Calculation method for bearing capacity of embedded steel beams under fatigue load

Abstract

The present invention relates to a method for calculating the bearing capacity of a reinforced beam under fatigue load, and belongs to the technical field of engineering structure reinforcement and reconstruction. It includes calculations of the bending bearing capacity of the normal section under static load, calculations of the fatigue stress of the normal section, calculations of the shear bearing capacity of the inclined section under static load, and calculations of the fatigue stress of the inclined section. The present invention takes into account the influences of concrete compressive strength, cross-sectional dimensions, location of embedded reinforcement drilling holes, tensile strength of steel bars in the original component, external load form, and the like, and provides a method for calculating the bearing capacity of a reinforced beam. The present invention adopts the generally recognized flat section assumption and stress equivalent graph for concrete structure bearing capacity calculation, establishes equilibrium equations of bending moment and force, combines the component failure forms defined in existing specifications, and derives according to the truss-arch theoretical mechanical model and the fatigue stress damage amount obtained from experiments. It has high reliability and can provide new ideas for the study of the bending and shear bearing capacity of embedded reinforced concrete beams.

Description

Technical Field

[0001] The invention relates to a method for calculating the bearing capacity of a reinforced beam under fatigue load, and belongs to the technical field of engineering structure reinforcement and reconstruction. Background Art

[0002] As a post-anchoring method, rebar planting technology is widely used in the reinforcement, renovation and maintenance of reinforced concrete structures. Rebar planting technology is to drill holes in the original concrete components, inject adhesives, and then implant steel bars. The steel bars of the rebar-planting overlap section are connected to the original components. The load is transferred to the original components through the bonding force between the adhesive and the implanted steel bars. After the adhesive of the rebar-planting overlap section is cured, the stirrups of the newly added components are tied and concrete is poured, so that the new and old concrete components bear the load as a whole.

[0003] The calculation of the design value of the anchor bearing capacity of the rebar in my country's "Concrete Structure Reinforcement Design Code" (GB50367-2013) is based on the tensile bearing capacity of a single rebar under static load and unidirectional pull-out state, as shown below:

[0004] The design value of anchorage depth and tensile bearing capacity of a single embedded reinforcement shall be calculated as follows:

[0005] (Formula 1)

[0006] (Formula 2)

[0007] In the formula, is the design value of tensile bearing capacity of embedded steel; is the design value of tensile strength of the steel bar used for embedding; is the cross-sectional area of ​​the steel bar; is the design value of the anchorage depth of the rebar; is the basic anchorage depth of the rebar; In order to consider the influence of various factors on the tensile bearing capacity of the embedded reinforcement, the correction factor of the anchoring depth needs to be increased; It is a correction factor to consider the displacement ductility coefficient. When the concrete strength is not higher than C30, for the 6 degree zone and the 7 degree zone, the first and second type sites, 1.10; for the third and fourth types of sites in the 7-degree area and the 8-degree area, is 1.25. When the concrete strength grade is higher than C30, The basic anchorage depth is 1.0. For details on the value of the basic anchorage depth, please refer to the "Specification for the Design of Reinforcement of Concrete Structures" (GB50367-2013). However, the "Specification for the Design of Reinforcement of Concrete Structures" (GB50367-2013) does not involve the calculation of the bearing capacity of concrete beams that are reinforced with embedded steel bars, and also lacks provisions for preventing fatigue damage of embedded steel bars.

[0008] Secondly, the "Concrete Structure Design Code" (GB50010-2010) provides the calculation method of the bending resistance of the normal section and the shear resistance of the inclined section of the cast-in-place concrete beam as follows:

[0009] The bending bearing capacity of the normal section is calculated as follows:

[0010] ) (Formula 3)

[0011] The shear bearing capacity of the inclined section is calculated as follows:

[0012] (Formula 4)

[0013] For detailed parameter values, please refer to the Code for Design of Concrete Structures (GB50010-2010).

[0014] Compared with cast-in-place components, rebar-embedded components have new and old concrete interfaces and rebar-rubber-rebar and rubber-mixed interfaces. The additional shear stress caused by the tension of rebar-embedded components also needs to be considered. Whether the calculation principle of cast-in-place components in the Code for Design of Concrete Structures (GB50010-2010) is applicable to rebar-embedded components needs further discussion. The use of chemical rebar-embedded technology in the reinforcement and reconstruction of engineering structures subjected to fatigue loads has broad application prospects, but the current research on the stress state of rebar-embedded structures under fatigue loads is not in-depth and lags behind engineering practice.

[0015] In addition, the bearing capacity test phenomenon of post-installed anchorage beams can be found in: Yan X, Liang L. Fatigue performance of post-installed anchorage beams[J]. Construction and Building Materials, 2019, 229(Dec.30). However, the article only describes the test phenomenon and does not provide the relevant calculation formula for bearing capacity.

[0016] In summary, in the existing specifications, only the bonding and tensile strength between a single embedded bar and the base material need to be considered for stress. However, when the embedded bar technology is used in post-anchored components, it is also necessary to consider whether the embedded bar and the steel bars in the original component can transmit stress reliably, as well as the complex stress conditions of the embedded bar in the component. In the embedded bar overlap section of the anchored component after the embedded bar is embedded, the embedded bar forms a whole common stress with the concrete of the original component through the adhesive, and the load is transmitted with the steel bars in the original component through the bonding force of the concrete. In addition to the strength of the material itself, the factors that affect the overlap force transmission performance of the embedded bar are the thickness of the concrete cover, the transverse reinforcement and the anchoring depth. The thickness of the concrete cover refers to the distance from the wall of the embedded bar drilling hole to the surface of the specimen. The increase in thickness means the better the concrete force transmission performance, which can improve the anti-splitting ability of the component. The transverse reinforcement can constrain the concrete in the overlap section, delay the appearance and development speed of cracks inside the specimen, improve the anti-splitting ability of the component, and then increase the bonding strength between the embedded bar and the concrete. Only when the anchoring depth reaches a certain value can sufficient bonding force be guaranteed to give full play to the strength of each material. When the anchorage depth of the rebar is sufficient and there is no bond slip failure between the adhesive and the concrete, the rebar transfers the load through bonding force. During the load transfer process, the ribbed steel bars cause radial thrust due to the squeezing of the transverse ribs, which causes additional shear stress in the rebar. Longitudinal splitting cracks are prone to occur around the rebar, requiring sufficient concrete cover thickness and lateral restraint.

[0017] The existing specifications do not involve the calculation method of the bearing capacity of rebar-embedded components. The relevant research analyzed the experimental phenomenon of the failure of rebar-embedded beams, but did not conduct theoretical analysis to provide a reference for calculation. The design of rebar-embedded components only proposes design regulations for the anchor depth of rebars in accordance with the "Specification for the Design of Reinforcement of Concrete Structures" (GB50367-2013). In fact, in addition to the influence of the anchor depth of rebars, the bearing capacity of rebar-embedded components also needs to consider the influence of concrete compressive strength, cross-sectional dimensions, rebar drilling location, tensile strength of steel bars in the original components, external load forms, etc. At present, there is no clear specification or research to provide a basis for calculating the bearing capacity of rebar-embedded components in the design stage.

[0018] When the reinforcement is overlapped at the bottom of the beam, in theory, the reinforcement is only subjected to tension, but according to the test, there are oblique cracks in the overlap section of the reinforcement, indicating that there is shear force here. Analyzing the test results and combining with existing research, it is believed that the oblique cracks in the overlap section of the beam are caused by two reasons: the additional shear stress generated by the tension of the reinforcement and the shear force generated by the external load.

[0019] Bridges, industrial plants and other structures are constantly subjected to multiple and large-amplitude fatigue loads during use. As a reinforcement and reconstruction technology with broad application prospects, the performance of embedded steel bars in such buildings has been less studied. Therefore, there are technical deficiencies in this field, and it is necessary to propose a calculation method for the bending and shear bearing capacity of embedded steel bars under fatigue loads to provide a theoretical basis for engineering design. Summary of the invention

[0020] In view of the above-mentioned defects in the prior art, the present invention proposes a method for calculating the bearing capacity of a rebar-embedded beam under fatigue load, which provides a theoretical basis for engineering design.

[0021] In order to achieve the above object, the technical solution adopted by the present invention is as follows:

[0022] The invention is applicable to a bearing capacity calculation method of a reinforced beam under fatigue load, which comprises four parts in sequence: calculation of the bending bearing capacity of a normal section under static load, calculation of fatigue stress of a normal section, calculation of shear bearing capacity of an inclined section under static load and calculation of fatigue stress of an inclined section.

[0023] Furthermore, the cross-section of the embedded beam is divided into three sections according to the area and position of the longitudinal steel bars, including: the original component section, i.e. the pre-cast section, the embedded steel bar overlap section and the newly added embedded steel bar component section, i.e. the post-cast section; the pre-cast section is provided with top stand bars wrapped by a concrete base material, bottom pre-set bars and stirrups connecting the two, and the above-mentioned stand bars, pre-set bars and stirrups constitute the original component steel bars; the post-cast section also includes stand bars and stirrups, as well as embedded steel bars at the bottom; the embedded steel bars of the post-cast section are overlapped with the pre-set bars of the pre-cast section to form an embedded steel bar overlap section.

[0024] Furthermore, the calculation of the bending bearing capacity of the normal section under static load includes the following steps:

[0025] S11. The following assumptions are made when calculating the bending resistance of the normal section of the reinforced beam under static load:

[0026] The cross-sectional strain conforms to the plane cross-sectional assumption; the triangular stress diagram of concrete in the compression zone is equivalent to a rectangle; the tensile effect of concrete in the tension zone is ignored;

[0027] S12. According to the concrete structure design theory, the theoretical values ​​of the bending bearing capacity of each section are given as follows:

[0028] Original component segment:

[0029] (Formula 5)

[0030] Added reinforcement component segment:

[0031] (Formula 6)

[0032] In the formula, is the equivalent rectangular stress diagram coefficient; is the design value of concrete axial compressive strength; is the width of the rectangular section; , It is the height of the concrete compression zone of the original component segment and the newly added reinforced component segment; is the diameter of the pre-set reinforcement; is the diameter of the rebar; is the design value of tensile strength of pre-set reinforcement; is the design value of the tensile strength of the embedded reinforcement; It is the distance from the outer edge of the compression zone to the center of gravity of the resultant force of the pre-set reinforcement; It is the distance from the outer edge of the compression zone to the center of gravity of the resultant force of the embedded reinforcement; is the original component segment bending moment; is the bending moment of the newly added reinforced bar segment;

[0033] S13. Although the embedded reinforcement overlap section has both pre-set reinforcement and embedded reinforcement, the main function of the embedded reinforcement is to transmit tension through bonding and anchoring, so that the first cast section and the later cast section are evenly stressed as a whole, and the effect on the bending bearing capacity of the embedded reinforcement overlap section is negligible; in the process of manufacturing embedded reinforcement beams, in order to prevent the drill bit from deviating and damaging the original component stirrups during drilling, the drilling position will be moved up, and the center of gravity of the embedded reinforcement will be higher than the center of gravity of the pre-set reinforcement, that is, Less than ,make Also less than , resulting in the overall bearing capacity of the planted beam being lower than that of the cast-in-place beam; compared with the bearing capacity calculation of the cast-in-place beam, when the area of ​​the planted reinforcement is the same as that of the original component reinforcement, the calculation formula for the bending capacity of the planted reinforcement beam is:

[0034] (Formula 7)

[0035] Furthermore, the normal section fatigue stress calculation includes the following steps:

[0036] S21. The fatigue stress calculation of the normal section is mainly based on the steel bar stress calculation:

[0037] The pre-cast steel bars in the first pouring section and the embedded steel bars in the last pouring section constitute the longitudinal tensile steel bars. The loss of steel bar stress caused by fatigue damage is introduced into the steel bar stress calculation. The longitudinal tensile steel bar stress can be expressed as:

[0038] (Formula 8)

[0039] (Formula 9)

[0040] (Formula 10)

[0041] (Formula 11)

[0042] (Formula 12)

[0043] In the formula, , are the maximum and minimum bending moments generated by the upper and lower limits of the cyclic load; , Reason , The resulting reinforcement stress; It is the distance from the outer edge of the compression zone to the center of gravity of the resultant force of the pre-set reinforcement; is the stress amplitude of the reinforcement; is the ratio of the elastic modulus of steel bar to the fatigue elastic modulus of concrete; is the section moment of inertia;

[0044] S22. The stress amplitude of the longitudinal tensile reinforcement in the tension zone shall satisfy:

[0045] (Formula 13)

[0046] In the formula, is the stress amplitude limit of the reinforcement;

[0047] S23. According to (Equation 14), the specimen is subjected to 2×10 6 The stress amplitude without fatigue failure in the secondary cyclic loading is less than 120Mpa;

[0048] (Formula 14)

[0049] In the formula, is the stress range of the steel bar; is the fatigue life of the specimen.

[0050] Furthermore, the calculation of the shear bearing capacity of the inclined section under static load includes the following steps:

[0051] S31. The location of the post-casting reinforcement is too close to the edge of the reinforcement beam, and the concrete substrate will be damaged in a wedge shape. At this time, the shear bearing capacity is calculated by the following formula:

[0052] (Formula 15)

[0053] (Formula 16)

[0054] (Formula 17)

[0055] (Formula 18)

[0056] (Formula 19)

[0057] (Formula 20)

[0058] In the formula, is the standard value of shear bearing capacity of concrete edge failure; The shear bearing capacity of the ideal wedge-shaped concrete when the edge of a single embedded vertical member is sheared; is the outer diameter of the rebar; is the effective length of the anchor bolt under shear load; is the standard value of concrete cube compressive strength; The lateral projection area of ​​the ideal edge failure of concrete when a single embedded reinforcement is subjected to shear; It is the lateral projection area of ​​the actual edge damage of the concrete; It is the distance from the embedded reinforcement to the edge of the concrete substrate in one direction; It is the distance from the embedded reinforcement to the edge of the concrete substrate in the other direction; is the anchor bolt spacing; is the thickness of the member in the anchoring direction; is the influence coefficient of edge distance on shear bearing capacity; is the influence coefficient of the ratio of edge distance to member thickness on shear bearing capacity;

[0059] In the formula, the shear capacity of the substrate is calculated with the diameter of the rebar, the anchorage depth, the strength of the concrete substrate, the distance between the rebar anchorage position and the two edges of the original component, the spacing between the rebars and the thickness of the original component in the anchorage direction as parameters; the rebar substrate in the post-casting section undergoes wedge-shaped splitting failure, the ideal wedge-shaped concrete body height penetrates the rebar anchorage depth, the thickness of the concrete substrate in the rebar anchorage direction is large, and the distance between the rebar drilling position and the edge of the original component is small. The concrete will not form a wedge-shaped body when it is sheared, and will only crack at the rebar end at the initial stage of loading. There are no wedge-shaped failure cracks at other positions of the rebar overlap section. Later, under the shear force of the external load, the crack develops toward the loading point; the concrete at the bottom of the rebar beam exits the shear resistance work after cracking due to the shear prying effect at the initial stage of loading, and the influence of the rebar shear prying effect on the concrete at the bottom of the beam is no longer calculated. The shear force applied by the external load is resisted by the concrete and stirrups within the effective height of the beam section;

[0060] S32. Give the theoretical calculation method of shear bearing capacity:

[0061] Assuming that the shear force is transmitted by the concrete when the embedded beam is not cracked, the diagonal units into which the concrete is divided after the diagonal cracks appear are regarded as diagonal compression web members. The angle between the diagonal compression web member and the axis of the specimen is , the longitudinal reinforcement is the tension lower chord, and the stirrups are the tension vertical bars, forming a truss model;

[0062] The effective cross-sectional area of ​​the concrete diagonal compression web in the truss model is:

[0063] (Formula 21)

[0064] In the formula, is the effective coefficient of the truss model, ; is the internal force arm of the longitudinal reinforcement; is the effective cross-sectional width;

[0065] According to the overlap section of the embedded reinforcement, the upward movement of the drilling position Less than ,Pick , the effective cross-sectional width is ,in, is the cross-section width, is the drilling diameter;

[0066] The shear bearing capacity of concrete is expressed as:

[0067] (Formula 22)

[0068] In the formula, is the compressive stress of the concrete diagonal web member;

[0069] According to the balance of forces:

[0070] (Formula 23)

[0071] (Formula 24)

[0072] In the formula, The longitudinal steel is subjected to tensile stress by the tension bar; is the cross-sectional area of ​​the stirrups; is the tensile strength of stirrups; is the stirrup spacing;

[0073] The shear capacity of stirrups in the truss model is expressed as:

[0074] (Formula 25)

[0075] In the formula, is the stirrup reinforcement ratio;

[0076] In the truss model, the ideal state is that the concrete reaches the compressive strength when the stirrups reach the tensile strength, and the concrete diagonal compression web members and the stirrup tension vertical members are destroyed at the same time;

[0077] In order to ensure that the specimen has a certain ductility surplus, it is necessary to combine the arch model for analysis. In the arch model, it is assumed that the concrete has not reached the compressive strength when the stirrups yield, and the residual strength of the concrete is ;

[0078] In the arch model, the arch section height is the sum of the concrete compression zone height and the protective layer thickness. Combining the arch model with the truss model, the arch section height can be calculated as:

[0079] (Formula 26)

[0080] After the stirrups yield, the stress of the concrete arch model is:

[0081] (Formula 27)

[0082] The shear bearing capacity of concrete arch is:

[0083] (Formula 28)

[0084] (Formula 29)

[0085] In the formula, is the angle between the concrete diagonal compression web member and the specimen axis in the arch model; is the shear span ratio;

[0086] set up , Substituting (Equation 26), (Equation 27), (Equation 29) into (Equation 28) we get:

[0087] (Formula 30)

[0088] Ideally, is 45°, substitute it into the above formula and simplify it to get:

[0089] (Formula 31)

[0090] The oblique compression field theory believes that the compressive strength of the concrete oblique compression web is less than the material compressive strength measured by the standard test block, and there is a compressive stress softening phenomenon. The calculation of the softening coefficient is given:

[0091] (Formula 32)

[0092] In the formula, is the compressive strength of the standard concrete cylinder; The softening coefficient is calculated as follows:

[0093] (Formula 33)

[0094] Combining (Equation 22), (Equation 25) and (Equation 31), the shear bearing capacity of the inclined section is obtained:

[0095] (Formula 34)

[0096] Arranged:

[0097] (Formula 35)

[0098] That is, the shear bearing capacity of the inclined section is calculated using section width, section height, concrete compressive strength, stirrup cross-sectional area, reinforcement ratio, drill hole diameter and shear span ratio as parameters.

[0099] Furthermore, anchor bolts are used to replace the embedded reinforcement in the post-casting section in the calculation of the shear bearing capacity of the inclined section under the static load.

[0100] Furthermore, the anchoring depth of the rebar in the post-casting section is greater than the post-anchored anchor bolts.

[0101] Furthermore, the inclined section fatigue stress calculation includes the following steps:

[0102] S41. First determine the shear force that the concrete of the embedded beam bears after cyclic loading:

[0103] (Formula 36)

[0104] In the formula, , Maximum and minimum shear values ​​for cyclic loading; is the design value of fatigue tensile strength of concrete;

[0105] The stirrup stress amplitude is expressed as:

[0106] (Formula 37)

[0107] (Formula 38)

[0108] (Formula 39)

[0109] Introducing the amount of damage caused by fatigue development:

[0110] (Formula 40)

[0111] (Formula 41)

[0112] In the formula, is the distance from the resultant force point of the compression zone to the resultant force point of the tension reinforcement, where the height of the compression zone is Calculate according to (Formula 11);

[0113] S42. The stress amplitude of the inclined section stirrups shall satisfy:

[0114] (Formula 42)

[0115] In the formula, is the limit value of fatigue stress amplitude of stirrups, which is selected with reference to fatigue stress amplitude of ordinary steel bars.

[0116] After adopting the above technical solution, the present invention has the following beneficial effects compared with the prior art:

[0117] Compared with the prior art, the present invention proposes a method for calculating the bearing capacity of reinforced beams under fatigue loads, which takes into account the effects of concrete compressive strength, cross-sectional dimensions, reinforcement drilling positions, tensile strength of steel bars in the original components, external load forms, etc., and provides calculation formulas for the bending bearing capacity and fatigue stress of the positive section of reinforced concrete beams, as well as the shear bearing capacity and fatigue stress of the oblique section. The equilibrium equations of bending moment and force are established by adopting the recognized flat section assumptions and stress equivalent graphs for concrete structure bearing capacity calculation; combined with the component failure forms specified in existing specifications, the deduction is performed based on the truss-arch theoretical mechanical model and the fatigue stress damage obtained from the experiment, which has high reliability and can provide new ideas for the study of the bending and shear bearing capacity of reinforced concrete beams. BRIEF DESCRIPTION OF THE DRAWINGS

[0118] Figure 1 It is a side cross-sectional schematic diagram of a reinforced beam of the present invention;

[0119] Figure 2 yes Figure 1 A top view cross-sectional schematic diagram of ;

[0120] Figure 3 It is a schematic diagram of the bending bearing capacity of the positive section of the implanted steel beam of the present invention;

[0121] Figure 4 It is a force analysis diagram of the bending bearing capacity of the positive section of the embedded steel beam of the present invention. DETAILED DESCRIPTION

[0122] The following is combined with Figure 1-4 The present invention is further described in detail with specific implementations to facilitate a clear understanding of the present invention, but they do not constitute a limitation on the present invention.

[0123] In the description of the present invention, it should be noted that the directions or positional relationships indicated by terms such as “upper”, “lower”, “front”, “back”, “left”, “right”, “vertical”, “inside” and “outside” are based on the directions or positional relationships shown in the accompanying drawings, and are only for the convenience of describing the present invention and simplifying the description, rather than indicating or implying that the device or element referred to must have a specific direction, be constructed and operated in a specific direction, and therefore cannot be understood as a limitation on the present invention.

[0124] In the description of the present invention, it should be noted that, unless otherwise clearly specified and limited, the terms "installed", "connected", and "connected" should be understood in a broad sense, for example, it can be a fixed connection, a detachable connection, or an integral connection; it can be a mechanical connection or an electrical connection; it can be a direct connection or an indirect connection through an intermediate medium. For ordinary technicians in this field, the specific meanings of the above terms in the present invention can be understood according to specific circumstances.

[0125] Example 1

[0126] The present embodiment is applicable to a method for calculating the bearing capacity of a rebar-embedded beam under fatigue load, and the method for calculating the bearing capacity of a rebar-embedded beam comprises four parts in sequence: calculation of the bending bearing capacity of the normal section under static load, calculation of the fatigue stress of the normal section, calculation of the shear bearing capacity of the inclined section under static load, and calculation of the fatigue stress of the inclined section.

[0127] like Figure 1-2 As shown, the cross section of the embedded beam 1 is divided into three sections according to the area and position of the longitudinal reinforcement, including: the original component section, i.e. the pre-cast section 2, the embedded reinforcement overlap section 3 and the newly added embedded reinforcement component section, i.e. the post-cast section 4. The pre-cast section 2 is provided with a top frame bar 8 wrapped by a concrete base material, a bottom pre-set bar 6 and stirrups 10 connecting the two. The above-mentioned frame bar 8, pre-set bar 6 and stirrups 10 constitute the original component reinforcement. The post-cast section 4 also includes frame bars 8 and stirrups 10, as well as embedded bars 7 at the bottom. The embedded bars 7 of the post-cast section 4 are overlapped with the pre-set bars 6 of the pre-cast section 2 to form an embedded reinforcement overlap section 3. A new and old concrete interface 5 is formed between the pre-cast section 2 and the post-cast section 4.

[0128] In this embodiment, Figure 3-4 As shown in Figure 2, the calculation of the bending bearing capacity of the normal section under static load includes the following steps:

[0129] S11. When the anchorage depth of the rebar beam meets certain requirements, the rebar and the original concrete will not produce bond failure when it is destroyed under static force. The longitudinal reinforcement at the bottom of the beam yields, and then the concrete in the compression zone is crushed, which is consistent with the failure mode of the integrally cast beam. Therefore, the following assumptions are used for the bending resistance calculation of the positive section of the rebar beam 1 under static load:

[0130] The cross-sectional strain conforms to the plane cross-sectional assumption. The triangular stress diagram of the concrete in the compression zone is equivalent to a rectangle. The tensile effect of the concrete in the tension zone is ignored.

[0131] S12. According to the concrete structure design theory, the theoretical values ​​of the bending bearing capacity of each section are given as follows:

[0132] Original component segment:

[0133] Formula 5

[0134] Added reinforcement component segment:

[0135] Formula 6

[0136] In the formula, Figure 3-4 As shown, is the equivalent rectangular stress diagram coefficient. It is the design value of concrete axial compressive strength. is the width of the rectangular section. , It is the height of the concrete compression zone of the original component section and the newly added reinforced component section. is the preset reinforcement diameter. is the diameter of the rebar. It is the design value of tensile strength of pre-set reinforcement. It is the design value of tensile strength of embedded reinforcement. It is the distance from the outer edge of the compression zone to the center of gravity of the resultant force of the pre-set reinforcement. It is the distance from the outer edge of the compression zone to the center of gravity of the resultant force of the embedded reinforcement. is the bending moment of the original component segment. It is the bending moment of the newly added reinforced bar segment.

[0137] S13. Although there are both pre-set bars 6 and embedded bars 7 in the embedded bar lap section 3, the main function of the embedded bar 7 is to transmit the tensile force through bonding and anchoring, so that the first cast section 2 and the post-cast section 4 are evenly stressed as a whole, and the effect on the bending bearing capacity of the cross section of the embedded bar lap section 3 is negligible. In the process of manufacturing the embedded bar beam 1, in order to prevent the drill bit from deviating and damaging the original component stirrup 10 during drilling, the drilling position will be moved up, and the center of gravity of the embedded bar 7 is higher than the center of gravity of the pre-set bar 6, that is, Less than ,make Also less than , resulting in the overall bearing capacity of the planted beam being lower than that of the cast-in-place beam. Comparing the bearing capacity calculation of the cast-in-place beam, when the area of ​​the planted reinforcement 7 steel bars is the same as the area of ​​the original component steel bars, the calculation formula for the bending capacity of the planted reinforcement beam is:

[0138] Formula 7

[0139] In this embodiment, the normal section fatigue stress calculation includes the following steps:

[0140] S21. The fatigue study of reinforced concrete beams shows that the fatigue failure morphology of beams is affected by stress level and reinforcement ratio. When the reinforcement ratio meets the requirements of properly reinforced beams, the specimens all suffer from bending fatigue failure in the positive section, and the failure is determined by the fracture of the tensile reinforcement. Some theories believe that the influence of stress amplitude cannot be ignored. When the fatigue stress amplitude is low, the failure morphology is bending fatigue failure, but when the stress amplitude is high, the specimen will suffer from shear fatigue failure. For embedded beams whose mechanical properties meet the requirements of properly reinforced beams in integrally cast beams, regardless of the stress level, the fatigue failure morphology of the specimens is bending fatigue failure, and the tensile pre-installed reinforcement on the outer side of the bottom of the beam is fatigue fractured during failure.

[0141] In engineering, according to the infinite life design concept, the fatigue stress of the component should be less than the fatigue limit value during the normal use stage, and the fatigue damage will not appear in the stage where the strain increases rapidly to the specimen failure, and the specimen life tends to ∞. For rectangular cross-section reinforced concrete bending members, the characteristic of the fatigue failure of the positive section is that a longitudinal steel bar is fatigued and fractured at the weaker vertical crack of the member. Therefore, the fatigue stress verification of the positive section is mainly based on the steel bar stress verification.

[0142] The pre-cast section 2 pre-cast reinforcement 6 and the post-cast section 4 embedded reinforcement 7 constitute the longitudinal tensile reinforcement. The loss of the steel bar stress caused by fatigue damage is introduced into the steel bar stress calculation. The longitudinal tensile reinforcement stress can be expressed as:

[0143] Formula 8

[0144] Formula 9

[0145] Formula 10

[0146] Formula 11

[0147] Formula 12

[0148] In the formula, , are the maximum and minimum bending moments generated by the upper and lower limits of the cyclic load. , Reason , The resulting steel bar stress. It is the distance from the outer edge of the compression zone to the center of gravity of the resultant force of the pre-set reinforcement 6. is the reinforcement stress amplitude. It is the ratio of the elastic modulus of steel bar to the fatigue elastic modulus of concrete. is the section moment of inertia.

[0149] S22. The stress amplitude of the longitudinal tensile reinforcement in the tension zone shall satisfy:

[0150] Formula 13

[0151] In the formula, is the stress amplitude limit of the steel bar.

[0152] S23. The allowable stress amplitude stipulated in the Code for Design of Steel Structures GB50017-2017 is based on the fact that the specimen is subjected to 2×10 6 According to formula 14, the specimen is given by the condition that it is not damaged after 2×10 6 The stress amplitude without fatigue failure in secondary cyclic loading is less than 120Mpa.

[0153] Formula 14

[0154] In the formula, is the stress range of the steel bar. is the fatigue life of the specimen.

[0155] In this embodiment, the calculation of the shear bearing capacity of the inclined section under static load includes the following steps:

[0156] S31. There is no calculation standard for the shear bearing capacity of the post-anchored concrete substrate. The specification only stipulates that structural measures should be taken to prevent it. For the shear of the substrate of the post-anchored anchor bolt, the specification provides the shear bearing capacity calculation of the concrete shear prying failure. By analyzing the shear calculation of the anchor bolt substrate, considering the differences in size and structure between the rebar and the anchor bolt, a calculation method suitable for describing the shear bearing capacity of the rebar substrate is found. The implantation position of the post-casting section 4 rebar 7 is too close to the edge of the rebar beam 1, and the concrete substrate will be damaged in a wedge shape. At this time, the shear bearing capacity is calculated by the following formula:

[0157] Formula 15

[0158] Formula 16

[0159] Formula 17

[0160] Formula 18

[0161] Formula 19

[0162] Formula 20

[0163] In the formula, It is the standard value of shear bearing capacity of concrete edge failure. When the edge of a single-bar embedded vertical member is subjected to shear, the ideal wedge-shaped concrete body destroys the shear bearing capacity. The outer diameter of the rebar. is the effective length of the anchor bolt under shear load. It is the standard value of concrete cube compressive strength. It is the lateral projection area of ​​the ideal edge failure of concrete when a single embedded rebar is subjected to shear. It is the lateral projection area of ​​the actual edge damage of concrete. It is the distance from the embedded reinforcement to the edge of the concrete substrate in one direction. It is the distance from the embedded reinforcement to the edge of the concrete substrate in the other direction. is the anchor bolt spacing. is the thickness of the component in the anchoring direction. is the influence coefficient of edge distance on shear bearing capacity. It is the influence coefficient of the ratio of edge distance to member thickness on the shear bearing capacity.

[0164] In the formula, the shear capacity of the substrate is calculated with the diameter of the rebar, the anchorage depth, the strength of the concrete substrate, the distance between the rebar anchorage position and the two edges of the original component, the spacing between the rebars and the thickness of the original component in the anchorage direction as parameters. The rebar substrate in the post-casting section undergoes wedge-shaped splitting failure. The ideal wedge-shaped concrete body height penetrates the rebar anchorage depth. The thickness of the concrete substrate in the rebar anchorage direction is large, and the distance between the rebar drilling position and the edge of the original component is small. The concrete will not form a wedge-shaped body when it is sheared. It will only crack at the rebar end 71 at the initial stage of loading. There are no wedge-shaped failure cracks at other locations of the rebar overlap section. Later, under the shear force of the external load, the crack develops toward the loading point. After the concrete at the bottom of the rebar beam cracks due to the shear prying effect at the initial stage of loading, it exits the shear resistance work. The influence of the rebar shear prying effect on the concrete at the bottom of the beam is no longer calculated. The shear force applied by the external load is resisted by the concrete and stirrups within the effective height of the beam section.

[0165] When calculating the standard value of the shear force of the concrete at the bottom of the beam to resist the shear and prying action of the embedded reinforcement, since the anchoring depth of the embedded reinforcement is large enough, the cracking of the concrete at the bottom of the beam will not form a wedge-shaped body, and the calculation of the projection area of ​​the wedge-shaped body in the shear resistance of the anchor base is not applicable to the embedded reinforcement components. The shear force borne by the concrete mainly depends on the tensile strength, the configuration of the shear stirrups and the longitudinal reinforcement. Therefore, under the premise that the anchoring depth of the embedded reinforcement is sufficient to cause ductile failure of the embedded reinforcement beam, the shear bearing capacity of the inclined section is given with reference to the shear bearing capacity of reinforced concrete. Most of the existing specifications are semi-empirical formulas, and no clear mechanical model is given. Now a mechanical model is established based on the truss-arch theory, and a theoretical calculation method for shear bearing capacity is given.

[0166] S32. Give the theoretical calculation method of shear bearing capacity:

[0167] The truss model assumes that the shear force is transmitted by the concrete when the embedded beam 1 is not cracked. After the diagonal crack appears, the diagonal units into which the concrete is divided are regarded as diagonal compression web members. The angle between the diagonal compression web member and the axis of the specimen is The longitudinal reinforcement is the tension lower chord, and the stirrup 10 is the tension vertical rod, forming a truss model.

[0168] The effective cross-sectional area of ​​the concrete diagonal compression web in the truss model is:

[0169] Formula 21

[0170] In the formula, is the effective coefficient of the truss model, . It is the internal force arm of the longitudinal reinforcement. is the effective section width.

[0171] According to the overlap section of the embedded reinforcement, the upward movement of the drilling position Less than ,Pick , the effective cross-sectional width is ,in, is the cross-section width, is the drilling diameter.

[0172] The shear bearing capacity of concrete is expressed as:

[0173] Formula 22

[0174] In the formula, is the compressive stress of the concrete diagonal web member.

[0175] According to the balance of forces:

[0176] Formula 23

[0177] Formula 24

[0178] In the formula, The longitudinal steel is subjected to tensile stress. is the cross-sectional area of ​​stirrups. is the tensile strength of stirrups. is the stirrup spacing.

[0179] The shear capacity of stirrups in the truss model is expressed as:

[0180] Formula 25

[0181] In the formula, is the stirrup reinforcement ratio.

[0182] In the truss model, the ideal state is that the concrete reaches the compressive strength when the stirrups reach the tensile strength, and the concrete diagonal compression web members and the stirrup tension vertical members are destroyed at the same time.

[0183] In order to ensure that the specimen has a certain ductility surplus, it is necessary to combine the arch model for analysis. In the arch model, it is assumed that the concrete has not reached the compressive strength when the stirrups yield, and the residual strength of the concrete is .

[0184] In the arch model, the arch section height is the sum of the concrete compression zone height and the protective layer thickness. Combining the arch model with the truss model, the arch section height can be calculated as:

[0185] Formula 26

[0186] After the stirrups yield, the stress of the concrete arch model is:

[0187] Formula 27

[0188] The shear bearing capacity of concrete arch is:

[0189] Formula 28

[0190] Formula 29

[0191] In the formula, It is the angle between the concrete diagonal compression web member and the specimen axis in the arch model. is the shear span ratio.

[0192] set up , Substituting equation 26, equation 27, and equation 29 into equation 28, we get:

[0193] Formula 30

[0194] Ideally, is 45°, substitute it into the above formula and simplify it to get:

[0195] Formula 31

[0196] The oblique compression field theory believes that the compressive strength of the concrete oblique compression web is less than the material compressive strength measured by the standard test block, and there is a compressive stress softening phenomenon. The calculation of the softening coefficient is given:

[0197] Formula 32

[0198] In the formula, is the compressive strength of the standard concrete cylinder. The softening coefficient is calculated as follows:

[0199] Formula 33

[0200] Combining equations 22, 25, and 31, the shear bearing capacity of the inclined section is obtained:

[0201] Formula 34

[0202] Arranged:

[0203] Formula 35

[0204] That is, the shear bearing capacity of the inclined section is calculated using section width, section height, concrete compressive strength, stirrup cross-sectional area, reinforcement ratio, drill hole diameter and shear span ratio as parameters.

[0205] In this embodiment, the fatigue stress calculation of the inclined section includes the following steps:

[0206] S41. "Concrete Structure Design Code" GB50010-2010 stipulates the calculation of the stress amplitude of stirrups in fatigue calculation of inclined sections. The shear force is generally borne by concrete and stirrups. First, it is necessary to determine the shear force borne by the concrete of the embedded beam after cyclic loading:

[0207] Formula 36

[0208] In the formula, , Maximum and minimum shear values ​​for cyclic loading. is the design value of concrete fatigue tensile strength.

[0209] The stirrup stress amplitude is expressed as:

[0210] Formula 37

[0211] Formula 38

[0212] Formula 39

[0213] Introducing the amount of damage caused by fatigue development:

[0214] Formula 40

[0215] Formula 41

[0216] In the formula, is the distance from the resultant force point of the compression zone to the resultant force point of the tension reinforcement, where the height of the compression zone is Calculate according to formula 11.

[0217] S42. The stress amplitude of the inclined section stirrups shall satisfy:

[0218] Formula 42

[0219] In the formula, is the limit value of fatigue stress amplitude of stirrups, which is selected with reference to fatigue stress amplitude of ordinary steel bars.

[0220] Example 2

[0221] In this embodiment, anchor bolts are used to replace the embedded steel bars 7 in the post-casting section 4 in Example 1 in the calculation of the shear bearing capacity of the inclined section under static load. The depth of the post-anchored anchor bolts is less than the anchorage depth of the embedded steel bars 7 in the post-casting section in Example 1. Other calculation methods are the same as in Example 1 and will not be described in detail here.

[0222] The above is only a preferred embodiment of the present invention, and does not impose any formal limitation on the structure of the present invention. The layout type and the number of uses of the present invention are not limited to this example, and can be optimized and selected according to the actual project. Any modification, equivalent change and decoration of the above embodiment based on the technical principle of the present invention that does not deviate from the content of the technical solution of the present invention are still within the scope of the technical solution of the present invention.

Claims

1. The method for calculating the bearing capacity of the embedded beam under fatigue load is characterized by: The bearing capacity calculation method of the embedded beam includes four parts: the calculation of the bending bearing capacity of the normal section under static load, the calculation of the fatigue stress of the normal section, the calculation of the shear bearing capacity of the oblique section under static load, and the calculation of the fatigue stress of the oblique section. The cross section of the embedded beam (1) is divided into three sections according to the area and position of the longitudinal steel bars, including: an original component section, namely the first-cast section (2), an embedded steel bar overlap section (3) and a newly added embedded steel bar component section, namely the post-cast section (4); the first-cast section (2) is provided with a top frame bar (8) wrapped by a concrete base material, a bottom pre-set bar (6) and stirrups (10) connecting the two, and the frame bar (8), pre-set bar (6) and stirrups (10) constitute the original component steel bars; the post-cast section (4) also includes the frame bar (8) and stirrups (10), and also includes embedded steel bars (7) at the bottom; the embedded steel bars (7) of the post-cast section (4) are overlapped with the pre-set bars (6) of the first-cast section (2) to form an embedded steel bar overlap section (3); The calculation of the bending bearing capacity of the normal section under static load includes the following steps: S11. The following assumptions are made when calculating the bending resistance of the positive section of the reinforced beam (1) under static load: The cross-sectional strain conforms to the plane cross-sectional assumption; the triangular stress diagram of concrete in the compression zone is equivalent to a rectangle; the tensile effect of concrete in the tension zone is ignored; S12. According to the concrete structure design theory, the theoretical values ​​of the bending bearing capacity of each section are given as follows: Original component segment: (Formula 5) Added reinforcement component segment: (Formula 6) In the formula, is the equivalent rectangular stress diagram coefficient; is the design value of concrete axial compressive strength; is the width of the rectangular section; , It is the height of the concrete compression zone of the original component segment and the newly added reinforced component segment; is the diameter of the pre-set reinforcement; is the diameter of the rebar; is the design value of tensile strength of pre-set reinforcement; is the design value of the tensile strength of the embedded reinforcement; It is the distance from the outer edge of the compression zone to the center of gravity of the resultant force of the pre-set reinforcement; It is the distance from the outer edge of the compression zone to the center of gravity of the resultant force of the embedded reinforcement; is the original component segment bending moment; is the bending moment of the newly added reinforced bar segment; S13. Although the embedded reinforcement overlap section (3) has both pre-set reinforcement (6) and embedded reinforcement (7), the main function of the embedded reinforcement (7) is to transmit tension through bonding and anchoring, so that the first cast section (2) and the post-cast section (4) are evenly stressed as a whole, and the effect on the bending bearing capacity of the cross section of the embedded reinforcement overlap section (3) is negligible; during the production process of the embedded reinforcement beam (1), in order to prevent the drill bit from deviating and damaging the original component stirrups (10) during drilling, the drilling position will be moved up, and the center of gravity of the embedded reinforcement (7) will be higher than the center of gravity of the pre-set reinforcement (6), that is, Less than ,make Also less than , resulting in the overall bearing capacity of the embedded beam being lower than that of the cast-in-place beam; compared with the bearing capacity calculation of the cast-in-place beam, when the area of ​​the embedded reinforcement (7) is the same as the area of ​​the original component reinforcement, the calculation formula for the bending capacity of the embedded beam is: (Formula 7).

2. The method for calculating the bearing capacity of a reinforced beam under fatigue load according to claim 1 is characterized in that: The normal section fatigue stress calculation comprises the following steps: S21. The fatigue stress calculation of the normal section is mainly based on the steel bar stress calculation: The pre-placed reinforcement (6) of the first pouring section (2) and the embedded reinforcement (7) of the later pouring section (4) constitute the longitudinal tensile reinforcement. The loss of the reinforcement stress caused by fatigue damage is introduced into the reinforcement stress calculation. The longitudinal tensile reinforcement stress can be expressed as: (Formula 8) (Formula 9) (Formula 10) (Formula 11) (Formula 12) In the formula, , are the maximum and minimum bending moments generated by the upper and lower limits of the cyclic load; , Reason , The resulting reinforcement stress; is the distance from the outer edge of the compression zone to the center of gravity of the resultant force of the pre-set reinforcement (6); is the stress amplitude of the reinforcement; is the ratio of the elastic modulus of steel bar to the fatigue elastic modulus of concrete; is the section moment of inertia; S22. The stress amplitude of the longitudinal tensile reinforcement in the tension zone shall satisfy: (Formula 13) In the formula, is the stress amplitude limit of the reinforcement; S23. According to (Equation 14), the specimen is subjected to 2×10 6 The stress amplitude without fatigue failure in the secondary cyclic loading is less than 120Mpa; (Formula 14) In the formula, is the stress range of the steel bar; is the fatigue life of the specimen.

3. The method for calculating the bearing capacity of a reinforced beam under fatigue load according to claim 2 is characterized in that: The calculation of the shear bearing capacity of the inclined section under static load comprises the following steps: S31. The implantation position of the rebar (7) in the post-casting section (4) is too close to the edge of the rebar beam (1), and the concrete base material will be damaged in a wedge shape. In this case, the shear bearing capacity is calculated by the following formula: (Formula 15) (Formula 16) (Formula 17) (Formula 18) (Formula 19) (Formula 20) In the formula, is the standard value of shear bearing capacity of concrete edge failure; It is the shear bearing capacity of the ideal wedge-shaped concrete when the edge of a single-root vertical member is sheared; is the outer diameter of the rebar; is the effective length of the anchor bolt under shear load; is the standard value of concrete cube compressive strength; The lateral projection area of ​​the ideal edge failure of concrete when a single embedded reinforcement is subjected to shear; It is the lateral projection area of ​​the actual edge damage of the concrete; It is the distance from the embedded reinforcement to the edge of the concrete substrate in one direction; It is the distance from the embedded reinforcement to the edge of the concrete substrate in the other direction; is the anchor bolt spacing; is the thickness of the member in the anchoring direction; is the influence coefficient of edge distance on shear bearing capacity; is the influence coefficient of the ratio of edge distance to member thickness on the shear bearing capacity; In the formula, the shear capacity of the base material is calculated with the diameter of the rebar, the anchorage depth, the strength of the concrete base material, the distance between the rebar anchorage position and the two edges of the original component, the spacing between the rebars and the thickness of the original component in the anchorage direction as parameters; the rebar base material in the post-casting section undergoes wedge-shaped splitting failure, the ideal wedge-shaped concrete body height penetrates the rebar anchorage depth, the thickness of the concrete base material in the rebar anchorage direction is large, and the distance between the rebar drilling position and the edge of the original component is small. The concrete will not form a wedge-shaped body when it is sheared, and will only crack at the rebar end (71) at the initial stage of loading. There is no wedge-shaped failure crack at other positions of the rebar overlap section. Later, under the shear force of the external load, the crack develops toward the loading point; the concrete at the bottom of the rebar beam cracks due to the shear prying effect at the initial stage of loading and then quits the shear resistance work. The influence of the shear prying effect of the rebar on the concrete at the bottom of the beam is no longer calculated. The shear force applied by the external load is resisted by the concrete and stirrups within the effective height of the beam section; S32. Give the theoretical calculation method of shear bearing capacity: Assuming that the shear force of the embedded beam (1) is transmitted by the concrete before cracking, the diagonal units divided by the concrete after the diagonal cracks appear are regarded as diagonal compression web members. The angle between the diagonal compression web member and the axis of the specimen is , the longitudinal reinforcement is the tension lower chord, and the stirrup (10) is the tension vertical rod, forming a truss model; The effective cross-sectional area of ​​the concrete diagonal compression web in the truss model is: (Formula 21) In the formula, is the effective coefficient of the truss model, ; is the internal force arm of the longitudinal reinforcement; is the effective cross-sectional width; According to the overlap section of the embedded reinforcement, the upward movement of the drilling position Less than ,Pick , the effective cross-sectional width is ,in, is the cross-section width, is the drilling diameter; The shear bearing capacity of concrete is expressed as: (Formula 22) In the formula, is the compressive stress of the concrete diagonal web member; According to the balance of forces: (Formula 23) (Formula 24) In the formula, The longitudinal steel is subjected to tensile stress by the tension bar; is the cross-sectional area of ​​the stirrups; is the tensile strength of stirrups; is the stirrup spacing; The shear capacity of stirrups in the truss model is expressed as: (Formula 25) In the formula, is the stirrup reinforcement ratio; In the truss model, the ideal state is that the concrete reaches the compressive strength when the stirrups reach the tensile strength, and the concrete diagonal compression web members and the stirrup tension vertical members are destroyed at the same time; In order to ensure that the specimen has a certain ductility surplus, it is necessary to combine the arch model for analysis. In the arch model, it is assumed that the concrete has not reached the compressive strength when the stirrups yield, and the residual strength of the concrete is ; In the arch model, the arch section height is the sum of the concrete compression zone height and the protective layer thickness. Combining the arch model with the truss model, the arch section height can be calculated as: (Formula 26) After the stirrups yield, the stress of the concrete arch model is: (Formula 27) The shear bearing capacity of concrete arch is: (Formula 28) (Formula 29) In the formula, is the angle between the concrete diagonal compression web member and the specimen axis in the arch model; is the shear span ratio; set up , Substituting (Equation 26), (Equation 27), (Equation 29) into (Equation 28) we get: (Formula 30) Ideally, is 45°, substitute it into the above formula and simplify it to get: (Formula 31) The oblique compression field theory believes that the compressive strength of the concrete oblique compression web is less than the material compressive strength measured by the standard test block, and there is a compressive stress softening phenomenon. The calculation of the softening coefficient is given: (Formula 32) In the formula, is the compressive strength of the standard concrete cylinder; The softening coefficient is calculated as follows: (Formula 33) Combining (Equation 22), (Equation 25) and (Equation 31), the shear bearing capacity of the inclined section is obtained: (Formula 34) Arranged: (Formula 35) That is, the shear bearing capacity of the inclined section is calculated using section width, section height, concrete compressive strength, stirrup cross-sectional area, reinforcement ratio, drill hole diameter and shear span ratio as parameters.

4. The method for calculating the bearing capacity of a rebar-embedded beam under fatigue load according to claim 3 is characterized in that: In the calculation of the shear bearing capacity of the inclined section under the static load, anchor bolts are used to replace the embedded reinforcement (7) in the post-cast section (4).

5. The method for calculating the bearing capacity of a reinforced beam under fatigue load according to claim 4 is characterized in that: The anchoring depth of the post-casting section embedded reinforcement (7) is greater than the post-anchoring anchor bolt.

6. The method for calculating the bearing capacity of a reinforced beam under fatigue load according to claim 5 is characterized in that: The inclined section fatigue stress calculation comprises the following steps: S41. First determine the shear force that the concrete of the embedded beam bears after cyclic loading: (Formula 36) In the formula, , Maximum and minimum shear values ​​for cyclic loading; is the design value of fatigue tensile strength of concrete; The stirrup stress amplitude is expressed as: (Formula 37) (Formula 38) (Formula 39) Introducing the amount of damage caused by fatigue development: (Formula 40) (Formula 41) In the formula, is the distance from the resultant force point of the compression zone to the resultant force point of the tension reinforcement, where the height of the compression zone is Calculate according to (Formula 11); S42. The stress amplitude of the inclined section stirrups shall satisfy: (Formula 42) In the formula, is the limit value of fatigue stress amplitude of stirrups, which is selected with reference to fatigue stress amplitude of ordinary steel bars.

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