Calculation method for checking reinforcement ratio of steel strand of full-prestressed concrete beam bridge

Abstract

The invention discloses a calculation method for checking the reinforcement ratio of a steel strand of a full-prestressed concrete beam bridge. The calculation method comprises the following steps: S1, counting section characteristics; s2, deriving an expression formula of the structure cracking bending moment about the steel strand area; s3, calculating the initial strain of the steel strand and the initial strain of the concrete at the position of the steel strand; s4, obtaining a corresponding steel strand area Ap1 when the stress of the steel strand reaches nominal yield strength in a structural limit state; s5, calculating the corresponding steel strand area Ap2 when the concrete micro-strain at the position of the steel strand in the structure limit state is 10000; s6, according to the linear relation, the relation among the stress of the steel strand, the distance from the steel strand to the center of the structural compression area and the area of the steel strand in the limit state is derived; s7, obtaining an expression formula of the ultimate bending moment design value of the structure about the steel strand area; and S8, checking the reinforcement ratio of the steel strand of the full-prestressed concrete beam bridge. According to the method, the reinforcement ratio of the steel strand of the full-prestressed concrete beam bridge can be effectively checked.

Description

Technical Field

[0001] The invention belongs to the technical field of bridge engineering in the transportation industry, and specifically relates to a calculation method for verifying the reinforcement ratio of steel strands of a full prestressed concrete beam bridge. Background Art

[0002] Prestressed concrete beams are one of the important transportation infrastructures for mankind. How to ensure their stability, safety and durability is an important issue. Controlling the reinforcement ratio is usually used as an effective structural means to limit the damage of prestressed concrete beams, and is widely used in prestressed concrete bridges. The configuration of steel strands in prestressed concrete beams is usually based on the "Highway Reinforced Concrete and Prestressed Concrete Bridge and Culvert Design Code JTG3362-2018" (hereinafter referred to as the code). The design value of the structural ultimate bending moment must be greater than or equal to the structural cracking moment for verification. According to the existing method of calculation, it can only be judged whether the reinforcement ratio of prestressed concrete bending members meets the conditions. According to the formula for the design value of the structural ultimate bending moment given in the code, the steel strand must be able to reach 1860MPa under the limit state. Then, according to the steel strand material partial factor of 1.47, it is reduced and the steel strand strength design value of 1260MPa is taken, which is inconsistent with the actual damage state of the structure.

[0003] Therefore, it is necessary to establish a calculation formula for the steel strand reinforcement ratio of fully prestressed concrete beams that is more in line with the actual damage state of the structure, so as to provide a reference for the future design of steel strand configuration and the verification of construction plans. Summary of the invention

[0004] The purpose of the present invention is to provide a calculation method for verifying the reinforcement ratio of steel strands of a fully prestressed concrete beam bridge, which can more reasonably verify the reinforcement ratio of steel strands of a fully prestressed concrete beam that conforms to the actual damage state of the structure.

[0005] The purpose of the present invention can be achieved through the following technical solutions.

[0006] A calculation method for checking the reinforcement ratio of steel strands of a fully prestressed concrete beam bridge comprises the following steps:

[0007] S1: Statistical section properties, including but not limited to: section inertia moment, distance from centroid to top and bottom edges, protective layer thickness, section elastic resistance moment, effective tensioning control stress of prestressed steel strands, and area moment of the area above the centroid axis of the full section to the centroid axis;

[0008] S2: Derivation of the formula for the structural cracking moment with respect to the area of ​​the steel strand:

[0009]

[0010] Where: M cr is the cracking moment of the normal section of the bending member, σ pc,bIn order to deduct all prestress losses, the prestressed steel bars generate concrete precompression stress at the crack-resistant edge of the component. γ is the plastic development coefficient, and f is tk is the standard tensile strength of concrete, W is the resistance moment of the lower edge of the section in the final state, A p is the area of ​​the steel strand, σ con is the tension stress of the steel strand, A1 is the cross-sectional area when the steel strand is tensioned, e is the distance from the cross-sectional centroid when the steel strand is tensioned, W1 is the resistance moment of the lower edge of the cross-sectional area when the steel strand is tensioned, and S is the area moment above the cross-sectional centroid axis in the final state;

[0011] S3: Calculate the initial strain of the steel strand and the initial strain of the concrete at the steel strand location based on the constitutive relationship;

[0012] S4: Calculate the height of the structural compression zone and the distance from the steel strand to the center of the structural compression zone when the stress of the steel strand reaches the nominal yield strength under the structural limit state according to the concrete strain diagram of the structural section at this time, and calculate the corresponding steel strand area A p1 , steel strand area A p1 is the upper limit of the preliminary steel strand area limit;

[0013] S5: Calculate the height of the structural compression zone and the distance from the steel strand to the center of the structural compression zone when the micro-strain of the concrete at the steel strand position is 10,000 under the structural limit state according to the concrete strain diagram of the structural section, and calculate the corresponding steel strand area A p2 and ultimate strength of steel strand, area of ​​steel strand A p2 It is the lower limit of the preliminary steel strand area limit range;

[0014] S6: Based on the linear relationship, the relationship between the stress of the steel strand in the limit state, the distance between the steel strand and the center of the structural compression zone and the area of ​​the steel strand are derived respectively;

[0015] S7: Get the formula for expressing the ultimate bending moment design value of the structure with respect to the area of ​​the steel strand;

[0016] S8: Based on the fact that the design value of the structural ultimate bending moment must be greater than or equal to the structural cracking bending moment, verify the reinforcement ratio of the steel strands in the fully prestressed concrete beam bridge.

[0017] The calculation of the present invention is more consistent with the actual failure form of the full prestressed concrete beam bridge, and reflects the mechanical meaning of the reinforcement ratio of the prestressed steel strand. The calculation result is more reasonable than the specification.

[0018] As a preferred calculation method of the present invention, in step S3, when calculating the initial strain of the steel strand and the initial strain of the concrete at the steel strand position, the concrete constitutive curve refers to the German Rüsch curve, and the constitutive relationship of the steel strand adopts the following formula:

[0019]

[0020] Where: f ps is the stress of the steel strand, E ps The elastic modulus of the steel strand is 1.92×10 5 MPa, ε is the strain of the steel strand, f py The yield stress is 1680.15MPa, and N, K, and Q are fitting constants, which are 7.344, 1.0618, and 0.01174, respectively.

[0021] When calculating the initial strain of the steel strand and the initial strain of the concrete at the steel strand position, the initial stress of the concrete at the steel strand position σ pe The calculation uses the following formula:

[0022]

[0023] Where: A pn is the area of ​​the original steel strand arrangement of the structure, I1 is the section moment of inertia when the steel strand is tensioned, G is the gravity effect, I is the section moment of inertia in the final state, and ω is the concrete density.

[0024] As a preferred design method of the present invention: in the steps S4 and S5, the calculation of the area of ​​the steel strand and the distance from the steel strand to the center of the structural compression zone are respectively calculated using the following formulas:

[0025] f pu A p =f ck βxb,

[0026]

[0027] Where: f pu is the actual damage value of the steel strand at the ultimate bending moment of the section, f ck is the standard compressive strength of concrete, β is the ratio of the height of the rectangular stress block in the compression zone to the actual height of the compression zone, x is the height of the compression zone, b is the section width, z is the distance from the steel strand to the center of the compression zone of the structure, and h0 is the effective height of the section.

[0028] According to the formula for the relative limit compression zone height of prestressed concrete beams in the "Highway Reinforced Concrete and Prestressed Concrete Bridge and Culvert Design Specification JTG3362-2018", the steel strand must reach a nominal yield strength of at least 0.85f at the structural limit. pk =1581MPa.

[0029] The present invention is analogous to the common concrete beam in the limit state, the concrete microstrain at the lower edge of the cross section cannot exceed 10000, and the concrete microstrain at the steel strand position of the prestressed concrete beam in the limit state cannot exceed 10000, so as to ensure that the deflection-span ratio of the structure in the limit state is reasonable.

[0030] As a preferred calculation method of the present invention, in step S6, the relationship between the stress of the steel strand, the distance from the steel strand to the center of the structural compression zone and the area of ​​the steel strand under the limit state is assumed according to a linear relationship:

[0031] f pu =a+gA p

[0032] z=c+dA p

[0033] Wherein: a, c, d, g are all constants.

[0034] As a preferred calculation method of the present invention: in step S7, the expression formula of the structural limit bending moment design value with respect to the steel strand area adopts the following formula:

[0035]

[0036] Where: M ud is the design value of the bending bearing capacity of the positive section of the flexural member, z is the distance between the steel strand and the center of the compression zone, λ is the ductility guarantee index, and with reference to reinforced concrete beams, λ is taken as 1.25.

[0037] As a preferred calculation method of the present invention: in step S8, the reinforcement ratio of the steel strands of the full prestressed concrete beam bridge is calculated using the following formula:

[0038]

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

[0040] 1. The present invention analyzes the actual strength of the steel strand damage instead of f pk =1860Mpa, in order to adapt to the actual failure mode of prestressed reinforced concrete beams when the prestressed steel bars are not necessarily broken at the ultimate failure. According to the calculation method of the present invention, it is possible to verify whether the reinforcement ratio of prestressed concrete bending members meets the conditions, and whether the structure needs to be equipped with ordinary steel bars. The steel strand configuration requirements if ordinary steel bars are not configured are obtained, and the "minimum reinforcement ratio of prestressed concrete beams" is corrected to "auxiliary reinforcement ratio of prestressed concrete beams". Whether to configure ordinary steel bars is determined based on the structure itself and the configuration of steel strands.

[0041] 2. Compared with the existing calculation method, the calculation method of the present invention can perform calculation according to the actual strength of steel strand failure, and provides a recommended range of values ​​for the actual strength of steel strand failure, thereby ensuring the rationality of the structural design.

[0042] 3. The ductility guarantee index λ of the present invention is taken as 1.25 according to the mechanical meaning of the structure, with reference to the reinforced concrete beam, instead of the partial material safety factor λ of the steel strand. p =1.47, that is, the strength of the prestressed steel strand at the ultimate failure is generally between 1581Mpa and 1860Mpa (0.85f pk ~1.0f pk ), then f pu / λ is between 1260Mpa and 1488Mpa, instead of directly taking the value f as in the standard formula pd =1260Mpa, which shows the mechanical meaning of the design value of the structural ultimate bending moment. BRIEF DESCRIPTION OF THE DRAWINGS

[0043] Figure 1 is a cross-sectional view of a prestressed concrete beam bridge in Example 1 of the present invention;

[0044] Figure 2 is the strain diagram when the stress of the steel strand is 1581 MPa under the structural limit state of Example 1;

[0045] Figure 3 is the strain diagram when the micro strain of concrete at the steel strand position under the ultimate limit state is 10000 in the ultimate limit state of the structure of Example 1;

[0046] Figure 4 is a curve diagram drawn according to the final calculation result of Example 1;

[0047] Figure 5 is a cross-sectional view of a prestressed concrete beam bridge in Example 2 of the present invention;

[0048] Figure 6 is the strain diagram when the stress of the steel strand is 1581 MPa under the structural limit state of Example 2;

[0049] Figure 7 It is a curve diagram drawn according to the final calculation result of Example 2. DETAILED DESCRIPTION

[0050] The technical solution of the present invention is described in detail below in conjunction with the accompanying drawings and embodiments so that those skilled in the art can better understand and implement the technical solution of the present invention.

[0051] Example 1

[0052] This embodiment uses Figure 1 The present invention is described in detail by taking the cross section of the prestressed concrete beam bridge shown in the figure as an example. The present invention provides a calculation method for verifying the reinforcement ratio of the steel strand of a full prestressed concrete beam bridge, comprising the following steps:

[0053] S1: Statistical section properties, including but not limited to: section inertia moment, distance from centroid to top and bottom edges, protective layer thickness, section elastic resistance moment, effective tensioning control stress of prestressed steel strands, area moment of the area above the centroid axis of the full section to the centroid axis, etc.;

[0054] In this embodiment, the beam length L = 30m, the prestressed concrete beam adopts C50 concrete, and the cross section is arranged with 4 bundles of 8-Φ15 prestressed steel tendons during design. The original steel strand area A of the structure is pn =32×140mm 2 , for reference Figure 1 , using post-tensioning method, the cross-sectional area A is 724040mm 2 , the final state section inertia moment I is 1.6496×10 11 mm 4 The distance between the centroid and the top and bottom edges is 496mm and 804mm respectively, and the thickness of the protective layer is a p is 140mm, the section height h is 1300mm, the effective tension control stress of the prestressed steel strand (the tension stress of the steel strand σ con ) is 1122MPa, and the resistance moment W of the lower edge of the final state section is 0.20528×10 9 mm 3 The area moment above the centroid axis of the final section (the area moment of the area above the centroid axis of the full section to the centroid axis) S is 0.15888×10 9 mm 3 , the distance e between the combined force point of the four strands and the centroid of the cross section p =664mm;

[0055] When the steel strand is tensioned, the cross-sectional area A1 is 589100mm 2 , the section moment of inertia I1 is 1.2885×10 11 mm 4 , the resistance moment W1 of the lower edge of the section is 0.16314×10 9 mm 3 , the distance between the combined force point of the 4 strands and the centroid of the section is e = 650mm, and the thickness of the flange plate is h f =(800×150+410×80 / 2+90×80) / 800=180mm.

[0056] S2: Derive the expression formula of the structural cracking moment with respect to the area of ​​the steel strand;

[0057]

[0058] Where: M cr is the cracking moment of the normal section of the bending member, σ pc,bTo deduct all prestress losses, the prestressed steel bars generate concrete prestress (MPa) at the crack-resistant edge of the component. γ is the plastic development coefficient, and f is tk is the standard tensile strength of concrete (MPa), W is the resistance moment of the lower edge of the final state section (mm 3 ), A p is the area of ​​the steel strand (mm 2 ), σ noc is the tension stress of the steel strand (MPa), A1 is the cross-sectional area of ​​the steel strand when tensioning (mm 2 ), e is the distance from the cross-section centroid when the steel strand is tensioned (mm), W1 is the resistance moment of the lower edge of the cross-section when the steel strand is tensioned (mm 3 ), S is the area moment above the centroid of the final state section (mm 3 ).

[0059] In this embodiment,

[0060]

[0061] Where: f tk The standard tensile strength of C50 concrete is 2.65MPa.

[0062] S3: Calculate the initial strain of the steel strand and the initial strain of the concrete at the steel strand position based on the constitutive relationship. The concrete constitutive curve is recommended to refer to the German Rüsch curve. The constitutive relationship of the steel strand adopts the following formula:

[0063]

[0064] Where: f ps is the stress of the steel strand, E ps The elastic modulus of the steel strand is 1.92×10 5 MPa, ε is the strain of the steel strand, f py The yield stress is 1680.15MPa, and N, K, and Q are fitting constants, which are 7.344, 1.0618, and 0.01174, respectively.

[0065] Initial concrete stress at the strand positionσ pe The calculation uses the following formula:

[0066]

[0067] Where: I1 is the moment of inertia of the section when the steel strand is tensioned (mm 4 ), G is the gravity effect, and ω is the concrete density.

[0068] In this embodiment, the German Rüsch curve of C50 concrete adopts the following formula:

[0069]

[0070] Where: c is the concrete stress, ε c is the concrete strain.

[0071]

[0072] Where: C50 concrete bulk density ω is 24kN / m 3 .

[0073]

[0074]

[0075] The initial microstrain ε0 of the steel strand is 6000, the initial stress of the concrete at the steel strand position is 20.9 MPa, and the initial microstrain ε c0 It is 810.

[0076] S4: According to the structural strain diagram, the stress of the steel strand in the structural limit state is calculated as f pu1 1581MPa (According to the formula for the relative limit compression zone height of prestressed concrete beams in the "Highway Reinforced Concrete and Prestressed Concrete Bridge and Culvert Design Code JTG3362-2018", the steel strand must at least reach the nominal yield strength of 0.85f at the structural limit pk =1581MPa), the corresponding height of the structural compression zone and the distance from the steel strand to the center of the structural compression zone, and the corresponding steel strand area A are calculated p1 , steel strand area A p1 is the upper limit of the preliminary steel strand area limit range; the steel strand area and the distance from the steel strand to the center of the structural compression zone are calculated using the following formulas:

[0077] f pu A p =f ck βxb,

[0078]

[0079] h0=ha p ,

[0080] Where: f pu is the actual damage value of the steel strand at the ultimate bending moment of the section (MPa), f ck is the standard compressive strength of concrete (MPa), C50 concrete is 32.4MPa, β is the ratio of the height of the rectangular stress block in the compression zone to the actual height of the compression zone, usually taken as 0.8, x is the height of the compression zone, b is the section width corresponding to the calculated stress, and h0 is the effective height of the section.

[0081] In this embodiment, the limit state structural strain reference Figure 2 , the height of the compression zone and the distance from the strand to the center of the compression zone of the structure:

[0082] ε t =10000-ε c0 -ε0=10000-810-6000=3190

[0083]

[0084] Where: ε cu is the ultimate compressive strain of concrete, the value of C50 concrete is 3300, ε t is the tensile strain of concrete at the steel strand under this limit state.

[0085]

[0086] Where βx = 0.8 × 590 = 180 + 292 = 472 mm,

[0087] Where: 1600mm is the flange width, 180mm is the web width.

[0088]

[0089] Calculate the corresponding steel strand area A p1 6979mm 2 The distance z1 from the steel strand to the center of the compression zone of the structure is 924mm.

[0090] S5: Calculate the height of the structural compression zone and the distance from the steel strand to the center of the structural compression zone when the micro-strain of the concrete at the steel strand position is 10,000 under the structural limit state according to the concrete strain diagram of the structural section, and calculate the corresponding steel strand area A p2 and ultimate strength of steel strand, area of ​​steel strand A p2 It is the lower limit of the preliminary steel strand area limit range;

[0091] In this embodiment, the limit state structural strain reference Figure 3 , the height of the compression zone and the distance from the strand to the center of the compression zone of the structure:

[0092]

[0093] The micro strain of the steel strand is 10000+ε c0 +ε0=10000+6000+810=16810.

[0094] At this time, the stress of the steel strand

[0095]

[0096] Where βx=0.8×288=180+50.4=230.4mm

[0097] Where: 1600mm is the flange width, 180mm is the web width.

[0098]

[0099] Calculate the corresponding steel strand area A p2 5353mm 2 , the distance z2 from the steel strand to the center of the structural compression zone is 1045mm and the ultimate strength of the steel strand is 1798MPa.

[0100] S6: Based on the linear relationship, the relationship between the stress of the steel strand under the limit state, the distance from the steel strand to the center of the structural compression zone and the area of ​​the steel strand are derived respectively. The linear relationship is assumed as follows:

[0101] f pu =a+eA p ,

[0102] z=c+dA p ,

[0103] Wherein: a, c, d, g are all constants.

[0104]

[0105] In this embodiment, it can be obtained that:

[0106] f pu =-0.13A p +2521,

[0107] z=-0.08A p +1470,

[0108] S7: The formula for expressing the ultimate bending moment design value of the structure with respect to the area of ​​the steel strand is obtained using the following formula:

[0109]

[0110] Where: M ud is the design value of the bending bearing capacity of the positive section of the flexural member, z is the distance between the steel strand and the center of the compression zone (mm), λ is the ductility guarantee index, and with reference to reinforced concrete beams, λ is taken as 1.25.

[0111] In this embodiment:

[0112]

[0113] Where: 5353≤Ap ≤6979,

[0114] S8: According to the design value of the structural ultimate bending moment, which must be greater than or equal to the structural cracking bending moment, the reinforcement ratio of the steel strands of the full prestressed concrete beam bridge is verified. The following formula is used to verify the reinforcement ratio of the steel strands of the full prestressed concrete beam bridge:

[0115]

[0116] In this embodiment:

[0117]

[0118] According to the formula, the comparison chart is obtained, refer to Figure 4 , that is, the solution range is 573≤A p ≤5668, and the initial steel strand area limit range 5353≤A p ≤6979, the intersection is 5353≤A p ≤5668. Conclusion obtained from empirical calculation: In this embodiment, the steel strand configuration area of ​​the prestressed concrete beam bridge is 4448mm 2 , ordinary steel bars need to be configured, otherwise the structure configuration is unreasonable. If the area of ​​the steel strand is changed to 5353≤A p ≤5668, only prestressed steel strands need to be configured, and ordinary steel bars are not needed.

[0119] Comparative Example 1: Existing Calculation Method

[0120] S1: Statistical cross-sectional properties, including but not limited to: cross-sectional inertia moment, distance between centroid and top edge and bottom edge, protective layer thickness, cross-sectional elastic resistance moment, effective tensioning control stress of prestressed steel strand, area moment above the cross-sectional centroid axis, etc., which are the same as in Example 1;

[0121] S2: Calculate the structural cracking moment:

[0122]

[0123] S3: Calculate the ultimate bending moment design value of the structure:

[0124]

[0125]

[0126] Where: f pd is the design strength of the steel strand, which is 1260MPa, f cd is the design value of the axial compressive strength of concrete, which is 22.4MPa for C50 concrete.

[0127] S4: Reinforcement ratio verification, M ud / Mcr ≥1 indicates that the structural reinforcement is reasonable:

[0128]

[0129] This indicates that the structural reinforcement is unreasonable. The existing calculation method can only determine whether the reinforcement ratio of prestressed concrete flexural members meets the conditions. However, according to the calculation method of the present invention, it is not only possible to determine whether the reinforcement ratio of prestressed concrete flexural members meets the conditions, but also to determine whether the structure needs to be configured with ordinary steel bars. As shown in Example 1, a clear conclusion is obtained through the calculation method of the present invention: Example 1 needs to configure ordinary steel bars, or the configuration area of ​​the steel strands is changed, and ordinary steel bars are not needed.

[0130] Example 2

[0131] S1: Statistical section properties, including but not limited to: section inertia moment, distance from centroid to top and bottom edges, protective layer thickness, section elastic resistance moment, effective tensioning control stress of prestressed steel strands, and area moment of the area above the centroid axis of the full section to the centroid axis;

[0132] like Figure 5 As shown, in this embodiment, the beam length L = 30m, the prestressed concrete composite beam adopts C150 concrete main beam + cast-in-place C50 concrete bridge deck, the elastic modulus of C150 is E1 = 42.5GPa, C50 is converted, and 4 bundles of 17-Φ15 prestressed steel tendons are arranged in the cross section during design, and the original steel strand area A of the structure is arranged pn =68×140mm 2 , using post-tensioning method, the cross-sectional area A is 1129853mm 2 , the final state section inertia moment I is 4.0848×10 11 mm 4 The distance between the centroid and the top and bottom edges is 630mm and 1060mm respectively, and the thickness of the protective layer is a p is 100 mm, the section height h is 1690 mm, and the effective tension control stress of the prestressed steel strand (σ con The tensile stress of the steel strand is 1186MPa, and the resistance moment W of the lower edge of the final state section is 0.38536×10 9 mm 3 The area moment above the centroid axis of the final section (the area moment of the area above the centroid axis of the full section to the centroid axis) S is 0.30658×10 9 mm 3 , the distance e between the combined force point of the four strands and the centroid of the cross section p =960mm;

[0133] When the steel strands are tensioned, no C50 concrete is poured, and the cross-sectional area A1 is 628853mm2 , the section moment of inertia I1 is 1.57×10 11 mm 4 , the resistance moment W1 of the lower edge of the section is 0.24455×10 9 mm 3 , the distance between the combined force point of the 4 strands and the centroid of the cross section is e = 540 mm, and the thickness of the top plate is h f =(2500×180+500×60 / 2+600×60) / 2500=200mm.

[0134] S2: Derive the expression formula of the structural cracking moment with respect to the area of ​​the steel strand;

[0135]

[0136] In this embodiment,

[0137]

[0138] Where: f tk The standard tensile strength of C150 concrete is 7.2MPa.

[0139] S3: Calculate the initial strain of the steel strand and the initial strain of concrete at the steel strand position based on the constitutive relationship. The constitutive curve of C50 concrete is recommended to refer to the German Rüsch curve, and the constitutive curve of C150 concrete is recommended to refer to the American standard curve. The constitutive relationship of the steel strand adopts the following formula:

[0140]

[0141] The initial stress of concrete at the strand position is calculated using the following formula:

[0142]

[0143] Where: A pn is the area of ​​the original steel strand arrangement, I1 is the section inertia moment when the steel strand is tensioned (mm 4 ), G is the gravity effect, and ω is the concrete density.

[0144] In this embodiment, the German Rüsch curve of C50 concrete adopts the following formula:

[0145]

[0146] The constitutive curve of C150 concrete adopts the following formula:

[0147]

[0148] Where: c is the concrete stress, εc is the concrete strain.

[0149]

[0150] Where: The bulk density of C50 and C150 concrete is 24kN / m 3 .

[0151]

[0152]

[0153] The initial microstrain ε0 of the steel strand is 6200, the initial stress of the concrete at the steel strand position is 35.7 MPa, and the initial microstrain ε c0 It is 835.

[0154] S4: According to the structural strain diagram, the stress of the steel strand under the structural limit state is calculated to be 1581MPa (according to the formula for the relative limit compression zone height of prestressed concrete beams in the "Highway Reinforced Concrete and Prestressed Concrete Bridge and Culvert Design Code JTG3362-2018", the steel strand must at least reach the nominal yield strength of 0.85f at the structural limit pk =1581MPa), the corresponding height of the structural compression zone and the distance from the steel strand to the center of the structural compression zone, and the corresponding steel strand area A are calculated p1 , steel strand area A p1 is the upper limit of the preliminary steel strand area limit range; the steel strand area and the distance from the steel strand to the center of the structural compression zone are calculated using the following formulas:

[0155] f pu A p =f ck βxb

[0156]

[0157] h0=ha p

[0158] Where: f ck is the standard compressive strength of concrete (MPa), C50 concrete is 32.4MPa, and C150 concrete is 124MPa.

[0159] In this embodiment, the limit state structural strain reference Figure 6 , the height of the compression zone and the distance from the strand to the center of the compression zone of the structure:

[0160] ε t =10000-ε c0 -ε0=10000-835-6200=2965

[0161]

[0162] Where: ε cu is the ultimate compressive strain of concrete, the value of C50 concrete is 3300, ε t is the tensile strain of concrete at the steel strand under this limit state.

[0163]

[0164] Where βx = 0.8 × 838 = 200 + 180 + 120 + 170.4 = 670.4 mm

[0165] In the formula: 2500mm is the width of the flange plate, 600mm is the width of the chamfer reinforcement layer, 420mm is the average width of the chamfer change layer, and 240mm is the thickness of the web.

[0166]

[0167] Calculate the corresponding steel strand area A p1 25880mm 2 The distance z1 from the steel strand to the center of the compression zone of the structure is 1255mm.

[0168] S5: Calculate the height of the structural compression zone and the distance from the steel strand to the center of the structural compression zone when the micro-strain of the concrete at the steel strand position is 10,000 under the structural limit state according to the concrete strain diagram of the structural section, and calculate the corresponding steel strand area A p2 and ultimate strength of strands;

[0169] In this embodiment, the limit state structural strain can also refer to the embodiment 1. Figure 3 , the height of the compression zone and the distance from the strand to the center of the compression zone of the structure:

[0170]

[0171] The micro strain of the steel strand is 10000+ε c0 +ε0=10000+6200+835=17650

[0172]

[0173]

[0174] Where βx = 0.8 × 395 = 200 + 116 = 316 mm

[0175] In the formula: 2500mm is the width of the flange plate, and 600mm is the width of the chamfer reinforcement layer.

[0176]

[0177] Calculate the corresponding steel strand area A p2 13790mm 2 , the distance z2 from the steel strand to the center of the structural compression zone is 1432mm and the ultimate strength of the steel strand is 1799MPa.

[0178] S6: Based on the linear relationship, the relationship between the stress of the steel strand under the limit state, the distance from the steel strand to the center of the structural compression zone and the area of ​​the steel strand are derived respectively. The linear relationship is assumed as follows:

[0179] f pu =a+gA p ,

[0180] z=c+dA p ,

[0181] Wherein: a, c, d, g are all constants.

[0182]

[0183] In this embodiment:

[0184] f pu =-0.02A p +2048,

[0185] z=-0.015A p +1634,

[0186] S7: The formula for expressing the ultimate bending moment design value of the structure with respect to the area of ​​the steel strand is obtained using the following formula:

[0187]

[0188] In this embodiment:

[0189]

[0190] Where: 13790≤A p ≤25880

[0191] S8: According to the design value of the structural ultimate bending moment, which must be greater than or equal to the structural cracking bending moment, the reinforcement ratio of the steel strands of the full prestressed concrete beam bridge is verified. The following formula is used to verify the reinforcement ratio of the steel strands of the full prestressed concrete beam bridge:

[0192]

[0193] In this embodiment:

[0194]

[0195] According to the formula, the comparison chart is obtained, refer to Figure 7 , that is, the solution range is 812≤4A p ≤11,9 and initial steel strand area limit range 13790≤A p ≤25880 has no intersection, so in this embodiment, the configuration of the steel strands of the prestressed concrete beam bridge is unreasonable. If the original steel strand area is 9452mm according to Example 2 2 When the structure is damaged, the steel strands have not reached the nominal yield load and have not fully played their role. Therefore, there are systematic deficiencies in the cross-section, and it is not possible to configure only prestressed steel strands. The cross-section needs to be modified.

[0196] Comparative Example 2: Existing Calculation Method

[0197] S1: Statistical cross-sectional properties, including but not limited to: cross-sectional inertia moment, distance between centroid and top edge, bottom edge, protective layer thickness, cross-sectional elastic resistance moment, effective tensioning control stress of prestressed steel strand, area moment above the cross-sectional centroid axis, etc., which are the same as those in Example 2;

[0198] S2: Calculate the structural cracking moment:

[0199]

[0200] S3: Calculate the ultimate bending moment design value of the structure:

[0201]

[0202] but

[0203] Where: f pd is the design strength of the steel strand, which is 1260MPa, f cd is the design value of the axial compressive strength of concrete, which is 22.4MPa for C50 concrete and 99MPa for C150 concrete.

[0204] S4: Reinforcement ratio verification, M ud / M cr ≥1 indicates that the structural reinforcement is reasonable:

[0205]

[0206] This indicates that the structural reinforcement is unreasonable. The existing calculation method can only determine whether the reinforcement ratio of prestressed concrete flexural members meets the conditions. However, according to the calculation method of the present invention, not only can it be determined whether the reinforcement ratio of prestressed concrete flexural members meets the conditions, but it can also be determined whether the structure needs to be configured with ordinary steel bars. As shown in Example 2, a clear conclusion is obtained through the calculation method of the present invention: Example 2 must be configured with ordinary steel bars, and the structure itself does not meet the requirement of not configuring ordinary steel bars.

[0207] The above embodiments are only preferred embodiments of the present invention, but cannot be used as limitations of the invention. Any modifications and improvements based on the concept of the present invention should fall within the protection scope of the present invention. The specific protection scope shall be subject to the claims.

Claims

1. A calculation method for verifying the reinforcement ratio of steel strands in a fully prestressed concrete beam bridge, characterized in that: The following steps are involved: S1: Statistical section properties, including but not limited to: section inertia moment, distance from centroid to top and bottom edges, protective layer thickness, section elastic resistance moment, effective tensioning control stress of prestressed steel strands, and area moment of the area above the centroid axis of the full section to the centroid axis; S2: Derivation of the formula for the structural cracking moment with respect to the area of ​​the steel strand: Where: M cr is the cracking moment of the normal section of the bending member, σ pc,b In order to deduct all prestress losses, the prestressed steel bars generate concrete precompression stress at the crack-resistant edge of the component. γ is the plastic development coefficient, and f is tk is the standard tensile strength of concrete, W is the resistance moment of the lower edge of the section in the final state, A p is the area of ​​the steel strand, σ con is the tension stress of the steel strand, A1 is the cross-sectional area when the steel strand is tensioned, e is the distance from the cross-sectional centroid when the steel strand is tensioned, W1 is the resistance moment of the lower edge of the cross-sectional area when the steel strand is tensioned, and S is the area moment above the cross-sectional centroid axis in the final state; S3: Calculate the initial strain of the steel strand and the initial strain of the concrete at the steel strand location based on the constitutive relationship; S4: Calculate the height of the structural compression zone and the distance from the steel strand to the center of the structural compression zone when the stress of the steel strand reaches the nominal yield strength under the structural limit state according to the concrete strain diagram of the structural section at this time, and calculate the corresponding steel strand area A p1 , steel strand area A p1 is the upper limit of the preliminary steel strand area limit; S5: Calculate the height of the structural compression zone and the distance from the steel strand to the center of the structural compression zone when the micro-strain of the concrete at the steel strand position is 10,000 under the structural limit state according to the concrete strain diagram of the structural section, and calculate the corresponding steel strand area A p2 and ultimate strength of steel strand, area of ​​steel strand A p2 It is the lower limit of the preliminary steel strand area limit range; S6: Based on the linear relationship, the relationship between the stress of the steel strand in the limit state, the distance between the steel strand and the center of the structural compression zone and the area of ​​the steel strand are derived respectively; S7: Get the formula for expressing the ultimate bending moment design value of the structure with respect to the area of ​​the steel strand; S8: Based on the fact that the design value of the structural ultimate bending moment must be greater than or equal to the structural cracking bending moment, verify the reinforcement ratio of the steel strands in the fully prestressed concrete beam bridge.

2. The calculation method for verifying the reinforcement ratio of steel strands of a fully prestressed concrete beam bridge according to claim 1 is characterized in that: In step S3, when calculating the initial strain of the steel strand and the initial strain of the concrete at the position of the steel strand, the concrete constitutive curve refers to the German Rüsch curve, and the constitutive relationship of the steel strand adopts the following formula: Where: f ps is the stress of the steel strand, E ps The elastic modulus of the steel strand is 1.92×10 5 MPa, ε is the strain of the steel strand, f py The yield stress is 1680.15MPa, and N, K, and Q are fitting constants, which are 7.344, 1.0618, and 0.01174, respectively.

3. The calculation method for verifying the reinforcement ratio of steel strands of a fully prestressed concrete beam bridge according to claim 2 is characterized in that: In step S3, when calculating the initial strain of the steel strand and the initial strain of the concrete at the steel strand position, the initial stress σ of the concrete at the steel strand position is pe The calculation uses the following formula: Where: A pn is the area of ​​the original steel strand arrangement of the structure, I1 is the section moment of inertia when the steel strand is tensioned, G is the gravity effect, I is the section moment of inertia in the final state, and ω is the concrete density.

4. The calculation method for verifying the reinforcement ratio of steel strands of a fully prestressed concrete beam bridge according to claim 3 is characterized by: In the steps S4 and S5, the calculation of the area of ​​the steel strand and the distance from the steel strand to the center of the structural compression zone is respectively performed using the following formulas: f pu A p =f ck βxb, Where: f pu is the actual damage value of the steel strand at the ultimate bending moment of the section, f ck is the standard compressive strength of concrete, β is the ratio of the height of the rectangular stress block in the compression zone to the actual height of the compression zone, x is the height of the compression zone, b is the section width, z is the distance from the steel strand to the center of the compression zone of the structure, and h0 is the effective height of the section.

5. The calculation method for verifying the reinforcement ratio of steel strands of a fully prestressed concrete beam bridge according to claim 1 is characterized in that: According to the meaning of the formula for the relative limit compression zone height of prestressed concrete beams in the "Highway Reinforced Concrete and Prestressed Concrete Bridge and Culvert Design Code JTG3362-2018", the steel strand must at least reach the nominal yield strength of 0.85f at the structural limit. pk =1581MPa.

6. The calculation method for verifying the reinforcement ratio of steel strands of a fully prestressed concrete beam bridge according to claim 1 is characterized in that: Analogously to the fact that the micro-strain of concrete at the lower edge of the cross section of an ordinary concrete beam under the ultimate state cannot exceed 10,000, the micro-strain of concrete at the steel strand position of a prestressed concrete beam under the ultimate state cannot exceed 10,000 either, in order to ensure a reasonable deflection-span ratio in the ultimate state of the structure.

7. The calculation method for verifying the reinforcement ratio of steel strands of a fully prestressed concrete beam bridge according to claim 4 is characterized in that: In step S6, the relationship between the stress of the steel strand, the distance from the steel strand to the center of the structural compression zone and the area of ​​the steel strand under the limit state is assumed to be a linear relationship: f pu =a+gA p , z=c+dA p , Wherein: a, c, d, g are all constants.

8. The calculation method for verifying the reinforcement ratio of steel strands of a fully prestressed concrete beam bridge according to claim 7 is characterized in that: In step S7, the formula for expressing the structural ultimate bending moment design value with respect to the steel strand area is as follows: Where: M ud is the design value of the bending bearing capacity of the positive section of the flexural member, z is the distance between the steel strand and the center of the compression zone, λ is the ductility guarantee index, and with reference to reinforced concrete beams, λ is taken as 1.

25.

9. The calculation method for verifying the reinforcement ratio of steel strands of a fully prestressed concrete beam bridge according to claim 8 is characterized in that: In step S8, the following formula is used to calculate the reinforcement ratio of the steel strands of the full prestressed concrete beam bridge:

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