A method for calculating cracking moment and flexural capacity of fabricated steel pipe concrete composite truss girder under negative bending moment

Abstract

This invention discloses a method for calculating the cracking moment and flexural bearing capacity of prefabricated steel-concrete composite truss girders under negative bending moment, belonging to the field of bridge engineering technology. The invention includes the following steps: determining the design parameters of the steel-concrete composite truss girder; calculating the curvature φ of the support section and the cracking moment M of the section in the negative bending zone; and calculating the flexural bearing capacity M in the negative bending zone based on the strain distribution and internal force equilibrium relationship of the positive section. u The theoretical calculation values ​​obtained by the method of this invention agree well with the experimental values. It can quickly and accurately obtain the cracking bending moment and flexural bearing capacity of prefabricated steel-concrete composite truss bridges under negative bending moment, thus improving the calculation efficiency and accuracy of the cracking bending moment and flexural bearing capacity of composite trusses.

Description

Technical Field

[0001] This invention belongs to the field of bridge engineering technology, specifically relating to a method for calculating the cracking bending moment and flexural bearing capacity of prefabricated steel-concrete composite trusses under negative bending moment. Background Technology

[0002] Cast-in-place reinforced concrete bridges have been the mainstream bridge structure in China's bridge construction over the past two decades. However, the construction of cast-in-place reinforced concrete bridges has problems such as high energy consumption, heavy pollution, and disruption of urban traffic. Moreover, the performance of concrete bridges deteriorates severely during service, requiring major repairs or replacement every two or three decades, making it difficult to reach the designed service life.

[0003] Steel-concrete composite beams are a type of bridge structure suitable for prefabricated, low-carbon construction. Steel, being a lightweight and homogeneous material, has more stable properties than concrete, effectively addressing the problems encountered during the construction and service of cast-in-place reinforced concrete bridges. Prefabricated steel-concrete composite trusses are another structural form within steel-concrete composite bridges. By fully utilizing the high force transmission efficiency of the truss system and the suitability of segmental manufacturing for steel trusses and assembly construction for precast concrete slabs, they are highly suitable for the industrialized construction of heavy-load, long-span bridges.

[0004] In the positive bending moment region, steel-concrete composite trusses fully utilize the material advantages of steel under tension and concrete under compression. However, in the negative bending moment region, the tensile cracking of the concrete slab and the compressive buckling of the lower flange of the steel beam have consistently hindered the application and development of composite trusses, severely affecting their serviceability limit state and ultimate limit state performance in the negative bending moment region. Current specifications lack calculation formulas for the cracking moment and flexural capacity of prefabricated steel-concrete composite trusses under negative bending moment, thus failing to effectively guide the design of such bridges and consequently posing potential structural safety hazards. Summary of the Invention

[0005] The purpose of this invention is to provide a method for calculating the cracking bending moment and flexural bearing capacity of prefabricated steel-concrete composite truss girders under negative bending moment, so as to accurately calculate the cracking bending moment and flexural bearing capacity of prefabricated steel-concrete composite truss bridges under negative bending moment, thereby guiding the design and ensuring the safety of bridge structures.

[0006] The technical solution of this invention is: a method for calculating the cracking bending moment and flexural bearing capacity of prefabricated steel-concrete composite truss beams under negative bending moment, comprising the following steps:

[0007] S1. Determine the design parameters of the steel-concrete composite truss: Determine the segment length L of the prefabricated steel-concrete composite truss in the negative moment zone; determine the distance x from the support section to the end section in the negative moment zone; determine the vertical distance y from the centroidal axis of the concrete slab to the top surface of the concrete slab. ctThe vertical distance y from the centroidal axis of the concrete slab to the bottom surface of the concrete slab cb The vertical distance y from the top surface of the steel truss to the central axis of the steel truss st Determine the spacing b of the shear studs and the shear stiffness K, and determine the elastic modulus E of the concrete slab and the steel truss. c and E s The area A of the concrete slab c Moment of inertia I of concrete slabs and steel trusses c and I s ;

[0008] S2. Calculate the curvature φ of the section at the support point in the negative bending zone and the cracking moment M of the section. The calculation formula is as follows:

[0009]

[0010] M=φ(E c I c +E s I s -E c A c yy ct )+E c A c (ε pc +ε tk (2)

[0011] In the formula, y = y cb +y st P is the concentrated force acting on the support section in the negative bending moment region, EI = E s I s +E c I c ;ε pc ε represents the prestressing strain generated by the prestressed steel strands on the top surface of the concrete slab. tk The ultimate tensile strain of concrete, represented by the mathematical symbol e, has a value of 2.718.

[0012] S3. Calculate the flexural capacity M in the negative bending zone based on the strain distribution and internal force equilibrium relationship of the positive section. u The calculation formula is as follows:

[0013] M u =N ut1 (y1-y c )+N ut2 (y2-y c (3)

[0014] In the formula, N ut1 It is the ultimate tensile bearing capacity of the steel reinforcement (including prestressed steel strands) in concrete slabs, N. ut2y1 is the ultimate tensile bearing capacity of the upper chord member; y2 is the vertical distance from the centroidal axis of the concrete slab to the bottom surface of the lower chord member of the steel truss; y1 is the vertical distance from the centroidal axis of the upper chord member of the steel truss to the bottom surface of the lower chord member of the steel truss. c It is the vertical distance from the centroidal axis of the lower chord of the steel truss to the bottom surface of the lower chord of the steel truss.

[0015] Furthermore, in step S2, formulas (1) and (2) are derived from the strain distribution diagram of the normal section considering the slippage of the steel-concrete interface in the prefabricated steel-concrete composite truss.

[0016] Furthermore, in step S3, formula (3) is derived using the plastic stress distribution method based on the strain distribution and internal force balance relationship of the prefabricated steel-concrete composite truss under the ultimate limit state of bearing capacity.

[0017] Further, in step S3, N ut1 N ut2 The calculation formula is as follows:

[0018] N ut1 =f st A st +σ p A p (4-1)

[0019] N ut1 =f st A st (4-2)

[0020] N ut2 =f y A s1 (5)

[0021] When a concrete slab contains both ordinary steel bars and prestressed steel strands, N is calculated using formula (4-1). ut1 When the concrete slab contains only ordinary steel reinforcement, N is calculated using formula (4-2). ut1 ;

[0022] In the formula, f st It is the yield strength of the steel reinforcement in the concrete slab, f y It is the yield strength of the upper chord steel, σ p It is the tensile stress of the prestressed steel strands in the concrete slab, A st A p A s1 These are the areas of the reinforcing steel bars in the concrete slab, the prestressed steel strands, and the top chord.

[0023] The beneficial effects of this invention are as follows: Based on the strain distribution relationship of the prefabricated steel-concrete composite truss considering the slippage at the steel-concrete interface, this invention derives a calculation formula for the cracking bending moment of the composite truss; based on the strain distribution and internal force balance relationship of the prefabricated steel-concrete composite truss under the ultimate limit state of bearing capacity, a calculation formula for the flexural bearing capacity of the composite truss under negative bending moment is obtained. The theoretical calculation values ​​obtained by this invention agree well with the experimental values, and can quickly and accurately obtain the cracking bending moment and flexural bearing capacity of prefabricated steel-concrete composite truss bridges under negative bending moment, thus improving the calculation efficiency and accuracy of the cracking bending moment and flexural bearing capacity of the composite truss. Attached Figure Description

[0024] Figure 1 This is a structural diagram of a prefabricated steel-concrete composite truss beam in a specific embodiment of the present invention;

[0025] Figure 2 This is a structural elevation view of the test specimen in the negative bending moment zone of the steel-concrete composite truss in a specific embodiment of the present invention.

[0026] Figure 3 This is a cross-sectional structural diagram of the test specimen in the negative bending moment zone of the steel-concrete composite truss in a specific embodiment of the present invention.

[0027] Figure 4 This is a flowchart illustrating a method for calculating the cracking moment and flexural bearing capacity of a prefabricated steel-concrete composite truss under negative bending moment in a specific embodiment of the present invention.

[0028] Figure 5 This is a diagram showing the strain distribution of the prefabricated steel-concrete composite truss considering steel-concrete interface slippage in a specific embodiment of the present invention.

[0029] Figure 6 This is a diagram showing the longitudinal normal stress distribution of the CTB1 section of the steel-concrete composite truss specimen under the ultimate limit state of bearing capacity in a specific embodiment of the present invention.

[0030] Figure 7 This is a diagram showing the longitudinal normal stress distribution of the CTB2 section of the steel-concrete composite truss specimen under the ultimate limit state of bearing capacity in a specific embodiment of the present invention.

[0031] In the diagram: 1-Precast concrete bridge deck; 2-Buried shear studs; 3-Concrete composite truss. Detailed Implementation

[0032] The present invention will be further described in detail below with reference to the accompanying drawings and specific embodiments.

[0033] Example 1

[0034] Prefabricated steel-concrete composite truss beams, such as Figure 1As shown, the prefabricated steel-concrete composite truss is assembled from a steel-concrete truss 3 and a precast concrete bridge deck 1 using bundled shear studs 2.

[0035] In this embodiment, a test study on the negative bending moment zone of the steel-concrete composite truss was first conducted. The specimen size and material properties used in the theoretical derivation formula were the same as those used in the experimental study.

[0036] Test study on negative bending moment zone of steel-concrete composite truss girder. Specimen dimensions are as follows: Figure 2 (Dimensions in the figure are in mm). Two specimens were designed. Specimen CTB1 uses a prestressed reinforced concrete bridge deck combined with locally released shear studs, while specimen CTB2 uses a conventional reinforced concrete bridge deck combined with standard shear studs. The cross-sections of specimens CTB1 and CTB2 are shown below. Figure 3 (The dimensions in the figure are in mm). The concrete properties of the specimen are shown in Table 1, and the steel plate and steel properties of the specimen are shown in Table 2.

[0037] Table 1

[0038]

[0039] Table 2

[0040]

[0041] A method for calculating the cracking moment and flexural bearing capacity of prefabricated steel-concrete composite truss beams under negative bending moment, such as... Figure 4 As shown, it includes the following steps:

[0042] S1. Determine the design parameters of the steel-concrete composite truss: Determine the segment length L of the prefabricated steel-concrete composite truss in the negative moment zone; determine the distance x from the support section to the end section in the negative moment zone; determine the vertical distance y from the centroidal axis of the concrete slab to the top surface of the concrete slab. ct The vertical distance y from the centroidal axis of the concrete slab to the bottom surface of the concrete slab cb The vertical distance y from the top surface of the steel truss to the central axis of the steel truss st Determine the spacing b of the shear studs and the shear stiffness K, and determine the elastic modulus E of the concrete slab and the steel truss. c and E s Moment of inertia I of concrete slabs and steel trusses c and I s .

[0043] S2. Calculate the curvature φ of the section at the support point in the negative bending zone and the cracking moment M of the section. The calculation formula is as follows:

[0044]

[0045] M=φ(E c Ic +E s I s -E c A c yy ct )+E c A c (ε pc +ε tk (2)

[0046] In the formula, y = y cb +y st P is the concentrated force acting on the support section in the negative bending moment region, EI = E s I s +E c I c ;ε pc ε represents the prestressing strain generated by the prestressed steel strands on the top surface of the concrete slab. tk This represents the ultimate tensile strain of the concrete.

[0047] Formulas (1) and (2) are derived from the strain distribution relationship of the prefabricated steel-concrete composite truss considering the steel-concrete interface slippage. The strain distribution relationship of the prefabricated steel-concrete composite truss considering the steel-concrete interface slippage under negative bending moment is established as follows: Figure 5 As shown;

[0048] The longitudinal normal stress distributions of the sections of prefabricated steel-concrete composite truss specimens CTB1 and CTB2 under the ultimate limit state of bearing capacity under negative bending moment are as follows: Figure 6 , Figure 7 As shown.

[0049] S3. Calculate the flexural capacity M in the negative bending zone based on the strain distribution and internal force equilibrium relationship of the positive section. u The calculation formula is as follows:

[0050] M u =N ut1 (y1-y c )+N ut2 (y2-y c (3)

[0051] In the formula, N ut1 It is the ultimate tensile bearing capacity of the steel reinforcement (including prestressed steel strands) in concrete slabs, N. ut2 y1 is the ultimate tensile bearing capacity of the upper chord member; y2 is the vertical distance from the centroidal axis of the concrete slab to the bottom surface of the lower chord member of the steel truss; y1 is the vertical distance from the centroidal axis of the upper chord member of the steel truss to the bottom surface of the lower chord member of the steel truss. c It is the vertical distance from the centroidal axis of the lower chord of the steel truss to the bottom surface of the lower chord of the steel truss;

[0052] Formula (3) is derived from the strain distribution and internal force balance relationship of the prefabricated steel-concrete composite truss under the ultimate limit state of bearing capacity, using the plastic stress distribution method.

[0053] N ut1 N ut2 The calculation formula is as follows:

[0054] N ut1 =f st A st +σ p A p (4-1)

[0055] N ut1 =f st A st (4-2)

[0056] N ut2 =f y A s1 (5)

[0057] When a concrete slab contains both ordinary steel bars and prestressed steel strands, N is calculated using formula (4-1). ut1 When the concrete slab contains only ordinary steel reinforcement, N is calculated using formula (4-2). ut1 ;

[0058] In the formula, f st It is the yield strength of the steel reinforcement in the concrete slab, f y It is the yield strength of the upper chord steel, σ p It is the tensile stress of the prestressed steel strands in the concrete slab, A st A p A s1 These are the areas of the reinforcing steel bars in the concrete slab, the prestressed steel strands, and the top chord.

[0059] In this embodiment, the experimental research method for the negative bending moment zone of the steel-concrete composite truss is as follows: In order to generate a negative bending moment, a concentrated force from bottom to top needs to be applied at the mid-span of the composite truss. During the test, a load-controlled loading method is adopted. When the estimated ultimate load is below 75% (the specimen has not entered the elastic-plastic stage), the load increment is 40kN per level. When the estimated ultimate load is above 75% (the specimen has entered the elastic-plastic stage), the load increment is 20kN per level, and the loading continues until the specimen fails.

[0060] In the experimental study of the negative bending moment zone of steel-concrete composite truss, the CTB1 concrete slab specimen was prestressed using prestressed steel strands. The prestressed strain ε generated on the top surface of the concrete under the prestressed stress was measured. pc The ultimate tensile strain ε of concrete is 0.000139. tk It is usually taken as 0.0001.

[0061] The stiffness K of a single locally released shear stud in specimen CTB1 can be obtained using the following formula:

[0062]

[0063]

[0064] In the formula, b is the longitudinal spacing of the shear studs, and A std It is the area of ​​the shear stud body, f cd It is the design value of the axial compressive strength of concrete;

[0065] b = 70mm, shear stud diameter 13mm, A std =133mm 2 f cd =22.4MPa, E c =3.78*10 4 MPa, substituting the above data into formulas (6) and (7) and considering that the shear studs are arranged in a double row in parallel, the stiffness of the shear studs for the local release effect of specimen CTB1 is K1=620N / mm.

[0066] The stiffness K of a single conventional shear stud in specimen CTB2 can be obtained using the following formula:

[0067]

[0068] In the formula, d s It is the diameter of the shear stud, E c It is the elastic modulus of concrete, f ck It is the standard value of concrete compressive strength;

[0069] d s =13mm, f ck =32.4MPa, E c =3.78*10 4 Substituting the above data into formula (8) and considering that the shear studs are arranged in a double row in parallel, the stiffness of the conventional shear studs in specimen CTB2 is K2=374054N / mm.

[0070] E s =2.06*10 11 Pa, E c =3.78*10 10 Pa, I s =0.00245m 4 I c =8.333E-05m 4 EI = E s I s +E c I c=508703266.5m 4 .

[0071] In this embodiment, A c =0.1m 2 A s =0.013308141m 2 ,

[0072] In this embodiment, y = y cb +y st =0.47m.

[0073] In this embodiment, L = 6.5m and x = 3.15m.

[0074] For specimen CTB1, α = 0.0039; for specimen CTB2, α = 0.095; and for specimens CTB1 and CTB2, β = 2.258E-09.

[0075] For CTB1 specimen, P = 280 kN; for CTB2 specimen, P = 110 kN.

[0076] Let K1, K2, b, EI, Substituting y, L, x, α, β, P into equations (1) and (2) yields the following results:

[0077] For CTB1 specimen, the curvature Φ = 7.886E-07 and the cracking moment M = 425kN*m;

[0078] For the CTB2 specimen, the curvature Φ = 3.098E-07 and the cracking moment M = 178kN*m.

[0079] In this embodiment, the comparison between the cracking bending moment test data and the theoretical calculation data is shown in Table 3. It can be found that the theoretical value is in good agreement with the test.

[0080] Table 3

[0081]

[0082] In this embodiment, f st =452N / mm 2 f y =355kN / mm 2 A st =3.14 / 4*10 2 *18=1413mm 2 A p =6 * 140 = 840 mm 2 , σ p =2354kN / mm2 A s1 =2*(120+124)*8=3904mm 2 Substituting the above data into equations (4-1), (4-2), and (5), we obtain the following results:

[0083] For CTB1 specimen, N ut1 =f st *A st +σ p *A p =1028.5kN;

[0084] For CTB2 specimen, N ut1 =f st *A st =639.0kN;

[0085] For CTB1 and CTB2 specimens, N ut2 =f y *A s1 =1385.9kN.

[0086] In this embodiment, y1 = 0.89m, y2 = 0.78m, y c =0.06m, substituting the above data into equation (3) yields the following result:

[0087] For CTB1 specimen, M u =N ut1 (y1-y c )+N ut2 (y2-y c )=1028.5*(0.89-0.06)+1385.9*(0.78-0.06)=1851.6kN*m;

[0088] For CTB2 specimen, M u =N ut1 (y1-y c )+N ut2 (y2-y c )=639.0*(0.89-0.06)+1385.9*(0.78-0.06)=1881.8kN*m.

[0089] In this embodiment, the comparison between the test data and theoretical calculation data of flexural bearing capacity is shown in Table 4. It can be found that the theoretical value is in good agreement with the test.

[0090] Table 4

[0091]

Claims

1. A method for calculating the cracking bending moment and flexural bearing capacity of prefabricated steel-concrete composite truss beams under negative bending moment, characterized in that... Includes the following steps: S1. Determine the design parameters of the steel-concrete composite truss: Determine the segment length L of the prefabricated steel-concrete composite truss in the negative moment zone; determine the distance x from the support section to the end section in the negative moment zone; determine the vertical distance y from the centroidal axis of the concrete slab to the top surface of the concrete slab. ct The vertical distance y from the centroidal axis of the concrete slab to the bottom surface of the concrete slab cb The vertical distance y from the top surface of the steel truss to the central axis of the steel truss st Determine the spacing b of the shear studs and the shear stiffness K, and determine the elastic modulus E of the concrete slab and the steel truss. c and E s The area A of the concrete slab c Moment of inertia I of concrete slabs and steel trusses c and I s ; S2. Calculate the curvature φ of the section at the support point in the negative bending zone and the cracking moment M of the section. The calculation formula is as follows: M=φ(E c I c +E s I s -E c A c yy ct )+E c A c (ε pc +ε tk ) (2) In the formula, y = y cb +y st P is the concentrated force acting on the support section in the negative bending moment region, EI = E s I s +E c I c ;ε pc ε represents the prestressing strain generated by the prestressed steel strands on the top surface of the concrete slab. tk This represents the ultimate tensile strain of the concrete. S3. Calculate the flexural capacity M in the negative bending zone based on the strain distribution and internal force equilibrium relationship of the positive section. u The calculation formula is as follows: M u =N ut1 (y1-y c )+N ut2 (y2-y c ) (3) In the formula, N ut1 It is the ultimate tensile bearing capacity of the steel reinforcement in a concrete slab, N. ut2 y1 is the ultimate tensile bearing capacity of the upper chord member; y2 is the vertical distance from the centroidal axis of the concrete slab to the bottom surface of the lower chord member of the steel truss; y1 is the vertical distance from the centroidal axis of the upper chord member of the steel truss to the bottom surface of the lower chord member of the steel truss. c It is the vertical distance from the centroidal axis of the lower chord of the steel truss to the bottom surface of the lower chord of the steel truss; N ut1 N ut2 The calculation formula is as follows: N ut1 =f st A st +s p A p (4-1) N ut1 =f st A st (4-2) N ut2 =f y A s1 (5) When a concrete slab contains both ordinary steel bars and prestressed steel strands, N is calculated using formula (4-1). ut1 When the concrete slab contains only ordinary steel reinforcement, N is calculated using formula (4-2). ut1 ; In the formula, f st It is the yield strength of the steel reinforcement in the concrete slab, f y It is the yield strength of the upper chord steel, σ p It is the tensile stress of the prestressed steel strands in the concrete slab, A st A p A s1 These are the areas of the reinforcing steel bars in the concrete slab, the prestressed steel strands, and the top chord.

2. The method for calculating the cracking moment and flexural bearing capacity of a prefabricated steel-concrete composite truss beam under negative bending moment according to claim 1, characterized in that: In step S2, formulas (1) and (2) are derived from the strain distribution diagram of the normal section considering the slippage of the steel-concrete interface in the prefabricated steel-concrete composite truss.

3. The method for calculating the cracking moment and flexural bearing capacity of a prefabricated steel-concrete composite truss beam under negative bending moment as described in claim 1 or 2, characterized in that: In step S3, formula (3) is derived using the plastic stress distribution method based on the strain distribution and internal force balance relationship of the prefabricated steel-concrete composite truss under the ultimate limit state of bearing capacity.