace the action of the discarded part of the body on the rest.
A rod in equilibrium under the action of forces F 1 , F 2 , F 3 , F 4 , F 5 (Fig. 86, A), mentally cut into two parts I and II (Fig. 86, b) and consider one of the parts, for example the left one.
Since the connections between the parts have been eliminated, the action of one of them on the other should be replaced by a system of internal forces in the section. Since the action is equal to the reaction and opposite in direction, the internal forces arising in the section balance the external forces applied to the left part.
Let's put it on point ABOUT coordinate system xyz. Let us decompose the main vector and the main moment into components directed along the coordinate axes:
Component N z - called longitudinal (normal) force, causing tensile or compressive deformation. Components Q x and Q y are perpendicular to the normal and tend to move one part of the body relative to another, they are called transverse forces. Moments M x and M y bend the body and are called bending . Moment M z twisting body is called torque . These forces and moments are internal force factors (Fig. 86, V).
The equilibrium conditions allow us to find the components of the main vector and the main moment of internal forces:
In particular cases, individual internal force factors may be equal to zero. Thus, under the action of a plane system of forces (for example, in the plane zy) force factors arise in its sections: bending moment M x, shear force Q y, longitudinal force N z. Equilibrium conditions for this case:
To determine internal power factors it is necessary:
1. Mentally draw a section at the point of the structure or rod that interests us.
2. Discard one of the cut off parts and consider the equilibrium of the remaining part.
3. Draw up equilibrium equations for the remaining part and determine from them the values and directions of internal force factors.
Internal force factors arising in the cross section of the rod determine the deformed state.
The section method does not allow one to establish the law of distribution of internal forces over a section.
Effective characteristics for assessing the load on parts will be the intensity of internal interaction forces - voltage And deformation .
Let's consider the cross section of the body (Fig. 87). Based on the previously accepted assumption that the bodies under consideration are solid, we can assume that the internal forces are continuously distributed over the entire section.
In the section we select an elementary area Δ A, and the resultant of the internal forces on this area will be denoted by Δ R. Ratio of resultant internal forces Δ R on site Δ A to the area of this site is called the average voltage on this site,
If the area ΔA is reduced (contracted to a point), then in the limit we obtain the voltage at the point
.
The force ΔR can be decomposed into components: normal ΔN and tangential ΔQ. Using these components, the normal σ and tangential τ stress are determined (Fig. 88):
To measure stress in the International System of Units (SI), newton per square meter is used, called pascal Pa (Pa = N/m2). Since this unit is very small and inconvenient to use, multiple units are used (kN/m2, MN/m2 and N/mm2). Note that 1 MN/m 2 = 1 MPa = 1 N/mm. This unit is most convenient for practical use.
In the technical system of units (MCGSS), kilogram-force per square centimeter was used to measure stress. The relationship between stress units in the International and Technical Systems is established on the basis of the relationship between force units: 1 kgf = 9.81 N 10 N. Approximately we can consider: 1 kgf/cm 2 = 10 N/cm 2 = 0.1 N/mm 2 = 0.1 MPa or 1 MPa = 10 kgf/cm2.
Normal and shear stresses are a convenient measure for assessing the internal forces of a body, since materials resist them in different ways. Normal stresses tend to bring together or remove individual particles of the body in the direction normal to the section plane, and shear stresses tend to move some particles of the body relative to others along the section plane. Therefore, shear stresses are also called shear stresses.
The deformation of a loaded body is accompanied by a change in the distances between its particles. The internal forces arising between the particles change under the influence of the external load until an equilibrium is established between the external load and the internal resistance forces. The resulting state of the body is called a stressed state. It is characterized by a set of normal and tangential stresses acting across all areas that can be drawn through the point in question. To study the state of stress at a point on a body means to obtain dependencies that make it possible to determine the stresses along any area passing through the specified point.
The stress at which destruction of the material occurs or noticeable plastic deformation occurs is called the limiting stress and is designated σ pre; τ prev. . These voltages are determined experimentally.
To avoid destruction of elements of structures or machines, the operating (design) stresses (σ, τ) arising in them should not exceed the permissible stresses, which are indicated in square brackets: [σ], [τ]. Allowable stresses are the maximum stress values that ensure safe operation of the material. Allowable stresses are assigned as a certain part of the experimentally found limiting stresses that determine the exhaustion of the material’s strength:
Where [ n] - the required or permissible safety factor, showing how many times the permissible stress should be less than the maximum.
The safety factor depends on the properties of the material, the nature of the acting loads, the accuracy of the calculation method used and the operating conditions of the structural element.
Under the influence of forces, displacements occur not only in the structure, but also in the material from which it is made (although in many cases such displacements are far beyond the capabilities of the naked eye and are detected using highly sensitive sensors and instruments).
To determine deformations at a point TO consider a small segment KL length s, emanating from this point in an arbitrary direction (Fig. 89).
As a result of deformation of the point TO And L will move to position TO 1 and L 2, respectively, and the length of the segment will increase by the amount Δs. Attitude
represents the average elongation along the segment s.
Reducing the segment s, bringing the point closer L to the point TO, in the limit we obtain linear deformation at the point TO towards KL:
If at point K we draw three axes parallel to the coordinate axes, then linear deformations in the direction of the coordinate axes X, at And z will be equal to ε x, ε y, ε z, respectively.
The deformation of a body is dimensionless and is often expressed as a percentage. Typically, deformations are small and under elastic conditions do not exceed 1–1.5%.
Let us consider a right angle formed in an undeformed body by segments OM And ON(Fig. 90). As a result of deformation under the influence of external forces, the angle MON will change and become equal to the angle M 1 O 1 N 1 . In the limit, the difference in angles is called the angular strain or shear strain at a point ABOUT in the plane MON:
In coordinate planes, angular deformations or shear angles are designated: γ xy, γ yx, γ xz.
At any point of the body, there are three linear and three angular components of deformation, which determine the deformed state at the point.
Section method allows you to determine the internal forces that arise in a rod that is in equilibrium under the action of an external load.
STEPS OF THE SECTION METHOD
Section method consists of four successive stages: cut, discard, replace, balance.
Let's cut it a rod that is in equilibrium under the action of a certain system of forces (Fig. 1.3, a) into two parts with a plane perpendicular to its z-axis.
Let's discard one of the parts of the rod and consider the remaining part.
Since we, as it were, cut an infinite number of springs connecting infinitely close particles of the body, now divided into two parts, at each point of the cross section of the rod it is necessary to apply elastic forces, which, during the deformation of the body, arose between these particles. In other words, we will replace the action of the discarded part by internal forces (Fig. 1.3, b).
INTERNAL FORCES IN THE METHOD OF SECTIONS
The resulting infinite system of forces, according to the rules of theoretical mechanics, can be brought to the center of gravity of the cross section. As a result, we obtain the main vector R and the main moment M (Fig. 1.3, c).
Let's decompose the main vector and the main moment into components along the x, y (major central axes) and z axes.
We get 6 internal power factors arising in the cross section of the rod during its deformation: three forces (Fig. 1.3, d) and three moments (Fig. 1.3, e).
Force N - longitudinal force
– transverse forces,
moment about the z axis () – torque
moments about the x, y axes () – bending moments.
Let us write the equilibrium equations for the remaining part of the body ( let's balance):
From the equations, the internal forces arising in the cross section of the rod under consideration are determined. 
12.Method of sections. The concept of internal efforts. Simple and complex deformations. Deformations of the body (structural elements) under consideration arise from the application of an external force. In this case, the distances between the particles of the body change, which in turn leads to a change in the forces of mutual attraction between them. Hence, as a consequence, internal efforts arise. In this case, internal forces are determined by the universal method of sections (or cutting method). Simple and complex deformations. Using the principle of superposition.
The deformation of a beam is called simple if only one of the above internal force factors occurs in its cross sections. Hereinafter, a force factor will be called any force or moment.
Lemma. If the beam is straight, then any external load (complex load) can be decomposed into components (simple loads), each of which causes one simple deformation (one internal force factor in any section of the beam).
The reader is invited to independently prove the lemma for any particular case of loading a beam (hint: in some cases it is necessary to introduce fictitious self-balanced loads).
There are four simple deformations of straight timber:
Pure tension – compression (N ≠ 0, Q y = Q z = M x = M y = M z =0);
Pure shift (Q y or Q z ≠ 0, N = M x = M y = M z = 0);
Pure torsion (M x ≠ 0, N = Q y = Q z = M y = M z = 0);
Pure bending (M y or M z ≠ 0, N = Q y = Q z = M x = 0).
Based on the lemma and the principle of superposition, the problems of strength of materials can be solved in the following sequence:
In accordance with the lemma, decompose a complex load into simple components;
Solve the obtained problems about simple deformations of a beam;
Summarize the results found (taking into account the vector nature of the parameters of the stress-strain state). In accordance with the superposition principle, this will be the desired solution to the problem.
13. The concept of tense internal forces. Relationship between stresses and internal forces.Mechanical stress is a measure of internal forces arising in a deformable body under the influence of various factors. Mechanical stress at a point on a body is defined as the ratio of internal force to unit area at a given point of the section under consideration.
Stresses are the result of the interaction of particles of a body when it is loaded. External forces tend to change the relative position of the particles, and the resulting stresses prevent the displacement of the particles, limiting it in most cases to a certain small value.
Q - mechanical stress.
F is the force generated in the body during deformation.
S - area.
There are two components of the mechanical stress vector:
Normal mechanical stress - applied to a single area of the section, normal to the section (indicated).
Tangential mechanical stress - applied to a single sectional area, in the sectional plane along a tangent (indicated).
The set of stresses acting along various areas drawn through a given point is called the stress state at the point.
In the International System of Units (SI), mechanical stress is measured in pascals.
14. Central tension and compression. Internal efforts. Voltages. Conditions of strength.Central tension (or central compression) This type of deformation is called in which only a longitudinal force (tensile or compressive) occurs in the cross section of the beam, and all other internal forces are equal to zero. Sometimes central tension (or central compression) is briefly called tension (or compression).
Rule of signs
Tensile longitudinal forces are considered to be positive, and compressive forces - negative.
Consider a straight beam (rod) loaded with force F
Rod stretching
Let us determine the internal forces in the cross sections of the rod using the section method.
Voltage is the internal force N per unit area A. Formula for normal tensile stresses σ
Since the transverse force during central tension-compression is zero2, then the shear stress = 0.
Tensile-compressive strength condition
max = | |
15. Central tension and compression. Condition of strength. Three types of problems in central tension (compression). The strength condition allows solving three types of problems:
1. Strength check (test calculation)
2. Selection of cross-section (design calculation)
3. Determination of carrying capacity (permissible load)
Objectives and methods of strength of materials
Strength of materials– the science of engineering methods for calculating the strength, rigidity and stability of structures, structures, machines and mechanisms.
Strength– the ability of a structure, its parts and components to withstand a certain load without collapsing.
Rigidity- the ability of a structure and its elements to resist deformation (changes in shape and size).
Sustainability- the ability of a structure and its elements to maintain a certain initial form of elastic equilibrium.
In order for structures as a whole to meet the requirements of strength, rigidity and stability, it is necessary to give their elements the most rational shape and determine the appropriate dimensions. Strength of materials solves these problems based on theoretical and experimental data.
In the strength of materials, methods of theoretical mechanics and mathematical analysis are widely used, data from sections of physics that study the properties of various materials, materials science and other sciences are used. In addition, the strength of materials is an experimental-theoretical science, since it widely uses experimental data and theoretical research.
Strength reliability models
Assessment of the strength reliability of a structural element begins with the selection calculation model(scheme). Model call a set of ideas, conditions and dependencies that describe an object or phenomenon.
Material models.
In calculations of strength reliability, the material of a part is represented as a homogeneous continuous medium, which makes it possible to consider the body as a continuous medium and apply methods of mathematical analysis.
Under homogeneity the material understands the independence of its properties from the size of the allocated volume.
The calculation model of the material is endowed with such physical properties as elasticity, plasticity and creep.
Elasticity– the property of a body (part) to restore its shape after removing the external load.
Plastic– the property of a body to retain after unloading, completely or partially, the deformation obtained during loading.
Creep– the property of a body to increase deformation over time under the action of external forces.
Form models.
In most cases, structures have a complex shape, the individual elements of which can be reduced to the main types:
1. The rod or timber called a body in which two sizes are small compared to the third.
Rods can have straight or curved axes, as well as constant or variable cross-section.
Straight rods include beams, axles, shafts; to curves - lifting hooks, chain links, etc.
2. Shell- a body bounded by two curved surfaces, the distance between which is small compared to other dimensions.
Shells can be cylindrical, conical, or spherical. Shells include thin-walled tanks, boilers, building domes, ship hulls, fuselage skins, wings, etc.
3. Plate- a body limited by two flat or slightly curved surfaces, having a small thickness.
The plates are flat bottoms and covers of tanks, ceilings of engineering structures, etc.
4. Array or massive body- a body in which all three sizes are of the same order.
These include: foundations of structures, retaining walls, etc.
Loading models.
Powers are a measure of the mechanical interaction of structural elements. Forces are external and internal.
External forces– these are the forces of interaction between the structural element under consideration and the bodies associated with it.
External forces can be volumetric or surface.
Volume forces These are the forces of inertia and gravity. They act on every infinitesimal element of volume.
Surface forces are detected during contact interaction of a given body with other bodies.
Surface forces can be concentrated or distributed.
R– concentrated force, N. It acts on a small part of the surface of the body.
q– intensity of distributed load, N/m.
External forces can be represented as a concentrated moment M(Nm) or distributed torque m(N·m/m).
Based on the nature of changes over time, loads are divided into static and variable.
Static called a load that slowly increases from zero to its nominal value and remains constant during operation of the part.
Variable called a load that changes periodically over time.
Models of destruction.
Loading models correspond to destruction models - equations (conditions) connecting the performance parameters of a structural element at the moment of destruction with parameters ensuring strength.
Depending on the loading conditions, fracture models are considered: static, low cycle And fatigue(multicycle).
Internal forces. Section method
The interaction between parts (particles) within a structural element is characterized by internal forces.
Inner forces represent the forces of interatomic interaction (bonds) that arise when external loads are applied to the body.
Practice shows that internal forces determine the strength reliability of a part (body).
To find internal forces use section method. To do this, mentally dissect the body into two parts, discard one part, and consider the other together with external forces. The internal forces are distributed over the section in a somewhat complex manner. Therefore, the system of internal forces is brought to the center of gravity of the section so that the main vector and the main moment can be determined M internal forces acting along the section. Then we decompose the main vector and the main moment into components along three axes and get internal power factors section: component N z called normal, or longitudinal force in cross section, strength Qx And Qy are called shear forces, moment M z(or M to) is called torque, and moments M x And M y - bending moments relative to the axes X And y, respectively.
Thus, if external forces are given, then internal force factors are calculated as algebraic sums of projections of forces and moments acting on the mentally cut-off part of the body.
After determining the numerical values of the internal forces, construct diagrams– graphs (diagrams) showing how internal forces change when moving from section to section.
As is known, there are forces external and internal. If we take an ordinary student ruler in our hands and bend it, we do this by applying external forces - our hands. If the hand effort is removed, the ruler will return to its original position on its own, under the influence of its internal forces (these are the forces of interaction between the particles of the element from the influence of external forces). The greater the external forces, the greater the internal ones, but the internal ones cannot constantly increase, they grow only to a certain limit, and when the external forces exceed the internal ones, it will happen destruction. Therefore, it is extremely important to be aware of the internal forces in a material in terms of its strength. Internal forces are determined using section method. Let's look at it in detail. Let's say the rod is loaded with some forces (top left figure). Cutting a rod with a cross section of 1–1 into two parts, and we will consider any of them - the one that seems simpler to us. Eg, discard the right side and consider the equilibrium of the left side (upper right figure).
The action of the discarded right part on the remaining left replace internal forces, there are infinitely many of them, since these are forces of interaction between particles of the body. It is known from theoretical mechanics that any system of forces can be replaced by an equivalent system consisting of a main vector and a main moment. Therefore, we will reduce all internal forces to the main vector R and the main moment M (Fig. 1.1, b). Since our space is three-dimensional, the main vector R can be expanded along the coordinate axes and get three forces - Q x, Q y, N z (Fig. 1.1, c). In relation to the longitudinal axis of the rod, the forces Q x, Q y are called transverse or shear forces (located across the axis), N z is called the longitudinal force (located along the axis).
The main moment M, when expanded along the coordinate axes, will also give three moments (Fig. 1.1, d) in accordance with the same longitudinal axis - two bending moments M x and M y and a torque T (can be designated as M k or M z).
Thus, in the general case of loading there is six components of internal forces, which are called internal force factors or internal forces. To determine them in the case of a spatial system of forces, six equilibrium equations, and in the case of a flat one – three.
To remember the sequence of the section method, you should use a mnemonic technique - remember the word ROSE from the first letters of the actions: R cut (by section), ABOUT discard (one of the parts), Z we replace (the action of the discarded part by internal forces), U we balance (i.e., using equilibrium equations we determine the value of internal forces).
The following types of deformations occur in practice. If, in the event of loading in an element under the influence of forces, one internal force factor arises, then such deformation is called simple or main. Simple deformations are tension-compression (longitudinal force occurs), shear (transverse force), bending (bending moment), torsion (torque). If an element simultaneously experiences several deformations (torsion with bending, bending with tension, etc.), then such deformation is called complex.
The interaction between parts of a structure (body) is characterized by internal forces that arise inside it under the influence of external loads.
Internal forces are determined using section method. The essence of the section method is as follows: if, under the action of external forces, the body is in a state of equilibrium, then any cut off part of the body, together with the external and internal forces exerted on it, will also be in equilibrium, therefore, the equilibrium equations are applicable to it. That is, they do not affect the conditions of equilibrium of the body, since they are self-balanced.
Let us consider a body to which a certain system of external forces F 1, F 2, ..., F n is applied, satisfying the equilibrium conditions, i.e. under the action of these external forces, the body is in a state of equilibrium. If necessary, then the support reactions are determined from the equilibrium equations (we take an object, discard the connections, replace the discarded connections with reactions, compose the equilibrium equations and ). Reactions may not be found if they are not among the external forces applied on one side of the sections under consideration.
We mentally dissect the body with an arbitrary section, discard the left part of the body and consider the balance of the remaining part.
If there were no internal forces, the remaining unbalanced part of the body would begin to move under the influence of external forces. To maintain balance, we replace the action of the thrown part of the body with internal forces applied to each particle of the body.
It is known from theoretical mechanics that any system of forces can be brought to any point in space in the form of the main vector of forces \vec(R) and the main moment of forces \vec(M) (Poinsot's theorem). The magnitude and direction of these vectors are unknown.
It is most convenient to determine these vectors through their projections on the x, y, z axes. $$\vec(R) = \vec(N) + \vec(Q_x)+\vec(Q_y), \ \ \vec(M) = \vec(M_k) + \vec(M_x)+\vec(M_y ) $$ or
The projections of the vectors \vec(R) and \vec(M) have the following names:
- N - longitudinal force,
- Q x and Q y are transverse (cutting) forces along the x and y axes, respectively,
- M k - torque (sometimes designated by the letter T),
- M x, M y - bending moments around the x and y axes, respectively
In the general case, to determine internal forces, we have 6 unknowns, which can be determined from 6 equilibrium equations.
where \sum F_i, \sum M(F)_i are external forces and moments acting on the remaining part of the body.
Having solved a system of 6 equations with 6 unknowns, we determine all internal efforts. Not all six internal
force factors simultaneously - this depends on the type of external load and the method of its application.
Example: for a rod
The general rule for determining any internal effort is:
The forces Q x , Q y , N are equal to the algebraic sum of the projections of all forces located on one side of the selected section, respectively, on the x, y or z axis.
Moments M x , M y , M k are equal to the algebraic sum of the moments of all forces located on one side of the selected section, respectively, relative to the x, y or z axes passing through the center of gravity of the selected section.
When using the above rule, it is necessary to adopt the rule of signs for internal efforts.
Rule of signs
- The normal tensile force (directed from the section) is considered positive, and the compressive force is considered negative.
- A torque in a section directed counterclockwise is considered positive, while a torque directed clockwise is considered negative.
- A positive bending moment corresponds to compressed fibers from above, a negative bending moment from below.
- It is convenient to determine the sign of the transverse force by the direction in which the resulting transverse load tries to rotate the cut-off part of the beam relative to the section under consideration: if clockwise, the force is considered positive, counterclockwise, negative.
1 The graph of changes in internal force along a given axis of the body is called a diagram.
