Method for calculating flexural bearing capacity and reinforcement ratio of concrete beam

Abstract

The application provides a calculation method of flexural bearing capacity and reinforcement ratio of a concrete beam, modifies existing theories, carries out test and comprehensive analysis on flexural performance of a basalt fiber hybrid steel fiber ultra-high performance concrete beam through establishment of a finite element model, deduces a calculation formula of flexural bearing capacity of the basalt fiber hybrid steel fiber ultra-high performance concrete beam, compares the calculation result with the measured value to verify the accuracy of the calculation formula, and further obtains calculation formulas of minimum reinforcement ratio and limit balanced reinforcement ratio of the basalt fiber hybrid steel fiber ultra-high performance concrete beam through the flexural performance test. The calculation method provides a reference for application of the basalt fiber hybrid steel fiber ultra-high performance concrete beam in engineering.

Description

Technical Field

[0001] This invention belongs to the field of building materials technology and relates to a method for calculating the parameters of concrete beams. Background Technology

[0002] With the increasing demands for concrete strength and corrosion resistance in modern construction, ultra-high performance concrete (UHVPC) is gaining more and more attention. However, steel fibers, one of the raw materials for UHVPC, are relatively expensive. If ordinary steel bars, which are cheaper, are used as raw materials for UHVPC, they cannot fully utilize the high strength, high toughness, and high durability characteristics of UHVPC. Therefore, both steel fiber reinforced concrete and ordinary reinforced concrete are not widely used in engineering projects.

[0003] Basalt fiber-reinforced composite material, abbreviated as BFRP in English, is a type of fiber-reinforced polymer. Due to its extremely high tensile strength, corrosion resistance, and fatigue resistance, it can replace traditional steel reinforcement. Furthermore, steel fibers in ultra-high performance concrete can effectively improve the defects of basalt reinforcement, such as wide cracks and large deformations caused by its low elastic modulus, highlighting the role of steel fibers. Some scholars have shown that waste tire steel fibers obtained by pyrolyzing waste tire steel wires can replace traditional industrial steel fibers. This not only optimizes the economy of concrete but also meets the current societal requirements for environmental friendliness, achieving sustainable development. Basalt-reinforced steel fiber hybrid ultra-high performance concrete beams can be used as materials for ultra-high performance concrete beams, but there is currently no clear calculation method for their flexural bearing capacity and reinforcement ratio. Summary of the Invention

[0004] To address the problems described in the background art, this invention proposes a method for calculating the flexural bearing capacity and reinforcement ratio of a concrete beam, wherein the concrete beam is a basalt-reinforced steel fiber ultra-high performance concrete beam.

[0005] The technical solution of the present invention includes the following steps:

[0006] Step 1: Measure the flexural capacity of the concrete beam and obtain the measured value;

[0007] Step 2: Establish the finite element model of the concrete beam: Use ABAQUS software to establish the model of the concrete beam and determine the material constitutive relation model of the concrete beam;

[0008] Step 3: Using the finite element model, apply boundary conditions and loads to simulate the bending test of the concrete beam, obtain the simulated value of the bending bearing capacity of the concrete beam, and compare the measured value with the simulated value. If the requirements are not met, repeat steps 2 to 3. If the requirements are met, the accuracy of the finite element model is feasible.

[0009] Step 4: Derive the calculation formula for the flexural bearing capacity of the concrete beam and compare the calculation results with the measured values. If the requirements are not met, repeat steps 2 to 4. If the requirements are met, output the calculation formula for the flexural bearing capacity.

[0010] Step 5: Based on the under-reinforced failure and over-reinforced failure that occurred in the simulated bending test of concrete beams, obtain the calculation formulas for the minimum reinforcement ratio and the boundary equilibrium reinforcement ratio.

[0011] Furthermore, in step two, the material constitutive relation model adopts the plastic damage model, and the compressive constitutive model is:

[0012]

[0013] Where, σ c For concrete compressive stress; f c ε is the axial compressive strength of concrete. c ε0 represents the compressive strain of the concrete; ε0 represents the concrete reaching the required strain f. c Compressive strain at time; ε cu This represents the ultimate compressive strain of the concrete.

[0014] The tension constitutive model is:

[0015]

[0016] Where, σ t For concrete tensile stress; f t E represents the tensile strength of concrete. c ε is the elastic modulus of concrete. t0 ε represents the peak strain of the concrete. tu This represents the ultimate tensile strain of the concrete.

[0017] The constitutive model of basalt ribs is:

[0018]

[0019] Where, σ f For basalt reinforcement strain; E f ε is the elastic modulus of basalt reinforcement; f ε fu These represent the strain of the basalt reinforcement and the ultimate tensile strain, respectively.

[0020] Furthermore, in step three, the ratio of the measured value to the simulated value of the flexural bearing capacity of the concrete beam is required to be ≥0.9 and ≤1.1.

[0021] Furthermore, in step four, the formula for calculating the flexural bearing capacity of the concrete beam is:

[0022]

[0023] Among them, M u For bending bearing capacity, f c denoted as axial compressive strength of concrete, b as beam width, x as the equivalent height of the rectangular stress distribution in the compression zone of the cross-section, h0 as the effective height of the cross-section, and f... t where h is the axial tensile strength of the concrete, h is the beam height, and a is the axial tensile strength of the concrete. f It is the distance from the concrete at the tension edge of the section to the point of resultant force of the basalt reinforcement.

[0024] Furthermore, in step four, the ratio of the calculated value to the measured value of the flexural bearing capacity of the concrete beam is required to be ≥0.9 and ≤1.1.

[0025] Furthermore, in step five, the minimum reinforcement ratio is:

[0026]

[0027] Where, ρ min For the minimum reinforcement ratio, f t f is the axial tensile strength of concrete. fy This represents the nominal yield strength value of the basalt reinforcement.

[0028] The limit equilibrium reinforcement ratio is:

[0029]

[0030] Where, ρ b For the limit equilibrium reinforcement ratio, A f Let b be the cross-sectional area of ​​the basalt reinforcement, h0 be the beam width, and h0 be the effective height of the cross-section. b The relative height of the compression zone is the boundary, α and β are the section design coefficients, and f c Where f is the axial compressive strength of concrete, k is the equivalent tensile stress coefficient, and f is the tensile stress. t f is the axial tensile strength of concrete. fy is the nominal yield strength value of the basalt reinforcement, and h is the beam height.

[0031] Compared with existing technologies, this invention provides a calculation method that modifies existing theories. By establishing a finite element model, it conducts experimental and comprehensive analysis of the flexural performance of basalt-reinforced steel fiber reinforced ultra-high performance concrete beams, deriving a calculation formula for the flexural bearing capacity of these beams. The calculation results are compared with measured values ​​to verify the accuracy of the formula. Furthermore, through flexural performance tests, calculation formulas for the minimum reinforcement ratio and the boundary equilibrium reinforcement ratio of basalt-reinforced steel fiber reinforced ultra-high performance concrete beams are also obtained. This invention's calculation method provides a reference for the engineering application of basalt-reinforced steel fiber reinforced ultra-high performance concrete beams. Attached Figure Description

[0032] Figure 1 This is a schematic diagram of a concrete beam flexural bearing capacity test.

[0033] Figure 2 This is a diagram showing the layout of the measuring points for the electromechanical dial indicator.

[0034] Figure 3 This is a schematic diagram showing the bonding positions of the steel strain gauge and the concrete strain gauge.

[0035] Figure 4 The constitutive relation model for concrete beam material is given, where Figure 4 (a) is the constitutive model of basalt ribs. Figure 4 (b) is the constitutive model of ordinary HPB300 steel bars.

[0036] Figure 5 A stress-strain model for a basalt-reinforced steel fiber reinforced ultra-high performance concrete beam is provided, in which... Figure 5 (a) is the cross-sectional form. Figure 5 (b) represents the strain distribution. Figure 5 (c) shows the stress distribution. Figure 5 (d) represents the equivalent rectangular stress distribution.

[0037] Wherein: Q - jack; J - electromechanical dial gauge; G - steel reinforcement strain gauge; H - concrete strain gauge; B - basalt reinforcement. Detailed Implementation

[0038] The embodiments of the present invention are described in detail below with reference to the accompanying drawings. However, these descriptions do not constitute a limitation of the present invention and are merely illustrative. Through these descriptions, the advantages of the present invention will become clearer. All modifications that can be directly derived or conceived by those skilled in the art from the disclosure of the present invention should be considered within the scope of protection of the present invention. The positional relationships described in the embodiments are consistent with those shown in the accompanying drawings. Other parts not described in detail in the embodiments are prior art.

[0039] Example

[0040] Five basalt-reinforced steel fiber reinforced ultra-high performance concrete beams were prepared. The specified dimensions of the concrete beams were 120mm×200mm×2000mm, with dimensional deviations within 2%. In accordance with the Class I environmental requirements of the standard, a protective layer thickness of 15mm was set, the stirrup spacing was 100mm, and 8 stirrups were set at each of the left and right ends. No stirrups were set in the pure bending section.

[0041] The flexural capacity and reinforcement ratio are calculated below.

[0042] 1. Testing the flexural capacity of concrete beams

[0043] According to the requirements of GB / T 50152-2012 "Standard for Test Methods of Concrete Structures" and the estimated ultimate load value, a 100T universal testing machine was used for the test. The test diagram is shown below. Figure 1 As shown.

[0044] The test method is as follows:

[0045] (1) Preloading: In order for the loading equipment to work properly and to ensure that the loading point is in full contact with the test beam, preloading is required. The basic principle of preloading is that it should not affect the load-bearing capacity of the test beam. Therefore, the maximum value of preloading should not exceed half of the theoretical calculation value of cracking load. It is loaded in two stages. If a gap is found between the loading point and the test beam, it should be filled with sand to fill the gap. After confirming that everything is normal, unload to zero.

[0046] (2) Formal loading: The loading method is adopted in stages. The loading rate is controlled at 0.2kN / s, and the load step is 10kN per stage. When the load approaches the theoretical value of cracking load, the load step is changed to 2kN per stage. After the beam cracks, the load step is changed to 5kN per stage. The duration of each load stage is 5min. During the load period, the evolution of cracks is observed and the width is measured in time.

[0047] (3) Failure Criteria: When the reading of dial gauge J increases rapidly and the load value stops increasing or increases slowly, it indicates that the test beam is nearing failure. At this time, reduce the loading rate and turn on the video recorder to record the entire process of beam failure. When the ultra-high performance concrete at the beam loading point is crushed or the lower concrete cracks excessively, it is considered that the beam has lost its load-bearing capacity, and the test ends.

[0048] The data collection method is as follows:

[0049] (1) Load: Load data are collected using the pressure sensor on the loading device, especially the cracking load and ultimate load.

[0050] (2) Deflection of the test beam: The deflection was measured using an electromechanical dial indicator J. Five dial indicators were placed: one at the midpoint of the lower end of the span, 300 mm to the left and right of the midpoint of the lower end of the span, and one at the top of each end support. The layout of the measuring points is shown in the diagram below. Figure 2 .

[0051] (3) Basalt reinforcement strain: The strain of the basalt reinforcement was measured using strain gauges G. The strain gauges were attached before the test beam was poured. The attachment locations of the strain gauges G are shown in [reference needed]. Figure 3 .

[0052] (4) Concrete strain: Concrete strain was measured using concrete strain gauges H. Five concrete strain gauges H were evenly spaced at the top and bottom of one side of the test beam at mid-span. The placement of the concrete strain gauges H is shown in the figure. Figure 3 .

[0053] (5) Cracks: The surface of the test beam is painted white with putty. A square grid with a side length of 50mm is drawn on any side with an ink line. During the load holding stage, the evolution process of the cracks is recorded with a black marker. The cracks are numbered according to the order in which they appear and the current load value is marked at the end of the crack. The crack width is measured with a crack observation instrument, and the crack width corresponding to each load level is recorded on the record table.

[0054] Cracking load F of a 5-basalt-reinforced steel fiber reinforced ultra-high performance concrete beam cr and ultimate load F u See Table 1.

[0055] Table 1 Test results of concrete beams

[0056]

[0057]

[0058] 2. Establish a finite element model of a basalt-reinforced steel fiber reinforced ultra-high performance concrete beam.

[0059] A finite element model is established using ABAQUS. In order to comprehensively analyze the crushing or tensile cracking behavior of concrete, the constitutive model of ultra-high performance concrete adopts a plastic damage model, which is characterized by its ability to simulate the brittle characteristics of concrete very well.

[0060] The constitutive model under compression is shown in equation (1), the constitutive model under tension is shown in equation (2), the constitutive model of basalt tendons is shown in equation (3), and the constitutive relation model is shown in equation (3). Figure 4 (a); The stirrups and reinforcing bars of the concrete beam are ordinary steel bars of type HPB300. The constitutive model is regarded as an ideal elastic-plastic model, see Figure 4 (b)

[0061]

[0062] In the formula, σ c For concrete compressive stress; f c ε is the axial compressive strength of concrete. c ε0 represents the compressive strain of the concrete; ε0 represents the concrete reaching the required strain f. c Compressive strain at time; ε cu This represents the ultimate compressive strain of concrete.

[0063]

[0064] In the formula, σ t For concrete tensile stress; f t E represents the tensile strength of concrete. c ε is the elastic modulus of concrete. t0 ε represents the peak strain of the concrete. tuThis represents the ultimate tensile strain of the concrete.

[0065]

[0066] In the formula, σ f For basalt reinforcement strain; E f ε is the elastic modulus of basalt reinforcement; f ε fu These represent the strain of the basalt reinforcement and the ultimate tensile strain, respectively.

[0067] 3. Verify the accuracy of the finite element model.

[0068] Using a finite element model, boundary conditions and loads are applied to simulate the bending test of a concrete beam, and the simulated value of the flexural bearing capacity of the concrete beam is obtained. The measured value is compared with the simulated value. In this embodiment, the ratio of the measured value to the simulated value is required to be ≥0.9 and ≤1.1.

[0069] Table 2 shows the simulated and measured values ​​of the ultimate bending bearing capacity of the 5-basalt-reinforced steel fiber reinforced ultra-high performance concrete beam. As can be seen from Table 2, the accuracy of the finite element model is feasible.

[0070] Table 2 Comparison of simulated and measured values ​​of ultimate flexural bearing capacity of concrete beams

[0071] Concrete beam number Measured value / kN Simulated value / kN Measured value / Simulated value ① 142.98 158.5 0.90 ② 145.70 156.8 0.93 ③ 152.60 153.2 1.00 ④ 138.42 148.3 0.93 ⑤ 132.84 144.8 0.92

[0072] 4. Calculation formula for the flexural bearing capacity of basalt-reinforced steel fiber reinforced ultra-high performance concrete beams

[0073] 4.1 Basic Assumptions

[0074] (1) The stress-strain model of the basalt-reinforced steel fiber reinforced ultra-high performance concrete beam is shown in Figure 5 As shown.

[0075] (2) The basalt-reinforced steel fiber ultra-high performance concrete beam conforms to the plane section assumption throughout the bending process.

[0076] (3) Stress-strain of basalt tendons

[0077] Based on material tests of basalt tendons, their stress-strain relationship is shown in equation (4):

[0078]

[0079] In the formula, σ f For basalt reinforcement tensile stress; E f ε is the elastic modulus of basalt reinforcement; f The strain is tensile strain in the basalt reinforcement.

[0080] (4) Referring to the ideal elastic-plastic steel fiber reinforced concrete constitutive model, the compressive stress-strain relationship of the basalt-reinforced steel fiber reinforced ultra-high performance concrete beam is shown in equation (5):

[0081]

[0082] In the formula, σ c The compressive stress of basalt-reinforced steel fiber reinforced ultra-high performance concrete beams; f c The axial compressive strength of basalt-reinforced steel fiber reinforced ultra-high performance concrete beams; ε c The compressive strain of the basalt-reinforced steel fiber reinforced ultra-high performance concrete beam is ε0; ε0 represents the compressive strain of the basalt-reinforced steel fiber reinforced ultra-high performance concrete beam reaching f. c The compressive strain at time ε0 is taken as ε0=0.0025+(f) according to the Technical Specification for Reactive Powder Concrete Structures (DBJ43 / T325-2017). cuk -100)×10 -5 ;ε cu The ultimate compressive strain of the basalt-reinforced steel fiber reinforced ultra-high performance concrete beam is taken as ε according to the "Technical Specification for Reactive Powder Concrete Structures" (DBJ43 / T325-2017). cu =0.0042-0.3×(f cuk -100)×10 -5 .

[0083] (5) Ignore the bond slip effect between basalt reinforcement and basalt reinforcement mixed with steel fiber ultra-high performance concrete beam.

[0084] 4.2 Equivalent stress diagram of the compression zone of a concrete beam's normal section

[0085] The resultant force of the basalt-reinforced steel fiber reinforced ultra-high performance concrete beam in the compression zone is:

[0086]

[0087] In the formula, C is the resultant force of the concrete in the compression zone, b is the beam width, and σ c The compressive stress of the basalt-reinforced steel fiber reinforced ultra-high performance concrete beam at a distance y from the neutral axis is ε. c The compressive strain of the basalt-reinforced steel fiber reinforced ultra-high performance concrete beam is shown.

[0088] The normal section of the concrete beam conforms to the plane section assumption, and the compressive strain ε of the concrete at point y on its neutral axis is... c for:

[0089]

[0090] In the formula, ε cuThe ultimate compressive strain of the basalt-reinforced steel fiber reinforced ultra-high performance concrete beam is taken as ε according to the "Technical Specification for Reactive Powder Concrete Structures" (DBJ43 / T325-2017). cu =0.0042-0.3×(f cuk -100)×10 -5 .

[0091] The distance from the neutral axis to the resultant force of the concrete in the compression zone is:

[0092]

[0093] Among them, y c x is the distance from the neutral axis to the resultant force of the concrete in the compression zone. c σ represents the height of the concrete in the compression zone of the test beam under ultimate stress. c The compressive stress of the basalt-reinforced steel fiber reinforced ultra-high performance concrete beam at a distance y from the neutral axis is ε. c The compressive strain of the basalt-reinforced steel fiber reinforced ultra-high performance concrete beam is shown.

[0094] The distance yu from the centroid of the area bounded by the stress-strain curves of the compression zone of the basalt-reinforced steel fiber reinforced ultra-high performance concrete beam to the neutral axis is given by:

[0095]

[0096] Where, ε cu The ultimate compressive strain of a basalt-reinforced steel fiber reinforced ultra-high performance concrete beam. σ c represents the compressive stress in the basalt-reinforced steel fiber reinforced ultra-high performance concrete beam at a distance y from the neutral axis, and ε represents the stress at which the beam is reinforced with steel fibers. c The compressive strain of the basalt-reinforced steel fiber reinforced ultra-high performance concrete beam is shown.

[0097] make The bending moment of the basalt-reinforced steel fiber reinforced ultra-high performance concrete beam in the compression zone can be obtained as follows:

[0098] M c =C(y0+h0-x) c )=k1f c bx c [h0-(1-k2)x c ]=αf c bβx c (10),

[0099] In the formula, C is the resultant force of the concrete in the compression zone, y0 is the distance from the neutral axis to the resultant force of the concrete in the compression zone, h0 is the effective height of the section, and x c f represents the height of the concrete in the compression zone of the test beam under ultimate condition. cα is the axial compressive strength of concrete, b is the beam width, and α and β are the section design coefficients.

[0100] Substituting (7) into (6), we can obtain the resultant force C of the concrete in the compression zone as:

[0101]

[0102] In the formula, C is the resultant force of the concrete in the compression zone, and ε u The ultimate compressive strain of the basalt-reinforced steel fiber reinforced ultra-high performance concrete beam is σ. c The compressive stress of the basalt-reinforced steel fiber reinforced ultra-high performance concrete beam at a distance y from the neutral axis is ε. c The compressive strain of the basalt-reinforced steel fiber reinforced ultra-high performance concrete beam is given by b, where b is the beam width and x is the cross-section of the beam. c f represents the height of the concrete in the compression zone of the test beam under ultimate condition. c This refers to the axial compressive strength of concrete.

[0103] Substituting (7) into (8), we can obtain the distance from the resultant force C of the concrete in the compression zone to the neutral axis as:

[0104]

[0105] In the formula, y c x is the distance from the neutral axis to the resultant force of the concrete in the compression zone. c σ represents the height of the concrete in the compression zone of the test beam under ultimate stress. c The compressive stress of the basalt-reinforced steel fiber reinforced ultra-high performance concrete beam at a distance y from the neutral axis is ε. c The compressive strain of the basalt-reinforced steel fiber reinforced ultra-high performance concrete beam is given by b, where b is the beam width and y is the cross-section of the beam. u Let ε be the distance from the centroid of the area bounded by the stress-strain curves in the compression zone of the concrete beam to the neutral axis. u The ultimate compressive strain of a basalt-reinforced steel fiber reinforced ultra-high performance concrete beam.

[0106] Depend on Figure 5 The stress-strain model of the basalt-reinforced steel fiber reinforced ultra-high performance concrete beam shown, and equations (10) and (11) can be obtained as follows:

[0107] β=2(1-k2) (13),

[0108] In the formula, β is the cross-sectional design coefficient.

[0109] After sorting, we can obtain:

[0110]

[0111] In the formula, α and β are the cross-sectional design coefficients.

[0112] From the constitutive model equation of the basalt-reinforced steel fiber reinforced ultra-high performance concrete beam in equation (5), we can obtain:

[0113]

[0114]

[0115] ε0 represents the f-value of basalt-reinforced steel fiber reinforced ultra-high performance concrete beams. c The compressive strain at time ε0 is taken as ε0=0.0025+(f) according to the Technical Specification for Reactive Powder Concrete Structures (DBJ43 / T325-2017). cuk -100)×10 -5 ;ε cu The ultimate compressive strain of the basalt-reinforced steel fiber reinforced ultra-high performance concrete beam is taken as ε according to the "Technical Specification for Reactive Powder Concrete Structures" (DBJ43 / T325-2017). cu =0.0042-0.3×(f cuk -100)×10 -5 The cubic compressive strength f of basalt-reinforced steel fiber reinforced ultra-high performance concrete beams cuk Substituting these values ​​into the above ranges yields ε0 and ε. cu In this embodiment, the ratio of waste tire steel fibers to traditional industrial steel fibers in the basalt-reinforced hybrid steel fiber ultra-high performance concrete beam is 6:4. This ratio is a common proportion for hybrid steel fibers. The measured cubic compressive strength f... cuk =159.8MPa, calculated ε0 = 0.003098, ε cu =0.04182.

[0116] Let ε0 = 0.003098, ε cu Substituting =0.04182 into equations (15) and (16), we get k1=0.743 and k2=0.593.

[0117] Substituting k1 = 0.743 and k2 = 0.593 into (13) and (14), we get α = 0.91 and β = 0.81.

[0118] The equivalent rectangular stress distribution height in the compression zone of the cross-section is x = 0.81x. c The distributed stress value is 0.91f. c .

[0119] 4.3 Equivalent stress diagram of the tension zone of a concrete beam's normal section

[0120] The height of the ultra-high performance concrete beam with basalt reinforcement and steel fiber in the tension zone is taken as the portion below the neutral axis of the cross section.

[0121]

[0122] In the formula, h is the beam height, x c β represents the concrete height in the compression zone of the test beam under ultimate condition, and β is the section design coefficient.

[0123] According to the equilibrium condition of the cross-sectional forces, we have:

[0124] α1f c bx = f fy A f +kf t bx t (18),

[0125] In the formula, f c denoted as y = axial compressive strength of concrete, b is the beam width, x is the equivalent height of the rectangular stress distribution in the compression zone of the cross-section, and f is the beam width. fy Let f be the nominal yield strength value of the basalt reinforcement. fy =0.8f fu ;f fu A represents the tensile strength of the basalt reinforcement. f denoted as , where is the cross-sectional area of ​​the basalt reinforcement; k is the tensile stress equivalence coefficient.

[0126] Moment equilibrium condition for taking moments about the point of application of the resultant force of the basalt reinforcement:

[0127]

[0128] In the formula, f c denoted as axial compressive strength of concrete, b as beam width, x as the equivalent height of the rectangular stress distribution in the compression zone of the cross-section, h0 as the effective height of the cross-section, and f... t a is the axial tensile strength of concrete. f The distance from the concrete at the tension edge of the cross section to the resultant point of the basalt reinforcement is given by α, where α is the cross section design factor, k is the tensile stress equivalent factor, and f is the tensile stress equivalent factor. t This represents the axial tensile strength of concrete.

[0129] Combining (17), (18), and (19), we get:

[0130]

[0131] Where α and β are the cross-sectional design coefficients, f c denoted as y = axial compressive strength of concrete, b is beam width, x is the equivalent height of the rectangular stress distribution in the compression zone of the cross-section, h is beam height, and f is... fy A represents the nominal yield strength of the basalt reinforcement. f Let a be the cross-sectional area of ​​the basalt reinforcement. f M is the distance from the concrete at the tension edge of the cross section to the resultant force point of the basalt reinforcement. u This refers to the bending bearing capacity.

[0132] Substituting the x value calculated in (20) into equation (18), we can obtain the tensile stress equivalent coefficient k of each basalt-reinforced steel fiber ultra-high performance concrete beam, as shown in Table 3.

[0133] Table 3 Equivalent coefficients k for tensile stress in various concrete beams

[0134]

[0135] In practical applications, for this basalt-reinforced steel fiber ultra-high performance concrete beam, the tensile stress equivalence coefficient can be uniformly taken as a safe value, k = 0.4.

[0136] 4.4 Formula for calculating the flexural capacity of a beam's normal section

[0137] Based on the above analysis, the formula for calculating the flexural bearing capacity of basalt-reinforced steel fiber reinforced ultra-high performance concrete beams can be derived as follows:

[0138]

[0139] Among them, M u For bending bearing capacity, f c denoted as axial compressive strength of concrete, b as beam width, x as the equivalent height of the rectangular stress distribution in the compression zone of the cross-section, h0 as the effective height of the cross-section, and f... t where h is the axial tensile strength of the concrete, h is the beam height, and a is the axial tensile strength of the concrete. f It is the distance from the concrete at the tension edge of the section to the point of resultant force of the basalt reinforcement.

[0140] The measured value of the ultimate bending bearing capacity of the 5-basalt-reinforced steel fiber ultra-high performance concrete beam was compared with the calculated value of formula (21). In this embodiment, the ratio of the calculated value to the measured value is required to be ≥0.9 and ≤1.1.

[0141] The measured and calculated values ​​of the ultimate bending bearing capacity of the 5-basalt-reinforced steel fiber ultra-high performance concrete beam are shown in Table 4. As can be seen from Table 4, the ratio of the calculated value to the experimental value of formula (21) is ≥0.9 and ≤1.1, with an average ratio of 0.962 and a standard deviation of 0.031. The calculated value is very close to the measured value and relatively stable, which is in line with engineering applications. The accuracy of the formula for calculating the bending bearing capacity is feasible.

[0142] Table 4 Comparison of Measured and Calculated Values ​​of Ultimate Bending Capacity of Concrete Beams

[0143] Concrete beam number Measured value / kN Calculated value / kN Measured value / Calculated value ① 142.98 117.95 0.93 ② 145.70 115.23 0.96 ③ 152.60 113.25 1.02 ④ 138.42 108.53 0.95 ⑤ 132.84 105.32 0.95 average value / / 0.962 Standard deviation / / 0.031

[0144] 5. Formula for calculating the reinforcement ratio of basalt-reinforced steel fiber reinforced ultra-high performance concrete beams

[0145] 5.1 Minimum reinforcement ratio

[0146] In bending tests, when a concrete beam fails due to under-reinforcement, it fails before it can fully utilize its superior load-bearing capacity. To avoid under-reinforcement failure, it is necessary to control the minimum reinforcement ratio.

[0147] Minimum reinforcement ratio:

[0148]

[0149] Where, ρ min For the minimum reinforcement ratio, f t f is the axial tensile strength of concrete. fy This represents the nominal yield strength value of the basalt reinforcement.

[0150] 5.2 Boundary Balance Reinforcement Ratio

[0151] In bending tests, when a concrete beam fails due to over-reinforcement, there are no obvious signs before failure, which is a brittle failure. To avoid over-reinforcement failure, it is necessary to control the limit equilibrium reinforcement ratio.

[0152] The relative height of the pressure zone ξ b for:

[0153]

[0154] Where β is the cross-sectional design coefficient, f fy E represents the nominal yield strength of the basalt reinforcement. s It is the elastic modulus.

[0155] When the reinforcement ratio of the test beam exceeds the limit equilibrium reinforcement ratio, the actual relative limit compression zone height ξ > ξ b The boundary between over-reinforced and appropriately reinforced beams is ξ=ξ b At this point, according to the equilibrium of forces in the cross section, we can obtain:

[0156]

[0157] Among them, f fy A represents the nominal yield strength of the basalt reinforcement. f Where is the cross-sectional area of ​​the basalt reinforcement, k is the tensile stress equivalence coefficient, and f is the cross-sectional area of ​​the basalt reinforcement. t denoted as axial tensile strength of concrete, b as beam width, h as beam height, and f as axial tensile strength. c α represents the axial compressive strength of concrete, and β represents the cross-sectional design coefficients.

[0158] so:

[0159]

[0160] Where, ξ b f represents the relative height of the pressure zone.fy A represents the nominal yield strength of the basalt reinforcement. f Where is the cross-sectional area of ​​the basalt reinforcement, k is the tensile stress equivalence coefficient, and f is the cross-sectional area of ​​the basalt reinforcement. t denoted as axial tensile strength of concrete, b as beam width, h as beam height, and f as axial tensile strength. c α represents the axial compressive strength of concrete, and β represents the cross-sectional design coefficients.

[0161] From equation (25), the limit equilibrium reinforcement ratio can be obtained as follows:

[0162]

[0163] Where, ρ b For the limit equilibrium reinforcement ratio, A f Let b be the cross-sectional area of ​​the basalt reinforcement, h0 be the beam width, and h0 be the effective height of the cross-section. b The height of the relative compression zone is the boundary, α is the section design factor, and f c Where f is the axial compressive strength of concrete, k is the equivalent tensile stress coefficient, and f is the tensile stress. t β is the axial tensile strength of concrete, β is the section design factor, and f is the cross-sectional design factor. fy is the nominal yield strength value of the basalt reinforcement, and h is the beam height.

[0164] Of course, the above calculations are for technical reference only. In practical applications, it is also necessary to determine the boundary equilibrium reinforcement ratio of basalt-reinforced steel fiber ultra-high performance concrete beams based on the actual cross-sectional structure of the concrete beams.

[0165] The preferred embodiments of the present invention have been described in detail above with reference to the accompanying drawings and specific examples. However, the present invention is not limited to the specific details in the above embodiments. Within the scope of the technical concept of the present invention, various simple modifications can be made to the technical solution of the present invention, and these simple modifications all fall within the protection scope of the present invention.

Claims

1. A method for calculating the flexural bearing capacity and reinforcement ratio of a concrete beam, characterized in that, Includes the following steps: Step 1: Measure the flexural capacity of the concrete beam and obtain the measured value; Step 2: Establish the finite element model of the concrete beam: Use ABAQUS software to establish the model of the concrete beam and determine the material constitutive relation model of the concrete beam; Step 3: Using the finite element model, apply boundary conditions and loads to simulate the bending test of the concrete beam, obtain the simulated value of the bending bearing capacity of the concrete beam, and compare the measured value with the simulated value. If the requirements are not met, repeat steps 2 to 3. If the requirements are met, the accuracy of the finite element model is feasible. Step 4: Derive the calculation formula for the flexural bearing capacity of the concrete beam and compare the calculation results with the measured values. If the requirements are not met, repeat steps 2 to 4. If the requirements are met, output the calculation formula for the flexural bearing capacity. The formula for calculating the flexural bearing capacity of the concrete beam is as follows: ， in, For bending bearing capacity, This refers to the axial compressive strength of concrete. For the width of the beam, This represents the equivalent height of the rectangular stress distribution in the compression zone of the cross-section. The effective height of the cross section, This refers to the axial tensile strength of concrete. For Liang Gao, The distance from the concrete at the tension edge of the section to the point of resultant force of the basalt reinforcement; Step 5: Based on the under-reinforced and over-reinforced failures that occurred in the simulated bending tests of concrete beams, obtain the calculation formulas for the minimum reinforcement ratio and the boundary equilibrium reinforcement ratio. The minimum reinforcement ratio is: ， in, To minimize the reinforcement ratio, This refers to the axial tensile strength of concrete. This represents the nominal yield strength value of the basalt reinforcement. The limit equilibrium reinforcement ratio is: ， in, To determine the boundary equilibrium reinforcement ratio, This represents the cross-sectional area of ​​the basalt reinforcement. For the width of the beam, The effective height of the cross section, The height of the relative compression zone is the boundary, and α and β are the section design coefficients. This refers to the axial compressive strength of concrete. The tensile stress equivalent coefficient is... This refers to the axial tensile strength of concrete. This represents the nominal yield strength value of the basalt reinforcement. For Liang Gao; The concrete beam is a basalt-reinforced steel fiber ultra-high performance concrete beam.

2. The method for calculating the flexural bearing capacity and reinforcement ratio of a concrete beam according to claim 1, characterized in that: In step two, the material constitutive relation model adopts the plastic damage model, and the compressive constitutive model is as follows: ， in, This refers to the compressive stress in the concrete. This refers to the axial compressive strength of concrete. This represents the compressive strain of the concrete. For concrete to reach Compressive strain at time; This represents the ultimate compressive strain of the concrete. The tension constitutive model is: ， in, This refers to the tensile stress in the concrete. It refers to the tensile strength of concrete; The elastic modulus of concrete; This represents the peak strain of the concrete. This represents the ultimate tensile strain of the concrete. The constitutive model of basalt ribs is: ， in, For basalt reinforcement strain; The elastic modulus of basalt reinforcement; , These represent the strain of the basalt reinforcement and the ultimate tensile strain, respectively.

3. The method for calculating the flexural bearing capacity and reinforcement ratio of a concrete beam according to claim 2, characterized in that: In step three, the ratio of the measured value to the simulated value of the flexural bearing capacity of the concrete beam is required to be ≥0.9 and ≤1.

1.

4. The method for calculating the flexural bearing capacity and reinforcement ratio of a concrete beam according to claim 1, characterized in that: In step four, the ratio of the calculated value to the measured value of the flexural bearing capacity of the concrete beam must be ≥0.9 and ≤1.1.

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