A design method for the single-column pier capping beam structure based on the space strut-and-tie model

Abstract

The present invention relates to a design method for a single-column pier capping beam structure based on a spatial tension-compression bar model. The method of the present invention first establishes a basic configuration of a spatial tension-compression bar model for a single-column pier capping beam; then the basic configuration is parametrically expressed by six structural dimensions in the single-column pier capping beam. Based on the principle of minimum energy, an optimal spatial tension-compression bar model with parametric expression for the single-column pier capping beam is established and the internal forces of each spatial member of the spatial tension-compression bar model are calculated; the calculation results of the internal forces of each spatial member are used to replace the plane design method in the current code method to carry out the reinforcement design of the capping beam structure. The present invention conducts structural design based on the optimal spatial tension-compression bar model. Since the spatial effect of structural force is considered, its principle is scientific and reasonable, which is of great significance for the safe and reasonable design of the structure; moreover, the present invention also greatly improves the design efficiency of the single-column pier capping beam structure and fills the gap in the existing code.

Description

Technical Field

[0001] The present invention relates to a design method for a single-column pier structure of a bridge; it belongs to the technical field of civil engineering applications. Background Art

[0002] During the construction of urban highway bridges and bridges in inland navigation channels, due to the influence of terrain, land occupation area, urban landscape, water passing section and navigation channel, etc. The lower piers often adopt the design scheme of single-column piers to achieve the purposes of reducing the land occupation area, improving the layout of the lower bridge structure, reducing the water blocking area, avoiding conflicts between the foundation and underground buildings, and increasing the width of the navigation channel. Correspondingly, in order to enhance the torsional resistance of the upper structure and meet the design load-bearing requirements, it is usually necessary to set a capping beam on the top of the pier column to facilitate the setting of double supports or multiple supports. However, due to various design problems, such diseases and problems as vertical cracks in the capping beam of the single-column pier have occurred during the construction and operation process.

[0003] An important factor for the occurrence of diseases such as cracks is limited by the bridge design specifications and design theories in our country, which have quite obvious limitations of the times, and the understanding of the complexity of its mechanical properties is insufficient, resulting in insufficient calculation of the tensile stress at the top of the double-support capping beam, thus causing cracking in the longitudinal direction of the bridge. Along with the development of the longitudinal cracks at the top, vertical cracks then appear along the transverse direction of the capping beam of the pier column.

[0004] Generally speaking, the capping beam of a single-column pier belongs to a deep flexural member, and the general flexural member method based on the plane section assumption is not applicable. The strut-and-tie model calculation method given in the new specification, namely "Code for Design of Highway Reinforced Concrete and Prestressed Concrete Bridges and Culverts (JTG 3362-2018)", is more applicable. However, the method in the "Code" is a plane analysis method and is only applicable to the capping beam of a single-column pier with a relatively wide transverse direction and a relatively narrow longitudinal direction; for the capping beam of a single-column pier with little difference in the transverse and longitudinal dimensions, there is no targeted calculation method in the code method. If the code method is used for calculation, only the spatial effect of the structure can be ignored, and similar code clauses can be used for calculation and reinforcement design for its transverse and longitudinal directions respectively. Undoubtedly, this method has unknowns, and its reliability and rationality are worthy of study. For such components, currently engineering technicians generally adopt the method of integrating the spatial stress analysis results of the solid finite element model, and the calculation results obtained in this way are relatively reliable. However, the solid finite element model method has a cumbersome calculation process and cannot directly carry out reinforcement design according to the simplified calculation formula or program, and its practicability is poor.

[0005] On the basis of the above, the present patent has invented a design method for the capping beam structure of a single-column pier based on the spatial strut-and-tie model. Summary of the Invention

[0006] In view of the deficiencies of the prior art, the method of the present invention first establishes a basic configuration of a spatial strut-and-tie model for a single-column pier capping beam; then, the basic configuration is parametrically expressed by six structural dimensions in the single-column pier capping beam (the transverse width b of the capping beam, the transverse and longitudinal thickness t of the capping beam, the height h of the capping beam, the diameter d of the pier column, the transverse center distance s of the supports in the capping beam, and the width a of the support steel plate). Based on the principle of minimum energy, an optimal spatial strut-and-tie model with parametric expression for the single-column pier capping beam is established; based on the optimal spatial strut-and-tie model, the internal forces of each spatial member in the spatial strut-and-tie model are calculated; the calculation results of the internal forces of each spatial member are used to replace the plane design method in the current code method to carry out the reinforcement design of the capping beam structure.

[0007] A design method for a single-column pier structure of a bridge, which establishes a spatial strut-and-tie model of the structure based on the structural dimension parameters of the single-column pier capping beam, calculates and establishes the configuration of the optimal spatial strut-and-tie model according to the minimum energy criterion, calculates the internal force of the tie rod in the model for reinforcement design, and realizes the spatial design of the single-column pier capping beam.

[0008] The main contents of the present invention include:

[0009] (1) Spatial strut-and-tie model of single-column pier capping beam

[0010] The strut-and-tie model is a simplified force flow analysis model abstracted from the concrete structure continuum, consisting of struts, ties and nodes, which can reflect the internal force transmission path of the structure and can be used to determine the structural dimensions of the concrete D region, the steel bar area of the section and the layout position of the steel bars. Through the established strut-and-tie model, the internal force transmission mechanism inside the concrete structure can be truly reflected, and this is used as the basis for the structural dimension determination and reinforcement design. The characteristics of the spatial strut-and-tie model of the single-column pier capping beam are as follows: ① In the single-column pier capping beam structure, the internal forces are spatially diffused, and in both the transverse and longitudinal directions of the bridge, the diagonal struts in the strut-and-tie model have inclination angles; ② In the transverse cross-section of the bridge, the horizontal tie rod is located at the top of the capping beam; in the longitudinal cross-section of the bridge, the horizontal tie rod is located near the middle of the capping beam.

[0011] (2) Calculation of internal forces of members based on the optimal spatial strut-and-tie model

[0012] The calculation process of the internal forces of the members based on the optimal spatial strut-and-tie model is briefly described as follows:

[0013] Taking the distance x from the upper middle horizontal tie rod in the longitudinal direction of the bridge to the top of the capping beam as an unknown quantity, the inclination angle θ of the strut A-G in the spatial strut-and-tie model in the transverse direction of the bridge and the inclination angle α in the longitudinal direction of the bridge are expressed, so as to express the strain energy of the spatial strut-and-tie model of the single-column pier capping beam.

[0014] Derive the strain energy expression of the spatial tension-compression bar model expressed by x. According to the minimum energy method, when its derivative is 0, the spatial tension-compression bar model is the optimal configuration.

[0015] After obtaining the spatial tension-compression bar model of this optimal configuration, solve the internal forces of the bars in the model according to the geometric conditions for the reinforcement design calculation.

[0016] The design method of the single-column pier capping beam structure based on the spatial tension-compression bar model of the present invention includes the following steps:

[0017] Step 1: Establish the basic configuration of the spatial tension-compression bar model of the single-column pier capping beam; the basic configuration includes: capping beam 1, pier column section 2, double bearing steel plate 3, bearing reaction 4, tension bar 5, compression bar 6. The structural dimensions of the capping beam 1 are the cross-bridge width b, the longitudinal-bridge thickness t, and the height h respectively; the structural dimensions of the pier column section 2 are the pier column diameter d and the pier column section height d; the double bearing steel plate 3 is a rectangular plate with a width of a. The double bearing steel plates are located on the central axis of the pier cap (1) in the longitudinal direction of the bridge. The center distance of the double bearing steel plates in the cross-bridge direction is s; the bearing reaction 4 is P d ; The tension bar 5 and the compression bar 6 are both connected by nodes A, B, C, D, E, F, G, H, A', B', C', D', E', F', G', H'; for example: the tension bar 5 includes the tension bar formed by connecting nodes A and D, the tension bar formed by connecting nodes A' and D', the tension bar formed by connecting nodes G and G', and the tension bar formed by connecting nodes H and H'; the compression bar 6 includes the compression bar formed by nodes A and G, the compression bar formed by nodes G and B, the compression bar formed by nodes B and C, the compression bar formed by nodes D and H, the compression bar formed by nodes H and E, the compression bar formed by nodes E and F, the compression bar formed by nodes G and H, the compression bar formed by nodes B and E, the compression bar formed by nodes A' and G', the compression bar formed by nodes G' and B', the compression bar formed by nodes B' and C', the compression bar formed by nodes D' and H', the compression bar formed by nodes H' and E', the compression bar formed by nodes E' and F', the compression bar formed by nodes G' and H', the compression bar formed by nodes B' and E', the compression bar formed by nodes A and A', the compression bar formed by nodes D and D', the compression bar formed by nodes B and B', and the compression bar formed by nodes E and E';

[0018] Nodes A, B, C, D, E, F, G, H and A', B', C', D', E', F', G', H' are symmetric on the central axis of the capping beam 1 in the longitudinal direction of the bridge; among them:

[0019] (1) The vertical positions of A, D, A', and D' are on the same plane at the top of the capping beam 1, and their plane positions correspond to the acting positions of the bearing reaction 4;

[0020] (2) The vertical positions of C, F, C', and F' are on the same plane at the bottom of Pier Segment 2, and their planar positions correspond to the positions at 1 / 4 radius of Pier Segment 1;

[0021] (3) The vertical positions of G, H, G', and H' are on a plane at a distance of x from the top of Cap Beam 1. Their planar positions are at an angle of θ in the transverse bridge direction view and at an angle of α in the longitudinal bridge direction view;

[0022] (4) B, E, B', and E' are respectively the intersections of struts G - B and B - C, struts H - E and E - F, struts G' - B' and B' - C', and struts H' - E' and E' - F'. The planar positions of B, E, B', and E' correspond to the nodes of C, F, C', and F'. Their vertical positions are related to the distance x from the top of Cap Beam 1 of the horizontal tie rod in the middle of the longitudinal bridge direction and are a solution function of x;

[0023] Step 2: Establish the strain energy formula determined by the model struts:

[0024] (1)

[0025] In the formula: V ε is the strain energy, K l is the strut stiffness, E l is the elastic modulus of the member material, l is the member length, A l is the cross-sectional area of the member, F l is the axial force of the member, l i , A li and F li are respectively the length, area, and axial force of the i-th member;

[0026] Among them, when calculating the strut area, it is necessary to define the strut width and thickness, and their definitions follow the following principles:

[0027] (1) Since the force flow emits from below the bearing steel plate 3, the top widths of struts A - B and A - A' are the entire bearing width a and the half bearing width a / 2 respectively;

[0028] (2) Starting from the horizontal strut B - E or B - B' downwards, the direction of the force flow is vertical and fills the entire pier cross-section. Therefore, the top width of strut B - C, that is, the bottom width of strut G - B, is d / 2, where d is the diameter of Pier Segment 2;

[0029] (3) The bottom width of strut A - G and the top width of strut G - B fill the entire cross-section in the longitudinal bridge direction width and are t / 2;

[0030] (4) For Pier Segment 2, since its cross-section is circular, the area A of its strut B - C bcIt can be expressed by the following formula:

[0031] (2)

[0032] Step 3: Taking the inclination angle θ of the compression bar A-G in the transverse bridge direction and the inclination angle α in the longitudinal bridge direction as parameters, calculate the internal force, length, and cross-sectional area of each compression bar in the spatial tension-compression bar model; where:

[0033] (1) The internal force of the compression bar A-G under the action of a unit concentrated load is calculated as follows: ; The length of the compression bar A-G is calculated as follows: ; The cross-sectional area of the compression bar A-G is calculated as follows: ; where θ and α are the inclination angles of the compression bar A-G in the transverse bridge direction and the longitudinal bridge direction respectively, t is the longitudinal bridge thickness of the capping beam 1, a is the width of the bearing steel plate 3, and d is the diameter of the pier column segment 2;

[0034] (2) The internal force of the compression bar G-B under the action of a unit concentrated load is calculated as follows: ; The length of the compression bar G-B is calculated as follows: ; The cross-sectional area of the compression bar G-B is calculated as follows: ; where θ and α are the inclination angles of the compression bar A-G in the transverse bridge direction and the longitudinal bridge direction respectively, t is the longitudinal bridge thickness of the capping beam 1, a is the width of the bearing steel plate 3, and d is the diameter of the pier column segment 2;

[0035] (3) The internal force of the compression bar B-C under the action of a unit concentrated load is 1; The length of the compression bar B-C is calculated as follows: ; The cross-sectional area of the compression bar B-C is calculated as follows: ; where θ and α are the inclination angles of the compression bar A-G in the transverse bridge direction and the longitudinal bridge direction respectively, t is the longitudinal bridge thickness of the capping beam 1, h is the height of the capping beam 1, a is the width of the bearing steel plate 3, and d is the diameter of the pier column segment 2;

[0036] (4) The internal force of the compression bar B-B’ under the action of a unit concentrated load is calculated as follows: ; The length of the compression bar B-B’ is calculated as follows: ; The cross-sectional area of the compression bar B-B’ is calculated as follows: ; where α is the inclination angle of the compression bar A-G in the longitudinal bridge direction, and d is the diameter of the pier column segment 2;

[0037] Step 4: Calculate the strain energy of the spatial tension-compression bar model according to formula (1), which is expressed as follows:

[0038] (3)

[0039] The strain energy is expressed as a function of a single unknown variable, i.e., the distance x from the top of the capping beam to the middle horizontal tie rod in the longitudinal direction of the bridge, and the following expression is obtained:

[0040] (4)

[0041] (5)

[0042] (6)

[0043] where θ and α are the inclination angles of the compression bar A-G in the transverse and longitudinal directions of the bridge respectively, t is the longitudinal thickness of the capping beam 1, h is the height of the capping beam 1, a is the width of the bearing steel plate (3), s is the center spacing of the double bearing steel plates 3, d is the diameter of the pier column section 2, and x is the distance from the top of the capping beam to the middle horizontal tie rod in the longitudinal direction of the bridge;

[0044] Step Five: According to the minimum energy criterion (maximum stiffness criterion) of the optimal simplified configuration of the tension and compression bar model, the strain energy of the optimal simplified configuration should take the minimum value, i.e., the derivative of V ε with respect to x is 0, which is expressed as follows:

[0045] (7)

[0046] Since V ε is a function of a single variable x, the unknown variable x is solved from Equation (7) and is a function of the six structural dimensions in the single-column pier capping beam. Its expression is simplified as follows:

[0047] (8)

[0048] Step Six: Calculate the internal forces of each bar according to x:

[0049] (9)

[0050] (10)

[0051] (11)

[0052] In the formula: F d is the bearing reaction force on each bearing; N y is the internal force of the diagonal compression bar A-G; the internal forces of the tie rods A-D and G-G' are T th，d , T ts，d respectively; the six structural dimensions in the pier capping beam are: the transverse width b of the capping beam, the longitudinal thickness t of the capping beam, the height h of the capping beam, the diameter d of the pier column section, the transverse bearing center spacing s of the capping beam, and the width a of the bearing steel plate;

[0053] Step 7: Based on the internal forces T of tie rods A-D and tie rod G-G’ th，d 、T ts，d perform reinforcement calculation:

[0054] (12)

[0055] (13)

[0056] In the formula: γ 0 is the structural importance coefficient, which is taken according to the current code; A s1 is the cross-sectional area of the transverse tension reinforcement at the top of the capping beam; A s2 is the splitting reinforcement area under the support.

[0057] In practical applications, according to the spatial force flow characteristics and topological optimization results of the single-column pier capping beam structure, the basic configuration of the spatial tension-compression bar model of the single-column pier capping beam is established, and its basic configuration is shown in Figure 1 .

[0058] In step 3 of the present invention, taking the inclination angle θ of the compression bar A-G in the transverse direction of the bridge and the inclination angle α in the longitudinal direction of the bridge as parameters, the internal forces, lengths and cross-sectional areas of each compression bar in the spatial tension-compression bar model are calculated as shown in Table 1:

[0059]

[0060] The spatial tension-compression bar model involved in the present invention can better describe the spatial force characteristics of the single-column pier capping beam under vertical loads. However, as the thickness-width ratio of the single-column pier capping beam increases, the spatial tension-compression bar model proposed in the present invention gradually approaches the plane tension-compression bar model method in the existing code.

[0061] The present invention calculates the model strain energy using the compression bars in the spatial tension-compression bar model. Its characteristics are: ① The compression bars have inclination angles both in the transverse direction and the longitudinal direction of the bridge, effectively reflecting the spatial characteristics of the model; ② The length, width and thickness of the compression bars are all known, and can be accurately expressed by a single unknown variable - the distance x from the horizontal tie rod in the middle in the longitudinal direction of the bridge to the top of the capping beam.

[0062] A design method for a single-column pier capping beam structure based on a spatial tension-compression bar model according to the present invention has calculation results different from those of the existing code method, and not only obtains the transverse reinforcement calculation results, but also obtains the longitudinal reinforcement calculation results.

[0063] A design method for a single-column pier capping beam structure based on a spatial tension-compression bar model according to the present invention is applicable to the structural design of double-bearing single-column pier capping beams.

[0064] The present invention relates to a design method for the single-column pier capping beam structure based on the space strut-and-tie model. The design method is applicable to the structural design of single-column pier capping beams with a width-to-thickness ratio less than or equal to 2 (preferably in the range of 1 to 1.5).

[0065] The present invention relates to a design method for the single-column pier capping beam structure based on the space strut-and-tie model. The design method is applicable to the structural design of hammerhead-shaped single-column pier capping beams.

[0066] The present invention relates to a design method for the single-column pier capping beam structure based on the space strut-and-tie model. The strut-and-tie model on which the design method is based is a space model.

[0067] The present invention relates to a design method for the single-column pier capping beam structure based on the space strut-and-tie model. The design method not only considers the effect of the structural lateral width but also the effect of the structural thickness.

[0068] Principle and advantages

[0069] The present invention takes the space strut-and-tie model of the single-column pier capping beam as the research object, and through constructing its optimal configuration, proposes a design method for the single-column pier capping beam structure based on the space strut-and-tie model.

[0070] The basic principles include:

[0071] (1) The strut-and-tie model is developed from the truss model and is a simplified force flow analysis model abstracted from the concrete structure continuum. It consists of struts, ties, and nodes, and can reflect the internal force transmission path of the structure, and can be used to determine the structural dimensions of the concrete D-region, the steel bar area of the cross-section, and the arrangement position of the steel bars.

[0072] (2) Based on the space strut-and-tie model of the single-column pier capping beam, the internal forces of the space model members can be calculated, so as to carry out the spatial reinforcement design calculation in the transverse and longitudinal directions of the bridge.

[0073] The main advantages include:

[0074] (1) Different from the existing code methods that can only carry out the plane transverse reinforcement design calculation, the present invention based on the space strut-and-tie model can carry out the spatial reinforcement design calculation in both the transverse and longitudinal directions of the bridge at the same time.

[0075] (2) The space strut-and-tie model fully considers the spatial mechanical properties of the single-column pier capping beam under vertical loads. The design method based on the space strut-and-tie model is more scientific, reasonable, and economical than the existing code methods based on the plane model.

[0076] (3) The present invention takes into account six structural dimension parameters of the single-column pier capping beam (the cross-bridge width b of the capping beam, the cross-longitudinal thickness t of the capping beam, the height h of the capping beam, the diameter d of the pier column, the cross-bridge center spacing s of the supports on the capping beam, and the width a of the bearing steel plate). Compared with the existing code methods, the design of the single-column pier capping beam by the present invention is more refined.

[0077] (4) The present invention can achieve parametric design of the single-column pier capping beam, which has important significance and practical value.

[0078] (5) Based on the space strut-and-tie model, the present invention has obvious advantages over the existing code design methods for single-column pier capping beams with small width-to-thickness ratios, hammerhead-shaped single-column pier capping beams and similar structures.

[0079] In summary: The present invention solves the problem that the current code can only perform plane calculations for the capping beam structure, and for the capping beam structure of single-column piers with spatial effects, the design calculation method is blank; the present invention conducts structural design based on the optimal space strut-and-tie model. Due to considering the spatial effect of structural forces, its principle is scientific and reasonable, which is of great significance for the safe and reasonable design of the structure; moreover, the present invention also greatly improves the design efficiency of the single-column pier capping beam structure. BRIEF DESCRIPTION OF THE DRAWINGS

[0080] Attached Figure 1 is the space strut-and-tie model of the single-column pier capping beam. Where θ is the inclination angle of the diagonal strut A-G in the cross-bridge direction; α is the inclination angle of the diagonal strut A-G in the longitudinal-bridge direction.

[0081] Attached Figure 2 is the schematic diagram of the width of the compression strut of the space strut-and-tie model in the cross-bridge direction and the longitudinal-bridge direction respectively.

[0082] Attached Figure 3 is the schematic diagram of the area of the B-C compression strut.

[0083] Figure 4 of the drawings is the design layout diagram of the test single-column pier capping beam, which consists of Figure 4(a) and Figure 4(b). Figure 4(a) is the elevation layout diagram, and Figure 4(b) is the side layout diagram.

[0084] Attached Figure 5 is the schematic diagram of the plane strut-and-tie model for cross-bridge direction calculation based on the existing code method.

[0085] Attached Figure 6 is the schematic diagram of the plane strut-and-tie model for cross-bridge direction calculation based on the existing code method.

[0086] Attached Figure 7 is the optimal space strut-and-tie model of the test capping beam.

[0087] Attached Figure 8 is the finite element model of the test capping beam. Detailed implementation mode

[0088] Appendix Figure 1-3 And Table 1 and Formulas (1)-(13) are the principles of this implementation mode and the present invention. Among them Figure 1 is the spatial strut-and-tie model of the single-column pier cap beam, where θ is the inclination angle of the diagonal strut A-G in the transverse bridge direction; α is the inclination angle of the diagonal strut A-G in the longitudinal bridge direction, and the inclination angles θ and α of the diagonal strut in the transverse bridge direction and the longitudinal bridge direction affect the configuration of the spatial strut-and-tie model. Figure 2 is the schematic diagram of the widths of the struts of the spatial strut-and-tie model in the transverse bridge direction and the longitudinal bridge direction respectively. Figure 2 In actual application: ① The top widths of the A-G struts in the transverse bridge direction and the longitudinal bridge direction are the full width a of the bearing and the half width a / 2 of the bearing respectively; ② The bottom width of the A-G strut in the longitudinal bridge direction is t / 2; ③ The bottom widths of the G-B struts in the transverse bridge direction and the longitudinal bridge direction are both d / 2. Figure 3 is the schematic diagram of the area of the B-C strut, and the calculation formula is shown in Equation (2).

[0089] The specific implementation includes the following steps:

[0090] Step 1: Establish the basic configuration of the spatial strut-and-tie model of the single-column pier cap beam. The basic configuration includes: the cap beam 1, the pier column section 2, the double-bearing steel plate 3, the bearing reaction force 4, the tie rod 5, and the strut 6. The structural dimensions of the cap beam 1 are the transverse bridge width b, the longitudinal bridge thickness t, and the height h; the structural dimensions of the pier column section 2 are the pier column diameter d and the pier column section height d; the double-bearing steel plate 3 is a rectangular plate with a width of a, and the double-bearing steel plates are both located on the central axis of the pier cap (1) in the longitudinal bridge direction, and the center distance of the double-bearing steel plates in the transverse bridge direction is s; the bearing reaction force 4 is P dThe pull rods 5 and the compression rods 6 are both composed of joints A, B, C, D, E, F, G, H, A', B', C', D', E', F', G', and H' connected together. The pull rod 5 includes the pull rod formed by connecting joints A and D, the pull rod formed by connecting joints A' and D', the pull rod formed by connecting joints G and G', and the pull rod formed by connecting joints H and H'. The compression rod 6 includes the compression rod formed by joints A and G, the compression rod formed by joints G and B, the compression rod formed by joints B and C, the compression rod formed by joints D and H, the compression rod formed by joints H and E, the compression rod formed by joints E and F, the compression rod formed by joints G and H, the compression rod formed by joints B and E, the compression rod formed by joints A' and G', the compression rod formed by joints G' and B', the compression rod formed by joints B' and C', the compression rod formed by joints D' and H', the compression rod formed by joints H' and E', the compression rod formed by joints E' and F', the compression rod formed by joints G' and H', the compression rod formed by joints B' and E', the compression rod formed by joints A and A', the compression rod formed by joints D and D', the compression rod formed by joints B and B', and the compression rod formed by joints E and E'. Joints A, B, C, D, E, F, G, H and A', B', C', D', E', F', G', H' are symmetric about the central axis of the capping beam 1 in the longitudinal direction of the bridge. Among them:

[0091] (1) The vertical positions of A, D, A', and D' are on the same plane at the top of the capping beam 1, and their planar positions correspond to the acting positions of the bearing reaction force 4.

[0092] (2) The vertical positions of C, F, C', and F' are on the same plane at the bottom of the pier column segment 2, and their planar positions correspond to the positions of 1 / 4 radius of the pier column segment.

[0093] (3) The vertical positions of G, H, G', and H' are on a plane at a distance of x from the top of the capping beam 1. Their planar positions are at an angle of θ in the transverse direction of the bridge and at an angle of α in the longitudinal direction of the bridge.

[0094] (4) B, E, B', and E' are respectively the intersections of the compression rods G - B and B - C, the compression rods H - E and E - F, the compression rods G' - B' and B' - C', and the compression rods H' - E' and E' - F'. The planar positions of B, E, B', and E' correspond to the nodes C, F, C', and F'. Their vertical positions are related to the distance x from the top of the capping beam 1 of the horizontal pull rod in the middle of the longitudinal direction of the bridge and are a solution function of x.

[0095] Step 2: Establish the strain energy formula determined by the model compression rod:

[0096] (1)

[0097] In the formula: Vε is the strain energy, and K l is the stiffness of the compression bar, and E l is the elastic modulus of the bar material, l is the length of the bar, and A l is the cross-sectional area of the bar, and F l is the axial force of the bar, and l i and A li and F li are respectively the length, area, and axial force of the i-th bar.

[0098] Among them, when calculating the area of the compression bar, it is necessary to define the width and thickness of the compression bar, and their definitions follow the following principles:

[0099] (1) Since the force flow emits from below the support steel plate 3, the top widths of the A-B compression bar and the A-A' compression bar are the entire width a of the support and the half width a / 2 of the support, respectively;

[0100] (2) From the horizontal compression bar B-E or B-B' downward, the direction of the force flow is vertical and fills the entire pier cross-section. Therefore, the top width of the B-C compression bar, that is, the bottom width of the G-B compression bar, is d / 2, where d is the diameter of the pier section 2;

[0101] (3) The bottom width of the A-G compression bar and the top width of the G-B compression bar fill the entire cross-section in the cross-bridge direction width and are t / 2;

[0102] (4) For the pier section 2, since its cross-section is circular, the area A of its B-C compression bar bc can be expressed by the following formula:

[0103] (2)

[0104] Step 3: Taking the inclination angle θ of the compression bar A-G in the cross-bridge direction and the inclination angle α in the longitudinal-bridge direction as parameters, calculate the internal force, length, and compression bar area of each compression bar in the spatial tension-compression bar model; among them:

[0105] (1) The internal force of the compression bar A-G under the action of a unit concentrated load, and its calculation method is: ; the length of the compression bar A-G, and its calculation method is: ; the area of the compression bar A-G, and its calculation method is: . Among them, θ and α are the inclination angles of the compression bar A-G in the cross-bridge direction and the longitudinal-bridge direction, respectively, t is the longitudinal-bridge thickness of the capping beam 1, a is the width of the support steel plate (3), and d is the diameter of the pier section 2.

[0106] (2) The internal force of the compression bar G-B under the action of a unit concentrated load, and its calculation method is: ; the length of the compression bar G-B, and its calculation method is: ; The compression bar area of the compression bar G - B, and its calculation method is as follows: . Where θ and α are the inclination angles of the compression bar A - G in the transverse direction and longitudinal direction of the bridge respectively, t is the longitudinal thickness of the capping beam 1, a is the width of the bearing steel plate (3), and d is the diameter of the pier column segment 2.

[0107] (3) The internal force of the compression bar B - C under the action of a unit concentrated load is 1; the length of the compression bar B - C, and its calculation method is as follows: ; The compression bar area of the compression bar B - C, and its calculation method is as follows: . Where θ and α are the inclination angles of the compression bar A - G in the transverse direction and longitudinal direction of the bridge respectively, t is the longitudinal thickness of the capping beam 1, h is the height of the capping beam 1, a is the width of the bearing steel plate (3), and d is the diameter of the pier column segment 2.

[0108] (4) The internal force of the compression bar B - B' under the action of a unit concentrated load, and its calculation method is as follows: ; The length of the compression bar B - B', and its calculation method is as follows: ; The compression bar area of the compression bar B - B', and its calculation method is as follows: . Where α is the inclination angle of the compression bar A - G in the longitudinal direction of the bridge, and d is the diameter of the pier column segment 2.

[0109] Step Four: Calculate the strain energy of the spatial tension - compression bar model according to formula (1), and it is expressed as the following formula:

[0110] (3)

[0111] Express the strain energy as a function of a single unknown variable - the distance x from the middle horizontal tie rod in the longitudinal direction of the bridge to the top of the capping beam, and the following expression is obtained:

[0112] (4)

[0113] (5)

[0114] (6)

[0115] Where θ and α are the inclination angles of the compression bar A - G in the transverse direction and longitudinal direction of the bridge respectively, t is the longitudinal thickness of the capping beam 1, h is the height of the capping beam 1, a is the width of the bearing steel plate (3), s is the center - to - center spacing of the double bearing steel plates 3, d is the diameter of the pier column segment 2, and x is the distance from the middle horizontal tie rod in the longitudinal direction of the bridge to the top of the capping beam.

[0116] Step Five: According to the minimum energy criterion (maximum stiffness criterion) of the optimal simplified configuration of the tension - compression bar model, the strain energy of the optimal simplified configuration should take the minimum value, that is, the derivative of V ε with respect to x is 0, and it is expressed as follows:

[0117] (7)

[0118] Since V ε is a function of the single variable x, the unknown x is solved according to Equation (7) and is a function of the six structural dimensions in the single-column pier cap beam. Its simplified expression is as follows:

[0119] (8)

[0120] Step 6: Calculate the internal forces of each member according to x:

[0121] (9)

[0122] (10)

[0123] (11)

[0124] In the formula: F d is the reaction force of each support; N y is the internal force of the diagonal compression bar A-G; the internal forces of the tie bars A-D and G-G' are T th，d , T ts，d respectively; the six structural dimensions in the column pier cap beam are: the cross-bridge width b of the cap beam, the longitudinal thickness t of the cap beam, the height h of the cap beam, the diameter d of the pier column section, the cross-bridge center distance s of the supports on the cap beam, and the width a of the support steel plate;

[0125] Step 7: Perform reinforcement calculation according to the internal forces T th，d of the tie bars A-D and G-G' ts，d :

[0126] (12)

[0127] (13)

[0128] In the formula: γ 0 is the structural importance coefficient, which is taken according to the current code; A s1 is the cross-sectional area of the transverse tension reinforcement at the top of the cap beam; A s2 is the cross-sectional area of the splitting reinforcement under the support.

[0129] In practical applications, according to the spatial force flow characteristics and topological optimization results of the single-column pier cap beam structure, the basic configuration of the spatial tension and compression bar model of the single-column pier cap beam is established. The basic configuration is shown in Figure 1 .

[0130] In the third step of the present invention, taking the inclination angle θ of the compression bars A-G in the transverse direction of the bridge and the inclination angle α in the longitudinal direction of the bridge as parameters, the internal forces, lengths and areas of the compression bars in the spatial tension-compression bar model are calculated as shown in Table 1:

[0131]

[0132] Engineering verification

[0133] For the design method of the single-column pier capping beam structure based on the spatial tension-compression bar model, the middle pier capping beam of a certain river-crossing continuous box girder bridge is taken as the analysis and test object. The structural calculations are carried out respectively based on the spatial tension-compression bar model method and the existing code method, and the tension bar forces and reinforcement areas in the transverse and longitudinal directions of the capping beam are calculated. And they are respectively compared with the tension bar forces and reinforcement areas in the transverse and longitudinal directions of the capping beam calculated by the finite element method. The correctness, effectiveness and rationality of the method of the present invention are verified.

[0134] The design layout drawing of the pier capping beam of the test bridge is shown in Figure 4 (unit: mm). Among them: the transverse width of the capping beam is 8 m, the longitudinal thickness is 6 m, the height is 6 m, and the diameter of the pier is 4 m. The size of the bearing steel plate is 1 m×1 m, the center distance between the double bearings is 4.5 m, and the standard combination value of the single bearing load is 68410 kN.

[0135] (1) Based on the existing code method

[0136] For the design calculation of the single-column pier capping beam in the "Code for Design of Highway Bridges and Culverts - Concrete and Prestressed Concrete Bridges and Culverts" (JTG 3362-2018) of our country, only the plane design in the transverse direction is considered, while its spatial effect is ignored. According to this current code, ignoring the spatial effect of the single-column pier capping beam, plane calculations are carried out for both the transverse and longitudinal directions. Among them, for the transverse calculation, the design provisions for the single-column bent cap in the code are adopted; for the longitudinal calculation, the relevant provisions of the anchorage splitting force under the anchor in the bridge end anchorage area, which is similar to the longitudinal force-bearing characteristics of the capping beam in the code, are borrowed for calculation.

[0137] ① Transverse calculation

[0138] (14)

[0139] In the formula: T th,d is the design value of the internal force of the transverse tension bar at the pier top; F d is the design value of the vertical load at the pier top, taken according to the basic combination; γ 0 is the structural importance coefficient; s is the center distance between the bearings; h is the height of the transverse variable-width section at the pier top. When h>b, h = b, where b is the transverse width of the top of the capping beam; b’ is the transverse width of the pier body or capping beam at a position with a height of h from the pier top; f sdis the design value of the tensile strength of ordinary steel bars, taking 0.75 times the yield strength of the steel bars; A s is the area of the ordinary steel bars in the tie rod, calculated according to the steel bars within the height range of 2h / 9 at the top of the capping beam. The schematic diagram of the plane tension and compression rod model for cross-bridge direction calculation is shown in Figure 5 .

[0140] ② Calculation in the longitudinal direction of the bridge

[0141] (15)

[0142] In the formula: T ts,d is the splitting force under the bearing; P d is the design value of the vertical load at the pier top, taken according to the basic combination; γ 0 is the structural importance coefficient; a is the width of the bearing steel plate; t is the thickness of the capping beam (in this component, the thickness t of the capping beam is for the end cross-section height h in the original code provisions for the end anchorage area); d b is the distance from the splitting force under the bearing to the top of the capping beam; e is the eccentricity between the prestressed anchoring force and the central axis of the component in the end anchorage area in the original code provisions. In this component, e = 0; α’ is the angle between the prestressed steel bars and the anchorage surface of the component in the end anchorage area in the original code provisions. In this component, α’ = 0; γ is the eccentricity of the prestressed anchoring force in the cross-section in the end anchorage area in the original code provisions, γ = 2e / t. In this component, γ = 0. The schematic diagram of the plane tension and compression rod model for longitudinal direction calculation is shown in Figure 6 .

[0143] (2)Based on the space tension and compression rod model method

[0144] Using the above formulas (7) - (8), the parameters of the space tension and compression rod model are found for the test single-column pier capping beam, and its optimal space tension and compression rod model is as shown in Figure 7 . Based on this space tension and compression rod model, the internal force T th,d of the cross-bridge direction pier top tie rod and the internal force T ts,d of the longitudinal direction tie rod under the bearing are calculated using the above formulas (10) - (11).

[0145] (3)Based on the finite element method

[0146] A finite element model of the test single-column pier capping beam was established using the large finite element software Abaqus. In the model mesh, two-node linear interpolation elements (T3D2) were used to simulate the steel bars. Since the stress state of concrete is relatively complex, eight-node linear interpolation solid elements with reduced integration (C3D8R) were used to simulate the concrete in the finite element simulation. The interaction between the steel bars and the concrete was simulated using the Embedded method. To accurately simulate the stress state of the pile cap and the boundary conditions of the pier, part of the pile foundation was considered in the finite element model. The boundary condition at the end of the pile foundation was fully fixed, and a concentrated force of 68410 kN was applied to the capping beam support. To balance the calculation accuracy and calculation efficiency, the mesh size of the finite element model in this paper was 400 mm. The finite element model is shown in Figure 8 . The model was subjected to linear elastic analysis and finite element stress integration, and the stress integration results can be approximated as tensile forces.

[0147] Comparison results

[0148] The method based on the existing code and the method of the space strut-and-tie model based on the present invention were respectively compared with the method based on finite element analysis. The comparison indicators were the cross-bridge and longitudinal tie forces or the cross-bridge and longitudinal reinforcement areas. The results are shown in Table 2 below.

[0149]

[0150] As can be seen from the above table, the results of the method of the present invention are in good agreement with the finite element method, and the maximum relative error between the two is 5.94%. In contrast, the maximum error between the existing code method and the finite element method is 66.89%. This shows that the structural design method of the single-column pier capping beam based on the space strut-and-tie model has high accuracy, verifying the correctness and effectiveness of the method described in this invention patent. The errors between the method of the present invention and the existing code method are -36.52% and -20.02% respectively, indicating that the structural design method of the single-column pier capping beam based on the space strut-and-tie model is more economical than the existing code method.

Claims

1. A design method for the single-column pier capping beam structure based on the spatial tension-compression bar model, characterized in that: Based on the structural dimension parameters of the single-column pier capping beam, a spatial tension-compression bar model of the structure is established. The optimal configuration of the spatial tension-compression bar model is calculated and established according to the minimum energy criterion. The internal forces of the tension bars and compression bars in the model are calculated to carry out the reinforcement design, realizing the spatial design of the single-column pier capping beam. The specific steps are as follows: Step 1: Establish the basic configuration of the spatial tension-compression bar model of the single-column pier capping beam; the basic configuration includes: capping beam (1), pier column section (2), double-bracket steel plate (3), support reaction force (4), tension bar (5), compression bar (6); The structural dimensions of the capping beam (1) are the cross-bridge width b, the longitudinal thickness t, and the height h; the structural dimensions of the pier column segment (2) are the pier column diameter d and the pier column segment height d; the double bearing steel plate (3) is a rectangular plate with a width of a. The double bearing steel plates are both located on the central axis of the capping beam (1) in the longitudinal direction of the bridge. The center spacing of the double bearing steel plates in the cross-bridge direction is s; the bearing reaction (4) is P d ; The tie rods (5) include the tie rods formed by connecting nodes A and D, the tie rods formed by connecting nodes A' and D', the tie rods formed by connecting nodes G and G', and the tie rods formed by connecting nodes H and H'; the struts (6) include the struts formed by nodes A and G, the struts formed by nodes G and B, the struts formed by nodes B and C, the struts formed by nodes D and H, the struts formed by nodes H and E, the struts formed by nodes E and F, the struts formed by nodes G and H, the struts formed by nodes B and E, the struts formed by nodes A' and G', the struts formed by nodes G' and B', the struts formed by nodes B' and C', the struts formed by nodes D' and H', the struts formed by nodes H' and E', the struts formed by nodes E' and F', the struts formed by nodes G' and H', the struts formed by nodes B' and E', the struts formed by nodes A and A', the struts formed by nodes D and D', the struts formed by nodes B and B', and the struts formed by nodes E and E'; The nodes A, B, C, D, E, F, G, H and A’, B’, C’, D’, E’, F’, G’, H’ are symmetric on the central axis of the capping beam (1) in the longitudinal direction of the bridge. Among them, the positions of the nodes connecting the tension bar (5) and the compression bar (6) have the following characteristics: (1) The vertical positions of A, D, A’, D’ are on the same plane at the top of the capping beam (1), and their plane positions correspond to the acting positions of the support reaction force (4). (2) The vertical positions of C, F, C’, F’ are on the same plane at the bottom of the pier column section (2), and their plane positions correspond to the positions of 1 / 4 radius of the pier column section. (3) The vertical positions of G, H, G’, H’ are on the plane at a distance of x from the top of the capping beam (1). Their plane positions are at an angle of θ in the transverse direction of the bridge and at an angle of α in the longitudinal direction of the bridge. (4) B, E, B’, E’ are the intersections of the compression bars G-B and B-C, the compression bars H-E and E-F, the compression bars G’-B’ and B’-C’, and the compression bars H’-E’ and E’-F’ respectively; the plane positions of B, E, B’, E’ correspond to the nodes C, F, C’, F’, and their vertical positions are related to the distance x from the top of the capping beam (1) of the horizontal tension bar in the middle of the longitudinal direction of the bridge and are a solution function of x. Step 2: Establish the strain energy formula determined by the compression bars in the model: Where: V ε is the strain energy, K l is the stiffness of the compression bar, E l is the elastic modulus of the bar material, l is the length of the bar, A l is the cross-sectional area of the bar, F l is the axial force of the bar, l i 、A li and F li are respectively the length, area and axial force of the i-th bar; Among them, when calculating the area of the compression bar, it is necessary to define the width and thickness of the compression bar, and their definitions follow the following principles: (1) Since the force flow emits from below the support steel plate (3), the top widths of the A-B compression bar and the A-A’ compression bar are the entire width a of the support and the half width a / 2 of the support respectively. (2) From the horizontal compression bar B-E or B-B’ downwards, the direction of the force flow is vertical and fills the entire pier cross-section. Therefore, the top width of the B-C compression bar, that is, the bottom width of the G-B compression bar, is d / 2, where d is the diameter of the pier column section (2). (3) The bottom width of the A-G compression bar and the top width of the G-B compression bar fill the entire cross-section in the longitudinal width direction and are t / 2. (4) For the pier column segment (2), since its cross-section is circular, the area A of the B-C strut bc can be expressed by the following formula: Step 3: Taking the inclination angle θ of the compression bar A-G in the transverse direction of the bridge and the inclination angle α in the longitudinal direction of the bridge as parameters, calculate the internal forces, lengths and areas of the compression bars in the spatial tension-compression bar model; among them: (1) The internal force of the compression bar A-G under the action of a unit concentrated load, and its calculation method is as follows: The length of the compression bar A-G, and its calculation method is as follows: The area of the compression bar A-G, and its calculation method is as follows: Where θ and α are the inclination angles of the compression bar A-G in the transverse direction and longitudinal direction of the bridge respectively, t is the longitudinal thickness of the capping beam (1), a is the width of the bearing steel plate (3), and d is the diameter of the pier column section (2); (2) The internal force of the compression bar G-B under the action of a unit concentrated load, and its calculation method is as follows: The length of the compression bar G-B, and its calculation method is as follows: The cross-sectional area of the compression bar G-B, and its calculation method is as follows: Where θ and α are the inclination angles of the compression bar A-G in the transverse direction and longitudinal direction of the bridge respectively, t is the longitudinal thickness of the capping beam (1), a is the width of the bearing steel plate (3), and d is the diameter of the pier column section (2); (3) The internal force of the compression bar B-C under the action of a unit concentrated load is 1; the length of the compression bar B-C, and its calculation method is as follows: The cross-sectional area of the compression bar B-C, and its calculation method is as follows: Where θ and α are the inclination angles of the compression bar A-G in the transverse direction and longitudinal direction of the bridge respectively, t is the longitudinal thickness of the capping beam (1), h is the height of the capping beam (1), a is the width of the bearing steel plate (3), and d is the diameter of the pier column segment (2); (4) The internal force of the strut B - B' under the action of a unit concentrated load, and its calculation method is: cotα; the length of the strut B - B', and its calculation method is: The cross-sectional area of the strut B - B', and its calculation method is: where θ and α are the inclination angles of the strut A - G in the transverse direction and longitudinal direction of the bridge respectively, and d is the diameter of the pier column section (2); Step 4: Calculate the strain energy V of the spatial tension-compression bar model according to formula (1) ε , as expressed in the following formula: Express the strain energy V ε as a function of a single unknown, the distance x from the top of the capping beam to the middle horizontal tie rod in the longitudinal direction of the bridge, to obtain the following expression: Where: θ and α are the angles of the compression struts A-G in the transverse direction and longitudinal direction of the bridge respectively, t is the longitudinal thickness of the capping beam (1), h is the height of the capping beam (1), a is the width of the bearing steel plate (3), s is the center spacing of the double bearing steel plates (3), d is the diameter of the pier column section (2), and x is the distance from the middle horizontal tie rod in the longitudinal direction of the bridge to the top of the capping beam; Step Five: According to the minimum energy criterion of the optimal simplified configuration of the tension-compression bar model, the strain energy of the optimal simplified configuration should take the minimum value, that is, the derivative of the strain energy V ε with respect to x is 0, which is expressed as follows: Since the strain energy V ε is a function of a single variable x, the unknown x is solved according to Equation (7) and is a function of the six structural dimensions in the single-column pier capping beam. Its expression is simplified as follows: x = f(a, b, d, h, s, t) (8) Step Six: Calculate the internal forces of each member according to x: Where: F d is the reaction force of each support; N y is the internal force of the diagonal compression bar A-G; the internal forces of the tension bars A-D and G-G' are T th，d and T ts，d respectively; the six structural dimensions in the pier cap beam are: the cross-bridge width b of the cap beam, the longitudinal thickness t of the cap beam, the height h of the cap beam, the diameter d of the pier column section, the cross-bridge center distance s of the supports, and the width a of the support steel plate; Step 7: Perform reinforcement calculation based on the internal forces T of tie rods A-D and tie rod G-G'. th，d , T ts，d ​ A s1 = γ 0 T th,d / f sd (12) A s2 = γ 0 T ts,d / f sd (13) In the formula: γ 0 is the structural importance coefficient, which is taken according to the current code; A s1 is the cross-sectional area of the transverse tension reinforcement at the top of the capping beam; A s2 is the cross-sectional area of the splitting reinforcement under the support; The design method is applicable to the structural design of single-column pier capping beams with a width-to-thickness ratio of 1 to 1.

5.

2. A structural design method for a single-column pier capping beam based on a spatial tension-compression strut model according to claim 1, characterized in that: The design method is applicable to the structural design of double-bearing single-column pier capping beams.

3. A structural design method for a single-column pier capping beam based on a spatial tension-compression strut model according to claim 1, characterized in that: The design method is applicable to the structural design of hammerhead-shaped single-column pier capping beams.

4. A structural design method for a single-column pier capping beam based on a spatial tension-compression strut model according to claim 1, characterized in that: The tension-compression strut model on which the design method is based is a spatial model.

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