A method for analyzing the bearing capacity of a reinforced ultra-high performance concrete rectangular section eccentric compression member

Abstract

The application relates to the field of analysis of the bearing capacity of eccentric compression components of super high performance concrete, and particularly discloses a method for analyzing the bearing capacity of eccentric compression components of reinforced super high performance concrete rectangular sections, which is based on different failure modes and stress states of reinforced UHPC rectangular section components under axial compression, small eccentric compression, large eccentric compression and pure bending, and establishes a compression-bending bearing capacity related equation reflecting the mutual relationship between the shaft force and the shaft bending moment of the rod end. The equation is a cubic function form, the expressions of the large eccentric compression and the small eccentric compression stages are the same, and the expressions are consistent with the axial compression point considering stability, so that the quantitative relationship between the shaft force and the bending moment of the component under different failure states such as axial compression, eccentric compression and pure bending is accurately described. The method is clear in logic, convenient in calculation, and suitable for the design and analysis of the reinforced UHPC rectangular section eccentric compression component.

Description

Technical Field

[0001] This invention relates to the field of bearing capacity analysis of ultra-high performance concrete eccentrically compressed members, and particularly to a method for analyzing the bearing capacity of rectangular cross-section reinforced ultra-high performance concrete eccentrically compressed members. Background Technology

[0002] Ultra-high performance concrete (UHPC) possesses ultra-high strength, high toughness, high durability, and high impact resistance, thus being regarded as a novel cement-based composite material. Previous research has primarily focused on its material properties. However, with a deeper understanding of UHPC's properties and its superior characteristics compared to ordinary concrete, UHPC has been gradually applied in practical engineering projects such as high-rise buildings and highway bridges. Although there is considerable research on UHPC material properties, research on the load-bearing capacity of UHPC components is still in its early stages, and most studies focus on single load-bearing capacities such as axial compressive, bending, and torsional resistance. Currently, no relevant equations have been established for the compressive-bending capacity of reinforced UHPC components. Here, the compressive-bending capacity equation refers to the relationship between the axial force and bending moment at both ends of a reinforced UHPC component when it reaches failure under eccentric compressive load.

[0003] The current Chinese standard "Code for Design of Ultra-High Performance Concrete Structures" T / CBMF 185-2022 (hereinafter referred to as the "Code") provides a simplified calculation diagram for rectangular cross-section eccentrically compressed members, as shown in the figure below. Figure 1 and Figure 2 As shown, the formula for calculating the compressive bearing capacity of its cross section is:

[0004] (6)

[0005] (7)

[0006] In the formula, N is the design value of axial pressure of the UHPC component; This is the design value for the axial compressive strength of the UHPC. The width of the cross section; and The heights of the equivalent rectangular stress diagrams of the UHPC in the compression and tension zones are respectively. and These are the design values ​​of the compressive strength of the longitudinal reinforcement and the longitudinal prestressed tendons, respectively. and These are the cross-sectional areas of the longitudinal reinforcement in the compression zone and tension zone, respectively. and These refer to the stresses of the longitudinal steel bars and longitudinal prestressed tendons on the tension side or the side with less compression, respectively. This represents the equivalent rectangular stress design value for the tension zone of the UHPC, and has... Where k represents the equivalent stress value in the tension zone of the UHPC and the design value of the axial tensile strength of the UHPC. The ratio is taken as 0.4; The prestressing tendon stress is the stress of the prestressing tendon when the normal stress of the concrete at the resultant point of the longitudinal prestressing tendon in the compression zone is zero. and These are the cross-sectional areas of the prestressing tendons in the compression and tension zones, respectively. This is the distance from the point of application of the axial force to the point of resultant force of the longitudinal reinforcement; The effective height of the cross-section; It is the distance from the resultant point of the longitudinal reinforcement in the tension zone to the tension edge of the section; This is the distance from the resultant point of the longitudinal reinforcement in the compression zone to the compression edge of the section; It is the distance from the resultant point of the longitudinal prestressing tendons in the compression zone to the compression edge of the section.

[0007] Equations (6) and (7) are the expressions for the compressive and flexural bearing capacities of eccentrically compressed UHPC rectangular cross-section members with prestressed tendons, respectively. However, in practical applications, the widely used UHPC members are mostly reinforced with ordinary steel bars and rarely with prestressed tendons. Based on equations (6) and (7), the eccentrically compressed UHPC rectangular cross-section members without prestressed tendons can be obtained (the simplified cross-section calculation diagram is shown in Figure 1). Figure 3 and Figure 4 The expressions for the compressive and flexural bearing capacity of the cross section (as shown) are as follows:

[0008] (8)

[0009] (9)

[0010] On the other hand, the expression for the compressive bearing capacity of the axially compressed UHPC reinforced concrete member given in the code is as follows:

[0011] (10)

[0012] In the formula, Indicates the compressive strength of reinforced UHPC members; The value is the stability coefficient of the reinforced UHPC axially compressed member, and is taken according to the relevant provisions of the national standard "Code for Design of Concrete Structures" GB50010-2010; This refers to the cross-sectional area of ​​all longitudinal reinforcing bars.

[0013] The following are shortcomings in the load-bearing capacity analysis of reinforced UHPC rectangular section eccentrically compressed members in the code: 1. As can be seen from formulas (8) and (9), the code provides expressions for the compressive and bending load-bearing capacity of the normal section of reinforced UHPC rectangular section eccentrically compressed members, but these expressions do not directly reflect the quantitative relationship between the axial force and bending moment at the member ends. 2. Compared with formula (10), formulas (8) and (9) do not contain a stability coefficient. This means that the stability effect was not considered. According to the "Code for Design of Concrete Structures" GB50010-2010, the stability coefficient of short members is... The stability coefficient of medium and long components Therefore, for medium-length eccentrically loaded members, the axial compression points calculated according to formulas (8) and (9) cannot match the axial compression points calculated by formula (10). 3. Although formulas (8) and (9) pass through the boundary points of eccentrically loaded members, they do not meet the theoretical requirement that the bending moment is the maximum at the boundary points.

[0014] In summary, it is necessary to provide functional relationships between axial force and bending moment that can reasonably reflect stress states such as axial compression, small eccentric compression, large eccentric compression, and pure bending, so as to facilitate efficient and accurate load-bearing capacity analysis of reinforced UHPC rectangular section eccentrically compressed members. Summary of the Invention

[0015] This invention aims to solve the aforementioned technical problems and provides a method for analyzing the bearing capacity of reinforced ultra-high performance concrete (UHPC) rectangular section eccentrically compressed members. This method can reasonably reflect the correlation between the axial force and bending moment at both ends of the UHPC member under eccentrically compressed load, facilitating efficient and accurate analysis of the bearing capacity of UHPC rectangular section eccentrically compressed members.

[0016] To achieve the above objectives, the technical solution adopted by this invention is as follows: a method for analyzing the bearing capacity of a reinforced ultra-high performance concrete (UHPC) rectangular section eccentrically compressed member. Based on the stress states of the symmetrically reinforced UHPC rectangular section eccentrically compressed member under four stress states—axial compression, small eccentric compression, large eccentric compression, and pure bending—a method is established to reflect the axial force at the member ends. and end moment The interrelationship between components during component failure is represented by a unified cubic function equation using both large and small eccentricity expressions:

[0017] (1)

[0018] In the formula, The coefficients are undetermined, and the solution process is as follows:

[0019] because - The curve must pass through the axial compression point. Pure bending point and boundary point And the bending moment is greatest at the limit point (i.e. That is, it must satisfy:

[0020] (11)

[0021] In the formula, , These are the compressive strength and flexural strength of the UHPC reinforced steel eccentrically compressed member, respectively, expressed as equations (10) and (3); , These are the axial force and bending moment at the limit failure of the UHPC reinforced steel member, respectively, and are called the limit axial force and limit bending moment, expressed as equations (4) and (5).

[0022] (10)

[0023] (3)

[0024] (4)

[0025] (5)

[0026] The undetermined coefficients can be obtained according to equation (11):

[0027] (2)

[0028] Compared with the prior art, the beneficial effects of the present invention are: (1) By establishing a cubic function form of the compression-bending bearing capacity correlation equation with unified expressions for large and small eccentric compression, the bearing capacity correlation characteristics between axial force and bending moment of reinforced UHPC members under the two failure states of large and small eccentric compression are accurately described. When analyzing the bearing capacity of the members, it is no longer necessary to judge large and small eccentricity, which significantly simplifies the analysis process. (2) The bearing capacity correlation equation is consistent with the axial compression point of axially compressed members and the pure bending point of pure bending members, and the bending moment reaches the maximum at the boundary point, which is theoretically consistent with the stress state of eccentrically compressed members. The bearing capacity correlation equation established by the present invention can be used to carry out the bearing capacity analysis of eccentrically compressed UHPC rectangular section members with steel reinforcement in a high efficiency and accuracy. Attached Figure Description

[0029] The specific embodiments of the present invention will be further described in detail below with reference to the accompanying drawings, wherein:

[0030] Figure 1 This is a schematic diagram of the stress on a UHPC rectangular section eccentrically compressed member with prestressed tendons.

[0031] Figure 2A schematic diagram of the cross-sectional parameters of a UHPC rectangular section eccentrically compressed member with prestressed tendons;

[0032] Figure 3 This is a schematic diagram of the stress on an eccentrically compressed UHPC rectangular section member without prestressing tendons.

[0033] Figure 4 A schematic diagram of the cross-sectional parameters of an eccentrically compressed UHPC rectangular section member without prestressing tendons;

[0034] Figure 5 A simplified calculation diagram of the cross-section of an eccentrically compressed member with a UHPC rectangular cross-section reinforced concrete.

[0035] Figure 6 This is a curve comparing the characteristic points of the "Regulations" with the bearing capacity correlation equation of this patent. Detailed Implementation

[0036] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only a part of the embodiments of the present invention, and not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those of ordinary skill in the art without creative effort are within the scope of protection of the present invention.

[0037] It should be noted that when a component is described as "fixed to" another component, it can be directly on the other component or may have a central component. When a component is described as "connected to" another component, it can be directly connected to the other component or may have a central component. When a component is described as "set on" another component, it can be directly set on the other component or may have a central component. When a component is described as "set in the middle," it is not simply set in the exact center, as long as it is not set within the area defined by both ends being in the middle. The terms "vertical," "horizontal," "left," "right," and similar expressions used in this document are for illustrative purposes only.

[0038] Unless otherwise defined, all technical and scientific terms used herein have the same meaning as commonly understood by one of ordinary skill in the art to which this invention pertains. The terminology used herein in the description of the invention is for the purpose of describing particular embodiments only and is not intended to be limiting of the invention. The term "and / or" as used herein includes any and all combinations of one or more of the associated listed items.

[0039] This application discloses a method for analyzing the bearing capacity of a reinforced ultra-high performance concrete rectangular section eccentrically compressed member. This application is based on the compressive bearing capacity and flexural bearing capacity expressions for prestressed reinforced UHPC rectangular section eccentrically compressed members in the "Code for Design of Ultra-High Performance Concrete Structures" T / CBMF 185-2022 (hereinafter referred to as the "Code"), namely equations (6) and (7). A simplified calculation diagram of the section is shown below. Figure 1 and Figure 2 As shown.

[0040] (6)

[0041] (7)

[0042] In the formula, N is the design value of axial pressure of the UHPC component; This is the design value for the axial compressive strength of the UHPC. The width of the cross section; and The heights of the equivalent rectangular stress diagrams of the UHPC in the compression and tension zones are respectively. and These are the design values ​​of the compressive strength of the longitudinal reinforcement and the longitudinal prestressed tendons, respectively. and These are the cross-sectional areas of the longitudinal reinforcement in the compression zone and tension zone, respectively. and These refer to the stresses of the longitudinal steel bars and longitudinal prestressed tendons on the tension side or the side with less compression, respectively. This represents the equivalent rectangular stress design value for the tension zone of the UHPC, and has... Where k represents the equivalent stress value in the tension zone of the UHPC and the design value of the axial tensile strength of the UHPC. The ratio is taken as 0.4; The prestressing tendon stress is the stress of the prestressing tendon when the normal stress of the concrete at the resultant point of the longitudinal prestressing tendon in the compression zone is zero. and These are the cross-sectional areas of the prestressing tendons in the compression and tension zones, respectively. This is the distance from the point of application of the axial force to the point of resultant force of the longitudinal reinforcement; The effective height of the cross-section; It is the distance from the resultant point of the longitudinal reinforcement in the tension zone to the tension edge of the section; This is the distance from the resultant point of the longitudinal reinforcement in the compression zone to the compression edge of the section; It is the distance from the resultant point of the longitudinal prestressing tendons in the compression zone to the compression edge of the section.

[0043] Considering that eccentrically compressed members, which are widely used in practical engineering applications, are mostly reinforced with ordinary steel bars and rarely with prestressed tendons, the UHPC rectangular section eccentrically compressed members in this application are only reinforced with ordinary steel bars and do not have prestressed tendons. The simplified calculation diagram of the section is as follows. Figure 3 and Figure 4 As shown. Based on equations (6) and (7), the formula for calculating the compressive bearing capacity of the eccentrically compressed UHPC rectangular section member without prestressing tendons can be obtained:

[0044] (8)

[0045] (9)

[0046] The relevant equations for the compression-bending capacity of reinforced UHPC eccentrically compressed members should accurately and reasonably reflect the relationship between the axial force and bending moment of the member section under axial compression, small eccentric compression, large eccentric compression, and pure bending conditions. Based on the stress state of symmetrically reinforced UHPC rectangular section eccentrically compressed members under these four stress states, an equation that can accurately reflect the axial force at the member ends is established. and end moment The equation relating these points to each other can be satisfied by the equations relating to the axial compression point, the pure bending point, and the boundary point, with the bending moment being the maximum at the boundary point (i.e., ...). The bearing capacity related equation is as follows. Furthermore, for ease of application, this bearing capacity related equation should not be used to determine whether the load is under large or small eccentric compression.

[0047] The axial compression point refers to the point where a member reaches its maximum compressive bearing capacity under pure axial compression. The formula for calculating the compressive bearing capacity of an axially compressed member in the code is as follows:

[0048] (12)

[0049] In the formula, The value is the stability coefficient of the reinforced UHPC axially compressed member, and is taken according to the relevant provisions of the national standard "Code for Design of Concrete Structures" GB50010-2010; This refers to the cross-sectional area of ​​all longitudinal reinforcing bars.

[0050] Therefore, the ultimate compressive bearing capacity of the normal section of the component under pure axial compression can be obtained. :

[0051] (10)

[0052] The pure bending point typically refers to the point where the tension reinforcement in the bending moment-curvature relationship initially yields. It marks the starting point for a member to transition from the cracked working stage to the failure stage. The formula for calculating the normal section bearing capacity of rectangular cross-section flexural members without considering prestressed tendons in the code is as follows:

[0053] (13)

[0054] The height x of the rectangular stress diagram of the UHPC in the compression zone is calculated according to equation (14):

[0055] (14)

[0056] According to equation (14), and considering symmetrical reinforcement, we have The height x of the UHPC rectangular stress diagram in the compression zone and the height x of the UHPC rectangular stress diagram in the tension zone can be obtained. Relationship:

[0057] (15)

[0058] Meanwhile, according to the relevant provisions of the "Code", for bending members, There is a relationship between x and x:

[0059] (16)

[0060] In the formula, The value represents the ratio of the height x of the rectangular stress diagram of the UHPC in the compression zone to the height of the neutral axis determined according to the plane section assumption, and is taken as 0.8.

[0061] According to equations (15) and (16), we can obtain:

[0062] (17)

[0063] (18)

[0064] Substituting equations (17) and (18) into equation (13), we can obtain the ultimate bending capacity of the cross section of the member under pure bending, i.e., the bending strength. expression:

[0065] (3)

[0066] The limit point refers to the specific stress state in a structural member where, upon reaching its ultimate limit state, the tensile reinforcement yields and the compression zone UHPC (Ultra-High Pressure Processing) crushes simultaneously. When a member exhibits limit failure, the height x of the rectangular stress diagram of the compression zone UHPC and the height x of the rectangular stress diagram of the tension zone UHPC are... They are respectively:

[0067] (19)

[0068] (20)

[0069] In the formula, To define the relative height of the compression zone, for ordinary reinforcing bars with a yield point, we have: ,in, This represents the design value of the tensile strength of the longitudinal reinforcement. This represents the maximum compressive strain of the UHPC. This is the elastic modulus of the reinforcing steel.

[0070] At the same time, there is a relation:

[0071] (twenty one)

[0072] In the formula, It is the distance from the point of application of the axial force to the point of resultant force of the longitudinal tensile reinforcement; This is the distance from the point of application of the axial force to the centroid of the cross section.

[0073] Substituting equations (19), (20), and (21) into equations (8) and (9), we can obtain the limit axial force of the UHPC rectangular section eccentrically compressed member with steel reinforcement. and limit bending moment :

[0074] (4)

[0075] (5)

[0076] In the formula: This is the design value for the axial compressive strength of the UHPC. This represents the equivalent rectangular stress design value for the tension zone of the UHPC, and has... Where k represents the equivalent stress value in the tension zone of the UHPC and the design value of the axial tensile strength of the UHPC. The ratio is taken as 0.4; The width of the cross section; The height of the cross section; The value represents the ratio of the height x of the rectangular stress diagram of the UHPC in the compression zone to the height of the neutral axis determined according to the plane section assumption, and is taken as 0.8. This represents the design value of the compressive strength of the longitudinal reinforcement. This represents the cross-sectional area of ​​the longitudinal reinforcement in the compression zone; This is the effective height of the cross-section, and it has... ; It is the distance from the resultant point of the longitudinal reinforcement in the tension zone to the tension edge of the section; This is the distance from the resultant point of the longitudinal reinforcement in the compression zone to the compression edge of the section; The height of the relative pressure zone is the boundary.

[0077] Based on the stress states of symmetrically reinforced UHPC rectangular section eccentrically compressed members under four stress states—axial compression, small eccentric compression, large eccentric compression, and pure bending—a cubic function equation reflecting the axial force at the member ends is established without distinguishing between large and small eccentric segments. and end moment The interrelationship between them when components fail:

[0078] (1)

[0079] In the formula, These are undetermined coefficients. Because... - The curve must pass through the axial compression point. Pure bending point and boundary point And the bending moment is greatest at the limit point (i.e. That is, it must satisfy:

[0080] (11)

[0081] The undetermined coefficients can be obtained according to equation (11):

[0082] (2)

[0083] For a given reinforced UHPC rectangular cross-section eccentrically compressed member, the UHPC strength grade is UHC120 and the cross-sectional dimensions are... =150mm×300mm, reinforcement ratio of 1.0053%, protective layer thickness of 24mm, the simplified cross-section calculation diagram is as follows. Figure 5 As shown. The NM curve is plotted using the three characteristic points calculated by formulas (10), (3), (4), and (5) of the "Code" and the bearing capacity related equation (1) of this patent, as shown. Figure 6 As shown. From Figure 6 It can be seen that the bearing capacity equation of this patent is consistent with the axial compression point, pure bending point, and boundary point, and meets the theoretical requirement that the boundary point is an extreme point; the bearing capacity equation of this patent is a smooth cubic parabola in both the large and small eccentric compression stages, and the flexural bearing capacity of the section is [not specified] in the small eccentric compression stage. With the cross-sectional compressive bearing capacity The flexural bearing capacity of the section decreases as the eccentric load increases. With the cross-sectional compressive bearing capacity It increases as it increases.

[0084] To verify the calculation accuracy of the bearing capacity-related equations proposed in this application, six sets of test data for reinforced UHPC rectangular cross-section eccentrically loaded columns were collected, and the calculation accuracy of the formulas in this application was compared and analyzed. The ultimate axial force test values ​​from the literature were also included. Substituting the relevant equations for the compressive-bending bearing capacity of this application, i.e., equation (1), the calculated value of the ultimate bending moment is obtained. Then compare it with the ultimate bending moment test value in the literature. (When not given in the literature, use) calculate, The distance from the point of application of the axial force to the centroid of the cross section is compared. The test parameters and calculation results are listed in Table 1.

[0085] Table 1 Comparison of Calculation Results of This Patent with Experimental Data in Literature

[0086]

[0087] [1] Ma Kaize, Ma Yudong, Liu Boquan. Experimental study on eccentric compression performance of ultra-high performance concrete columns [J]. Engineering Science and Technology, 2018, 50(3): 201-208.

[0088] As can be seen from Table 1, the ultimate bending moment calculated based on the bearing capacity-related equations of this application is... Compared with the ultimate bending moment obtained from experiments in the literature The fit is good.

[0089] Therefore, the advantages of this patent over previous technologies are mainly reflected in two aspects: (1) By establishing a cubic function form of the compression-bending bearing capacity correlation equation with unified expressions for large and small eccentricities, the bearing capacity correlation characteristics between axial force and bending moment of reinforced UHPC members under the two failure states of large and small eccentric compression are accurately described. When analyzing the bearing capacity of the members, it is no longer necessary to judge large and small eccentricities, which significantly simplifies the analysis process. (2) The bearing capacity correlation equation matches the axial compression point of axially compressed members and the pure bending point of pure bending members, and the bending moment reaches its maximum at the boundary point, which is theoretically consistent with the stress state of eccentrically compressed members. The bearing capacity correlation equation established by this invention can efficiently and accurately carry out the bearing capacity analysis of eccentrically compressed UHPC rectangular section members.

[0090] The above embodiments are only used to illustrate the technical solutions of the present invention and are not intended to limit them. Any modifications or equivalent substitutions that do not depart from the spirit and scope of the present invention should be covered within the scope of the technical solutions of the present invention.

Claims

1. A method for analyzing the bearing capacity of a reinforced ultra-high performance concrete rectangular section eccentrically loaded member, characterized in that, Based on the stress states of a symmetrically reinforced UHPC rectangular section eccentrically compressed member under axial compression, small eccentric compression, large eccentric compression, and pure bending, a system reflecting the axial force at the member ends is established. and end moment Equations relating the two: (1) In the formula, the coefficients for: (2) In the formula, , These are the compressive strength and flexural strength of the reinforced UHPC member, respectively. , These represent the limit axial force and limit bending moment at the point of limit failure in a reinforced UHPC eccentrically loaded member.

2. The method for analyzing the bearing capacity of a reinforced ultra-high performance concrete rectangular section eccentrically loaded member according to claim 1, characterized in that, The bending strength Boundary axial force and limit bending moment They are represented as follows: (3) (4) (5) In the formula: This is the design value for the axial compressive strength of the UHPC. This represents the equivalent rectangular stress design value for the tension zone of the UHPC, and has... Where k represents the equivalent stress value in the tension zone of the UHPC and the design value of the axial tensile strength of the UHPC. The ratio is taken as 0.4; The cross-sectional area; The width of the cross section; The height of the cross section; The value represents the ratio of the height x of the rectangular stress diagram of the UHPC in the compression zone to the height of the neutral axis determined according to the plane section assumption, and is taken as 0.

8. This represents the design value of the compressive strength of the longitudinal reinforcement. This represents the cross-sectional area of ​​the longitudinal reinforcement in the compression zone. This is the effective height of the cross-section, and it has... ; It is the distance from the resultant point of the longitudinal reinforcement in the tension zone to the tension edge of the section; This is the distance from the resultant point of the longitudinal reinforcement in the compression zone to the compression edge of the section; To define the relative height of the compression zone, for ordinary reinforcing bars with a yield point, we have: ,in, This represents the design value of the tensile strength of the longitudinal reinforcement. The maximum compressive strain of UHPC; This is the elastic modulus of the reinforcing steel.

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