Calculation of plates by limit states. Two groups of limit states. Limit State Groups

Building structures must, first of all, have sufficient reliability - that is, the ability to perform certain functions under appropriate conditions for a certain period of time. The termination of the performance of at least one of the functions provided for it by a building structure is called a failure.

Thus, failure is understood as the possibility of the occurrence of such a random event, the result of which is social or economic losses. It is believed that the structure at the moment preceding the failure passes into the limit state.

Limit states are such states, upon the occurrence of which the structure ceases to satisfy the requirements for it, i.e., it loses its ability to resist external loads or receives unacceptable movements or local damage.

The reasons for the onset of limit states in building structures can be overloads, the low quality of the materials from which they are made, and more.

The main difference between the method under consideration and the previous methods of calculation (calculation by allowable stresses) is that here the limiting states of structures are clearly established and instead of a single safety factor k a system of design coefficients is introduced into the calculation, guaranteeing a structure with a certain security against the onset of these states under the most unfavorable (but really possible) conditions. Currently, this method of calculation is accepted as the main official one.

Reinforced concrete structures can lose the required performance for one of two reasons:

1. As a result of the exhaustion of the bearing capacity (destruction of the material in the most loaded sections, loss of stability of individual elements or the entire structure as a whole);

2. As a result of excessive deformations (deflections, vibrations, settling), as well as due to the formation of cracks or their excessive opening.

In accordance with the above two reasons that can cause a loss of performance of structures, the standards establish two groups of their limit states:

By bearing capacity (first group);

By suitability for normal operation (second group).

The task of the calculation is to prevent the occurrence of any limit state in the considered structure during the period of manufacture, transportation, installation and operation.

Calculations for the limit states of the first group should ensure during the operation of the structure and for other stages of work its strength, shape stability, position stability, endurance, etc.

Calculations for the limiting states of the second group are performed in order to prevent, during the operation of the structure and at other stages of its operation, excessive crack opening in width, leading to premature corrosion of the reinforcement, or their formation, as well as excessive movements.

Estimated Factors

These are loads and mechanical characteristics of materials (concrete and reinforcement). They have statistical variability or spread of values. Limit state calculations take into account (in an implicit form) the variability of loads and mechanical characteristics of materials, as well as various unfavorable or favorable operating conditions for concrete and reinforcement, conditions for the manufacture and operation of elements of buildings and structures.

Loads, mechanical characteristics of materials and design coefficients are normalized. When designing reinforced concrete structures, the values of loads, resistances of concrete and reinforcement are set according to the chapters of SNiP 2.01.07-85 * and SP 52-101-2003.

Classification of loads. Normative and calculated loads

Loads and impacts on buildings and structures, depending on the duration of their action, are divided into permanent and temporary. The latter, in turn, are divided into long-term, short-term and special.

are the weight of the bearing and enclosing structures of buildings and structures, the weight and pressure of soils, the impact of prestressing reinforced concrete structures.

include: the weight of stationary equipment on floors - machine tools, apparatus, engines, containers, etc .; pressure of gases, liquids, bulk solids in containers; floor loads from stored materials and rack equipment in warehouses, refrigerators, granaries, book storages, archives and similar premises; temperature technological effects from stationary equipment; the weight of the water layer on water-filled flat surfaces, etc.

These include: the weight of people, repair materials in the areas of maintenance and repair of equipment, snow loads with a full standard value, wind loads, loads arising during the manufacture, transportation and installation of structural elements, and some others.

include: seismic and explosive impacts; loads caused by sharp disturbances in the technological process, temporary malfunction or breakdown of equipment, etc.

Loads in accordance with SNiP 2.01.07-85 * are also divided into normative and calculated.

Regulatory loads are called loads or impacts that are close in magnitude to the largest possible during the normal operation of buildings and structures. Their values are given in the norms.

Unfavorable load variability is estimated by the load safety factor γ f.

The design value of the load g for calculating the structure for strength or stability is determined by multiplying its standard value g p by the coefficient γ f , usually greater than 1

The values are differentiated depending on the nature of the loads and their magnitude. So, for example, when taking into account the own weight of concrete and reinforced concrete structures = 1.1; when taking into account the own weight of various screeds, backfills, heaters, performed in the factory, = 1.2, and at the construction site = 1.3. Load safety factors for uniformly distributed loads should be taken:

1.3 - with a full standard value of less than 2 kPa (2 kN / m 2);

1.2 - at a full standard value of 2 kPa (2 kN / m 2) and more. The coefficient of reliability for the load for its own weight when calculating the structure for position stability against ascent, overturning and sliding, as well as in other cases when a decrease in mass worsens the working conditions of the structure, is taken equal to 0.9.

Calculations for the limit states of the second group are carried out according to standard loads or according to calculated ones, taken with γ f = 1.

Buildings and structures are subjected to the simultaneous action of various loads. Therefore, the calculation of a building or structure as a whole, or of its individual elements, must be carried out taking into account the most unfavorable combinations of these loads or the forces caused by them. Unfavorable, but really possible combinations of loads during design are selected in accordance with the recommendations of SNiP 2.01.07-85*.

Depending on the composition of the considered loads, combinations are distinguished:

- main, including permanent, long-term and short-term loads

T \u003d ΣT post + ψ 1 ΣT long + ψ 2 ΣT multiple,

where T = M, T, Q;

ψ - combination coefficient (if 1 short-term load is taken into account, then ψ 1 \u003d ψ 2 \u003d 1.0, if the combination includes 2 or more short-term loads, then ψ 1 \u003d 0.95, ψ 2 \u003d 0.9);

- special, including, in addition to permanent, long-term and short-term loads, a special load (ψ 1 \u003d 0.95, ψ 2 \u003d 0.80).

1. The essence of the method

The method of calculation of structures by limit states is a further development of the method of calculation by destructive forces. When calculating by this method, the limit states of structures are clearly established and a system of design coefficients is introduced that guarantees the structure against the onset of these states under the most unfavorable combinations of loads and at the lowest values of the strength characteristics of materials.

The stages of destruction, but the safety of the structure under load is evaluated not by one synthesizing safety factor, but by a system of design coefficients. Structures designed and calculated using the limit state method are somewhat more economical.

2. Two groups of limit states

The limit states are the states in which structures cease to meet the requirements imposed on them during operation, i.e., they lose their ability to resist external loads and influences or receive unacceptable movements or local damage.

Reinforced concrete structures must meet the requirements of the calculation for two groups of limit states: for bearing capacity - the first group of limit states; according to suitability for normal operation - the second group of limit states.

loss of stability of the structure shape (calculation for the stability of thin-walled structures, etc.) or its position (calculation for overturning and sliding retaining walls, eccentrically loaded high foundations; calculation for the ascent of buried or underground reservoirs, etc.);

fatigue failure (fatigue calculation of structures under the influence of a repetitive movable or pulsating load: crane beams, sleepers, frame foundations and ceilings for unbalanced machines, etc.);

destruction from the combined effect of force factors and adverse influences external environment (periodic or constant exposure to an aggressive environment, the action of alternate freezing and thawing, etc.).

The calculation for the limit states of the second group is performed to prevent:

the formation of excessive or prolonged opening of cracks (if the formation or prolonged opening of cracks is permissible under the operating conditions);

excessive movements (deflections, angles of rotation, skew angles and vibration amplitudes).

The calculation of the limit states of the structure as a whole, as well as its individual elements or parts, is carried out for all stages: manufacturing, transportation, installation and operation; wherein calculation schemes must meet the accepted constructive solutions and each of the above steps.

3. Estimated factors

Design factors - loads and mechanical characteristics of concrete and reinforcement (tensile strength, yield strength) - have statistical variability (scatter of values). Loads and actions may differ from the given probability of exceeding the average values, and the mechanical characteristics of the materials may differ from the given probability of falling average values. Limit state calculations take into account the statistical variability of loads and mechanical characteristics of materials, non-statistical factors and various unfavorable or favorable physical, chemical and mechanical conditions for the operation of concrete and reinforcement, the manufacture and operation of elements of buildings and structures. Loads, mechanical characteristics of materials and design coefficients are normalized.

The values of loads, resistance of concrete and reinforcement are set according to the chapters of SNiP "Loads and effects" and "Concrete and reinforced concrete structures".

4. Classification of loads. Regulatory and design loads

Depending on the duration of the action, the load is divided into permanent and temporary. Temporary loads, in turn, are divided into long-term, short-term, special.

Loads from the weight of the bearing and enclosing structures of buildings and structures, the mass and pressure of soils, and the impact of prestressing reinforced concrete structures are constant.

Long-term loads are from the weight of stationary equipment on floors - machine tools, apparatus, engines, tanks, etc.; pressure of gases, liquids, bulk solids in containers; loads in warehouses, refrigerators, archives, libraries and similar buildings and structures; statutory part of the live load in residential buildings, service and household premises; long-term temperature technological effects from stationary equipment; loads from one overhead or one overhead crane, multiplied by the coefficients: 0.5 for medium-duty cranes and 0.7 for heavy-duty cranes; snow loads for III-IV climatic regions with coefficients of 0.3-0.6. The specified values of the crane, some temporary and snow loads form part of their total value and are introduced into the calculation taking into account the duration of the action of loads of these types on displacements, deformations, and cracking. The full values of these loads are short-term.

Short-term are the loads from the weight of people, parts, materials in the areas of maintenance and repair of equipment - walkways and other areas free from equipment; part of the load on the floors of residential and public buildings; loads arising during the manufacture, transportation and installation of structural elements; loads from overhead and overhead cranes used in the construction or operation of buildings and structures; snow and wind loads; temperature climatic effects.

Special loads include: seismic and explosive effects; loads caused by malfunction or breakdown of equipment and a sharp violation technological process(for example, with a sharp increase or decrease in temperature, etc.); the impact of uneven deformations of the base, accompanied by a fundamental change in the structure of the soil (for example, deformations of subsiding soils during soaking or permafrost soils during thawing), etc.

The normative loads are set by the norms according to a predetermined probability of exceeding the average values or according to the nominal values. Regulatory constant loads are taken according to the design values of geometric and design parameters and average density values. Regulatory temporary technological and installation loads are set at the highest values provided for normal operation; snow and wind - according to the average of annual unfavorable values or according to unfavorable values corresponding to a certain average period of their repetition.

Design loads for designing structures for strength and stability are determined by multiplying the standard load by the load safety factor Vf, usually greater than one, for example g=gnyf. Reliability coefficient from the weight of concrete and reinforced concrete structures Yf = M; from the weight of structures made of concrete on light aggregates (with an average density of 1800 kg / m3 or less) and various screeds, backfills, heaters, performed in the factory, Yf = l.2, at installation yf = \.3; from various temporary loads depending on their value yf = it 2. 1.4. The coefficient of overload from the weight of structures when calculating the stability of the position against ascent, overturning and sliding, as well as in other cases when a decrease in mass worsens the conditions for the operation of the structure, is taken 7f = 0.9. When calculating structures at the stage of construction, the calculated short-term loads are multiplied by a factor of 0.8. The design loads for the calculation of structures for deformations and displacements (for the second group of limit states) are taken equal to the standard values with the coefficient Yf -1-

combination of loads. Structures must be designed for various combinations of loads or the corresponding forces if the calculation is carried out according to an inelastic scheme. Depending on the composition of the loads taken into account, there are: the main combinations, consisting of permanent, long-term and short-term loads or forces from nx; special combinations consisting of permanent, long-term, possible short-term and one of the special loads or efforts from them.

^ve groups of basic combinations of loads are considered. When calculating structures for the main combinations of the first group, constant, long-term and one short-term loads are taken into account; in the calculation of structures for the main combinations of the second group, constant, long-term and two (or more) short-term loads are taken into account; while the values of short-term

loads or corresponding forces should be multiplied by a combination factor equal to 0.9.

When calculating structures for special combinations, the values of short-term loads or the corresponding forces should be multiplied by a combination factor equal to 0.8, except for the cases specified in the design standards for buildings and structures in seismic regions.

The norms also allow to reduce live loads when calculating beams and crossbars, depending on the area of the loaded floor.

5. The degree of responsibility of buildings and structures

The degree of responsibility of the building and structures when the structures reach the limit states is determined by the amount of material and social damage. When designing structures, one should take into account the reliability factor for the purpose of the unitary enterprise, the value of which depends on the class of responsibility of buildings or structures. The limiting values of the bearing capacity, the calculated values of resistances, the limiting values of deformations, crack openings, or the calculated values of loads, forces or other influences should be multiplied by this coefficient according to the purpose.

Experimental studies carried out at factories of prefabricated reinforced concrete products showed that for heavy concrete and concrete on porous aggregates, the variation coefficient U

0.135, which is accepted in the norms.

In mathematical statistics, using pa or neither, the probability of repeating values of temporary resistance less than V is estimated. If we accept x = 1.64, then repetition of values is likely<В не более чем у 5 % (и значения В не менее чем у 95 %) испытанных образцов. При этом достигается нормированная обеспеченность не менее 0,95.

When controlling the class of concrete in terms of axial tensile strength, the normative resistance of concrete to axial tensile Rbtn is taken equal to its guaranteed strength (class) on. axial stretch.

The design resistance of concrete for the calculation for the first group of limit states is determined by dividing the standard resistances by the corresponding reliability factors for concrete in compression ybc = 1.3 prn tension ^ = 1.5, and in the control of tensile strength yy = 1.3. Design resistance of concrete to axial compression

The calculated compressive strength of heavy concrete of classes B50, B55, B60 is multiplied by coefficients that take into account the peculiarity of the mechanical properties of high-strength concrete (reduction of creep deformations), respectively, equal to 0.95; 0.925 and 0.9.

The values of the design resistance of concrete with rounding are given in App. I.

When calculating structural elements, the design resistances of concrete Rb and Rbt are reduced, and in some cases they are increased by multiplying by the corresponding coefficients of the concrete working conditions uy, taking into account the characteristics of concrete properties: the duration of the load and its repeated repetition; conditions, nature and stage of operation of the structure; method of its manufacture, cross-sectional dimensions, etc.

The design compressive resistance of reinforcement Rsc used in the calculation of structures for the first group of limit states, when the reinforcement is bonded to concrete, is taken equal to the corresponding design tensile strength of reinforcement Rs, but not more than 400 MPa (based on the ultimate compressibility of concrete tub). When calculating structures for which the design resistance of concrete is taken for a long-term load, taking into account the coefficient of working conditions y&2 classes A-I V, At-IVC; /? dC \u003d 500 MPa with reinforcement of classes A-V, At-V, A-VI, At-VI, V-I, Vr-P, K-7, K-19 (since the ultimate compressibility of concrete increases slightly with prolonged action of the load) . In this case, special design requirements must be observed for the installation of transverse reinforcement, which protects the longitudinal compressed reinforcement from buckling, with a step of no more than 500 mm or no more than twice the width of a given face of the element. In the absence of adhesion of reinforcement to concrete Rsc-0.

When calculating structural elements, the design resistances of reinforcement are reduced or in some cases increased by multiplying by the corresponding coefficients of working conditions ySi, taking into account the possibility of incomplete use of its strength characteristics due to uneven distribution of stresses in the cross section, low strength of concrete, anchoring conditions, the presence of bends , the nature of the steel tensile diagram, the change in its properties depending on the operating conditions of the structure, etc.

When calculating the elements for the action of a transverse force, the design resistances of the transverse reinforcement are reduced by introducing the coefficient of working conditions -um ^ OD, which takes into account the uneven distribution of stresses in the reinforcement along the length of the inclined section. In addition, for welded transverse reinforcement made of wire of classes Вр-I and rod reinforcement of class A-III, the coefficient Vs2=0.9 is introduced, which takes into account the possibility of brittle fracture of the welded joint of clamps. The values of the design resistance of transverse reinforcement when calculating the shear force Rsw, taking into account the coefficients yst, are given in Table. 1 and 2 app. v.

In addition, the calculated resistances Rs, Rsc and Rsw should be multiplied by the coefficients of operating conditions: Ys3, 7 * 4 - with repeated application of the load (see Chapter VIII); ysb^lx/lp or uz

1x/1ap - in the zone of stress transfer and in the zone of anchoring of non-tensioned reinforcement without anchors; 7 ^ 6 - when working 'high-strength reinforcement at stresses above the conditional yield strength (7o.2.

The design resistance of reinforcement for the calculation for the second group of limit states is set at a reliability factor for reinforcement 7s = 1, i.e. are taken equal to the standard values Rs, ser = Rsn and are taken into account with the coefficient of reinforcement operating conditions

The crack resistance of a reinforced concrete structure is its resistance to cracking in stage I of the stress-strain state or the resistance to opening cracks in stage II of the stress-strain state.

Different requirements are imposed on the crack resistance of a reinforced concrete structure or its parts in the calculation, depending on the type of reinforcement used. These requirements apply to normal cracks and cracks inclined to the longitudinal axis of the element and are divided into three categories:

The opening of cracks under the action of constant, long-term and short-term loads is considered short; continuous crack opening is considered under the action of only constant and long-term loads. The maximum width of crack opening (accr - short and accr2 long), which ensures the normal operation of buildings, corrosion resistance of reinforcement and durability of the structure, depending on the category of requirements for crack resistance, should not exceed 0.05-0.4 mm (Table II .2).

Prestressed elements under liquid or gas pressure (tanks, pressure pipes, etc.), in a fully tensioned section with rod or wire reinforcement, as well as in a partially compressed section with wire reinforcement with a diameter of 3 mm or less, must meet the requirements of the First categories. Other prestressed elements, depending on the design conditions and the type of reinforcement, must meet the requirements of the second or third category.

The procedure for taking into account loads in the calculation for crack resistance depends on the category of requirements for crack resistance: with the requirements of the first category, the calculation is carried out according to the design loads with a safety factor for the load yf>l (as in the calculation for strength); under the requirements of the second and third categories, the calculation is carried out for the action of loads with the coefficient V / \u003d b The calculation for the formation of cracks to determine the need for checking for short-term opening of cracks for the requirements of the second category, the calculation for the formation of cracks is performed for the action of design loads with the coefficient yf>U checks for crack opening under the requirements of the third category are performed under the action of loads with a coefficient Y / -1. In the calculation of crack resistance, the joint action of all loads, except for special ones, is taken into account. Special loads are taken into account in the calculation of the formation of cracks in cases where cracks lead to a catastrophic situation. The calculation for closing cracks under the requirements of the second category is carried out for the action of constant and long-term loads with a coefficient y / -1. The procedure for accounting for loads is given in Table. P.Z. At the end sections of prestressed elements within the length of the zone of stress transfer from reinforcement to concrete 1P, cracking is not allowed under the combined action of all loads (except for special ones) entered into the calculation with the coefficient Y / = L THIS requirement is due to the fact that premature cracking in concrete at the end sections of the elements - can lead to pulling out of the reinforcement from the concrete under load and sudden failure.

increase in deflection. The effect of these cracks is taken into account in structural calculations. For elements operating under S& conditions of action of repeated loads and calculated for endurance, the formation of such cracks is not allowed.

Limit states of the first group. Strength calculations proceed from stage III of the stress-strain state. The section of the structure has the necessary strength if the forces from the design loads do not exceed the forces perceived by the section at the design resistances of the materials, taking into account the coefficient of working conditions. The force from design loads T (for example, bending moment or longitudinal force) is a function of standard loads, safety factors and other factors C (design model, dynamic factor, etc.).

Limit states of the second group. The calculation for the formation of cracks, normal and inclined to the longitudinal axis of the element, is carried out to check the crack resistance of elements to which the requirements of the first category are imposed, and also to establish whether cracks appear in elements whose crack resistance is imposed by the requirements of the second and third categories. It is believed that cracks normal to the longitudinal axis do not appear if the force T (bending moment or longitudinal force) from the action of loads does not exceed the force TSgf, which can be perceived by the section of the element

It is considered that cracks inclined to the longitudinal axis of the element do not appear if the main tensile stresses in concrete do not exceed the design values,

The calculation for crack opening, normal and inclined to the longitudinal axis, consists in determining the crack opening width at the level of tension reinforcement and comparing it with the maximum opening width. Data on the maximum crack opening width are given in Table. II.3.

Displacement calculation consists in determining the deflection of the element from loads, taking into account the duration of their action and comparing it with the ultimate deflection.

Limit deflections are set by various requirements: technological, due to the normal operation of cranes, technological installations, machines, etc.; constructive, due to the influence of neighboring elements that limit deformations, the need to withstand specified slopes, etc.; aesthetic.

Limit deflections of prestressed elements can be increased by the height of the bend, if this is not limited by technological or design requirements.

The procedure for taking into account loads when calculating deflections is as follows: when limited by technological or design requirements - for the action of permanent, long-term and short-term loads; when limited by aesthetic requirements - to the action of constant and long-term loads. In this case, the load safety factor is taken as Yf

Limit deflections established by the norms for various reinforced concrete elements are given in Table II.4. The limiting deflections of the consoles, related to the outreach of the console, are taken twice as large.

In addition, an additional volatility calculation must be performed for non-neighboring elements. reinforced concrete slabs ceilings, flights of stairs, landings, etc.: additional deflection from a short-term concentrated load of 1000 N with the most unfavorable scheme of its application should not exceed 0.7 mm.

Limit State Calculation Method

Chapter 2. Experimental foundations of the theory of resistance of reinforced concrete and methods for calculating reinforced concrete structures

Limit State Calculation Method

When calculating by this method, the structure is considered in its design limit state. For the design limit state, such a state of the structure is taken in which it ceases to satisfy the operational requirements imposed on it, i.e., either loses the ability to resist external influences, or receives unacceptable deformation or local damage.

For steel structures, two design limit states are established:

the first design limit state, determined by the bearing capacity (strength, stability or endurance); this limit state must be met by all steel structures;
the second design limit state, determined by the development of excessive deformations (deflections and displacements); this limit state must be satisfied by structures in which the magnitude of deformations can limit the possibility of their operation.

The first design limit state is expressed by the inequality

where N is the design force in the structure from the sum of the effects of design loads P in the most unfavorable combination;

Ф - the bearing capacity of the structure, which is a function of the geometric dimensions of the structure, the design resistance of the material R and the coefficient of working conditions m.

The maximum loads established by the norms (SNiP) that are allowed during normal operation of structures are called standard loads P n (see Appendix I, Loads and load factors).

The design loads P, for which the structure is calculated (according to the limit state), are taken somewhat higher than the normative ones. The design load is defined as the product of the standard load by the overload factor n (greater than one), taking into account the danger of exceeding the load in comparison with its standard value due to possible load variability:

The values of the coefficients p are given in the table Regulatory and design loads, overload factors.

Thus, structures are considered under the influence of not operational (normative), but design loads. From the impact of design loads in the structure, design forces are determined (axial force N or moment M), which are found by general rules strength of materials and structural mechanics.

The right side of the main equation (1.I)- bearing capacity of the structure Ф - depends on the ultimate resistance of the material to force effects, characterized by the mechanical properties of the material and called the normative resistance R n, as well as on the geometric characteristics of the section (sectional area F, modulus W, etc.).

For structural steel, the normative resistance is assumed to be equal to the yield strength,

(for the most common building steel grade St. 3 σ t \u003d 2,400 kg / cm 2).

The design resistance of steel R is taken as a voltage equal to the standard resistance multiplied by the coefficient of uniformity k (less than one), taking into account the danger of reducing the resistance of the material compared to its standard value due to the variability of the mechanical properties of the material

For ordinary low-carbon steels, k = 0.9, and for high-quality steels (low-alloyed) k = 0.85.

Thus, the calculated resistance R- this is the stress equal to the smallest possible value of the yield strength of the material, which is taken for the design as the limit.

In addition, for the safety of the structure, all possible deviations from normal conditions caused by the features of the structure's operation (for example, conditions that contribute to the appearance of increased corrosion, etc.) must be taken into account. To do this, the coefficient of working conditions m is introduced, which for most structures and connections is assumed to be equal to one (see Coefficients of working conditions m appendix).

Thus, the main calculation equation (1.I) will have the following form:

when checking the structure for strength under the action of axial forces or moments

where N and M are design axial forces or moments from design loads (taking into account overload factors); F nt - net cross-sectional area (minus holes); W nt - net section modulus (minus holes);

when checking the structure for stability

where F br and W br - area and moment of resistance of the gross section (excluding holes); φ and φ b - coefficients that reduce the design resistance to values that provide a stable balance.

Usually, when calculating the intended design, the section of the element is first selected and then the stress from the design forces is checked, which should not exceed the design resistance multiplied by the operating conditions coefficient.

Therefore, along with formulas of the form (4.I) and (5.I), we will write these formulas in working form through the calculated stresses, for example:

where σ is the design stress in the structure (from design loads).

The coefficients φ and φ b in formulas (8.I) and (9.I) are more correctly written on the right side of the inequality as coefficients that reduce the calculated resistances to critical stresses. And only for the convenience of conducting the calculation and comparing the results, they are written in the denominator of the left side of these formulas.

* The values of standard resistances and uniformity coefficients are given in the "Building Norms and Rules" (SNiP), as well as in the "Norms and specifications design of steel structures” (NITU 121-55).

"Design of steel structures",

There are several categories of voltages: basic, local, additional and internal. Basic stresses are stresses that develop inside the body as a result of balancing the effects of external loads; they count. With an uneven distribution of the power flow over the cross section, caused, for example, by a sharp change in the cross section or the presence of a hole, local stress concentration occurs. However, in plastic materials, which include building steel, ...

When calculating the allowable stresses, the structure is considered in its working condition under the action of loads allowed during the normal operation of the structure, i.e., standard loads. The structural strength condition is that the stresses in the structure from standard loads do not exceed the allowable stresses established by the norms, which are some part of the ultimate stress of the material accepted for building steel ...

Limit State Analysis Method - Steel Structure Analysis Method - Design Fundamentals - Steel Structure Design

When calculating by this method, the structure is considered in its design limit state. Such a state is taken as the design limit state ...

Two groups of limit states

The calculation for the limit states of the first group is performed to prevent:

Brittle, ductile or other type of fracture (strength calculation, taking into account, if necessary, the deflection of the structure before destruction);

Loss of stability of the structure shape (calculation for the stability of thin-walled structures, etc.) or its position (calculation for overturning and sliding of retaining walls, eccentrically loaded high foundations; calculation for the ascent of buried or underground reservoirs, etc.);

Fatigue failure (fatigue analysis of structures under the influence of a repetitive movable or pulsating load: crane beams, sleepers, frame foundations and ceilings for unbalanced machines, etc.);

Destruction from the combined effect of force factors and adverse environmental influences (periodic or constant exposure to an aggressive environment, the action of alternate freezing and thawing, etc.).

The calculation for the limit states of the second group is performed to prevent:

Formation of excessive or prolonged crack opening (if the formation or prolonged crack opening is permissible under operating conditions);

Excessive movements (deflections, angles of rotation, skew angles and vibration amplitudes).

The calculation of the limit states of the structure as a whole, as well as its individual elements or parts, is carried out for all stages: manufacturing, transportation, installation and operation; at the same time, design schemes must comply with the adopted design solutions and each of the listed stages.

The values of loads, resistance of concrete and reinforcement are set according to the chapters of SNiP "Loads and effects" and "Concrete and reinforced concrete structures".

Classification of loads. Regulatory and design loads

Depending on the duration of the action, the load is divided into permanent and temporary. Temporary loads, in turn, are divided into long-term, short-term, special.

Loads from the weight of the bearing and enclosing structures of buildings and structures, the mass and pressure of soils, and the impact of prestressing reinforced concrete structures are constant.

Long-term loads are from the weight of stationary equipment on floors - machine tools, apparatus, engines, tanks, etc.; pressure of gases, liquids, bulk solids in containers; loads in warehouses, refrigerators, archives, libraries and similar buildings and structures; part of the temporary load established by the norms in residential buildings, office and amenity premises; long-term temperature technological effects from stationary equipment; loads from one overhead or one overhead crane, multiplied by the coefficients: 0.5 for medium-duty cranes and 0.7 for heavy-duty cranes; snow loads for III-IV climatic regions with coefficients of 0.3-0.6. The indicated values of crane, some temporary and snow loads are part of their total value and are entered into the calculation taking into account the duration of the action of these types of loads on displacements, deformations, and cracking. The full values of these loads are short-term.

Special loads include: seismic and explosive effects; loads caused by a malfunction or breakdown of equipment and a sharp violation of the technological process (for example, with a sharp increase or decrease in temperature, etc.); the impact of uneven deformations of the base, accompanied by a fundamental change in the structure of the soil (for example, deformations of subsiding soils during soaking or permafrost soils during thawing), etc.

Average density values. Normative temporary; technological and installation loads are set according to the highest values provided for normal operation; snow and wind - according to the average of annual unfavorable values or according to unfavorable values corresponding to a certain average period of their repetition.

The design loads for calculating structures for strength and stability are determined by multiplying the standard load by the load safety factor Yf, usually greater than one, for example G= Gnyt. Reliability coefficient from the weight of concrete and reinforced concrete structures Yf = M; on the weight of structures made of concrete on light aggregates (with an average density of 1800 kg / m3 or less) and various screeds, backfills, heaters, performed in the factory, Yf = l,2, on installation Yf = l>3; from various live loads depending on their value Yf = l. 2. 1.4. The coefficient of overload from the weight of structures when calculating the stability of the position against ascent, overturning and sliding, as well as in other cases when a decrease in mass worsens the working conditions of the structure, is taken yf = 0.9. When calculating structures at the stage of construction, the calculated short-term loads are multiplied by a factor of 0.8. The design loads for the calculation of structures for deformations and displacements (for the second group of limit states) are taken equal to the standard values with the coefficient Yf = l-

Two groups of basic load combinations are considered. When calculating structures for the main combinations of the first group, constant, long-term and one short-term loads are taken into account; in the calculation of structures for the main combinations of the second group, constant, long-term and two (or more) short-term loads are taken into account; in this case, the values of short-term loads or the corresponding efforts should be multiplied by a combination factor equal to 0.9.

Load reduction. When calculating columns, walls, foundations multi-storey buildings temporary loads on floors can be reduced, taking into account the degree of probability of their simultaneous action, by multiplying by a coefficient

Where a - is taken equal to 0.3 for residential buildings, office buildings, dormitories, etc. and equal to 0.5 for various halls: reading rooms, meetings, trade, etc.; m is the number of loaded floors over the considered section.

The norms also allow to reduce live loads when calculating beams and crossbars, depending on the area of the loaded floor.

Reinforced concrete

Precast concrete and reinforced concrete: features and production methods

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RECONSTRUCTION OF INDUSTRIAL BUILDINGS

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rollers (flattening machine) diameter from 400 mm.,

dryer (flow) food electric,

conveyors, conveyors, screws.

Two groups of limit states

The limiting states are considered to be the states in which the structures cease to satisfy the requirements imposed on them during operation, i.e., they lose

Fundamentals of calculation for limit states. Calculation of structural elements of a solid section.

In accordance with the standards in force in Russia, wooden structures must be calculated using the limit state method.

Limiting states are such states of structures in which they cease to meet the requirements of operation. The external cause that leads to the limit state is the force effect (external loads, reactive forces). Limit states can occur under the influence of operating conditions wooden structures, as well as the quality, dimensions and properties of materials. There are two groups of limit states:

1 - according to the bearing capacity (strength, stability).

2 - by deformations (deflections, displacements).

First group limit states is characterized by loss of bearing capacity and complete unsuitability for further operation. Is the most responsible. In wooden structures, the following limit states of the first group may occur: destruction, buckling, overturning, unacceptable creep. These limit states do not occur if the following conditions are met:

those. when normal stresses ( σ ) and shear stresses ( τ ) do not exceed some limit value R, called design resistance.

Second group limit states are characterized by such signs, in which the operation of structures or structures, although difficult, is not completely excluded, i.e. design becomes unsuitable for normal operation. The suitability of a structure for normal use is usually determined by deflections

This means that bending elements or structures are suitable for normal use when the maximum value of the ratio of deflection to span is less than the maximum allowable relative deflection [ f/ l] (according to SNiP II-25-80).

The purpose of structural analysis is to prevent the occurrence of any of the possible limit states, both during transportation and installation, and during the operation of structures. The calculation for the first limit state is made according to the calculated values of the loads, and for the second - according to the normative ones. Standard values of external loads are given in SNiP "Loads and impacts". Design values are obtained taking into account the load safety factor γ n. Structures rely on an unfavorable combination of loads (dead weight, snow, wind), the probability of which is taken into account by the combination coefficients (according to SNiP "Loads and Impacts").

The main characteristic of materials, according to which their ability to resist forces is evaluated, is regulatory resistance R n . The normative resistance of wood is calculated from the results of numerous tests of small samples of clean (without defects) wood of the same species, with a moisture content of 12%:

R n = , Where

is the arithmetic mean of the tensile strength,

V- variation coefficient,

t- an indicator of reliability.

Regulatory resistance R n is the minimum probabilistic tensile strength of pure wood, obtained by static processing of the results of tests of standard samples of small size for short-term loading.

Design resistance R - This maximum voltage, which can withstand the material in the structure without collapsing, taking into account all the adverse factors in the operating conditions that reduce its strength.

In the transition from normative resistance R n to the calculated R it is necessary to take into account the effect on the strength of wood of a long-term load, defects (knots, oblique layer, etc.), the transition from small standard samples to elements building dimensions. The combined influence of all these factors is taken into account by the safety factor for the material ( To). The calculated resistance is obtained by dividing R n on the safety factor for the material:

To dl=0.67 - duration factor under the combined action of permanent and temporary loads;

To one = 0.27 ÷ 0.67 - uniformity coefficient, depending on the type of stress state, taking into account the effect of defects on the strength of wood.

Minimum value To one taken in tension, when the influence of defects is especially great. Design resistances To are given in table. 3 SNiP II-25-80 (for coniferous wood). R wood of other species is obtained using conversion factors, also given in SNiP.

The safety and strength of wood and wooden structures depend on temperature and humidity conditions. Humidification contributes to the decay of wood, and elevated temperature (beyond a known limit) reduces its strength. Accounting for these factors requires the introduction of coefficients for working conditions: m V ≤1, m T ≤1.

In addition, SNiP assumes taking into account the ply factor for glued elements: m sl = 0.95÷1.1;

beam coefficient for high beams, more than 50 cm high: m b ≤1;

bending coefficient for bent glued elements: m Mr≤1, etc.

The modulus of elasticity of wood, regardless of the species, is taken equal to:

The design characteristics of construction plywood are also given in SNiP; moreover, when checking stresses in plywood elements, as for wood, the coefficients of the working conditions are introduced m. In addition, for the design resistance of wood and plywood, a coefficient is introduced m dl=0.8 if the total design force from permanent and temporary loads exceeds 80% of the total design force. This factor is in addition to the reduction included in the material safety factor.

Lecture No. 2 Fundamentals of calculation for limit states

Lecture No. 2 Fundamentals of calculation for limit states. Calculation of structural elements of a solid section. In accordance with the standards in force in Russia, wooden structures must be calculated according to

Limit State Design

Limit States are conditions in which the structure can no longer be used as a result of external loads and internal stresses. In structures made of wood and plastics, two groups of limit states can occur - the first and second.

The calculation of the limit states of structures in general and its elements should be carried out for all stages: transportation, installation and operation - and should take into account all possible combinations of loads. The purpose of the calculation is to prevent neither the first nor the second limit states in the processes of transportation, assembly and operation of the structure. This is done on the basis of taking into account the normative and design loads and resistances of materials.

The limit state method is the first step in ensuring the reliability of building structures. Reliability refers to the ability of an object to maintain the quality inherent in the design during operation. The specificity of the theory of reliability of building structures is the need to take into account random values of loads on systems with random strength indicators. characteristic feature limit state method is that all the initial values operated in the calculation, random in nature, are represented in the norms by deterministic, scientifically based, normative values, and the effect of their variability on the reliability of structures is taken into account by the corresponding coefficients. Each of the reliability factors takes into account the variability of only one initial value, i.e. is private. Therefore, the method of limit states is sometimes called the method of partial coefficients. Factors, the variability of which affects the level of reliability of structures, can be classified into five main categories: loads and impacts; geometric dimensions of structural elements; the degree of responsibility of structures; mechanical properties of materials; working conditions of the structure. Consider these factors. The possible deviation of standard loads up or down is taken into account by the load safety factor 2, which, depending on the type of load, has a different value greater or less than one. These coefficients, along with the standard values, are presented in the chapter SNiP 2.01.07-85 Design standards. "Loads and influences". The probability of the joint action of several loads is taken into account by multiplying the loads by the combination factor, which is presented in the same chapter of the standards. Possible unfavorable deviation of the geometric dimensions of structural elements is taken into account by the accuracy factor. However, this ratio is pure form not acceptable. This factor is used when calculating the geometric characteristics, taking the design parameters of sections with a minus tolerance. In order to reasonably balance the costs of buildings and structures for various purposes, a reliability coefficient for the purpose is introduced< 1. Степень капитальности и ответственности зданий и сооружений разбивается на три класса ответственности. Этот коэффициент (равный 0,9; 0,95; 1) вводится в качестве делителя к значению расчетного сопротивления или в качестве множителя к значению расчетных нагрузок и воздействий.

The main parameter of the resistance of a material to force impacts is the normative resistance established by regulatory documents based on the results of statistical studies of the variability of the mechanical properties of materials by testing material samples according to standard methods. A possible deviation from the normative values is taken into account by the material safety factor ym > 1. It reflects the statistical variability of the material properties and their difference from the properties of the tested standard samples. The characteristic obtained by dividing the standard resistance by the coefficient m is called the design resistance R. This main wood strength characteristic is standardized by SNiP P-25-80 “Design standards. Wooden structures”.

The unfavorable influence of the environment and the operating environment, such as: wind and installation loads, section height, temperature and humidity conditions, are taken into account by introducing coefficients of working conditions m. The coefficient m can be less than one if this factor or a combination of factors reduces the bearing capacity of the structure, and more units, otherwise. For wood, these coefficients are presented in SNiP 11-25-80 “Design standards.

Regulatory limit values of deflections meet the following requirements: a) technological (ensuring the conditions for normal operation of machinery and handling equipment, instrumentation, etc.); b) constructive (ensuring the integrity of structural elements adjacent to each other, their joints, the presence of a gap between the supporting structures and the structures of partitions, half-timbered houses, etc., ensuring the specified slopes); c) aesthetic and psychological (providing favorable impressions from appearance structures, preventing the perception of danger).

The magnitude of the ultimate deflections depends on the span and the type of applied loads. For wooden structures covering buildings from the action of permanent and temporary long-term loads, the maximum deflection ranges from (1/150) - i to (1/300) (2). The strength of wood is also reduced under the influence of some chemicals from biodestruction, introduced under pressure in autoclaves to a considerable depth. In this case, the operating condition coefficient tia = 0.9. The influence of stress concentration in the calculated sections of tensioned elements weakened by holes, as well as in bent elements from round timber with undercutting in the calculated section, reflects the coefficient of the working condition m0 = 0.8. The deformability of wood in the calculation of wooden structures for the second group of limit states is taken into account by the basic modulus of elasticity E, which, when the force is directed along the wood fibers, is taken to be 10,000 MPa, and across the fibers, 400 MPa. When calculating the stability, the modulus of elasticity is assumed to be 4500 MPa. The basic shear modulus of wood (6) in both directions is 500 MPa. The Poisson's ratio of wood across the fibers with stresses directed along the fibers is taken equal to pdo o \u003d 0.5, and along the fibers with stresses directed across the fibers, n900 \u003d 0.02. Since the duration and level of loading affects not only the strength, but also the deformation properties of wood, the value of the elastic modulus and shear modulus is multiplied by the coefficient τi = 0.8 when calculating structures in which the stresses in the elements arising from permanent and temporary long-term loads, exceed 80% of the total voltage from all loads. When calculating metal-wood structures, the elastic characteristics and design resistances of steel and joints of steel elements, as well as reinforcement, are taken according to the chapters of SNiP for the design of steel and reinforced concrete structures.

Of all sheet structural materials using wood raw materials, only plywood is recommended to be used as elements of load-bearing structures, the basic design resistances of which are given in Table 10 of SNiP P-25-80. Under the appropriate operating conditions of glued plywood structures, the calculation for the first group of limit states provides for the multiplication of the basic design resistances of plywood by the coefficients of operating conditions tv, tj, tn and tl. When calculating for the second group of limit states, the elastic characteristics of plywood in the plane of the sheet are taken according to Table. 11 SNiP P-25-80. Elastic modulus and shear modulus for structures in various conditions operation, as well as those subjected to the combined effects of permanent and temporary long-term loads, should be multiplied by the corresponding coefficients of the operating conditions adopted for wood

First group the most dangerous. It is determined by unsuitability for service, when the structure loses its bearing capacity as a result of destruction or loss of stability. This does not happen until the maximum normal O or shearing t stresses in its elements do not exceed the calculated (minimum) resistances of the materials from which they are made. This condition is written by the formula

The limit states of the first group include: destruction of any kind, general loss of stability of the structure or local loss of stability of a structural element, violation of the joints that turn the structure into a variable system, the development of unacceptable residual deformations. The calculation of the bearing capacity is carried out according to the probable worst case, namely: according to the highest load and the lowest resistance of the material, found taking into account all the factors influencing it. Unfavorable combinations are given in the rules.

Second group less dangerous. It is determined by the unsuitability of the structure for normal operation, when it bends to an unacceptable value. This does not happen until its maximum relative deflection /// does not exceed the maximum allowable values. This condition is written by the formula

The calculation of wooden structures according to the second limit state for deformations applies mainly to bending structures and aims to limit the magnitude of deformations. The calculation is carried out on the standard loads without multiplying them by the reliability factors, assuming the elastic work of the wood. The calculation for deformations is carried out according to the average characteristics of the wood, and not according to the reduced ones, as when checking the bearing capacity. This is explained by the fact that the increase in deflection in some cases, when using lower quality wood, does not pose a threat to the integrity of structures. This also explains the fact that the calculation of deformations is carried out for normative, and not for design loads. As an illustration of the limit state of the second group, one can give an example when, as a result of an unacceptable deflection of the rafters, cracks appear in roofing. The flow of moisture in this case disrupts the normal operation of the building, leads to a decrease in the durability of wood due to its moisture, but the building continues to be used. The calculation for the second limit state, as a rule, is of subordinate importance, because the main thing is to ensure the bearing capacity. However, deflection limits are of particular importance for structures with yielding bonds. Therefore, the deformation of wooden structures (composite racks, composite beams, plank-nail structures) must be determined taking into account the influence of the compliance of the bonds (SNiP P-25-80. Table 13).

loads, acting on structures are determined by the Building Regulations and Rules - SNiP 2.01.07-85 "Loads and Impacts". When calculating structures made of wood and plastics, mainly the constant load from the own weight of structures and other building elements is taken into account. g and short-term loads from the weight of snow S, wind pressure W. Loads from the weight of people and equipment are also taken into account. Each load has a standard and design value. The normative value is conveniently denoted by the index n.

Regulatory loads are the initial values of loads: Live loads are determined as a result of processing data of long-term observations and measurements. Permanent loads are calculated from the dead weight and volume of structures, other elements of the building and equipment. Regulatory loads are taken into account when calculating structures for the second group of limit states - for deflections.

Design loads are determined on the basis of normative ones, taking into account their possible variability, especially upwards. For this, the values of standard loads are multiplied by the load safety factor y, whose values are different for different loads, but they are all greater than unity. Distributed load values are given in terms of kilopascals (kPa), which corresponds to kilonewtons per square meter(kN/m). Most calculations use linear load values (kN/m). Design loads are used in the calculation of structures for the first group of limit states, for strength and stability.

g”, acting on the structure, consists of two parts: the first part is the load from all elements of the enclosing structures and materials supported by this structure. The load from each element is determined by multiplying its volume by the density of the material and by the spacing of structures; the second part is the load from the own weight of the main supporting structure. In the preliminary calculation, the load from the own weight of the main supporting structure can be determined approximately, given the actual dimensions of the sections and volumes of the structural elements.

is equal to the product of the normative factor by the load reliability factor y. For load from own weight of structures y= 1.1, but for loads from insulation, roofing, vapor barrier and others y= 1.3. Permanent load from conventional pitched roofs with an angle of inclination A it is convenient to refer to their horizontal projection by dividing it by cos A.

The normative snow load s H is determined based on the normative weight of the snow cover so, which is given in the norms of loads (kN / m 2) of the horizontal projection of the coating, depending on the snow region of the country. This value is multiplied by the coefficient p, which takes into account the slope and other features of the shape of the coating. Then the standard load s H = s 0 p<х > 25° p == (60° - a°)/35°. This. the load is uniform and can be two-sided or one-sided.

With vaulted roofs on segmented trusses or arches, a uniform snow load is determined taking into account the coefficient p, which depends on the ratio of the span length / to the height of the vault /: p = //(8/).

With the ratio of the height of the arch to the span f/l= 1/8 snow load can be triangular with a maximum value of s” on one leg and 0.5 s” on the other and zero value at the ridge. The coefficients p, which determine the values of the maximum snow load at the ratios f/l= 1/8, 1/6 and 1/5, respectively equal to 1.8; 2.0 and 2.2. The snow load on arched pavements can be defined as gable, considering conventionally the pavement to be gable along the planes passing through the chords of the axes of the floor at the arches. The calculated snow load is equal to the product of the standard load and the load safety factor 7- For most lightweight wooden and plastic structures with a ratio of standard constant and snow loads g n /s H < 0,8 коэффициент y= 1.6. For large ratios of these loads at =1,4.

The load from the weight of a person with a load is taken equal to - normative R"= 0.1 kN and calculated R = p and y = 0.1 1.2 = 1.2 kN. wind load. Normative wind load w consists of pressure sh’+ and suction w n - wind. The initial data in determining the wind load are the values of wind pressure directed perpendicular to the surfaces of the coating and walls of buildings Wi(MPa), depending on the wind region of the country and accepted according to the norms of loads and impacts. Regulatory wind loads w” are determined by multiplying the normal wind pressure by the coefficient k, taking into account the height of buildings, and the aerodynamic coefficient With, considering its shape. For most buildings made of wood and plastics, the height of which does not exceed 10 m, k = 1.

Aerodynamic coefficient With depends on the shape of the building, its absolute and relative dimensions, slopes, relative heights of coatings and wind direction. On most pitched roofs, the angle of inclination of which does not exceed a = 14 °, the wind load acts in the form of suction W-. At the same time, it basically does not increase, but reduces the forces in structures from constant and snow loads, and in the calculation it may not be taken into account in the margin of safety. The wind load must be taken into account when calculating the pillars and walls of buildings, as well as when calculating triangular and lancet structures.

The calculated wind load is equal to the standard multiplied by the safety factor y= 1.4. Thus, w = = w”y.

Regulatory resistances wood R H(MPa) are the main characteristics of the strength of wood areas clean from defects. They are determined by the results of numerous laboratory short-term tests of small standard samples of dry wood with a moisture content of 12% for tension, compression, bending, crushing and chipping.

95% of the tested wood samples will have a compressive strength equal to or greater than its standard value.

The values of standard resistances given in app. 5 are practically used in laboratory control of wood strength in the process of manufacturing wooden structures and in determining the bearing capacity of operating load-bearing structures during their examinations.

Design resistances wood R(MPa) - these are the main characteristics of the strength of real wood elements of real structures. This wood has natural blemishes and works under stress for many years. Design resistances are obtained on the basis of standard resistances, taking into account the reliability factor for the material at and loading duration factor t al according to the formula

Coefficient at much more than unity. It takes into account the decrease in the strength of real wood as a result of the heterogeneity of the structure and the presence of various defects that do not exist in laboratory samples. Basically, the strength of wood is reduced by knots. They reduce working area sections, cutting and pushing its longitudinal fibers, create an eccentricity of longitudinal forces and a slope of the fibers around the knot. The inclination of the fibers causes the wood to stretch across and at an angle to the fibers, the strength of which in these directions is much lower than along the fibers. Wood defects reduce the tensile strength of wood by almost half and by about one and a half times in compression. Cracks are most dangerous in areas where wood is chipped. With an increase in the size of the sections of the elements, the stresses during their destruction decrease due to the greater heterogeneity of the distribution of stresses over the sections, which is also taken into account when determining the design resistances.

Loading duration factor t dl<С 1- Он учитывает, что древесина без пороков может неограниченно долго выдерживать лишь около половины той нагрузки, которую она выдерживает при кратковременном нагружении в процессе испытаний. Следовательно, ее длительное R in resistance I yL almost W^ half the short-term /tg.

The quality of wood naturally affects the magnitude of its calculated resistances. Wood of the 1st grade - with the least flaws has the highest design resistance. The design resistance of wood of the 2nd and 3rd grades is lower, respectively. For example, the calculated resistance of pine and spruce wood of the 2nd grade to compression is obtained from the expression

The calculated resistance of pine and spruce wood to compression, tension, bending, chipping and crushing are given in App. 6.

Working conditions coefficients T to the design resistance of wood, the conditions in which wooden structures are manufactured and operate are taken into account. Breed factor T" takes into account the different strength of wood of different species, which differ from the strength of pine and spruce wood. The load factor t takes into account the short duration of the action of wind and installation loads. When crushed t n= 1.4, for other types of voltages t n = 1.2. The height coefficient of sections during bending of wood of glued-wood beams with a section height of more than 50 cm / 72b decreases from 1 to 0.8, with a section height of 120 cm - even more. The layer thickness coefficient of glued wood elements takes into account the increase in their compressive and bending strength as the thickness of the glued boards decreases, as a result of which the homogeneity of the structure of the glued wood increases. Its values are within 0.95. 1.1. The bending coefficient m rH takes into account the additional bending stresses that occur when the boards bend during the manufacture of bent glued wood elements. It depends on the ratio of the radius of the bend to the thickness of the h/b boards and has a value of 1.0. 0.8 as this ratio increases from 150 to 250. Temperature coefficient m t takes into account the decrease in the strength of wood structures operating at temperatures from +35 to +50 °C. It decreases from 1.0 to 0.8. Moisture coefficient t ow takes into account the decrease in the strength of wood structures operating in a humid environment. At air humidity in rooms from 75 to 95% t vl = 0.9. Outdoors in dry and normal areas t ow = 0.85. With constant moisture and in water t ow = 0.75. Stress concentration factor t k = 0.8 takes into account the local reduction in the strength of wood in the areas of tie-ins and holes in tension. The load duration coefficient t dl = 0.8 takes into account the decrease in the strength of wood as a result of the fact that long-term loads sometimes make up more than 80% of the total amount of loads acting on the structure.

Modulus of elasticity of wood determined in short-term laboratory tests, E cr= 15-10 3 MPa. When taking into account deformations under long-term loading, when calculating by deflections £ = 10 4 MPa (Appendix 7).

The normative and design resistances of construction plywood were obtained by the same methods as for wood. In this case, its sheet form and an odd number of layers with mutually perpendicular direction of the fibers were taken into account. Therefore, the strength of plywood in these two directions is different and along the outer fibers it is somewhat higher.

The most widely used in constructions is seven-layer plywood of the FSF brand. Its calculated resistances along the fibers of the outer veneers are: tensile # f. p = 14 MPa, compression #f. c \u003d 12 MPa, bending out of plane /? f.„ = 16 MPa, chipping in the plane # f. sk \u003d 0.8 MPa and cut /? f. cf - 6 MPa. Across the fibers of the outer veneers, these values are respectively equal to: I f_r= 9 MPa, compression # f. c \u003d 8.5 MPa, bending # F.i \u003d 6.5 MPa, chipping R$. CK= 0.8 MPa, cut # f. cf = = 6 MPa. The elastic and shear moduli along the outer fibers are, respectively, E f = 9-10 3 MPa and b f = 750 MPa and across the outer fibers £ f = 6-10 3 MPa and G$ = 750 MPa.

Limit State Design

Limit State Design Limit states are states at which the structure can no longer be used as a result of external loads and internal

Since 1955, the calculation of reinforced concrete structures in our country has been carried out according to the method of limit states.

· The limit is understood such a state of the structure, after reaching which further operation becomes impossible due to the loss of the ability to resist external loads or the receipt of unacceptable movements or local damage. In accordance with this, two groups of limit states are established: the first - by bearing capacity; the second - for suitability for normal use.

· Calculation for the first group of limit states is carried out in order to prevent the destruction of structures (strength analysis), loss of stability of the structure shape (buckling analysis) or its position (overturning or sliding analysis), fatigue failure (endurance analysis).

· Calculation for the second group of limit states It aims to prevent the development of excessive deformations (deflections), exclude the possibility of cracking in concrete or limit the width of their opening, and also ensure, if necessary, the closure of cracks after removing part of the load.

The calculation for the first group of limit states is the main one and is used in the selection of sections. The calculation for the second group is made for those structures that, being strong, lose their performance due to excessive deflections (beams, large spans at a relatively low load), cracking (tanks, pressure pipelines) or excessive crack opening, leading to premature corrosion of reinforcement .

The loads acting on the structure and the strength characteristics of the materials from which the structure is made are variable and may differ from average values. Therefore, to ensure that during the normal operation of the structure none of the limit states occurs, a system of design coefficients is introduced that takes into account possible deviations (in an unfavorable direction) of various factors affecting the reliable operation of structures: 1) load safety factors γ f , taking into account the variability of loads or impacts; 2) safety factors for concrete γ b and reinforcement γ s . taking into account the variability of their strength properties; 3) reliability coefficients for the purpose of the structure γ n , taking into account the degree of responsibility and capitalization of buildings and structures; 4) coefficients of working conditions γ bi and γ si , allowing to evaluate some features of the work of materials and structures in general, which cannot be reflected in the calculations in a direct way.

Estimated coefficients are established on the basis of probabilistic-statistical methods. They provide the required reliability of structures for all stages: manufacturing, transportation, erection and operation.

Thus, the main idea of the limit state calculation method is to ensure that even in those rare cases when the maximum possible loads act on the structure, the strength of concrete and reinforcement is minimal, and the operating conditions are the most unfavorable, the structure does not collapse and would not receive unacceptable deflections or cracks. At the same time, in many cases, it is possible to obtain more economical solutions than in the calculation by previously used methods.

Loads and impacts . When designing, one should take into account the loads arising during the construction and operation of structures, as well as during the manufacture, storage and transportation of building structures.

The calculations use the normative and design values of loads. The highest values of loads established by the norms that can act on the structure during its normal operation are called normative *. The actual load due to various circumstances may differ from the normative to a greater or lesser extent. This deviation is taken into account by the load safety factor.

The calculation of structures is carried out for design loads

where q n - standard load; γ f - load safety factor corresponding to the considered limit state.

When calculating for the first group of limit states γ f take: for constant loads γ f = 1.1...1.3; temporary γ f \u003d 1.2 ... 1.6, when calculating for position stability (overturning, sliding, ascent), when reducing the weight of the structure worsens its working conditions, take

The calculation of structures for the second group of limit states, taking into account the lower risk of their occurrence, is carried out for design loads at γ f = l. The exception is structures belonging to category I of crack resistance (see § 7.1), for which γ f >l.

Loads and impacts on buildings and structures can be permanent and temporary. The latter, depending on the duration of action, are divided into long-term, short-term and special.

Constant loads include the weight of parts of structures, including the weight of load-bearing and enclosing structures; weight and pressure of soils (embankments, backfills); prestressing effect.

Temporary long-term loads include: the weight of stationary equipment - machine tools, motors, containers, conveyors; weight of liquids and solids filling equipment; load on floors from stored materials and shelving in warehouses, refrigerators, book depositories, libraries and utility rooms.

In those cases where it is required to take into account the influence of the duration of the action of loads on deformations and the formation of cracks, long-term loads include a part of short-term ones. These are loads from cranes with a reduced standard value, determined by multiplying the full standard value of the vertical load from one crane in each span by a factor: 0.5 - for crane operating mode groups 4K-6K; 0.6 - for groups of crane operation mode 7K; 0.7 - for groups of operation mode of cranes 8K*; snow loads with a reduced standard value, determined by multiplying the full standard value (see §11.4) by a factor of 0.3 - for snow region III, 0.5 - for region IV, 0.6 - for regions V, VI; loads from people, equipment on floors of residential and public buildings with reduced standard values. These loads are referred to as long-term loads due to the fact that they can act for a time sufficient for creep deformations to appear, increasing the deflection and crack opening width.

Short-term loads include: loads from the weight of people, equipment on the floors of residential and public buildings with full standard values; loads from cranes with full standard value; snow loads with full standard value; wind loads, as well as loads arising from the installation or repair of structures.

Special loads occur during seismic, explosive or emergency impacts.

Buildings and structures are subjected to the simultaneous action of various loads, so their calculation should be carried out taking into account the most unfavorable combination of these loads or the forces caused by them. Depending on the composition of the loads taken into account, there are: the main combinations, consisting of permanent, long-term and short-term loads; special combinations consisting of permanent, long-term, short-term and one of the special loads.

Live loads are included in combinations as long-term - when taking into account the reduced standard value, as short-term - when taking into account the full standard value.

The probability of the simultaneous occurrence of the greatest loads or efforts is taken into account by the combination coefficients ψ 1 and ψ 2 . If the main combination includes a constant and only one temporary load (long-term and short-term), then the combination coefficients are taken equal to 1, when two or more temporary loads are taken into account, the latter are multiplied by ψ 1 \u003d 0.95 for long-term loads and ψ 1 \u003d 0.9 at short-term, since it is considered unlikely that they simultaneously reach the maximum calculated values.

* Groups of crane operation modes depend on crane operation conditions, lifting capacity and are accepted in accordance with GOST 25546-82.

When calculating structures for a special combination of loads, including explosive effects, it is allowed not to take into account short-term loads.

The values of the design loads should also be multiplied by the reliability factor for the purpose of the structures, taking into account the degree of responsibility and capitalization of buildings and structures. For class I structures (objects of particularly important national economic importance) γ n =1, for class II structures (important national economic objects) γ n =0.95, for class III structures (having limited national economic significance) γ n =0.9, for temporary structures with a service life of up to 5 years γ n =0.8.

Normative and design resistance of concrete. The strength characteristics of concrete are variable. Even samples from the same batch of concrete will show different strengths during testing, which is explained by the heterogeneity of its structure and different test conditions. The variability of the strength of concrete in structures is also affected by the quality of equipment, the qualifications of workers, the type of concrete, and other factors.

Rice. 2.3. Distribution curves:

F m and F - average and calculated values

efforts from external load;

F um and F u - the same, bearing capacity

Of all the possible strength values, it is necessary to enter into the calculation one that ensures the safe operation of structures with the necessary reliability. Methods of probability theory help to establish it.

The variability of the strength properties, as a rule, obeys the Gauss law and is characterized by a distribution curve (Fig. 2.3, a), which connects the strength characteristics of concrete with the frequency of their repetition in experiments. Using the distribution curve, you can calculate the average value of the compressive strength of concrete:

where n 1 , n 2 ,.., n k is the number of experiments in which the strength R 1 , R 2 ,…, R k was recorded, n is the total number of experiments. The spread of strength (deviation from the average) is characterized by the standard deviation (standard)

or coefficient of variation ν = σ/R m . In formula (2.8) Δ i = R i - R m .

Having calculated σ, it is possible to find the strength value R n using the methods of probability theory, which will have a given reliability (security):

where æ is the reliability index.

The higher æ (see Fig. 2.3, a), the more samples show strength R m - æσ and more, the higher the reliability. If we take R n = R m - σ as the minimum strength entered into the calculation (i.e., setting æ = 1), then 84% of all samples (they can be cubes, prisms, eights) will show the same or greater strength ( reliability 0.84). At æ \u003d 1.64-95% of the samples will show strength R n \u003d R m - 1.64 σ and more, and at æ \u003d 3 - 99.9% of the samples will have a strength not lower than R n \u003d R m -3 σ. Thus, if we enter into the calculation the value of R m -Зσ, then only in one case out of a thousand, the strength will be lower than the accepted one. Such a phenomenon is considered almost unbelievable.

According to the norms, the main characteristic controlled at the factory is concrete class "B" *, representing the strength of a concrete cube with a rib of 15 cm with a reliability of 0.95. The strength corresponding to the class is determined by formula (2.9) with æ = 1.64

The value of ν can vary within wide limits.

The manufacturer needs to provide the strength R n corresponding to the class of concrete, taking into account the coefficient ν, determined for specific production conditions. In enterprises with good organized production(producing concrete with high uniformity) the actual coefficient of variation will be small, the average strength of concrete [see. formula (2.10)] can be taken lower, thus saving cement. If the concrete produced by the enterprise has a large variability of strength (large coefficient of variation), then it is necessary to increase the strength of concrete R m to ensure the required values of R n , which will cause excessive consumption of cement.

* Until 1984, the main characteristic of the strength of concrete was its brand, which was defined as the average value of the compressive strength of concrete R m in kgf / cm 2.

The normative resistance of concrete prisms to axial compression R b,n (prism strength) is determined by the normative value of cubic strength, taking into account dependence (1.1), linking prismatic and cubic strength. The values of R b, n are given in table. 2.1.

The normative resistance of concrete to axial tension R bt,n in cases where the tensile strength of concrete is not controlled, are determined by the normative value of the cubic strength, taking into account dependence (1.2), which relates the tensile strength to the compressive strength. The values of R bt, n are given in Table. 2.1.

If the tensile strength of concrete is controlled by direct testing of samples in production, then the standard axial tensile strength is taken equal to

and characterizes the class of concrete in terms of tensile strength.

The design resistances of concrete for the limit states of the first group R b and R bt are determined by dividing the standard resistances by the corresponding reliability factors of concrete in compression γ bc or in tension γ bt:

For heavy concrete γ bc = 1.3; γ bt = 1.5.

These coefficients take into account the possibility of reducing the actual strength compared to the standard due to the difference in the strength of concrete in real structures from the strength in samples and a number of other factors depending on the conditions of manufacture and operation of structures.

Table 2.1.

Strength and deformation characteristics of heavy concrete

Compressive strength class of concrete	Normative resistances and design resistances of concrete for calculation by limit states of group II, MPa		Design resistance of concrete in the calculation for the limit states of group I, MPa		The initial modulus of elasticity of concrete in compression E b 10 -3 , MPa
Compressive strength class of concrete	compression R bn , R b,ser	stretching R btn , R bt,ser	compression R b	tension R bt	natural curing	heat treated
7.5V 10V 12.5V 15V 20V 25V 30V 35V 40V 45V 50V 55V60	5,50 7,50 9,50 11,0 15,0 18,5 22,0 25,5 29,0 32,0 36,0 39,5 43,0	0,70 0,85 1,00 1,15 1,40 1,60 1,80 1,95 2,10 2,20 2,30 2,40 2,50	4,50 6,00 7,50 8,50 11,5 14,5 17,0 19,5 22,0 25,0 27,5 30,0 33,0	0,480 0,570 0,660 0,750 0,900 1,05 1,20 1,30 1,40 1,45 1,55 1,60 1,65	16,0 18,0 21,0 23,0 27,0 30,0 32,5 34,5 36,0 37,5 39,0 39,5 40,0	14,5 16,0 19,0 20,5 24,5 27,0 29,0 31,0 32,5 34,0 35,0 35,5 36,0

Design concrete resistances for group II limit states Rb,ser and Rbt,ser are determined with safety factors γbc = γbt = 1, i.e. are taken equal to standard resistances. This is explained by the fact that the onset of limit states of group II is less dangerous than group I, since, as a rule, it does not lead to the collapse of structures and their elements.

When calculating concrete and reinforced concrete structures, the design resistances of concrete, if necessary, are multiplied by the coefficients of working conditions γ bi, taking into account: the duration and repeatability of the load, manufacturing conditions, the nature of the structure, etc. For example, in order to take into account the decrease in the strength of concrete that occurs with a continuous load, the coefficient γ b 2 = 0.85 ... 0.9 is introduced, when taking into account loads of short duration - γ b 2 = 1.1.

■ Regulatory and design resistances of reinforcement . The normative resistances of reinforcement R sn are taken equal to the smallest controlled values: for bar reinforcement, high-strength wire and reinforcing ropes - yield strength, physical σ y , or conditional σ 0.2; for ordinary reinforcing wire - a voltage of 0.75 of the tensile strength, since GOST does not regulate the yield strength for this wire.

The values of normative resistances R sn are taken in accordance with the current standards for reinforcing steels, as well as for concrete, with a reliability of 0.95 (Table 2.2).

The design tensile resistances of reinforcement R s and R s,ser for the limit states of groups I and II (Table 2.2) are determined by dividing the standard resistances by the corresponding reliability factors for reinforcement γ s:

The safety factor is set to exclude the possibility of destruction of the elements in case of excessive convergence of R s and R sn . It takes into account the variability of the area cross section rods, early development of plastic deformations of reinforcement, etc. Its value for bar reinforcement of classes A-I, A-II is 1.05; classes A-III - 1.07 ... 1.1; classes A-IV, A-V-1.15; classes A-VI - 1.2; for wire fittings of classes Bp-I, B-I - 1.1; classes B-II, Bp-II, K-7, K-19-1.2.

When calculating for the limit states of group II, the value of the safety factor for all types of reinforcement is assumed to be equal to one, i.e. calculated resistances R s , s er are numerically different from the normative ones.

When assigning the design compressive resistance of reinforcement R sc, not only the properties of steel are taken into account, but also the ultimate compressibility of concrete. Taking ε bcu = 2X 10 -3 , the elastic modulus of steel E s = 2 10 -5 MPa, it is possible to obtain the highest stress σ sc achieved in the reinforcement before the destruction of concrete from the condition of joint deformations of concrete and reinforcement σ sc = ε bcu E s = ε s E s . According to the norms, the design resistance of the reinforcement to compression R sc is taken equal to R s if it does not exceed 400 MPa; for reinforcement with a higher R s value, the design resistance R sc is assumed to be 400 MPa (or 330 MPa when calculated in the compression stage). With prolonged action of the load, the creep of concrete leads to an increase in the compressive stress in the reinforcement. Therefore, if the design resistance of concrete is taken taking into account the coefficient of working conditions γ b 2 \u003d 0.85 ... 0.9 (i.e., taking into account the prolonged effect of the load), then it is allowed subject to the appropriate design requirements increase the value of R sc up to 450 MPa for steels of class A-IV and up to 500 MPa for steels of classes At-IV and above.

When calculating structures according to group I of limit states, the design resistances of reinforcement, if necessary, are multiplied by the coefficients of working conditions γ si , taking into account the uneven distribution of stresses in the cross section, the presence of welded joints, repeated loading, etc. For example, the operation of high-strength reinforcement at stresses above the conditional yield strength is taken into account by the coefficient of working conditions γ s6 , the value of which depends on the class of reinforcement and varies from 1.1 to 1.2 (see § 4.2).

Table 2.2.

Strength and deformation characteristics

reinforcing steels and ropes.

fittings		Normative R sn and design resistances in the calculation for the limit states of group II R s , ser , MPa	Design resistance of reinforcement, MPa, when calculating according to the limit state of group I			elasticity E s , 10 5 MPa
			stretching
			longitudinal and transverse when calculating inclined sections for the action of the bending moment R s	transverse when calculating inclined sections for the action of a transverse force R sw
Rod
A-I	6…40	235	225	175	225	2,1
A-II	10…80	295	280	225	280	2,1
A-III	6…8	390	355	285	355	2,0
	10…40	390	365	290	365	2,0
A-IV	10…28	590	510	405	400	1,9
A-V	10…32	785	680	545	400	1,9
A-VI	10…28	980	815	650	400	1,9
A-IIIc (with elongation and tension control)	20…40	540	490	390	200	1,8
Wire
VR-I	3...5	410...395	375...360	270...260	375...360	1,7
B-II	3...8	1490...1100	1240...915	990...730	400	2,0
VR-II	3...8	1460...1020	1215...850	970...680	400	2,0
Rope
K-7	6...15	1450...1290	1210...1080	965...865	400	1,8
K-19	14	1410	1175	940	400	1,8

Note. In the table, bar reinforcement classes mean all types of reinforcement of the corresponding class, for example, under class A-V also mean A t -V, A t -VCK, etc.

■ The main provisions of the calculation.

When calculating for the I group of limit states (bearing capacity), the condition must be met

The left side of the expression (2.14) is the design force equal to the practically possible maximum force in the section of the element with the most unfavorable combination of design loads or actions; it depends on the efforts caused by design loads q at γ f >1, combination coefficients and reliability coefficients for the purpose of structures γ n . The design force F should not exceed the design load-bearing capacity of the section F u , which is a function of the design resistances of materials and operating conditions coefficients γ bi , γ si , taking into account unfavorable or favorable operating conditions of structures, as well as the shape and size of the section.

The curves (Fig. 2.3,b) of the distribution of forces from external load 1 and bearing capacity 2 depend on the variability of the factors discussed above and obey the Gauss law. The fulfillment of condition (2.14), expressed graphically, guarantees the required bearing capacity of the structure.

When calculating for the II group of limit states:

· by displacements - it is required that the deflections from the standard load f do not exceed the limit values of the deflections f u established by the standards for this structural element f ≤ f u . The value of f u is taken by ;

· on formation of cracks - the force from the design or normative load must be less than or equal to the force at which cracks appear in the section F ≤ F crc ;

· according to the opening of normal and oblique cracks - the width of their opening at the level of tensile reinforcement should be less than their limiting opening established by the norms a cr c , u a crc ≤ a cr c , u = 0.l...0.4 mm.

In necessary cases, it is required that the cracks formed from the full load be reliably closed (clamped) under the action of its long part. In these cases, a calculation is made for the closure of cracks.

SELF-CHECK QUESTIONS:

1. Stages of the stress-strain state of bent reinforced concrete elements. Which of these stages are used in the calculation of strength, crack resistance, deflections?

2. Features of the stress-strain state of prestressed structures.

3. The main provisions of the methods for calculating sections for allowable stresses and breaking loads. disadvantages of these methods.

4. The main provisions of the calculation by the method of limit states.

Groups of limit states.

5. What are the goals of calculation for groups I and II of limit states?

6. Classification of loads and their design combinations.

7. Normative and design loads. Reliability factors

by loads. To what extent do they vary?

8. Normative resistance of concrete. How is it related to average

strength? With what security is it assigned?

9. How is the design resistance of concrete determined for groups I and II

limit states? What is the purpose of introducing reliability coefficients and coefficients of working conditions?

10. How is the standard resistance of reinforcement assigned for various steels?

11. Calculated reinforcement resistance, safety factors

and working conditions.

12. Write down in general terms the conditions precluding the onset

limit states of groups I and II, and explain their meaning.

At this stage, we already understand that the calculations of building structures are carried out in accordance with some standards. What - it is impossible to say unambiguously, because in different countries different design standards are used.

So, in the CIS countries, various versions of the standards are used, based on Soviet SNiPs and GOSTs; in Europe, they mainly switched to Eurocode (Eurocode, EN), and in the USA ASCE, ACI, etc. are used. Obviously, your project will be tied to the standards of the country where this project was ordered from or where it will be implemented.

If the norms are different, then the calculations are different?

This question worries novice calculators so much that I have separated it into a separate paragraph. Indeed: if you open some foreign design standards and compare them, for example, with SNiP, you may get the impression that the foreign design system is based on completely different principles, methods, and approaches.

However, it should be understood that design standards cannot contradict the fundamental laws of physics and must be based on them. Yes, they can use different physical characteristics, coefficients, even models of the work of certain building materials, but they are all united by a common scientific base based on the strength of materials, structural and theoretical mechanics.

This is what the strength test of a metal structure element under tension looks like according to the Eurocode:

\[\frac(((N_(Ed))))(((N_(t,Rd)))) \le 1,0.\quad (1)\]

And here is what a similar check looks like for one of latest versions SNiP:

\[\frac(N)(((A_n)(R_y)(\gamma _c))) \le 1,0.\quad (2)\]

It is easy to guess that in both the first and second cases, the force from the external load (in the numerator) should not exceed the force that characterizes the bearing capacity of the structure (in the denominator). This good example a common, scientifically based approach to the design of buildings and structures by engineers from different countries.

Limit state concept

One day (actually, many years ago), scientists and research engineers noticed that it was not entirely correct to design an element based on a single test. Even for relatively simple designs, there can be a lot of options for each element, and Construction Materials in the process of wear change their characteristics. And if we consider the emergency and repair states of the structure, then this leads to the need to streamline, segment, classify all possible states of the structure.

This is how the concept of “limiting state” was born. A laconic interpretation is given in the Eurocode:

limit state - such a state of a structure at which the structure does not meet the appropriate design criteria

It can be said that the limit state occurs when the work of the structure under load goes beyond the scope of design decisions. For example, we designed a steel frame frame, but at a certain point in its operation, one of the racks lost stability and bent - there is a transition to the limit state.

The method of calculation of building structures by limit states is dominant (it replaced the less “flexible” method of allowable stresses) and is used today both in the regulatory framework of the CIS countries and in the Eurocode. But how can an engineer use this abstract concept in concrete calculations?

Limit State Groups

First of all, you need to understand that each of your calculations will relate to one or another limit state. The calculator simulates the work of the structure not in some abstract, but in the limit state. That is, all the design characteristics of the structure are selected based on the limit state.

At the same time, you do not need to constantly think about the theoretical side of the issue - all the necessary checks are already placed in the design standards. By performing checks, you thereby prevent the occurrence of the limit state for the designed structure. If all checks are satisfied, then we can assume that the limit state will not occur until the end life cycle structures.

Since in real design an engineer deals with a series of checks (for stresses, moments, forces, deformations), all these calculations are conditionally grouped, and they are already talking about groups of limit states:

limit states of group I (in Eurocode - by bearing capacity)
limit states of group II (in the Eurocode - according to serviceability)

If the first limit state occurs, then:

construction destroyed
the structure has not yet been destroyed, but the slightest increase in load (or a change in other operating conditions) leads to destruction

The conclusion is obvious: further operation of a building or structure that is in the first limit state is impossible. no way:

Figure 1. Destruction of a residential building (first limit state)

If the structure has passed into the second (II) limit state, then its operation is still possible. However, this does not mean at all that everything is in order with it - individual elements can receive significant deformations:

deflections
section rotations
cracks

As a rule, the transition of a structure to the second limit state requires any restrictions in operation, for example, reducing the load, reducing the speed of movement, etc.:

Figure 2. Cracks in the building concrete (second limit state)

In terms of strength of materials

At the "physical level", the onset of a limit state means, for example, that the stresses in a structural element (or group of elements) exceed a certain allowable threshold, called the design resistance. These may be other factors of the stress-strain state - for example, bending moments, transverse or longitudinal forces that exceed the bearing capacity of the structure in the limit state.

Checks for the first group of limit states

To prevent the onset of limit state I, the design engineer must check the characteristic sections of the structure:

strength
for sustainability
endurance

All load-bearing structural elements, without exception, are checked for strength, regardless of the material from which they are made, as well as the shape and size of the cross section. This is the most important and mandatory check, without which the calculator does not have the right to restful sleep.

The stability check is performed for compressed (centrally, eccentrically) elements.

Fatigue testing should be carried out on members that operate under cyclic loading and unloading conditions to prevent fatigue effects. This is typical, for example, for the spans of railway bridges, since during the movement of trains the loading and unloading stages of work constantly alternate.

As part of this course, we will get acquainted with the basic strength tests of reinforced concrete and metal structures.

Checks for the second group of limit states

To prevent the onset of the II limit state, the design engineer is obliged to check the characteristic sections:

on deformations (displacements)
for crack resistance (for reinforced concrete structures)

Deformations should be associated not only with linear displacements of the structure (deflections), but also with the angles of rotation of the sections. Ensuring crack resistance is an important step in the design of reinforced concrete structures from both conventional and prestressed reinforced concrete.

Examples of calculations for reinforced concrete structures

As an example, let's consider what checks need to be performed when designing structures from ordinary (non-stressed) reinforced concrete according to the standards,.

Table 1. Grouping of calculations by limit states:
M - bending moment; Q - transverse force; N - longitudinal force (compressive or tensile); e - application eccentricity longitudinal force; T is the torque; F - external concentrated force (load); σ- normal voltage; a - crack opening width; f - deflection of the structure

Please note that for each group of limit states, a whole series of checks are performed, and the type of check (formula) depends on the stress-strain state of the structural element.

We have already come close to learning how to calculate building structures. At the next meeting, we will talk about the loads, and immediately proceed to the calculations.

The calculation for the limit states of the first group is performed to prevent:

Brittle, ductile or other type of fracture (strength calculation, taking into account, if necessary, the deflection of the structure before destruction);

Fatigue failure (fatigue analysis of structures under the influence of a repetitive movable or pulsating load: crane beams, sleepers, frame foundations and ceilings for unbalanced machines, etc.);

The calculation for the limit states of the second group is performed to prevent:

Formation of excessive or prolonged crack opening (if the formation or prolonged crack opening is permissible under operating conditions);

Excessive movements (deflections, angles of rotation, skew angles and vibration amplitudes).

The calculation of the limit states of the structure as a whole, as well as its individual elements or parts, is carried out for all stages: manufacturing, transportation, installation and operation; at the same time, design schemes must comply with the adopted design solutions and each of the listed stages.

Estimated factors

The values of loads, resistance of concrete and reinforcement are set according to the chapters of SNiP "Loads and effects" and "Concrete and reinforced concrete structures".

Classification of loads. Regulatory and design loads

Depending on the duration of the action, the load is divided into permanent and temporary. Temporary loads, in turn, are divided into long-term, short-term, special.

Loads from the weight of the bearing and enclosing structures of buildings and structures, the mass and pressure of soils, and the impact of prestressing reinforced concrete structures are constant.

Long-term loads are from the weight of stationary equipment on floors - apparatuses, engines, tanks, etc.; pressure of gases, liquids, bulk solids in containers; loads in warehouses, refrigerators, archives, libraries and similar buildings and structures; part of the temporary load established by the norms in residential buildings, office and amenity premises; long-term temperature technological effects from stationary equipment; loads from one overhead or one overhead crane, multiplied by the coefficients: 0.5 for medium-duty cranes and 0.7 for heavy-duty cranes; snow loads for III-IV climatic regions with coefficients of 0.3-0.6. The indicated values of crane, some temporary and snow loads are part of their total value and are entered into the calculation taking into account the duration of the action of these types of loads on displacements, deformations, and cracking. The full values of these loads are short-term.

The design loads for calculating structures for strength and stability are determined by multiplying the standard load by the load safety factor Yf, usually greater than one, for example G= Gnyt. Reliability coefficient from the weight of concrete and reinforced concrete structures Yf = M; on the weight of structures made of concrete on light aggregates (with an average density of 1800 kg / m3 or less) and various screeds, backfills, heaters, performed in the factory, Yf = l,2, on installation Yf = l>3; from various live loads depending on their value Yf = l. 2...1.4. The coefficient of overload from the weight of structures when calculating the stability of the position against ascent, overturning and sliding, as well as in other cases when a decrease in mass worsens the working conditions of the structure, is taken yf = 0.9. When calculating structures at the stage of construction, the calculated short-term loads are multiplied by a factor of 0.8. The design loads for the calculation of structures for deformations and displacements (for the second group of limit states) are taken equal to the standard values with the coefficient Yf = l-

Load reduction. When calculating columns, walls, foundations of multi-storey buildings, temporary loads on floors can be reduced, taking into account the degree of probability of their simultaneous action, by multiplying by a coefficient

T) = a + 0.6/Km~, (II-11)

The norms also allow to reduce live loads when calculating beams and crossbars, depending on the area of the loaded floor.