Calculation of a design aimed at preventing limit states of the first group is expressed by the inequality:
N ≤ Ф, (2.1)
Where N– force in the element under consideration (longitudinal force, bending moment, transverse force) from the action of the maximum design values of loads; F– load-bearing capacity of the element.
To check the limit states of the first group, the maximum design values of loads F m are used, determined by the formula:
F m = F 0 g fm ,
Where F 0- characteristic value of the load, g fm,– reliability factor for the maximum load value, taking into account possible load deviation in an unfavorable direction. Characteristic load values F 0 and coefficient values g fm determined in accordance with DBN. Sections 1.6 – 1.8 of this methodological development are devoted to these issues.
When calculating loads, as a rule, the reliability coefficient for the purpose of the structure is taken into account g n, the values of which, depending on the responsibility class of the structure and the type of design situation, are given in Table. 2.3. Then the expression for determining the maximum load values will take the form:
F m = F 0 g fm ∙g n
The right side of inequality (1.1) can be represented as:
Ф = S R y g c ,(2.2)
Where Ry– design resistance of steel, established by the yield strength; S– geometric characteristics of the section (under tension or compression S represents the cross-sectional area A, during bending – moment of resistance W); g c– coefficient of operating conditions of the structure, the values of which, depending on the material of the structure, are established by the relevant standards. For steel structures values g c are given in table. 2.4.
Substituting value (2.2) into formula (2.1), we obtain the condition
N ≤ S R y g c
For stretched elements with S=A
N ≤ A R y g c
Dividing the left and right sides of the inequality by the area A, we obtain the condition for the strength of a tensile or compressed element:
For bendable elements when S = W, Then
M ≤ W R y g c
From the last expression follows a formula for checking the strength of a bending element
The formula for checking the stability of a compressed element is:
Where φ – coefficient longitudinal bending, depending on the flexibility of the rod
Table 2.4 – Operating conditions coefficient g c
| Structural elements | g with |
| 1. Solid beams and compressed elements of floor trusses under the halls of theaters, clubs, cinemas, under the premises of shops, archives, etc. under a temporary load that does not exceed the weight of the floor 2. Columns public buildings and water tower supports. 3. Columns of one-story industrial buildings with overhead cranes 4. Compressed main elements (except for supporting ones) of a composite T-section lattice from the corners of welded roof trusses and floors when calculating the stability of these with flexibility l ≥ 60 5. Tightenings, rods, guys, hangers in the calculations for strength in unweakened sections 6. Structural elements made of steel with a yield strength of up to 440 N/mm 2, bearing a static load, in calculations for strength in a section weakened by bolt holes (except for friction connections) 8. Compressed elements from single angles attached by one shelf (for unequal corners - a smaller shelf) with the exception of lattice elements of spatial structures and flat trusses from single angles 9 Base plates made of steel with a yield strength of up to 390 N/mm 2, bearing a static load, thickness, mm: a) up to 40 inclusive b) from 40 to 60 inclusive c) from 60 to 80 inclusive | 0,90 0,95 1,05 0,80 0,90 1,10 0,75 1,20 1,15 1,10 |
| Notes: 1. Coefficients g c< 1 при расчете одновременно учитывать не следует. 2. При расчетах на прочность в сечении, ослабленном отверстиями для болтов, коэффициенты gWith pos. 6 and 1, 6 and 2, 6 and 5 should be considered simultaneously. 3. When calculating the base plates, the coefficients given in pos. 9 and 2, 9 and 3, should be considered simultaneously. 4. When calculating connections, coefficients g c for elements given in pos. 1 and 2 should be taken into account together with the coefficient g V. 5. In cases not specified in this table, the calculation formulas should take g with =1 |
When calculating structures operating under repeated loading conditions (for example, when calculating crane beams), a cyclic design load is used to determine the forces, the value of which is determined by the formula.
Strength calculations can be carried out using one of two methods - according to the limit state, or according to permissible stresses. The calculation method for permissible stresses was adopted when calculating the machine building structures, and the basics of its use are given in the Strength of Materials course. When calculating building structures, a limit state calculation method has been adopted, which is more advanced than the calculation method based on permissible stresses.
Ultimate stress state– a state when a tense state arises at a point, leading to the emergence of a new process. For example, to the development plastic deformation, to the formation of cracks, etc. Different PNS occur with different types of loading.
Limit state- a condition in which the structure loses its functionality or its condition becomes undesirable. Efforts that cause a limiting state are called limiting.
It is necessary to distinguish between limit states and limit stress states. These concepts do not always coincide. Examples:
An increase in stress during beam bending to the yield point leads to the achievement of the PNS at points as far as possible from the neutral line. A further increase in the load leads to the stresses reaching the level of the yield strength in the entire section - the limit state in the section; qualitative changes occur in the structure, displacements increase sharply, since a plastic hinge is formed in the most loaded section.
An increase in tensile stresses leads to the sequential appearance of the following limiting stress states: a) the beginning of uniform plastic deformation; b) cervical formation; c) destruction.
Limit state calculation method
In accordance with GOST 27751-88 "Reliability of building structures and foundations. Basic provisions for calculation" limit states are divided into two groups:
the first group includes limit states that lead to complete unsuitability for use of structures, foundations (buildings or structures as a whole) or to a complete (partial) loss of the load-bearing capacity of buildings and structures as a whole;
the second group includes limit states that impede the normal operation of structures (foundations) or reduce the durability of buildings (structures) compared to the intended service life.
Limit states of the first group are characterized by:
failure of any nature (for example, plastic, brittle, fatigue);
loss of shape stability, leading to complete unsuitability for use;
loss of position stability;
transition to a changeable system;
qualitative change in configuration;
other phenomena in which there is a need to stop operation (for example, excessive deformations as a result of creep, plasticity, shear in joints, crack opening, and also the formation of cracks).
Limit states of the second group are characterized by:
achieving maximum deformations of the structure (for example, maximum deflections, rotations) or maximum deformations of the base;
reaching maximum levels of vibrations of structures or foundations;
formation of cracks;
reaching maximum crack openings or lengths;
loss of shape stability, leading to difficulty in normal operation;
other phenomena in which there is a need to temporarily limit the operation of a building or structure due to an unacceptable reduction in its service life (for example, corrosion damage).
The first limit state for tensioned and compressed elements is expressed by the relation:
Where 
– design resistance according to the yield strength;
– yield strength;
– reliability coefficient for the material (γ C >1);
– design tensile strength;
- tensile strength;
– operating conditions coefficient (γ C<1);
-reliability coefficient for structural elements calculated for strength using calculated resistances R u ;
– cross-sectional area of the stretched (compressed) element.
For bendable elements:
|
|
Formally, we can take the value on the right side of inequalities (2 .0), (2 .0), (2 .0) as the permissible stress; the calculation methods for the limit state and permissible stresses are the same, however, when calculating using limit states, the general and unchanged the safety factor is replaced by several variables. This allows, when calculating based on the limit state, to design operationally equal-strength structures.
When determining the design resistances for welds R W, the following are taken into account: the main material of the welded structure, auxiliary materials used in welding (brands of coated electrodes, electrode wires), the presence or absence of physical methods for controlling the weld.
20.12.2018
The calculation of structures based on limit states is based on clearly established two groups of limit states of structures, which must be prevented using a system of design coefficients; their introduction guarantees that limit states will not occur under unfavorable combinations of loads and at the lowest values of the strength characteristics of materials. When limiting states occur, structures no longer meet operational requirements - they collapse or lose stability under the influence of external loads and impacts, or unacceptable movements or cracks develop in them. For the purpose of more adequate and economical calculation, limit states are divided into two fundamentally different groups - the more responsible first (structures are destroyed when the conditions of this group occur) and the less responsible second (structures no longer meet the requirements of normal operation, but are not destroyed, they can be repaired). This approach made it possible to differentiate loads and strength characteristics of materials: in order to protect against the onset of limit states, in calculations for the first group, loads are assumed to be somewhat overestimated, and the strength characteristics of materials are assumed to be underestimated compared to calculations for the second group. This allows us to avoid the occurrence of limit states of group I.
The more important first group includes limit states in terms of load-bearing capacity, the second - in terms of suitability for normal operation. The limiting states of the first group include brittle, ductile or other types of failure; loss of stability of the structure’s shape or position; fatigue failure; destruction from the combined influence of force factors and unfavorable influences of the external environment (aggressiveness of the environment, alternating freezing and thawing, etc.). Perform strength calculations, taking into account, if necessary, the deflection of the structure before failure; calculations for overturning and sliding of retaining walls and eccentrically loaded high foundations; calculation for the ascent of buried or underground tanks; endurance calculations for structures subject to repeated moving or pulsating loads; stability calculations for thin-walled structures, etc. Recently, to the calculations for the first group, a new calculation was added for the progressive collapse of tall buildings under impacts not provided for by normal operating conditions.
The limiting states of the second group include unacceptable width and prolonged opening of cracks (if they are acceptable under operating conditions), unacceptable movements of structures (deflections, rotation angles, skew angles and vibration amplitudes). Calculations for limit states of structures and their elements are performed for the stages of manufacturing, transportation, installation and operation. Thus, for an ordinary bending element, the limit states of group I will be the exhaustion of strength (fracture) along the normal and inclined sections; limit states of group II - formation and opening of cracks, deflection (Fig. 3.12). In this case, the permissible crack opening width under long-term load is 0.3 mm, since at this width the cracks self-heal by the growing crystalline intergrowth in the cement stone. Since every tenth of a millimeter of permissible crack opening significantly affects the consumption of reinforcement in structures with conventional reinforcement, an increase in the permissible crack opening width by even 0.1 mm plays a very important role in saving reinforcement.
Factors included in the calculation of limit states (design factors) are loads on structures, their dimensions, and mechanical characteristics concrete and reinforcement. They are not constant and are characterized by scattering of values (statistical variability). The calculations take into account the variability of loads and mechanical characteristics of materials, as well as factors of a non-statistical nature, and various operating conditions of concrete and reinforcement, manufacturing and operation of elements of buildings and structures. All calculated factors and calculated coefficients are normalized in the relevant SP.
Limit states require further in-depth study: thus, in the calculations, normal and inclined sections in one element are separated (a unified approach is desirable), an unrealistic mechanism of destruction in an inclined section is considered, secondary effects in an inclined crack are not taken into account (dowel effect of working reinforcement and interlocking forces in an inclined crack (see Fig. 3.12, etc.)).
The first design factor is the loads, which are divided into standard and design, and according to the duration of action - into permanent and temporary; the latter can be short-term or long-term. More rarely occurring special loads are considered separately. Constant loads include the self-weight of structures, the weight and pressure of the soil, and prestressing forces of reinforcement. Long-term loads are the weight of stationary equipment on floors, the pressure of gases, liquids, bulk solids in containers, the weight of contents in warehouses, libraries, etc.; the part of the temporary load established by the standards in residential buildings, office and domestic premises; long-term temperature technological effects from equipment; snow loads for III...VI climatic regions with coefficients 0.3...0.6. These load values are part of their full value; they are entered into the calculation taking into account the influence of the duration of the load on displacement, deformation, and crack formation. Short-term loads include part of the load on the floors of residential and public buildings; weight of people, parts, materials in equipment maintenance and repair areas; loads arising during the manufacture, transportation and installation of structural elements; snow and wind loads; temperature climatic influences.
Special loads include seismic and explosive effects; loads caused by equipment malfunction and disruption of the technological process; uneven deformations of the base. Standard loads are established by standards based on a predetermined probability of exceeding average values or based on nominal values. Standard constant loads are taken based on the design values of the geometric and structural parameters of the elements and on the average values of the material density. Standard 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 repetitions. The magnitude of the design loads when calculating structures for group I of limit states is determined by multiplying the standard load by the load reliability factor уf, as a rule, уf > 1 (this is one of the factors preventing the occurrence of a limit state). Coefficient уf = 1.1 for dead weight reinforced concrete structures; уf = 1.2 for the dead weight of structures made of concrete with light aggregates; уf = 1.3 for various temporary loads; but уf = 0.9 for the weight of structures in cases where a decrease in mass worsens the operating conditions of the structure - in calculating stability against floating, overturning and sliding. When calculating according to the less dangerous group II of limit states, уf = 1.
Since the simultaneous action of all loads with maximum values is almost impossible, for greater reliability and efficiency, structures rely on different combinations loads: they can be basic (including constant, long-term and short-term loads), and special (including constant, long-term, possible short-term and one of the special loads). In the main combinations, when taking into account at least two temporary loads, their calculated values (or the corresponding efforts) are multiplied by the combination coefficients: for long-term loads w1 = 0.95; for short-term w2 = 0.9; with one temporary load w1 = w2 = 1. For three or more short-term loads, their calculated values are multiplied by combination coefficients: w2 = 1 for the first short-term load in terms of importance; w2 = 0.8 for the second; w2 = 0.6 for the third and all others. In special combinations of loads, w2 = 0.95 for long-term loads, w2 = 0.8 for short-term loads, except in cases of designing structures in seismic areas. For the purpose of economical design, taking into account the degree of probability of simultaneous loads, when calculating columns, walls, foundations multi-storey buildings temporary loads on floors can be reduced by multiplying by coefficients: for residential buildings, hostels, office premises and so on. with cargo area A > 9 m2
For reading rooms, meetings, shopping and other equipment maintenance and repair areas in production premises with cargo area A > 36 m2
where n - total number floors, temporary loads from which are taken into account when calculating the section in question.
The calculations take into account the degree of responsibility of buildings and structures; it depends on the degree of material and social damage when structures reach limit states. Therefore, when designing, the reliability coefficient for the intended purpose уn is taken into account, which depends on the responsibility class of buildings or structures. The maximum values of load-bearing capacity, calculated values of resistance, maximum values of deformations, crack opening are divided by the reliability coefficient for the intended purpose, and the calculated values of loads, forces and other influences are multiplied by it. Based on the degree of responsibility, buildings and structures are divided into three classes: Class I. уn = 1 - buildings and structures with high national economic or social significance; main buildings of thermal power plants, nuclear power plants; TV towers; covered athletic facilities with stands; buildings of theaters, cinemas, etc.; Class II yn = 0.95 - less significant buildings and structures not included in classes I and III; III class yn = 0.9 - warehouses, one-story residential buildings, temporary buildings and structures.
For a more economical and reasonable design of reinforced concrete structures, three categories of requirements for crack resistance have been established (resistance to crack formation in stage I or resistance to crack opening in stage II of the stress-strain state). The requirements for the formation and opening of cracks normal and inclined to the longitudinal axis of the element depend on the type of reinforcement used and operating conditions. In the first category, the formation of cracks is not allowed; in the second category, short-term crack openings limited in width are allowed, subject to their subsequent reliable closure; in the third category, short-term and long-term crack openings limited in width are allowed. Short-term opening refers to the opening of cracks under the action of constant, long-term and short-term loads; to long-term - crack opening under the action of only constant and long-term loads.
The maximum crack opening width аcrc, which ensures normal operation of buildings, corrosion resistance of reinforcement and durability of the structure, depending on the category of crack resistance requirements, should not exceed 0.1...0.4 mm (see Table 3.1).
Prestressed elements under liquid or gas pressure (tanks, pressure pipes, etc.) with a fully stretched section with rod or wire reinforcement, as well as with 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 operating conditions of the structure and the type of reinforcement, must meet the requirements of the second or third category. Structures without prestressing with rod reinforcement of class A400, A500 must meet the requirements of the third category (see Table 3.1).
The procedure for taking into account loads when calculating structures for crack resistance depends on the category of requirements (Table 3.2). In order to prevent prestressing reinforcement from being pulled out of concrete under load and sudden destruction of structures, the formation of cracks at the ends of elements within the length of the zone of stress transfer from reinforcement to concrete is not allowed under the combined action of all loads (except special ones) introduced into the calculation with the coefficient уf = 1 Cracks that arise during manufacturing, transportation and installation in a zone that will subsequently be compressed under load lead to a decrease in the forces of crack formation in the zone stretched during operation, an increase in the opening width and an increase in deflections. The influence of these cracks is taken into account in the calculations. The most important strength calculations for a structure or building are based on the III stage of the stress-strain state.
Structures have the required strength if the forces from the design loads (bending moment, longitudinal or transverse force, etc.) do not exceed the forces perceived by the section at the design resistance of the materials, taking into account the operating conditions coefficients. The magnitude of forces from design loads is influenced by standard loads, safety factors, design schemes, etc. The magnitude of the force perceived by the section of the calculated element depends on its shape, section dimensions, strength of concrete Rbn, reinforcement Rsn, reliability coefficients for materials ys and уb and coefficients of operating conditions of concrete and reinforcement уbi and уsi. Strength conditions are always expressed by inequalities, and the left side ( external influence) cannot significantly exceed right side(internal efforts); It is recommended to allow an excess of no more than 5%, otherwise the project will become uneconomical.
Limit states of the second group. Calculation of the formation of cracks, normal and inclined to the longitudinal axis of the element, is performed to check the crack resistance of elements that are subject to the requirements of the first category (if the formation of cracks is unacceptable). This calculation is also carried out for elements whose crack resistance is subject to requirements of the second and third categories, in order to establish whether cracks appear, and if they appear, proceed to the calculation of their opening.
Cracks normal to the longitudinal axis do not appear if the bending moment from external loads does not exceed the moment of internal forces
Cracks inclined to the longitudinal axis of the element (in the support zone) do not appear if the main tensile stresses in the concrete do not exceed the calculated values. When calculating the opening of cracks, normal and inclined to the longitudinal axis, determine the opening width of the cracks at the level of tensile reinforcement so that it is no more than the maximum opening width established by the standards
When calculating displacements (deflections), the deflection of elements due to loads is determined, taking into account the duration of their action fссs, so that it does not exceed the permissible deflection fcrc,ult. Maximum deflections are limited by aesthetic and psychological requirements (so that it is not visually noticeable), technological requirements (to ensure normal operation different technological installations, and etc.), design requirements(taking into account the influence of neighboring elements that limit deformations), physiological requirements, etc. (Table 3.3). It is advisable to increase the maximum deflections of prestressed elements, established by aesthetic and psychological requirements, by the height of the deflection due to prestressing (construction elevation), if this is not limited by technological or design requirements. When calculating deflections, if they are limited by technological or design requirements, the calculation is carried out under the action of constant, long-term and short-term loads; when limited by aesthetic requirements, structures are designed to withstand constant and long-term loads. The maximum deflections of the consoles, related to the console overhang, are increased by 2 times. Standards establish maximum deflections according to physiological requirements. A fragility calculation must also be performed for flights of stairs, platforms, etc., so that the additional deflection from a short-term concentrated load of 1000 N under the most unfavorable scheme of its application does not exceed 0.7 mm.
In the III stage of the stress-strain state, in sections normal to the longitudinal axis of the elements being bent and eccentrically compressed with relatively large eccentricities, with a two-digit stress diagram, the same bending stress-strain state is observed (Fig. 3.13). The forces perceived by the section normal to the longitudinal axis of the element are determined from the calculated resistances of the materials, taking into account the operating conditions coefficients. In this case, it is assumed that the concrete of the stretched zone does not work (obt = O); the stresses in the concrete of the compressed zone are equal to Rb with a rectangular stress diagram; stresses in longitudinal tensile reinforcement are equal to Rs; The longitudinal reinforcement in the compressed zone of the section experiences stress Rsc.
In terms of strength moment external forces There should not be more than a moment perceived by internal forces in compressed concrete and in tensile reinforcement. Strength condition relative to the axis passing through the center of gravity of the tensile reinforcement
where M is the moment of external forces from design loads (in eccentrically compressed elements - the moment of external longitudinal force relative to the same axis), M = Ne (e is the distance from the force N to the center of gravity of the section of tensile reinforcement); Sb is the static moment of the cross-sectional area of concrete in the compressed zone relative to the same axis; zs is the distance between the centers of gravity of tensile and compressed reinforcement.
The stress in prestressed reinforcement located in a zone compressed by loads, osc, is determined by work. In elements without prestress osc = Rsc. The height of the compressed zone x for sections operating in case 1, when the ultimate resistance is reached in tensile reinforcement and compressed concrete, is determined from the equilibrium equation of ultimate forces
where Ab is the cross-sectional area of concrete in the compressed zone; for N they take a minus sign for eccentric compression, a + sign for tension, N = 0 for bending.
The height of the compressed zone x for sections operating in case 2, when fracture occurs brittlely in compressed concrete, and stresses in tensile reinforcement do not reach the limit value, are also determined from equation (3.12). Ho in this case, the calculated resistance Rs is replaced by voltage os< Rs. Опытами установлено, что напряжение os зависит от относительной высоты сжатой зоны e = x/ho. Его можно определить по эмпирической формуле
where co = xo/ho is the relative height of the compressed zone under stress in the reinforcement os = osp (os = O in elements without prestress).
When os = osp (or when os = 0), the actual relative height of the compressed zone is e = 1, and co can be considered as the coefficient of completeness of the actual stress diagram in concrete when replacing it with a conventional rectangular diagram; in this case, the concrete force of the compressed zone is Nb = w*ho*Rb (see Fig. 3.13). The value of co is called a characteristic of the deformative properties of concrete in the compressed zone. The limiting relative height of the compressed zone plays a large role in strength calculations, since it limits the optimal case of failure when the tensile and compressed zones simultaneously exhaust their strength. The limiting relative height of the compressed zone eR = xR/h0, at which tensile stresses in the reinforcement begin to reach the limiting values Rs, is found from the dependence eR = 0.8/(1 + Rs/700), or from Table. 3.2. In the general case, the strength of a section normal to the longitudinal axis is calculated depending on the value of the relative height of the compressed zone. If e< eR, высоту сжатой зоны определяют из уравнения (3.12), если же e >eR, strength is calculated. The stresses of high-strength reinforcement os in the limit state can exceed the nominal yield strength. According to experimental data, this can happen if e< eR. Превышение оказывается тем большим, чем less value e, The experimental dependence has the form
When calculating the strength of sections, the design resistance of the reinforcement Rs is multiplied by the coefficient of operating conditions of the reinforcement
where n is the coefficient taken equal to: for fittings of classes A600 - 1.2; A800, Vr1200, Vr1500, K1400, K1500 - 1.15; A1000 - 1.1. 4 is determined at ys6 = 1.
The standards establish the maximum percentage of reinforcement: the cross-sectional area of longitudinal tensile reinforcement, as well as compressed reinforcement, if required by calculation, as a percentage of the cross-sectional area of concrete, us = As/bh0 is taken to be no less than: 0.1% - for bending, eccentrically tensile elements and eccentrically compressed elements with flexibility l0/i< 17 (для прямоугольных сечений l0/h < 5); 0,25 % - для внецентренно сжатых элементов при гибкости l0/i >87 (for rectangular sections l0/h > 25); for intermediate values of element flexibility, the value us is determined by interpolation. The maximum percentage of reinforcement for bending elements with single reinforcement (in the tensile zone) is determined from the equilibrium equation of the ultimate forces at a height of the compressed zone equal to the boundary one. For rectangular section
Limit percentage of reinforcement taking into account eR value, for prestressed elements
For elements without prestress
The maximum percentage of reinforcement decreases with increasing reinforcement class. Sections of bending elements are considered over-reinforced if their percentage of reinforcement is higher than the limit. A minimum percentage of reinforcement is necessary to absorb shrinkage, temperature and other forces not taken into account by the calculation. Typically umin = 0.05% for longitudinal tensile reinforcement of bending elements of rectangular cross-section. Stone and reinforced masonry structures are calculated similarly to reinforced concrete structures according to two groups of limit states. Calculation according to group I should prevent the structure from destruction (calculation based on load-bearing capacity), from loss of stability of shape or position, fatigue failure, destruction due to the combined action of force factors and the influence of the external environment (freezing, aggression, etc.). Calculation according to group II is aimed at preventing the structure from unacceptable deformations, excessive opening of cracks, and peeling of the masonry lining. This calculation is performed when cracks are not allowed in structures or their opening is limited (tank linings, eccentrically compressed walls and pillars at large eccentricities, etc.), or the development of deformation from conditions is limited collaboration(wall filling, frame, etc.).
Building structures must, first of all, have sufficient reliability - that is, the ability to perform certain functions in appropriate conditions for a certain period of time. The termination of a building structure’s performance of at least one of its intended functions is called failure.
Thus, by refusal we understand the possibility of such an occurrence random event, the result of which is social or economic losses. It is believed that the structure at the moment preceding failure goes into a limiting state.
Limit states are those states, upon the occurrence of which the structure ceases to satisfy the requirements imposed on it, that is, it loses the ability to resist external loads or receives unacceptable movements or local damage.
The reasons for the occurrence of limit states in building structures can be overloads, low quality of the materials from which they are made, and others.
The main difference between the method under consideration and previous methods of calculation (calculation based on permissible stresses) is that here the limit 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 the design with a certain security against the onset of these states under the most unfavorable (but realistically possible) conditions. Currently, this calculation method is accepted as the main official one.
Reinforced concrete structures may lose the necessary performance qualities for one of two reasons:
1. As a result of exhaustion of bearing capacity (material failure in the most loaded sections, loss of stability individual elements or the entire structure as a whole);
2. Due to excessive deformations (deflections, vibrations, settlement), as well as due to the formation of cracks or their excessive opening.
According to the above two reasons that can cause loss performance qualities structures, the standards establish two groups of their limit states:
By bearing capacity (first group);
According to suitability for normal use (second group).
The purpose of the calculation is to prevent the occurrence of any limit state in the structure under consideration during the period of manufacture, transportation, installation and operation.
Calculations for limit states of the first group must ensure during the operation of the structure and for other stages of operation its strength, stability of shape, stability of position, endurance, etc.
Calculations for limit states of the second group are performed to prevent excessive crack opening during the operation of the structure and at other stages of its operation, leading to premature corrosion of the reinforcement, or their formation, as well as excessive movements.
Calculation factors
These are loads and mechanical characteristics of materials (concrete and reinforcement). They have statistical variability or spread of values. In calculations for limit states, they take into account (implicitly) 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 established according to chapters of SNiP 2.01.07-85* and SP 52-101-2003.
Classification of loads. Standard and design 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 load-bearing and enclosing structures of buildings and structures, the weight and pressure of soils, the effect of prestressing of reinforced concrete structures. include: the weight of stationary equipment on floors - machines, apparatus, engines, containers, etc.; pressure of gases, liquids, granular bodies in containers; loads on floors from stored materials and racking equipment in warehouses, refrigerators, granaries, book depositories, archives and similar premises; temperature technological influences from stationary equipment; weight of the water layer on water-filled flat surfaces, etc.These include: the weight of people, repair materials in equipment maintenance and repair areas, snow loads with the 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 sudden disruptions in the technological process, temporary malfunction or breakdown of equipment, etc.Loads in accordance with SNiP 2.01.07-85* are also divided into standard and design.
Standard loads or impacts are called those that are close in magnitude to the greatest possible during normal operation of buildings and structures. Their values are given in the standards.
The variability of loads in an unfavorable direction is assessed by the load reliability coefficient γf.
The calculated 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 dead weight of concrete and reinforced concrete structures = 1.1; when taking into account your own weight various ties, backfills, insulation, performed in the factory = 1.2, and at the construction site = 1.3. Load reliability factors for uniformly distributed loads should be taken:
1.3 - with a full standard value of less than 2 kPa (2 kN/m2);
1.2 - with a full standard value of 2 kPa (2 kN/m2) or more. The load reliability coefficient for its own weight when calculating the structure for position stability against floating, capsizing and sliding, as well as in other cases when a decrease in mass worsens the operating conditions of the structure, is taken equal to 0.9.
Calculations for limit states of the second group are carried out using standard loads or calculated ones taken with γ f = 1.
Buildings and structures are subject to the simultaneous action of various loads. Therefore, the calculation of a building or structure as a whole, or its individual elements, must be carried out taking into account the most unfavorable combinations of these loads or forces caused by them. Unfavorable, but real possible combinations loads during design are selected in accordance with the recommendations of SNiP 2.01.07-85*.
Depending on the composition of the loads taken into account, combinations are distinguished:
- basic, including permanent, long-term and short-term loads
Т = ΣТ post + ψ 1 ΣТ duration + ψ 2 ΣТ multiple,
where T = M, T, Q;
ψ - combination coefficient (if 1 short-term load is taken into account, then ψ 1 = ψ 2 = 1.0, if the combination includes 2 or more short-term loads, then ψ 1 = 0.95, ψ 2 = 0.9);
- special, including, in addition to constant, long-term and short-term loads, a special load (ψ 1 = 0.95, ψ 2 = 0.80).
Limit states- these 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 arise - the first and second.
Calculation of limit states of structures as a whole and its elements must be carried out for all stages: transportation, installation and operation - and must take into account all possible combinations of loads. The purpose of the calculation is to prevent either the first or the second limit states during the processes of transportation, assembly and operation of the structure. This is done based on taking into account the standard and design loads and resistances of materials.
The limit state method is the first step in ensuring the reliability of building structures. Reliability is 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 loads on systems with random strength indicators. Characteristic feature The method of limit states is that all initial values operated in the calculation, random in nature, are represented in the standards by deterministic, scientifically based, normative values, and the influence of their variability on the reliability of structures is taken into account by the corresponding coefficients. Each of the reliability coefficients takes into account the variability of only one initial value, i.e. is of a private nature. Therefore, the limit state method is sometimes called the partial coefficient method. Factors whose variability affects the level of reliability of structures can be classified into five main categories: loads and impacts; geometric dimensions of structural elements; degree of responsibility of structures; mechanical properties of materials; operating conditions of the structure. Let's consider the listed factors. A 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 standard values, are presented in chapter SNiP 2.01.07-85 Design Standards. "Loads and impacts". The probability of the combined 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 coefficient. However, this coefficient is pure form not acceptable. This factor is used when calculating geometric characteristics, taking design parameters sections with minus tolerance. In order to reasonably balance the costs of buildings and structures for various purposes a reliability coefficient is introduced for the intended purpose< 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 a material’s resistance to force influences is the standard resistance set regulatory documents according to the results statistical research variability mechanical properties materials by testing material samples using standard methods. A possible deviation from standard values is taken into account by the reliability coefficient for the material ym > 1. It reflects the statistical variability of the properties of materials and their difference from the properties of tested standard samples. The characteristic obtained by dividing the standard resistance by the coefficient m is called the design resistance R. This main characteristic of wood strength is standardized by SNiP P-25-80 “Design Standards. Wooden Structures”.
The unfavorable influence of the environmental and operating environment, such as: wind and installation loads, section height, temperature and humidity conditions, are taken into account by introducing operating conditions coefficients t. Coefficient t may be less than one if this factor or a combination of factors reduces bearing capacity structures, and more than one in the opposite case. For wood, these coefficients are presented in SNiP 11-25-80 “Design standards.
Standard limit values of deflections meet the following requirements: a) technological (ensuring conditions for normal operation of machinery and handling equipment, instrumentation, etc.); b) structural (ensuring the integrity of adjacent structural elements, their joints, the presence of a gap between load-bearing structures and partition structures, half-timbering, etc., ensuring specified slopes); c) aesthetic and psychological (providing favorable impressions from appearance structures, preventing the feeling of danger).
The magnitude of the maximum deflections depends on the span and the type of applied loads. For wooden structures covering buildings against permanent and temporary long-term loads maximum deflection ranges from (1/150)-i to (1/300) (2). The strength of wood is also reduced under the influence of certain chemicals from biological damage, embedded 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 design sections of tensile elements weakened by holes, as well as in bending elements made of round timber with trimming in the design section, is reflected by the operating condition coefficient t0 = 0.8. The deformability of wood when calculating 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 400 MPa across the fibers. When calculating stability, the elastic modulus was 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 assumed to be equal to pdo o = 0.5, and along the fibers with stresses directed across the fibers, n900 = 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 modulus of elasticity and shear modulus is multiplied by the coefficient mt = 0.8 when calculating structures in which stresses in elements arising from permanent and temporary long-term loads exceed 80% of the total voltage from all loads. When calculating metal-wood structures, elastic characteristics and calculated resistances steel and connections of steel elements, as well as reinforcement, are accepted 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 for use as elements load-bearing structures, the basic design resistances of which are given in Table 10 of SNiP P-25-80. Under appropriate operating conditions for glue-plywood structures, calculations based on the first group of limit states provide for multiplying the basic design resistances of plywood by the operating conditions coefficients TV, TY, TN and TL. When calculating according to 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. Modulus of elasticity and shear modulus for structures located in different conditions operation, as well as those exposed to the combined influence of constant and temporary long-term loads, should be multiplied by the corresponding coefficients of operating conditions adopted for wood
First group most dangerous. It is determined by unsuitability for use when a structure loses its load-bearing capacity as a result of destruction or loss of stability. This does not happen while the maximum normal O or shearing stresses in its elements do not exceed the calculated (minimum) resistance of the materials from which they are made. This condition is written by the formula
a,t
The limiting states of the first group include: destruction of any kind, general loss of stability of a structure or local loss of stability of a structural element, violation of joints that turn the structure into a variable system, development of residual deformations of unacceptable magnitude. The load-bearing capacity calculation is carried out based on the probable worst case, namely: 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 norms.
Second group less dangerous. It is determined by the unsuitability of the structure for normal operation when it bends to an unacceptable amount. This does not happen until the maximum relative deflection of its /// does not exceed the maximum permissible values. This condition is written by the formula
G/1<. (2.2)
The calculation of wooden structures according to the second limit state for deformations applies mainly to bendable structures and is aimed at limiting the magnitude of deformations. The calculations are based on standard loads without multiplying them by safety factors, assuming elastic operation of the wood. The calculation for deformations is carried out based on the average characteristics of the wood, and not on the reduced ones, as when checking the load-bearing capacity. This is explained by the fact that an increase in deflection in some cases, when low-quality wood is used, does not pose a danger to the integrity of the structures. This also explains the fact that deformation calculations are carried out for standard, and not for design, loads. To illustrate the limiting state of the second group, we can give an example when, as a result of unacceptable deflection of the rafters, cracks appear in the roofing. The leakage of moisture in this case disrupts the normal operation of the building, leading to a decrease in the durability of the wood due to its moisture, but at the same time the building continues to be used. Calculation based on the second limit state, as a rule, has a subordinate meaning, because the main thing is to ensure load-bearing capacity. However, limitations on deflections are especially important for structures with ductile connections. Therefore, deformations of wooden structures (composite posts, composite beams, board and nail structures) must be determined taking into account the influence of the compliance of the connections (SNiP P-25-80. Table 13).
Loads, acting on structures are determined by Building Codes and Regulations - SNiP 2.01.07-85 “Loads and Impacts”. When calculating structures made of wood and plastics, mainly the constant load from the dead 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. It is convenient to denote the standard value with the index n.
Standard loads are the initial values of the loads: Temporary loads are determined as a result of processing data from long-term observations and measurements. Constant loads are calculated based on the dead weight and volume of structures, other building elements and equipment. Standard 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 upward. To do this, the values of standard loads are multiplied by the load safety factor y, the values of which are different for different loads, but all of them are greater than unity. Distributed load values are given in kilopascals (kPa), which corresponds to kilonewtons per square meter (kN/m). Most calculations use linear load values (kN/m). Design loads are used when calculating 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 the structures; the second part is the load from the own weight of the main supporting structure. In a preliminary calculation, the load from the dead weight of the main supporting structure can be determined approximately, given the actual dimensions of the sections and the volumes of the structural elements. equal to the product of the standard multiplied by the load reliability factor u. For loading from the dead weight of structures y= 1.1, and for loads from insulation, roofing, vapor barrier and others y = 1.3. Constant load from conventional pitched surfaces with an angle of inclination A it is convenient to refer to their horizontal projection by dividing it by cos A.The standard snow load s H is determined based on the standard weight of the snow cover so, which is given in load standards (kN/m 2) of the horizontal projection of the cover 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- For gable roofs with a ^ 25°, p = 1, for a > 60° p = 0, and for intermediate slope angles of 60° >*<х > 25° p == (60° - a°)/35°. This. the load is uniform and can be two- or one-sided.
With vaulted coverings along segmental trusses or arches, the 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 arch /: p = //(8/).
When the ratio of the height of the arch to the span f/l= A 1/8 snow load can be triangular with a maximum value at one support s" and 0.5 s" at the other and zero value at the ridge. Coefficients p that determine 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 lancet-shaped coverings can be determined as on gable roofs, considering the roof to be conditionally gable along planes passing through the chords of the axes of the floor at the arches. The design snow load is equal to the product of the standard load and the load safety factor 7- For most light wooden and plastic structures with the 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 assumed to be equal - standard R"= 0.1 kN and design R= p and y = 0.1 1.2 = 1.2 kN. Wind load. Standard wind load w consists of pressure w"+ and suction w n - wind. The initial data when determining the wind load are the values of wind pressure directed perpendicular to the surfaces of the roofing and walls of buildings Wi(MPa), depending on the wind region of the country and accepted according to the norms of loads and impacts. Standard 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, taking into account its shape. For most wood and plastic buildings whose height 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 coverings 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 generally does not increase, but rather reduces the forces in structures from constant and snow loads and may not be taken into account in the safety factor when calculating. Wind load must be taken into account when calculating the pillars and walls of buildings, as well as when calculating triangular and lancet-shaped structures.
The calculated wind load is equal to the standard load multiplied by the safety factor y= 1.4. Thus, w = = w"y.
Regulatory resistance wood R H(MPa) are the main characteristics of the strength of wood in areas free 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 the appendix. 5 are practically used in laboratory testing of wood strength during the manufacturing of wooden structures and in determining the load-bearing capacity of operating load-bearing structures during their inspections.
Calculated resistances wood R(MPa) are the main characteristics of the strength of real wood elements of real structures. This wood has natural defects and works under loads for many years. Calculated resistances are obtained based on standard resistances taking into account the reliability coefficient for the material at and loading duration coefficient t al according to the formula
R= R H m a Jy.
Coefficient at significantly more than one. It takes into account the decrease in the strength of real wood as a result of heterogeneity of structure and the presence of various defects that do not occur in laboratory samples. Basically, the strength of wood is reduced by knots. They reduce the working cross-sectional area by cutting and spreading its longitudinal fibers, creating eccentricity of longitudinal forces and inclination of the fibers around the knot. The inclination of the fibers causes 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 strength of wood in tension by almost half and by about one and a half times in compression. Cracks are most dangerous in areas where wood is being chipped. As the cross-sectional sizes of the elements increase, the stresses upon their destruction decrease due to the greater heterogeneity of the stress distribution across the sections, which is also taken into account when determining the design resistances.
Load duration coefficient 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 am almost ^^ half the short-term /tg.
The quality of wood naturally affects the values of its calculated resistances. 1st grade wood - with the least defects, has the highest calculated resistance. The calculated resistances of wood of the 2nd and 3rd grades are respectively lower. For example, the calculated compression resistance of pine and spruce wood of the 2nd grade is obtained from the expression
%. = # s n t dl /y = 25-0.66/1.25 = 13 MPa.
The calculated resistances of pine and spruce wood to compression, tension, bending, chipping and crushing are given in the appendix. 6.
Working conditions coefficients T The design resistance of wood takes into account the conditions in which wooden structures are manufactured and operated. Breed coefficient T" takes into account the different strength of wood of different species, different from the strength of pine and spruce wood. The load factor t„ takes into account the short duration of wind and installation loads. When crushed tn= 1.4, for other types of voltages t n = 1.2. The section height coefficient when bending wood of glued-wood beams with a section height of more than 50 cm /72b decreases from 1 to 0.8, and even more with a section height of 120 cm. The thickness coefficient of layers of glued wood elements takes into account the increase in their strength in compression and bending as the thickness of the boards being glued decreases, as a result of which the homogeneity of the structure of glued wood increases. Its values are in the range of 0.95...1.1. The bending coefficient m rH takes into account the additional bending stresses that arise when the boards bend during the production of bent glued-wood elements. It depends on the ratio of the bending radius to the thickness of the r/b boards and has values of 1.0...0.8 as this ratio increases from 150 to 250. Temperature coefficient m t takes into account the reduction in the strength of wood in structures operating at temperatures from +35 to +50 °C. It decreases from 1.0 to 0.8. Humidity coefficient t ow takes into account the decrease in the strength of wood in structures operating in a humid environment. When indoor air humidity is from 75 to 95%, tvl = 0.9. Outdoors in dry and normal areas t ow = 0.85. With constant hydration and in water t ow = 0.75. Stress concentration factor t k = 0.8 takes into account the local reduction in wood strength in areas with cut-ins and holes during tension. The load duration coefficient t dl = 0.8 takes into account the decrease in wood strength as a result of the fact that long-term loads sometimes account for more than 80% of the total 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 standard and calculated resistances of building plywood were obtained using the same methods as for wood. In this case, its sheet shape and an odd number of layers with mutually perpendicular fiber directions were taken into account. Therefore, the strength of plywood in these two directions is different and along the outer fibers it is slightly higher.
The most widely used in structures is seven-layer plywood of the FSF brand. Its calculated resistances along the fibers of the outer veneers are equal to: tensile # f. p = 14 MPa, compression #f. c = 12 MPa, bending out of plane /? f.„ = 16 MPa, shearing in plane # f. sk = 0.8 MPa and shear /? f. avg - 6 MPa. Across the fibers of the outer veneers, these values are respectively equal to: tensile I f_r= 9 MPa, compression # f. s = 8.5 MPa, bending # F.i = 6.5 MPa, shearing R$. CK= 0.8 MPa, cut # f. av = = 6 MPa. The moduli of elasticity and shear along the outer fibers are equal, respectively, Ё f = 9-10 3 MPa and b f = 750 MPa and across the outer fibers £ f = 6-10 3 MPa and G$ = 750 MPa.