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 calculation methods (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 the load-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. Due to excessive deformations (deflections, vibrations, settlement), as well as due to the formation of cracks or their excessive opening.
In accordance with these two reasons, which can cause a loss of performance qualities of structures, the standards establish two groups of them: 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 the 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 prestress 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 the own weight of various screeds, backfills, insulation, carried out 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 realistically 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 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).
What are limit states and how to deal with them in relation to structural calculations? Everyone knows that there are two groups of limit states: the first and the second. What does this division mean?
The name itself " limit state" means that for any structure, under certain conditions, a state occurs in which a certain limit is exhausted. Conventionally, for the convenience of calculations, two such limits were derived: the first limit state is when the limit of strength, stability and endurance of the structure is exhausted; the second limit state is when the deformations of the structure exceed the maximum permissible (the second limit state for reinforced concrete also includes restrictions on the occurrence and opening of cracks).
Before proceeding to the analysis of calculations for the first and second limit states, it is necessary to understand which part of the structural calculation is generally divided into these two parts. Any calculation begins with collecting the load. Then follows the choice of a design scheme and the calculation itself, as a result of which we determine the forces in the structure: moments, longitudinal and transverse forces. And only after the efforts have been determined, we proceed to calculations for the first and second limit states. Usually they are performed in exactly this sequence: first the first, then the second. Although there are exceptions, more about them below.
It cannot be said what is more important for any structure: strength or deformability, stability or crack resistance. It is necessary to carry out calculations based on two limit states and find out which of the limitations is the most unfavorable. But each type of structure has its own special points that are useful to know to make it easier to navigate the environment of limit states. In this article we will use examples to analyze limit states for various types reinforced concrete structures.
Calculation of beams, slabs and other bending elements based on the first and second limit states
So, you need to calculate a bending element, and you are wondering where to start the calculation, and how to understand if everything has been calculated? Everyone recommends making calculations not only for the first, but also for the second limit state. But what is it? Where are the specifics?
To calculate bending elements, you will need the “Manual for the design of concrete and reinforced concrete structures made of heavy concrete without prestressing reinforcement (to SNiP 2.03.01-84)” and SNiP 2.03.01-84 itself “Concrete and reinforced concrete structures”, necessarily with modification 1 (very important for calculations for the second group of limit states).
Open section 3 of the manual “Calculation of reinforced concrete elements according to limit states of the first group”, namely “Calculation of reinforced concrete elements according to strength” (starting from paragraph 3.10). Now you need to find out what stages it consists of:
- this is the part of the calculation in which we check whether our structure will withstand the influence of a bending moment. The combination of two is checked important factors: element section size and area of longitudinal reinforcement. If the check shows that the moment acting on the structure is less than the maximum permissible, then everything is fine and you can move on to the next stage.
2) Calculation of sections inclined to the longitudinal axis of the element- This is a calculation of a structure under the action of lateral force. To check, it is important for us to establish the cross-sectional dimensions of the element and the area of the transverse reinforcement. The same as at the previous stage of calculation, if the current shear force less than the maximum permissible, the strength of the element is considered ensured.
Both stages, along with examples, are discussed in detail in the manual. These two calculations are comprehensive strength calculations for classical flexural elements. If there are any special conditions (repeated loads, dynamics), they must be taken into account when calculating strength and endurance (often taken into account by introducing coefficients).
1) Calculation of reinforced concrete elements for the formation of cracks- this is the very first stage in which we find out whether cracks form in our element when exposed to forces acting on it. Cracks will not form if our maximum torque Mr less torque Mcrc, which causes cracking.
2) Calculation of reinforced concrete elements based on crack opening– this is the next stage at which we check the size of the crack opening in the structure and compare it with the permissible dimensions. Pay attention to paragraph 4.5 of the manual, which stipulates in which cases this calculation does not need to be performed - we don’t need any extra work. If the calculation is necessary, then you need to perform two parts:
a) calculation for the opening of cracks normal to the longitudinal axis of the element– we carry it out according to clauses 4.7-4.9 of the manual ( with mandatory consideration of amendment 1 to SNiP, because the calculation there is already radically different);
b) calculation for the opening of cracks inclined to the longitudinal axis of the element– it must be carried out according to clause 4.11 of the manual, also taking into account change 1.
Naturally, if according to the first stage of the calculation, cracks do not form, then we skip stage 2.
3) Determination of deflection– this is the last stage of calculation for the second limit state for bendable reinforced concrete elements, it is performed in accordance with clauses 4.22-4.24 of the manual. In this calculation, we need to find the deflection of our element and compare it with the deflection normalized by DSTU B.V.1.2-3:2006 “Deflections and displacements”.
If all these parts of the calculations are completed, consider that the calculation of the element for both the first and second limit states is completed. Of course, if there are any design features (trimming on the support, holes, concentrated loads, etc.), then the calculation needs to be supplemented taking into account all these nuances.
Calculation of columns and other centrally and eccentrically compressed elements based on the first and second limit states
The stages of this calculation are not particularly different from the stages of calculation of bending elements, and the literature is the same.
The calculation for the limit state of the first group includes:
1) Calculation of sections normal to the longitudinal axis of the element– this calculation, just like for bending elements, determines the required cross-sectional size of the element and its longitudinal reinforcement. But unlike the calculation of bending elements, where the strength of the section is checked against the action of the bending moment M, in this calculation the maximum vertical force N and the eccentricity of the application of this force “e” are highlighted (when multiplied, however, they still give the same bending moment). The manual describes in detail the calculation methodology for all standard and non-standard sections (starting from paragraph 3.50).
Feature of this calculation is that it is necessary to take into account the influence of element deflection, and the influence of indirect reinforcement is also taken into account. The deflection of the element is determined when calculating according to the second group of limit states, but when calculating according to the first limit state, it is allowed to simplify the calculation by introducing a coefficient in accordance with clause 3.54 of the manual.
2) Calculation of sections inclined to the longitudinal axis of the element– this calculation for the action of lateral force according to clause 3.53 of the manual is similar to the calculation of bending elements. As a result of the calculation, we obtain the area of transverse reinforcement in the structure.
The calculation for the limit state of the second group consists of the following steps:
1) Calculation of reinforced concrete elements for the formation of cracks.
2) Calculation of reinforced concrete elements based on crack opening.
These two stages are absolutely similar to the calculation of bending elements - there are maximum forces, it is necessary to determine whether cracks are formed; and if they are formed, then, if necessary, make a calculation for the opening of cracks, normal and inclined to the longitudinal axis of the element.
3) Determination of deflection. In exactly the same way as for bending elements, it is necessary to determine the deflection for eccentrically compressed elements. Limit deflections, as always, can be found in DSTU B V.1.2-3:2006 “Deflections and displacements”.
Calculation of foundations based on the first and second limit states
The calculation of foundations is fundamentally different from the above calculations. As always, when calculating foundations, it is necessary to start with the collection of loads or with the calculation of the building frame, as a result of which the main loads on the foundation N, M, Q are determined.
After the loads have been collected and the type of foundation has been selected, it is necessary to proceed to the calculation of the soil foundation under the foundation. This calculation, like any other calculations, is divided into calculations for the first and second limit states:
1) ensuring the bearing capacity of the foundation base - the strength and stability of the foundations is checked (first limit state) - calculation example strip foundation ;
2) calculation of the foundation based on deformations - determination of the design resistance of the foundation soil, determination of settlement, determination of foundation roll (second limit state).
The “Manual on the design of foundations of buildings and structures (to SNiP 2.02.01-83)” will help you understand this calculation.
As you already understood from the wording, when determining the size of the base of the foundation (whether it is a strip or a columnar foundation), we first of all perform the calculation of the soil base, and not the foundation. And in this calculation (except for rocky soils), it is much more important to calculate the foundation based on deformations - everything that is listed in paragraph 2 above. Calculation based on the first limit state is often not required at all, because Preventing deformations is much more important; they occur much earlier than the loss of bearing capacity of the soil. In what cases the calculation should be performed using the first group of limit states can be found in paragraph 2.259 of the manual.
Now let's look at the calculation of the base based on deformations. Most often, designers estimate the design resistance of the soil, compare it with the load on the soil from the building, selecting the required area of the foundation, and stop there. This is the wrong approach, because... only part of the work has been completed. The foundation calculation is considered complete when all the steps listed in paragraph 2 have been completed.
Determining the settlement of foundations is very important. This is especially important under different loads or uneven soils, when there is a risk of uneven settlement of the foundations (this is described in detail in this article “What you need to know about strip monolithic foundations”). To be sure of the continued integrity of the building structures, you should always check the difference in foundation settlements according to Table 72 of the manual. If the difference in settlement is higher than the maximum permissible, there is a risk of cracks in structures.
The foundation roll must be determined in the presence of bending moments acting on the foundation. The roll also needs to be checked when there is an uneven load on the ground - it also affects the deformation of the soil base.
But after the foundation has been calculated according to the second and possibly the first limit state and the dimensions of the foundation base have been determined, you need to move on to the next stage: the calculation of the foundation itself.
When calculating the foundation, we determined the pressure under the base of the foundation. This pressure is applied to the sole as a load (directed from bottom to top), and the support is a column or wall resting on the foundation (a kind of upside down). It turns out that in each direction from the support we have a console (usually these consoles are the same), and they need to be calculated taking into account a uniformly distributed load equal to the pressure under the base of the foundation. You can clearly understand the principle of calculation using the example of a columnar foundation using the “Manual for the design of foundations on a natural foundation for columns of buildings and structures (to SNiP 2.03.01-84 and SNiP 2.02.01-83)” - all stages of the calculation are outlined there in examples, both for the first and second limit states. Based on the results of the console calculation, we first determine the height of its section and reinforcement (this is a calculation based on the first limit state), then we check the crack resistance (this is a calculation based on the second limit state).
We need to act in exactly the same way in the case of calculating a strip foundation: having the protrusion of the sole in one direction from the wall and the pressure under this sole, we calculate the cantilever slab (with pinching on the support), the length of the console is equal to the protrusion of the sole, the width is taken for convenience of calculation equal to one meter, the load on the console is equal to the pressure under the base of the foundation. We find the maximum moment and shear force in the console and perform calculations for the first and second limit states - exactly as described in the calculation of bending elements.
Thus, when calculating foundations, we go through two cases of calculation based on the limit states of the first and second groups: first when calculating the foundation, then when calculating the foundation itself.
Conclusions. For any calculation, it is important to follow the sequence:
1) Collection of loads.
2) Selection of design scheme.
3) Determination of forces N, M and Q.
4) Calculation of the element based on the first limit state (strength and stability).
5) Calculation of the element based on the second limit state (deformability and crack resistance).
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Comments
0 #15 Irina 10/17/2018 19:39
Quote:
And you are also wrong.I also know that in the past, proginas were praised according to normative standards
Here is a quote from SNiP 1985:
Quote:
Quote:The calculated value of the load should be determined as the product of its standard value by the load reliability factor SNiP 2.01.07-85* Loads and impacts (with Amendments No. 1, 2), corresponding to the limit state under consideration and accepted: a)* when calculating strength and stability - in accordance with clauses 2.2, 3.4, 3.7, 3.11, 4.8, 6.11, 7.3 and 8.7; b) when calculating endurance - equal to one; c) in calculations for deformations - equal to one, unless other values are established in the design standards for structures and foundations; d) when calculating for other types of limit states - according to the design standards for structures and foundations.
I recommend that you, as well as the Russian-speaking Valery (if you are different Valeries), read the articleFrom me I would like to discuss how it is possible, obviously, until the new standards are updated, to show the standard (characteristic) values of vantagement, but, nevertheless, it is necessary to adjust to the different values, but without coefficients for CC1...CC3. If it’s not so, then it’s still written down.
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 limit states occur, structures no longer meet operational requirements; they collapse or lose stability under the influence of external loads and influences, 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 fracture; 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 necessary cases deflection of the structure before destruction; 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, a new calculation for progressive collapse was added to the calculations for the first group tall buildings under influences 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.
The factors included in the calculation of limit states (design factors) are the loads on structures, their dimensions, and the mechanical characteristics of concrete and reinforcement. They are not constant and are characterized by scattering of values (statistical variability). The calculations take into account load variability and mechanical characteristics materials, as well as non-statistical factors, 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 dead 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 live load established by the standards in residential buildings, in office and domestic premises; long-term temperature technological effects from equipment; snow loads for III...VI climatic regions with coefficients of 0.3...0.6. These load values are part of them full meaning, they are introduced into the calculation taking into account the influence of the duration of loads on displacements, deformations, 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 violation 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 permanent 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 highest values intended for normal use; 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 the dead weight of 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 are designed for different combinations of loads: they can be basic (including constant, long-term and short-term loads), and special (including constant, long-term, possible short-term loads). 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 action of loads, when calculating columns, walls, foundations of multi-story buildings, temporary loads on floors can be reduced by multiplying by coefficients: for residential buildings, dormitories, office premises, etc. 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 openings 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 of high economic or social importance; main buildings of thermal power plants, nuclear power plants; television towers; indoor sports 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. joint action all loads (except for special ones) entered into the calculation with the coefficient уf = 1. Cracks that arise during manufacturing, transportation and installation in an area that will subsequently be compressed under load lead to a decrease in the forces of crack formation in the area stretched during operation and an increase in the opening width and increased 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.
The designs have 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, concrete strength Rbn, reinforcement Rsn, safety factors for materials ys and уb and coefficients of operating conditions for concrete and reinforcement уbi and уsi. Strength conditions are always expressed by inequalities, and the left side (external influence) cannot significantly exceed the right side (internal forces); 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 the normal operation of various technological installations, 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 the 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 the stresses in the 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 general case Calculation of the strength of a section normal to the longitudinal axis is performed 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 under 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 due to joint work conditions is limited (wall filling, frame, etc.) .d.).
When calculating using this method, the structure is considered in its design limit state. The design limit state is taken to be the state of the structure in which it ceases to meet the operational requirements imposed on it, i.e., it 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 load-bearing capacity ( , stability or endurance); all steel structures must satisfy this limit state;
- the second design limit state, determined by the development of excessive deformations (deflections and displacements); This limiting state must be satisfied by structures in which the magnitude of the deformations may limit the possibility of their operation.
The first calculated limit state is expressed by the inequality
where N is the design force in the structure from the sum of the effects of the design loads P in the most unfavorable combination;
Ф is the load-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 operating conditions coefficient m.
The design loads P for which the structure is calculated (based on the limit state) are taken to be slightly higher than the normative ones. The design load is defined as the product of the standard load by the overload factor n (greater than unity), taking into account the danger of exceeding the load compared to its standard value due to possible load variability:
The values of coefficients p are given in the table Standard and design loads, overload factors.
Thus, structures are considered under the influence of design loads rather than operational (standard) loads. From the influence of design loads in the structure, the design forces (axial force N or moment M) are determined, which are found by general rules strength of materials and structural mechanics.
Right side of the main equation (1.I)- load-bearing capacity of the structure F - depends on the maximum resistance of the material to force influences, characterized by the mechanical properties of the material and called the standard resistance R n, as well as on the geometric characteristics of the section (sectional area F, moment of resistance W, etc.).
For building steel, the standard resistance is accepted equal to the limit fluidity,
(for the most common building steel grade St. 3 σ t = 2,400 kg/cm 2).
The design resistance of steel R is taken to be a voltage equal to the standard resistance multiplied by the uniformity coefficient k (less than unity), taking into account the risk of a decrease in the resistance of the material compared to its standard value due to variability in the mechanical properties of the material
For ordinary low-carbon steels k = 0.9, and for high-quality steels (low-alloy) k = 0.85.
Thus, the calculated resistance R- this is a stress equal to the lowest possible value of the yield strength of the material, which is accepted for the structure as the limiting value.
Thus, the main design equation (1.I) will have the following form:
- when testing a structure for strength under the action of axial forces or moments
where N and M are the calculated axial forces or moments from the calculated loads (taking into account load factors); F nt - net cross-sectional area (excluding holes); W nt is the moment of resistance of the net section (minus the holes);
- when checking the structure for stability
where F br and W br are the area and moment of resistance of the gross section (without deducting holes); φ and φ b are coefficients that reduce the design resistance to values that ensure stable equilibrium.
Usually, when calculating the intended structure, the cross-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 in terms of calculated stresses, for example:
- when testing for strength
- when checking for stability
where σ is the design stress in the structure (based on the design loads).
It is more correct to write the coefficients φ and φ b in formulas (8.I) and (9.I) on the right side of the inequality as coefficients that reduce the calculated resistance to critical stresses. And only for the sake of convenience of calculations and comparison of 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 “ Building codes and rules" (SNiP), as well as in the "Norms and technical conditions for the design of steel structures" (NiTU 121-55).
"Design of steel structures"
K.K. Mukhanov
There are several categories of voltages: main, local, additional and internal. Fundamental stresses are stresses that develop within the body as a result of balancing the effects of external loads; they are taken into account in the calculation. When the power flow is unevenly distributed over the cross-section, caused, for example, by a sharp change in the cross-section or the presence of a hole, a local stress concentration occurs. However, in plastic materials, which include construction steel,…
When calculating the permissible stresses, the structure is considered in its operating condition under the influence of loads allowed during normal operation of the structure, i.e., standard loads. The condition for the strength of the structure is that the stresses in the structure from standard loads do not exceed the permissible stresses established by the standards, which represent a certain part of the maximum stress of the material accepted for building steel...
At this stage, we already understand that calculations of building structures are carried out in accordance with some standards. It’s impossible to say for sure which ones, since different countries Different design standards are used.
Thus, in the CIS countries, different versions of standards are used, based on Soviet SNiPs and GOSTs; in European countries they have predominantly switched to Eurocode (EN), and in the USA ASCE, ACI, etc. are used. Obviously, your project will be tied to the standards of the country from which this project was ordered 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 highlighted it in 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 operation of certain building materials, but they are all united by a common scientific base based on the strength of materials, construction and theoretical mechanics.
This is what checking the strength of a metal structure element undergoing tension looks like according to Eurocode:
\[\frac(((N_(Ed))))(((N_(t,Rd)))) \le 1.0.\quad (1)\]
And here’s what a similar check looks like using one of them: latest versions SNiP:
\[\frac(N)(((A_n)(R_y)(\gamma _c))) \le 1.0.\quad (2)\]
It is not difficult to guess that in both the first and second cases the effort from external load(in the numerator) should not exceed the effort characterizing the load-bearing capacity of the structure (in the denominator). This is a clear example of a general, scientifically based approach to the design of buildings and structures by engineers from different countries.
Limit state concept
One day (many years ago, actually) scientists and research engineers noticed that it was not entirely correct to design an element based on a single test. Even for comparatively simple designs, there can be a lot of options for the operation of each element, and building materials during wear, they change their characteristics. And if we also consider the emergency and repair conditions of the structure, this leads to the need for ordering, segmentation, and classification of all possible states of the structure.
This is how the concept of “limit state” was born. A laconic interpretation is given in the Eurocode:
limit state - a state of a structure in which the structure does not meet the appropriate design criteria
We can say that the limit state occurs when the operation of a structure under load goes beyond the design solutions. For example, we designed a steel frame frame, but at a certain moment in its operation one of the racks lost stability and bent - there is a transition to a limit state.
The method of calculating building structures using limit states is dominant (it replaced the less “flexible” method of permissible 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 designer models the operation of the structure not in some abstract state, but in a limiting state. That is, all 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 included in the design standards. By performing checks, you thereby prevent the occurrence of a 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 talk about groups of limit states:
- limit states of group I (in Eurocode - according to bearing capacity)
- limit states of group II (in Eurocode - according to serviceability)
If the first limiting state has occurred, then:
- the structure is destroyed
- the structure has not yet been destroyed, but the slightest increase in load (or change in other operating conditions) leads to destruction
The conclusion is obvious: further operation of a building or structure in the first limit state is impossible under no circumstances:
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 that everything is fine with it - individual elements may receive significant deformations:
- deflections
- section rotations
- cracks
As a rule, the transition of a structure to the second limit state requires some restrictions in operation, for example, reducing the load, reducing the speed, etc.:
Figure 2. Cracks in the concrete of a building (second limit state)
In terms of material strength
On " 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 permissible threshold, called design resistance. These may also be other factors of the stress-strain state - for example, bending moments, transverse or longitudinal forces exceeding limit state load-bearing capacity of the structure.
Checks for the first group of limit states
To prevent the occurrence of the first limit state, the design engineer is obliged to check the characteristic sections of the structure:
- for strength
- for sustainability
- for endurance
Without exception, all load-bearing structural elements are tested for strength, regardless of the material from which they are made, as well as shape and size. cross section. This is the most important and mandatory check, without which the accountant does not have the right to restful sleep.
The stability check is performed for compressed (central, eccentric) elements.
Fatigue testing should be carried out on elements that are subject to cyclic loading and unloading to prevent fatigue effects. This is typical, for example, for the spans of railway bridges, since when trains move, the loading and unloading stages of work constantly alternate.
In this course we will become familiar with the basic strength tests of reinforced concrete and metal structures.
Checks for the second group of limit states
To prevent the occurrence of the second limit state, the design engineer is obliged to check the characteristic sections:
- for deformation (displacement)
- for crack resistance (for reinforced concrete structures)
Deformations should be associated not only with linear movements of the structure (deflections), but also with the angles of rotation of the sections. Ensuring crack resistance is important stage in the design of reinforced concrete structures from both ordinary and prestressed reinforced concrete.
Examples of calculations for reinforced concrete structures
As an example, let us consider what checks must be performed when designing structures made of ordinary (unstressed) reinforced concrete according to the standards.
Table 1. Grouping of calculations by limit states:
M - bending moment; Q - shear force; N - longitudinal force (compressive or tensile); e - eccentricity of application of longitudinal force; T - torque; F - external concentrated force (load); σ - normal stress; a is the 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 in which the structural element is.
We have already come close to learning how to calculate building structures. At our next meeting, we’ll talk about loads and immediately begin calculations.
