A calculation method for checking the reinforcement ratio of steel strands in fully prestressed concrete beam bridges

Abstract

The present invention discloses a calculation method for checking the reinforcement ratio of steel strands in a fully prestressed concrete beam bridge, comprising the following steps: S1: statistically analyzing cross-sectional characteristics; S2: deriving an expression formula for the structural cracking bending moment with respect to the area of ​​the steel strands; S3: calculating the initial strain of the steel strands and the initial strain of concrete at the position of the steel strands; S4: obtaining the corresponding steel strand area A when the stress of the steel strands reaches the nominal yield strength under the structural limit state. p1 ; S5: Calculate the steel strand area A when the concrete microstrain at the steel strand position is 10000 under the structural limit state p2 S6: Based on the linear relationship, the relationship between the stress of the steel strand under the ultimate limit state, the distance between the steel strand and the center of the structural compression zone, and the steel strand area is derived. S7: The formula for expressing the ultimate bending moment design value of the structure with respect to the steel strand area is obtained. S8: The reinforcement ratio of the steel strand in a fully prestressed concrete beam bridge is verified. This invention can effectively verify the reinforcement ratio of the steel strand in a fully prestressed concrete beam bridge.

Description

Technical Field

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

[0002] Prestressed concrete beams are one of the most important transportation infrastructures for mankind. How to ensure their stability, safety and durability is an important issue. Controlling the reinforcement ratio is usually adopted as an effective structural means to limit the damage of prestressed concrete beams, and it 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), and the design value of the structural ultimate bending moment must be greater than or equal to the structural cracking bending moment for verification. According to the existing calculation method, it can only be determined 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, the design value of the steel strand strength is taken as 1260MPa, 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, providing a reference for future steel strand configuration design and construction plan verification. Summary of the Invention

[0004] The purpose of the present invention is to provide a calculation method for checking the reinforcement ratio of steel strands in a fully prestressed concrete beam bridge, which can more reasonably check the reinforcement ratio of steel strands in 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 verifying the reinforcement ratio of steel strands in a fully prestressed concrete beam bridge comprises the following steps:

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

[0008] S2: Derive 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,bTo deduct all prestress losses, the prestressed steel bars generate concrete precompression stress at the cracking edge of the component. γ is the plastic development coefficient, and f is the tk is the standard tensile strength of concrete, W is the resistance moment of the lower edge of the final state section, 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 of ​​the steel strand when tensioning, 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 tensioning the steel strand, 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 corresponding 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 based on 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 range;

[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 based on the concrete strain diagram of the structural section, and calculate the corresponding steel strand area A. p2 and ultimate strength of steel strand, steel strand area A p2 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: Obtain 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 mode of the fully 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 strand stress, 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 1680.15 MPa, 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 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 Code 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 fact that the micro-strain of the concrete at the lower edge of the cross section of an ordinary concrete beam under the ultimate state cannot exceed 10,000, and the micro-strain of the concrete at the steel strand position of a prestressed concrete beam under the ultimate state cannot exceed 10,000, so as to ensure a reasonable deflection-span ratio in the ultimate state of the structure.

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

[0031] f pu =a+gA p

[0032] z=c+dA p

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

[0034] As a preferred calculation method of the present invention: in step S7, the expression formula of the structural ultimate 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 flexural bearing capacity of the normal section of the flexural member, z is the distance between the steel strand and the center of the compression zone, and λ is the ductility guarantee index. 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 fully 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 instead of the f pk =1860Mpa to accommodate the actual failure mode of prestressed reinforced concrete beams, where the prestressed steel bars may not necessarily break at ultimate failure. Verification using the calculation method of this invention not only determines whether the reinforcement ratio of prestressed concrete flexural members meets the requirements, but also determines whether the structure requires conventional reinforcement. The required steel strand configuration is determined if conventional reinforcement is not required. The "minimum reinforcement ratio for prestressed concrete beams" is revised to the "auxiliary reinforcement ratio for prestressed concrete beams," and the decision on whether to configure conventional reinforcement is made based on the structure itself and the steel strand configuration.

[0041] 2. Compared with the existing calculation method, the calculation method of the present invention can perform calculations based on 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 structural design.

[0042] 3. The ductility guarantee index λ of the present invention is based on the mechanical meaning of the structure and is taken as 1.25 with reference to reinforced concrete beams, rather than 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 microstrain of concrete at the steel strand position is 10,000 under the ultimate limit state of the structure in Example 1;

[0046] Figure 4 is a curve diagram drawn according to the final calculation results 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 results 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 using the cross section of a prestressed concrete beam bridge as an example. The present invention provides a method for calculating the reinforcement ratio of steel strands in a fully prestressed concrete beam bridge, comprising the following steps:

[0053] S1: Statistical cross-sectional properties, including but not limited to: cross-sectional inertia moment, distance from centroid to top and bottom edges, cover thickness, cross-sectional elastic resistance moment, effective tensioning control stress of prestressed steel strands, area moment of the area above the centroidal axis of the full cross-sectional area about the centroidal 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. The original arrangement of the steel strand area A 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 of the final section (the area moment of the area above the centroid of the full section to the centroid) S is 0.15888×10 9 mm 3 , the distance e between the resultant 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 resultant force point of the 4 strands and the centroid of the section is e = 650 mm, 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,bThe concrete precompressive stress (MPa) generated by the prestressed steel bars at the cracking edge of the component is deducted for the total prestress loss, γ is the plastic development coefficient, and f is the 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 ​​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 tensioned (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 location 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 uses 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 1680.15 MPa, and N, K, and Q are fitting constants, which are 7.344, 1.0618, and 0.01174, respectively.

[0065] Initial stress of concrete at the strand position σ 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 for C50 concrete adopts the following formula:

[0069]

[0070] Where: σ 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: Calculate the stress of the steel strand in the ultimate limit state according to the structural strain diagram 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 reach a nominal yield strength of at least 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), which is 32.4 MPa for C50 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, which is 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, which is 3300 for C50 concrete, 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=472mm,

[0087] Where: 1600mm is the flange width and 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 924 mm.

[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 based on the concrete strain diagram of the structural section, and calculate the corresponding steel strand area A. p2 and ultimate strength of steel strand, steel strand area A p2 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 = 230.4 mm

[0097] Where: 1600mm is the flange width and 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 1045 mm and the ultimate strength of the steel strand is 1798 MPa.

[0100] 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. The linear relationship is assumed as follows:

[0101] f pu =a+eA p ,

[0102] z=c+dA p ,

[0103] Where: a, c, d, and 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 flexural 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), and λ is the ductility guarantee index. With reference to reinforced concrete beams, λ is taken as 1.25.

[0111] In this embodiment:

[0112]

[0113] Where: 5353≤Ap ≤6979,

[0114] 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, the reinforcement ratio of the steel strands of the fully prestressed concrete beam bridge is verified. The following formula is used to verify the reinforcement ratio of the steel strands of the fully 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 is 5353≤A p ≤6979, the intersection is 5353≤A p ≤5668. Conclusion obtained through 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 If the load is less than or equal to 5668, only prestressed steel strands need to be configured, and ordinary steel bars are not required.

[0119] Comparative Example 1: Existing Calculation Method

[0120] S1: Statistical cross-sectional properties, including but not limited to: cross-sectional inertia moment, distance from centroid to top and bottom edges, protective layer thickness, cross-sectional elastic resistance moment, effective tensioning control stress of prestressed steel strands, area moment above the cross-sectional centroid axis, etc., 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 1260 MPa, 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 possible to verify not only whether the reinforcement ratio of prestressed concrete flexural members meets the conditions, but also 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 requires the configuration of ordinary steel bars, or if the steel strand configuration area is changed, then ordinary steel bars are not needed.

[0130] Example 2

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

[0132] like Figure 5 As shown in the figure, 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, and C50 is converted. When designing, the cross section is arranged with 4 bundles of 17-Φ15 prestressed steel tendons, and the original steel strand area of ​​the structure is A 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 1186 MPa, 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 of the final section (the area moment of the area above the centroid of the full section to the centroid) S is 0.30658×10 9 mm 3 , the distance e between the resultant force point of the four strands and the centroid of the cross section p =960mm;

[0133] When the steel strands were tensioned, no C50 concrete was poured and the cross-sectional area A1 was 628853 mm2 , 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 four 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 7.2 MPa.

[0139] 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. 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 uses 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 of the structure, 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.

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

[0145]

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

[0147]

[0148] Where: σ 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: Based on the structural strain diagram, the stress of the steel strand under the structural limit state is calculated to be 1581 MPa (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), which is 32.4MPa for C50 concrete and 124MPa for C150 concrete.

[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, which is 3300 for C50 concrete, 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] Where: 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 1255 mm.

[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 based on 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] Where: 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 1432 mm and the ultimate strength of the steel strand is 1799 MPa.

[0178] 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. The linear relationship is assumed as follows:

[0179] f pu =a+gA p ,

[0180] z=c+dA p ,

[0181] Where: a, c, d, and 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: 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, the reinforcement ratio of the steel strands of the fully prestressed concrete beam bridge is verified. The following formula is used to verify the reinforcement ratio of the steel strands of the fully 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 yet 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 from centroid to top and bottom edges, cover thickness, cross-sectional elastic resistance moment, effective tensioning control stress of prestressed steel strands, area moment above the cross-sectional centroid axis, etc., the same as 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 1260 MPa, f cd is the design value of the axial compressive strength of concrete, which is 22.4 MPa for C50 concrete and 99 MPa 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 they cannot be used as limitations of the invention. Any modifications and improvements based on the concept of the present invention should fall within the scope of protection of the present invention. The specific scope of protection 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 cross-sectional properties, including but not limited to: cross-sectional inertia moment, distance from centroid to top and bottom edges, cover thickness, cross-sectional elastic resistance moment, effective tensioning control stress of prestressed steel strands, and area moment of the area above the centroidal axis of the full cross-sectional area about the centroidal axis; S2: Derive the formula for the structural cracking moment with respect to the area of ​​the steel strand: ， ， Where: is the cracking moment of the normal section of the flexural member, In order to deduct all prestress losses, the concrete precompression stress generated by the prestressed steel bars at the anti-cracking edge of the component is is the plastic development coefficient, is the standard tensile strength of concrete, is the resistance moment of the lower edge of the cross section in the final state, is the area of ​​the steel strand, To release the tensile stress of the steel strand, is the cross-sectional area of ​​the tensioned steel strand, is the distance from the cross-section centroid when the steel strand is tensioned, is the resistance moment of the lower edge of the cross section when the steel strand is tensioned, is the area moment above the centroid of the final state section; 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 strand to the center of the structural compression zone when the strand stress reaches the nominal yield strength under the structural limit state based on the concrete strain diagram of the structural section at this time, and calculate the corresponding strand area. , steel strand area is the upper limit of the preliminary steel strand area limit range; S5: Calculate the height of the structural compression zone and the distance from the strand to the center of the structural compression zone when the microstrain of the concrete at the strand position is 10,000 under the structural limit state based on the concrete strain diagram of the structural section, and calculate the corresponding strand area. and ultimate strength of steel strand, area of ​​steel strand 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: Obtain 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 in 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 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: ， Where: f ps is the strand stress, E ps The elastic modulus of the steel strand is 1.92×10 5 MPa, is the strand strain, f py The yield stress is 1680.15MPa, N 、 K 、 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 in 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 of the concrete at the steel strand position is The calculation uses the following formula: ， ， Where: is the area of ​​the original steel strand arrangement of the structure, is the section inertia moment of the steel strand when tensioned, G is the gravity effect, I is the final state section moment of inertia, is the concrete bulk density, A is the cross-sectional area of ​​the beam, L For the beam length.

4. The calculation method for verifying the reinforcement ratio of steel strands in a fully prestressed concrete beam bridge according to claim 3 is characterized by: In steps S4 and S5, the area of ​​the steel strand and the distance from the steel strand to the center of the structural compression zone are calculated using the following formulas: ， ， Where: is the actual damage value of the steel strand at the ultimate bending moment of the section, is the standard compressive strength of concrete, 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 cross-section width, z is the distance from the steel strand to the center of the structural compression zone, is the effective height of the section.

5. The calculation method for verifying the reinforcement ratio of steel strands in a fully prestressed concrete beam bridge according to claim 1 is characterized in that: 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 reach a nominal yield strength of at least 0.85 at the structural limit. f pk =1581 MPa.

6. The calculation method for verifying the reinforcement ratio of steel strands in 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 cannot exceed 10,000 under the ultimate state, the micro-strain of concrete at the steel strand position of a prestressed concrete beam under the ultimate state cannot exceed 10,000 either, 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 in 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 linear: ， ， Where: a, c, d, and g are all constants.

8. The calculation method for verifying the reinforcement ratio of steel strands in a fully prestressed concrete beam bridge according to claim 7 is characterized in that: In step S7, the formula for expressing the design value of the structural ultimate bending moment with respect to the area of ​​the steel strand is as follows: ， Where: is the design value of the bending bearing capacity of the normal section of the flexural member, is the distance between the steel strand and the center of the compression zone, For ductility assurance indicators, refer to reinforced concrete beams. Take it as 1.

25.

9. The calculation method for verifying the reinforcement ratio of steel strands in 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 fully prestressed concrete beam bridge: 。

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