Post-tensioned prestressed fiber reinforced rib beam reinforcement method for crack control under hot and humid environment

Abstract

This invention provides a method for reinforcing post-tensioned prestressed fiber-reinforced beams to control cracks in humid and hot environments. The specific steps are as follows: First, the area of ​​the bonded auxiliary fiber-reinforced reinforcement is obtained based on the lower limit of its reinforcement area. Second, a bond-slip model of the bonded auxiliary fiber-reinforced reinforcement and concrete is established under standard and humid and hot environments. The bond-slip constant and the amplification factor of the beam crack width under humid and hot environments are solved, and the height of the compression zone of the fiber-reinforced beam is obtained based on the amplification factor. Third, the area of ​​the i-th layer of unbonded prestressed fiber-reinforced reinforcement is obtained based on the minimum reinforcement area of ​​the i-th layer of unbonded prestressed fiber-reinforced reinforcement in the fiber-reinforced beam. Fourth, bonded auxiliary fiber-reinforced reinforcement and unbonded prestressed fiber-reinforced reinforcement are arranged, concrete is poured, and prestressing is applied to obtain the post-tensioned prestressed fiber-reinforced beam. This invention accurately and conveniently calculates the limit value of the reinforcement amount, avoiding repeated verification of the maximum crack width of the beam.

Description

Technical Field

[0001] This invention relates to the field of prestressed bridge design technology, and in particular to a method for reinforcing post-tensioned prestressed fiber-reinforced beams to control cracks in humid and hot environments. Background Technology

[0002] Currently, steel corrosion is a significant cause of insufficient durability in concrete structures. In coastal and saline areas of my country, bridge corrosion is a frequent problem, severely impacting structural lifespan and increasing maintenance and reconstruction costs. Considering the corrosion-resistant properties of fiber-reinforced bars, their use as a substitute for steel reinforcement in humid and highly corrosive island environments significantly improves bridge structural durability, resulting in substantial overall benefits.

[0003] Compared to steel reinforcement, fiber-reinforced bars (FRPs) have higher strength and lower elastic modulus. Beams reinforced with FRPs exhibit rapid deflection and crack width development after cracking, and the design of such members generally prioritizes serviceability. Using FRPs as prestressing tendons in beam structures allows their high-strength characteristics to be fully utilized, improving the serviceability of beam members. While FRPs exhibit brittle fracture characteristics, post-tensioned unbonded prestressing technology allows the deformation of the prestressed FRPs to be unrestricted by cross-sectional strain compatibility, delaying fracture and thus ensuring the ductile failure mode of the beam member.

[0004] Compared to normal environments, under long-term humid and hot conditions, the deterioration of the fiber-reinforced reinforcing bar ribs leads to a degradation of their interfacial bond performance with concrete, causing cracks in fiber-reinforced beams to develop more rapidly. Currently, research has been conducted both domestically and internationally on calculation methods for post-tensioned prestressed fiber-reinforced beams under normal conditions, and corresponding calculation programs have been provided. However, there are few reports on design methods for post-tensioned prestressed fiber-reinforced beams under humid and hot conditions. Therefore, this invention proposes a design method for post-tensioned prestressed fiber-reinforced beams under humid and hot conditions, making the engineering application of fiber-reinforced bars in island and reef environments more practical. Summary of the Invention

[0005] To address the problems existing in the prior art, this invention provides a method for reinforcing post-tensioned prestressed fiber-reinforced beams to control cracks in humid and hot environments. Bond-slip models under standard and humid and hot environments are established through bond-slip tests on center pull-out specimens of fiber-reinforced concrete composed of bonded auxiliary fiber-reinforced bars and concrete. The deformation difference between the bonded auxiliary fiber-reinforced bars and concrete is solved using the bond-slip differential equation, quantitatively describing the impact of the degradation of interfacial bonding performance caused by humid and hot environments on crack propagation. Based on the maximum crack width requirement under normal serviceability limits, the lower limits of the reinforcement amounts of bonded auxiliary fiber-reinforced bars and unbonded prestressed fiber-reinforced bars are calculated, avoiding repeated verification of the normal serviceability limits during the design process and reducing the waste of fiber-reinforced bar materials.

[0006] This invention provides a method for reinforcing post-tensioned prestressed fiber-reinforced beams to control cracks in humid and hot environments. The specific implementation steps include:

[0007] S1. Based on the dimensions and material properties of the fiber-reinforced beam, obtain the lower limit A of the reinforcement area for bonded fiber-reinforced reinforcement. bf,min The specific expression is as follows:

[0008]

[0009] Among them, f t d represents the tensile strength of concrete. b denoted as , where is the distance from the resultant force point of the bonded fiber-reinforced reinforcement to the compression edge of the section; , where is the width of the fiber-reinforced beam; and , where is the distance from ... bfu The tensile strength of the bonded fiber-reinforced rib;

[0010] S2, based on the lower limit A of the reinforcement area with bonded auxiliary fiber reinforcement. bf,min Determine the diameter d of the bond-supported fiber reinforcement bars in the fiber-reinforced beam. f And the quantity, and obtain the area A of the bond-assisted fiber reinforcement. bf ;

[0011] S3. Conduct bond-slip tests on the center pull-out specimens of fiber-reinforced concrete consisting of bonded auxiliary fiber-reinforced bars and concrete under standard and humid environments, and establish a bond-slip model under standard and humid environments.

[0012] S4. Based on the bond-slip model obtained in S3, solve for the bond-slip constant υ of the fiber-reinforced beam under standard and humid / heated conditions. N With υ TH And the amplification factor λ of the crack width in the fiber-reinforced beam under humid and hot conditions compared to standard conditions, the expression for the amplification factor λ of the crack width in the fiber-reinforced beam is:

[0013]

[0014] In the formula, w N With w TH The average crack widths under standard and humid / hot environments are respectively, l j This represents the average crack spacing.

[0015] S5. Based on the amplification factor λ of the crack width in the fiber-reinforced beam under humid and hot conditions compared to the standard conditions obtained in S4, the height c of the compression zone of the fiber-reinforced beam based on the amplification factor λ is obtained. w ;

[0016] S6. Based on the principle of minimizing the number of unbonded prestressed fiber-reinforced reinforcement layers in fiber-reinforced beams, obtain the minimum reinforcement area A of the i-th layer of unbonded prestressed fiber-reinforced reinforcement in the fiber-reinforced beam. fpi,min The specific expression is:

[0017]

[0018] In the formula, f pe For effective prestress, k is the deflection coefficient of the simply supported beam, η is the elongation coefficient of the unbonded prestressed fiber-reinforced tendon, and E pf e represents the elastic modulus of unbonded prestressed fiber-reinforced tendons. i ε is the distance from the point of application of the tensile force in the i-th layer of unbonded prestressed fiber-reinforced tendon to the centroid of the cross-section. bf,w For the strain of bonded fiber-reinforced ribs, d p E is the distance from the resultant point of the unbonded prestressed fiber-reinforced tendon to the compression edge of the section. bf For the elastic modulus of bonded fiber-reinforced ribs, M d Use bending moment in the design of fiber-reinforced beams;

[0019] S7. Based on the minimum reinforcement area A of the i-th layer of unbonded prestressed fiber-reinforced tendons in the fiber-reinforced beam. fpi,min The area A of the i-th layer of unbonded prestressed fiber reinforcement in the fiber-reinforced beam is obtained. fpi .

[0020] Preferably, in step S3, the slippage of the bond-assisted fiber reinforcement loading end in the bond-slip model is 0.21 mm.

[0021] Preferably, in step S3, the standard environment refers to an IA environmental action level, and the humid and hot environment refers to a temperature between 35℃ and 50℃ and a relative humidity ≥50%.

[0022] Preferably, in step S3, the specific expression of the bond-slip model is:

[0023]

[0024]

[0025] In the formula, τ N With τ TH The bond stresses τ between bond-assisted fiber-reinforced reinforcements and concrete under standard and humid / hot environments are respectively. 0.21,N With τ 0.21,TH The bonding stresses, s, are the bonding stresses corresponding to a slippage of 0.21 mm in the standard environment and the humid heat environment, respectively. 0.21This indicates a slip of 0.21 mm, where s is the slip amount.

[0026] Preferably, in step S4, the adhesive slip constant υ of the standard environment N and the bond slip constant υ in the humid and hot environment TH The expressions are as follows:

[0027]

[0028]

[0029] In the formula, A fb1 A represents the area of ​​a single bonded auxiliary fiber reinforced bar. c1 L represents the effective tensile area of ​​the concrete corresponding to a single bonded auxiliary fiber reinforced bar. per E is the perimeter of a single bonded auxiliary fiber reinforced bar. c This refers to the elastic modulus of concrete.

[0030] Preferably, in step S5, the height c of the compression zone of the fiber-reinforced rib beam is... w The specific expression is as follows:

[0031]

[0032] in:

[0033]

[0034]

[0035]

[0036]

[0037]

[0038] In the formula, d2 is the distance from the edge of the tensile zone of the concrete to the neutral axis, and d b1 c is the distance from the resultant point of the bonded fiber-reinforced rib to the neutral axis. c c is the tensile influence coefficient of the concrete between cracks. p w is the ratio of the maximum crack width to the average crack width. lim This is the limit value for the crack width.

[0039] Compared with the prior art, the present invention has the following advantages:

[0040] 1. This invention utilizes the bond-slip differential equation to solve for the deformation difference between the bonded auxiliary fiber-reinforced reinforcement and the surrounding concrete, rationally quantifying the impact of interface bond performance degradation caused by humid and hot environments on crack propagation. It establishes a reinforcement method for post-tensioned prestressed fiber-reinforced beams to control cracks under humid and hot environments, which is beneficial for promoting the application of fiber-reinforced beam components in humid and hot environments such as islands and reefs.

[0041] 2. This invention is based on the bond-slip test results of bond-assisted fiber reinforced bars and concrete. Compared with the current beam component test, the center pull-out specimen used in the bond-slip test has the characteristics of small size, low load level, relatively simple test process and easy control of test environment temperature and humidity. The universal testing machine can meet the test loading requirements, and has the advantages of small test site and low loading capacity of loading instruments.

[0042] 3. This invention addresses the design of post-tensioned prestressed fiber-reinforced beams in humid and hot environments. Using normal service performance as the control condition and based on the crack width limit under normal serviceability conditions, it can accurately and conveniently calculate the lower limit values ​​for the amount of bonded auxiliary fiber reinforcement and unbonded prestressed fiber reinforcement in post-tensioned prestressed fiber-reinforced beams under humid and hot environments. This avoids repeated calculations of the maximum crack width of the beam, reduces waste of fiber reinforcement material, and provides strong theoretical guidance for the bending design of post-tensioned prestressed fiber-reinforced beams under humid and hot environments. Attached Figure Description

[0043] Figure 1a This is a structural diagram of a post-tensioned prestressed fiber-reinforced beam in the reinforcement method for crack control of post-tensioned prestressed fiber-reinforced beams under humid and hot conditions according to the present invention.

[0044] Figure 1b This is a cross-sectional view of the post-tensioned prestressed fiber-reinforced beam in the reinforcement method for crack control of post-tensioned prestressed fiber-reinforced beams under humid and hot conditions according to the present invention.

[0045] Figure 2 This is a diagram showing the cross-sectional parameters of a post-tensioned prestressed fiber-reinforced beam under normal service limit conditions, as described in the reinforcement method for crack control in a humid and hot environment according to the present invention.

[0046] Figure 3 This invention provides a constitutive model for unbonded prestressed fiber-reinforced tendons and bonded auxiliary fiber-reinforced tendons in the reinforcement method for post-tensioned prestressed fiber-reinforced beams with crack control under humid and hot conditions.

[0047] Figure 4 This invention provides a constitutive model of concrete in the reinforcement method for post-tensioned prestressed fiber-reinforced beams designed for crack control under humid and hot conditions.

[0048] Figure 5aThis is a schematic diagram of the center pull-out specimen of the post-tensioned prestressed fiber-reinforced beam reinforcement method for crack control in humid and hot environments according to the present invention, which includes a bond-assisted fiber-reinforced reinforcement bond slip test with concrete.

[0049] Figure 5b This is a cross-sectional view of the center pull-out specimen in the bond-slip test of the bond-assisted fiber-reinforced reinforcement and concrete in the reinforcement method of post-tensioned prestressed fiber-reinforced beams for crack control under humid and hot conditions according to the present invention.

[0050] Figure 6 The diagram shows the loading device for the bond-assisted fiber-reinforced beam bond slip test in the method for crack control in humid and hot environments of the present invention.

[0051] Figure 7 The present invention provides a bond-slip constitutive relationship curve between bond-assisted fiber-reinforced reinforcement and concrete in the reinforcement method for post-tensioned prestressed fiber-reinforced beams for crack control under humid and hot conditions.

[0052] Figure 8 This is a diagram of a crack width analysis model based on bond-slip in the post-tensioned prestressed fiber-reinforced beam reinforcement method for crack control in humid and hot environments according to the present invention.

[0053] Figure 9 This is a flowchart illustrating the reinforcement process of the post-tensioned prestressed fiber-reinforced beam reinforcement method for crack control in humid and hot environments, as described in this invention.

[0054] Key reference numerals:

[0055] 1. Bonded auxiliary fiber reinforced bar, 2. Unbonded prestressed fiber reinforced bar, 3. PVC pipe, 4. Anchor, 5. Concrete, 6. Free end of bonded auxiliary fiber reinforced bar, 7. Loading end of bonded auxiliary fiber reinforced bar, 8. Loading end steel plate, 9. Displacement gauge, 10. Electro-hydraulic servo universal testing machine. Detailed Implementation

[0056] To provide a detailed description of the technical content, objectives, and effects of this invention, the following description will be provided in conjunction with the accompanying drawings.

[0057] Reinforcement methods for post-tensioned prestressed fiber-reinforced beams to control cracks in humid and hot environments, such as... Figure 9 As shown, the specific implementation steps are as follows:

[0058] S1. Based on the dimensions and material properties of the fiber-reinforced beam, obtain the minimum reinforcement area A for the 5-beam reinforced with bond-assisted fiber reinforcement. bf,min The specific expression is as follows:

[0059]

[0060] Among them, ft d represents the tensile strength of concrete 5. b Let f be the distance from the resultant point of the bonded fiber-reinforced bar 1 to the compression edge of the section, b be the width of the fiber-reinforced beam, and f be the distance from the resultant point of the bonded fiber-reinforced bar 1 to the compression edge of the section. bfu The tensile strength of the bonded auxiliary fiber reinforced bar 1.

[0061] S2, based on the minimum reinforcement area A of the bonded auxiliary fiber reinforced bar 1. bf,min Determine the diameter d of the bond-supported auxiliary fiber reinforcement 1 in the fiber-reinforced beam. f And the quantity, and obtain the area A of the bonded auxiliary fiber reinforced rib 1. bf .

[0062] S3. Conduct bond-slip tests on the center pull-out specimens of fiber-reinforced concrete consisting of bonded auxiliary fiber-reinforced bars 1 and concrete 5 under standard environmental and humid heat conditions, and establish standard environmental and humid heat conditions.

[0063] The bond-slip model under thermal conditions is specifically expressed as follows:

[0064]

[0065]

[0066] In the formula, τ N With τ TH The bond stresses τ between the bond-supported fiber-reinforced reinforcement 1 and concrete 5 are respectively under standard and humid / hot environments. 0.21,N With τ 0.21,TH The bonding stresses, s, are the bonding stresses corresponding to a slippage of 0.21 mm in the standard environment and the humid heat environment, respectively. 0.21 This indicates a slip of 0.21 mm, where s is the slip amount.

[0067] Specifically, in the bond-slip model, the slip of the bond-assisted fiber-reinforced rib loading end 7 is 0.21 mm; the standard environment refers to the IA environmental action level, and the humid and hot environment refers to the temperature between 35℃ and 50℃ and the relative humidity ≥50%.

[0068] S4. Based on the bond-slip model obtained in S3, solve for the bond-slip constant υ of the fiber-reinforced beam under standard and humid / heated conditions. N With υ TH The specific expression is:

[0069]

[0070]

[0071] In the formula, E bfE represents the elastic modulus of the bonded fiber-reinforced rib 1. c Let A be the elastic modulus of concrete 5. fb1 A represents the area of ​​a single bonded auxiliary fiber reinforced rib 1. c1 L represents the effective tensile area of ​​concrete 5 corresponding to a single bonded auxiliary fiber reinforced bar 1. per The perimeter is the length of a single bonded auxiliary fiber reinforced bar 1.

[0072] The amplification factor λ for crack width in fiber-reinforced beams under humid and hot conditions compared to standard conditions is expressed as follows:

[0073]

[0074] In the formula, w N With w TH The average crack widths under standard and humid / hot environments are respectively, l j This represents the average crack spacing.

[0075] S5. Based on the amplification factor λ of the crack width in the fiber-reinforced beam under humid and hot conditions compared to the standard conditions obtained in S4, the height c of the compression zone of the fiber-reinforced beam based on the amplification factor λ is obtained. w The specific expression is as follows:

[0076]

[0077] in:

[0078]

[0079]

[0080]

[0081]

[0082] In the formula, d p M is the distance M from the resultant point of the unbonded prestressed fiber-reinforced rib 2 to the compression edge of the section. d For the design of fiber-reinforced beams, the bending moment ε is used. bf,w For the strain of the bonded fiber-reinforced rib 1, c c c is the tensile influence coefficient of the concrete between cracks. p d1 is the ratio of the maximum crack width to the average crack width, and d2 is the distance from the edge of the tension zone of concrete 5 to the neutral axis. b1 w is the distance from the resultant point of the bonded fiber-reinforced rib 1 to the neutral axis. lim This is the limit value for the crack width.

[0083] Furthermore, the height c of the compression zone of the fiber-reinforced beam w In the coefficients d2 / d b1 The expression for the ratio is:

[0084]

[0085] In the formula, h is the height of the fiber-reinforced beam, and d c1 The thickness of the concrete protective layer 5.

[0086] S6. Based on the principle of minimizing the number of unbonded prestressed fiber-reinforced reinforcement layers in the fiber-reinforced beam, obtain the minimum reinforcement area A of the i-th layer of unbonded prestressed fiber-reinforced reinforcement 2 in the fiber-reinforced beam. fpi,min The specific expression is:

[0087]

[0088] In the formula, f pe For effective prestressing, k is the deflection coefficient of the simply supported beam, which is determined based on the load distribution; η is the elongation coefficient of the unbonded prestressed fiber-reinforced bar 2, which is determined based on the shape and load distribution, and its value ranges from 4 to 8; E pf e is the elastic modulus of the unbonded prestressed fiber-reinforced rib 2; i denoted as , which is the distance from the point of application of the tensile force of the i-th layer of unbonded prestressed fiber-reinforced rib 2 to the centroid of the cross section.

[0089] In a preferred embodiment of the present invention, the elongation coefficient η of the unbonded prestressed fiber reinforced tendon 2 is expressed as follows:

[0090]

[0091] In the formula, L is the calculated span of the fiber-reinforced beam, and δ m Let f(x) be the mid-span deflection of the fiber-reinforced beam, and f(x) be the second derivative of the deflection curve equation of the fiber-reinforced beam.

[0092] S7. Based on the minimum reinforcement area A of the i-th layer of unbonded prestressed fiber-reinforced reinforcement 2 in the fiber-reinforced beam. fpi,min The area A of the i-th layer of unbonded prestressed fiber-reinforced tendons 2 in the fiber-reinforced beam is obtained. fpi .

[0093] Furthermore, in order to obtain a post-tensioned prestressed fiber-reinforced beam, firstly, bonded auxiliary fiber-reinforced bars 1 are arranged, then unbonded prestressed fiber-reinforced bars 2 are inserted into PVC pipes 3, and anchorages 4 are bonded to both ends of the unbonded prestressed fiber-reinforced bars 2 with epoxy resin. Finally, concrete 5 is poured, and the unbonded prestressed fiber-reinforced bars 2 are prestressed, thereby obtaining a post-tensioned prestressed fiber-reinforced beam.

[0094] Specifically, post-tensioned prestressed fiber-reinforced beams, such as Figure 1a As shown, the structure includes bonded auxiliary fiber reinforcement 1, i (i≥1) layers of unbonded prestressed fiber reinforcement 2, PVC pipe 3 for achieving unbonded prestressed fiber reinforcement 2 with surrounding concrete, anchorages 4 for tensioning prestress set at both ends of unbonded prestressed fiber reinforcement 2, and concrete beam 5.

[0095] Both the bonded auxiliary fiber reinforcement 1 and the unbonded prestressed fiber reinforcement 2 used in post-tensioned prestressed fiber-reinforced beams adopt linear elastic constitutive relation models, such as... Figure 3 As shown, the constitutive relation model for concrete 5 uses a bilinear model, as follows: Figure 4 As shown.

[0096] The following describes, in conjunction with embodiments, a method for reinforcing post-tensioned prestressed fiber-reinforced beams for crack control in humid and hot environments according to the present invention:

[0097] In this embodiment, the fiber-reinforced beam is subjected to two symmetrical concentrated loads, and bonded auxiliary fiber-reinforced bars 1 and unbonded prestressed fiber-reinforced bars 2 are only configured in the tension zone. Specifically, as Figure 1b and Figure 2 As shown, the fiber-reinforced beam has a height h = 300 mm, width b = 200 mm, calculated span L = 3000 mm, and shear span a = 1100 mm. The thickness d of the concrete cover... c1 =30mm, bonded auxiliary fiber reinforced rib 1 and unbonded prestressed fiber reinforced rib 2 are selected from the same type of straight carbon fiber reinforced rib, and their tensile strength f bfu =f pfu =1800MPa, elastic modulus E bf =E pf =180Gpa, effective prestress f pe =720MPa. Concrete compressive strength f' c =40MPa, tensile strength f t =4MPa, elastic modulus E c =26GPa. Design using bending moment M d =35kN·m, crack width limit w lim =0.5mm. The distance d from the center of gravity of the pre-set bonded fiber reinforcement 1 to the bottom surface of concrete 5 is... c =45mm, with bonded auxiliary fiber reinforcement 1, distance d from the resultant point of the cross-section to the compression edge. b =255mm, distance d from the resultant point of unbonded prestressed fiber reinforced ribs to the compression edge of the section. p=225mm, the eccentricity e1 of the resultant force point of the unbonded prestressed fiber-reinforced rib 2 is 75mm. The specific implementation process is as follows:

[0098] S1. Based on the dimensions and material properties of the fiber-reinforced beam, obtain the minimum reinforcement area A of the bonded auxiliary fiber-reinforced bar 1. bf,min The specific expression is as follows:

[0099]

[0100] Among them, f t d represents the tensile strength of concrete 5. b Let f be the distance from the resultant point of the bonded fiber-reinforced bar 1 to the compression edge of the section, b be the width of the fiber-reinforced beam, and f be the distance from the resultant point of the bonded fiber-reinforced bar 1 to the compression edge of the section. bfu The tensile strength of the bonded auxiliary fiber reinforced bar 1.

[0101] S2, based on the minimum reinforcement area A of the bonded auxiliary fiber reinforced bar 1. bf,min Select 4 rods with a diameter d f A bonded auxiliary fiber reinforced rib 1 with a diameter of 8 mm was obtained, and the area A of the bonded auxiliary fiber reinforced rib 1 was calculated. bf =200.96mm 2 .

[0102] S3. Conduct bond-slip tests on center pull-out specimens of fiber-reinforced concrete consisting of bonded auxiliary fiber-reinforced bars 1 and concrete 5 under standard and humid / hot environments: First, select concrete 5 with dimensions of 200mm×200mm×200mm. Place the loading end steel plate 8 at a distance of 30mm from the bottom surface of concrete 5. The anchorage length of the fiber-reinforced bars is 3d. f 5d f and 8d f Three levels were established, and three test blocks were made at each level. PVC pipe 3 was used to isolate the unanchored sections of the fiber-reinforced bars from the concrete 5. Figure 5a and Figure 5b As shown. Next, the specimen was loaded using an electro-hydraulic servo universal testing machine 10, as follows. Figure 6 As shown, the loading rate is 0.8 mm / min. During the loading process, displacement gauge 9 is used to measure the displacement of the top surface of concrete 5, the free end 6 of the bonded auxiliary fiber reinforced bar, and the steel plate 8 at the loading end. The slippage s of the loading end 7 of the bonded auxiliary fiber reinforced bar is calculated by the following formula:

[0103]

[0104] In the formula, s s The average value of the readings of displacement gauge 9 at 8 points on the loading end steel plate; s cThe average reading of displacement gauge 9 on the top surface of concrete 5; F1 is the pull-out force; l b This is the distance between the loading end steel plate 8 and the bottom surface of the concrete 5.

[0105] Based on the above experiments, a bond-slip model was established before the anchor end slippage reached 0.21 mm under standard and humid conditions. Figure 7 As shown, the specific expression is:

[0106]

[0107]

[0108] In the formula, τ N With τ TH The bond stresses between the bond-supported fiber-reinforced reinforcement 1 and concrete 5 are respectively under standard and humid / hot environments. 0.21 This indicates a slip of 0.21 mm, where s is the slip amount.

[0109] S4. Based on the bond-slip model obtained in S3, solve for the bond-slip constant υ of the fiber-reinforced beam under standard and humid / heated conditions. N With υ TH The specific expression is:

[0110]

[0111]

[0112] In the formula, E bf E represents the elastic modulus of the bonded fiber-reinforced rib 1. c Let A be the elastic modulus of concrete 5. fb1 A represents the area of ​​a single bonded auxiliary fiber reinforced rib 1. c1 L represents the effective tensile area of ​​concrete 5 corresponding to a single bonded auxiliary fiber reinforced bar 1. per The perimeter is the length of a single bonded auxiliary fiber reinforced bar 1.

[0113] Calculate the average crack spacing l j The specific expression is:

[0114]

[0115] In the formula, d c h is the distance from the center of gravity of the bonded auxiliary fiber-reinforced bar 1 to the bottom surface of the concrete 5, and h is the height of the fiber-reinforced bar beam.

[0116] Using a crack width analysis model based on bond-slip, such as Figure 8As shown, the amplification factor λ for crack width in fiber-reinforced beams under humid and hot conditions compared to standard conditions is calculated, and the specific expression is as follows:

[0117]

[0118] In the formula, w N With w TH The average crack widths under standard and humid / hot environments are respectively, l j This represents the average crack spacing.

[0119] S5. Based on the fact that the height h of the fiber-reinforced beam in this embodiment is 300mm, the height c of the compression zone of the fiber-reinforced beam is... w In the coefficients d2 / d b1 The ratio is:

[0120] When h < 400 mm

[0121] In the formula, d2 is the distance from the edge of the tension zone of concrete 5 to the neutral axis, and d b1 d is the distance d from the resultant point of the bonded fiber-reinforced rib 1 to the neutral axis. c1 The thickness of the concrete protective layer 5.

[0122] Based on the amplification factor λ of the crack width in the fiber-reinforced beam under humid and hot conditions compared to the standard conditions obtained in S4, the height c of the compression zone of the fiber-reinforced beam based on the amplification factor λ is obtained. w = 87.35mm, the specific calculation process is as follows:

[0123]

[0124] in:

[0125]

[0126]

[0127]

[0128]

[0129]

[0130] In the formula, d p M is the distance M from the resultant point of the unbonded prestressed fiber-reinforced rib 2 to the compression edge of the section. d For the design of fiber-reinforced beams, the bending moment ε is used. bf,w For the strain of the bonded fiber-reinforced rib 1, c c The tensile influence coefficient of the concrete between cracks is taken as 0.85 in this embodiment.p The ratio of the maximum crack width to the average crack width is taken as 1.90 in this embodiment. lim This is the limit value for the crack width.

[0131] S6. Based on the principle of minimizing the number of unbonded prestressed fiber-reinforced reinforcement layers in the fiber-reinforced beam, obtain the minimum reinforcement area A of the first layer of unbonded prestressed fiber-reinforced reinforcement 2 in the fiber-reinforced beam. fp1,min The specific expression is:

[0132]

[0133] In the formula, f pe For effective prestress, k is the deflection coefficient of the simply supported beam, η is the elongation coefficient of the unbonded prestressed fiber-reinforced tendon 2, and E pf e1 is the elastic modulus of the unbonded prestressed fiber reinforced tendon 2, and e1 is the distance from the point of application of the tensile force of the first layer of unbonded prestressed fiber reinforced tendon 2 to the centroid of the cross section.

[0134] The expression for calculating the deflection coefficient k of a simply supported beam is as follows:

[0135]

[0136] In the formula, L is the calculated span of the fiber-reinforced beam, and a is the shear span of the fiber-reinforced beam.

[0137] The expression for the elongation coefficient η of the unbonded prestressed fiber-reinforced tendon 2 is as follows:

[0138]

[0139] In the formula, δ m Let f(x) be the mid-span deflection of the fiber-reinforced beam, and f(x) be the second derivative of the deflection curve equation of the fiber-reinforced beam.

[0140] S7. Based on the minimum reinforcement area A of the first layer of unbonded prestressed fiber-reinforced tendons 2 in the fiber-reinforced beam. fp1,min One carbon fiber reinforcing bar with a diameter of 16 mm was selected as the unbonded prestressed fiber reinforcing bar 2, and the area of ​​the unbonded prestressed fiber reinforcing bar 2, Afp1, was 200.96 mm2.

[0141] The embodiments described above are merely preferred embodiments of the present invention and are not intended to limit the scope of the present invention. Various modifications and improvements made by those skilled in the art to the technical solutions of the present invention without departing from the spirit of the present invention should fall within the protection scope defined by the claims of the present invention.

Claims

1. A method for reinforcing post-tensioned prestressed fiber-reinforced beams to control cracks in humid and hot environments, characterized in that, The specific implementation steps include: S1. Based on the dimensions and material properties of the fiber-reinforced beam, obtain the lower limit A of the reinforcement area for bonded fiber-reinforced reinforcement. bf,min The specific expression is as follows: ； Among them, f t d represents the tensile strength of concrete. b denoted as , where is the distance from the resultant force point of the bonded fiber-reinforced reinforcement to the compression edge of the section; , where is the width of the fiber-reinforced beam; and , where is the distance from ... bfu The tensile strength of the bonded fiber-reinforced rib; S2, based on the lower limit A of the reinforcement area with bonded auxiliary fiber reinforcement. bf,min Determine the diameter d of the bond-supported fiber reinforcement bars in the fiber-reinforced beam. f And the quantity, and obtain the area A of the bond-assisted fiber reinforcement. bf ; S3. Conduct bond-slip tests on the center pull-out specimens of fiber-reinforced concrete consisting of bonded auxiliary fiber-reinforced bars and concrete under standard and humid environments, and establish a bond-slip model under standard and humid environments. S4. Based on the bond-slip model obtained in S3, solve for the bond-slip constant υ of the fiber-reinforced beam under standard and humid / heated conditions. N With υ TH And the amplification factor λ of the crack width in the fiber-reinforced beam under humid and hot conditions compared to standard conditions, the expression for the amplification factor λ of the crack width in the fiber-reinforced beam is: ； In the formula, w N with w TH The average crack widths under standard and humid / hot environments are respectively, l j This represents the average crack spacing. S5. Based on the amplification factor λ of the crack width in the fiber-reinforced beam under humid and hot conditions compared to the standard conditions obtained in S4, the height c of the compression zone of the fiber-reinforced beam based on the amplification factor λ is obtained. w ; S6. Based on the principle of minimizing the number of unbonded prestressed fiber-reinforced reinforcement layers in fiber-reinforced beams, obtain the minimum reinforcement area A of the i-th layer of unbonded prestressed fiber-reinforced reinforcement in the fiber-reinforced beam. fpi,min The specific expression is: ； In the formula, f pe For effective prestress, k is the deflection coefficient of the simply supported beam, η is the elongation coefficient of the unbonded prestressed fiber-reinforced tendon, and E pf e represents the elastic modulus of unbonded prestressed fiber-reinforced tendons. i ε is the distance from the point of application of the tensile force in the i-th layer of unbonded prestressed fiber-reinforced tendon to the centroid of the cross-section. bf,w For the strain of bonded fiber-reinforced ribs, d p E is the distance from the resultant point of the unbonded prestressed fiber-reinforced tendon to the compression edge of the section. bf For the elastic modulus of bonded fiber-reinforced ribs, M d Use bending moment in the design of fiber-reinforced beams; S7. Based on the minimum reinforcement area A of the i-th layer of unbonded prestressed fiber-reinforced tendons in the fiber-reinforced beam. fpi,min The area A of the i-th layer of unbonded prestressed fiber-reinforced tendons in the fiber-reinforced beam is obtained. fpi .

2. The method for reinforcing post-tensioned prestressed fiber-reinforced beams with crack control under humid and hot conditions according to claim 1, characterized in that, In step S3, the slip amount of the bond-slip model at the loading end of the bond-assisted fiber reinforcement is 0.21 mm.

3. The method for reinforcing post-tensioned prestressed fiber-reinforced beams with crack control under humid and hot conditions according to claim 1, characterized in that, In step S3, the standard environment refers to environmental action level IA, and the hot and humid environment refers to a temperature between 35℃ and 50℃ and a relative humidity ≥50%.

4. The method for reinforcing post-tensioned prestressed fiber-reinforced beams with crack control under humid and hot conditions according to claim 1 or 3, characterized in that, In step S3, the specific expression of the bond-slip model is as follows: ； ； In the formula, τ N With τ TH The bond stresses τ between bond-assisted fiber-reinforced reinforcements and concrete under standard and humid / hot environments are respectively. 0.21,N With τ 0.21,TH The bonding stresses, s, are the bonding stresses corresponding to a slippage of 0.21 mm in the standard environment and the humid heat environment, respectively. 0.21 This indicates a slip of 0.21 mm, where s is the slip amount.

5. The method for reinforcing post-tensioned prestressed fiber-reinforced beams with crack control under humid and hot conditions according to claim 4, characterized in that, In step S4, the adhesive slip constant υ of the standard environment N and the bond slip constant υ in the humid and hot environment TH The expressions are as follows: ； ； In the formula, A fb1 A represents the area of ​​a single bonded auxiliary fiber reinforced bar. c1 L represents the effective tensile area of ​​the concrete corresponding to a single bonded auxiliary fiber reinforced bar. per E is the perimeter of a single bonded auxiliary fiber reinforced bar. c This refers to the elastic modulus of concrete.

6. The method for reinforcing post-tensioned prestressed fiber-reinforced beams with crack control under humid and hot conditions according to claim 5, characterized in that, In step S5, the height c of the compression zone of the fiber-reinforced beam is... w The specific expression is as follows: ； in: ； ； ； ； ； In the formula, d2 is the distance from the edge of the tensile zone of the concrete to the neutral axis, and d b1 c is the distance from the resultant point of the bonded fiber-reinforced rib to the neutral axis. c c is the tensile influence coefficient of the concrete between cracks. p w is the ratio of the maximum crack width to the average crack width. lim This is the limit value for the crack width.

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