Determination method of the calculation length coefficient of bridge piers

Abstract

The present invention discloses a method for determining the calculation length coefficient of a bridge pier. According to the actual dimensions of the bridge structure, an integral bridge model 1 including the upper main girder structure and the lower bridge pier structure is established; the buckling analysis is carried out on the model 1 to calculate the stability coefficient λ1 of the target bridge pier, and the single vertical load P applied to the top of the target bridge pier is extracted; then, according to the actual dimensions of the target bridge pier, a model 2 including only the target bridge pier is established; the buckling analysis is carried out on the model 2 to calculate the stability coefficient of the target bridge pier λ 2 ; the calculation length coefficient of the target bridge pier is determined according to the formula #imgabs0#. The advantages of the present invention are that the analogy method is adopted, various interference factors such as self-weight and the position of the analysis section can be excluded, it is applicable to bridge piers with regular sections and complex sections, the calculation method is simple, the calculation accuracy is high, and the engineering practicability is strong.

Description

Technical Field

[0001] The present invention relates to the technical field of bridge engineering, and particularly to a method for determining the calculation length coefficient of bridge piers. Background Art

[0002] Bridge piers are important components in bridge structures. In engineering design, the calculation length coefficient μ is introduced to consider the second-order effects of axially loaded members such as bridge piers, bridge towers, and columns. In bridge design, formula (1) is usually used to solve the calculation length of bridge piers:

[0003] l 0 = μl Formula (1)

[0004] In formula (1), l 0 is the distance between two adjacent inflection points (where the bending moment is zero) on the deflection curve after the axially loaded member buckles, that is, the free length, which is also the calculation length of the bridge pier; l is the geometric length of the axially loaded member. The calculation length coefficient μ is an important design parameter affecting the safety and economy of bridges. It is affected by various factors such as boundary conditions, structural stiffness, and structural self-weight, and the determination method is relatively complex. Currently, there are significant differences in the methods for determining the calculation length of bridge piers among various national codes, and even among different design codes in different fields in China. For example, in codes such as the "Code for Design of Concrete Structures" (GB50010-2010), the calculation length coefficient of axially loaded members is directly given based on engineering design experience and bridge structural forms. In codes such as the "Code for Design of Highway Reinforced Concrete and Prestressed Concrete Bridges and Culverts" (JTG3362-2004), the calculation length coefficient of axially loaded members under ideal boundary conditions is derived based on the Euler formula: when both ends of the axially loaded member are fixed, it is taken as 0.5; when one end is fixed and the other end is a non-movable hinge, it is taken as 0.7; when both ends are non-movable hinges, it is taken as 1.0; when one end is fixed and the other end is free, it is taken as 2.0. In the "Code for Design of Highway Reinforced Concrete and Prestressed Concrete Bridges and Culverts" (JTG3362-2018), AASHTO LFRD 2014 in the United States, and the European code EN1992-1-1:2004, etc., the bridge pier is separated from the frame structure, the boundary conditions are determined according to the deformation of the buckling pattern, and then elastic stability calculations are carried out. Finally, the calculation length coefficient is inversely calculated through the Euler formula. There are also cases where the bridge pier is regarded as a member in a single-span or multi-span bridge for buckling analysis, and the calculation length coefficient of the bridge pier is inversely calculated through the Euler formula.

[0005] The above methods for determining the calculation length coefficient of bridge piers have the following problems and defects:

[0006] 1. The calculation method is complex and not convenient for application in engineering design. For example, the methods provided in Codes for Design of Highway Reinforced Concrete and Prestressed Concrete Bridges and Culverts (JTG 3362-2018) etc. require establishing a calculation model of the component, calculating multiple parameters including displacement, rotation angle, axial force, etc. and several intermediate parameters, and some of the parameters are semi-theoretical and semi-empirical formulas, so the calculation accuracy cannot be guaranteed.

[0007] 2. Most methods are applicable to piers with regular cross-sections to a certain extent, but have poor applicability to piers with complex cross-sections and may even obtain incorrect calculation results.

[0008] 3. Whether to consider the self-weight load, the calculation position of the buckling load P cr value is not clear, resulting in a large difference in the buckling load P obtained by different designers even for the same structure. cr Ultimately, the calculated length coefficient μ also has a large difference. Summary of the Invention

[0009] The object of the present invention is to provide a method for determining the calculated length coefficient of a pier.

[0010] To achieve the above object, the present invention adopts the following technical solutions:

[0011] The method for determining the calculated length coefficient of the pier according to the present invention includes the following steps:

[0012] S1. According to the actual dimensions of the bridge structure, establish an integral bridge model 1 including the upper main girder structure and the lower pier structure;

[0013] The boundary conditions of the model 1 include: establishing a support constraint between the main girder and the target pier; establishing a "pile-soil" constraint between the target pier and the pile foundation;

[0014] The loads of the model 1 include: self-weight load G, secondary permanent load and vehicle load.

[0015] S2. Conduct a buckling analysis on the model 1, calculate the stability coefficient λ1 of the target pier, and extract the single vertical load P applied to the top of the target pier;

[0016] S3. According to the actual dimensions of the target pier, establish a model 2 including only the target pier;

[0017] The boundary conditions of the model 2 include: no constraint at the upper end of the target pier; establishing a "pile-soil" constraint at the lower end with the pile foundation;

[0018] The loads of the model 2 include: the single vertical load P applied to the top of the target pier, without considering the self-weight load G of the target pier;

[0019] S4. Perform a buckling analysis on Model 2 to calculate the stability coefficient of the target pier. λ 2 ;

[0020] S5. Determine the calculated length coefficient of the target pier according to the formula. ;

[0021] .

[0022] Furthermore, the buckling analysis is an ideal linear elastic bifurcation point stability analysis.

[0023] Furthermore, the "pile - soil" constraint means establishing the "pile - soil" constraint according to the "m" method based on the actual soil layer parameter information.

[0024] Furthermore, the vehicle load is a uniform line load applied based on the influence line of the target pier.

[0025] The advantages of the present invention are as follows: By using the analogy method, various interference factors such as self - weight and the position of the analysis section can be excluded, and it is applicable to piers with regular sections and complex sections. The calculation method is simple, the calculation accuracy is high, and the engineering practicability is strong. BRIEF DESCRIPTION OF THE DRAWINGS

[0026] Figure 1 is a schematic diagram of the bridge structure described in the present invention.

[0027] Figure 2 is a schematic diagram of the simplified force - bearing structure of the pier and the coordinate system described in the present invention.

[0028] Figure 3 is a flow chart of the method described in the present invention.

[0029] Figure 4 is a schematic diagram of the three - column bent pier described in the present invention.

[0030] Figure 5 is a schematic diagram of the beam bridge described in the present invention.

[0031] Figure 6 is a design parameter and layout diagram of the bearings of the beam bridge described in the present invention.

[0032] Figure 7 is a schematic diagram of Model 2 of the target pier of the method described in the present invention. DETAILED DESCRIPTION OF THE EMBODIMENTS

[0033] The technical solutions in the embodiments of the present invention will be clearly and completely described below. Obviously, the described embodiments are only a part of the embodiments of the present invention, rather than all the embodiments. All other embodiments obtained by those of ordinary skill in the art based on the embodiments of the present invention without creative efforts shall fall within the protection scope of the present invention.

[0034] Embodiment 1, Explanation of the principle on which the present invention is based

[0035] As Figure 1 shown in the bridge structure diagram, the main beam 1 is erected on the top of the pier 3 through the bearing 2, and the pier is located on the foundation pile 4. The pier 3 is a flexural-compression member in the bridge structure. Vertically, the pier 3 is mainly subjected to the concentrated load at the top of the column, and axially, it is mainly subjected to the uniformly distributed self-weight load.

[0036] For Figure 1 the shown results, the pier 3 can be regarded as fixed and constrained at the bottom of the foundation pile 4, and its imaginary fixed point 5 can be regarded as being located at 2 / α below the ground or the lowest scour line, that is, h = 2 / α in the figure. Where α is the deformation coefficient when calculating the pile foundation by the m method. That is, it is simplified to the force-bearing structure of a compression bar with a fixed bottom and a free upper end and no initial displacement for the pier 3. Taking the imaginary fixed point 5 of this structure as the coordinate origin, a coordinate system is established, as Figure 2 shown.

[0037] Under the action of the vertical concentrated axial compressive load P and the axial self-weight uniformly distributed load q at the top of the pier, the y-direction deformation of this structure, that is, the y-direction deformation of the pier 3, can be calculated using the differential equation of the elastic curve, as shown in formula (2):

[0038] Formula (2)

[0039] Where, E is the elastic modulus of the pier concrete; I is the sectional moment of inertia of the pier, l is the calculated length of the pier, P is the axial compressive load at the top of the pier, q is the axial uniformly distributed load of the pier, that is, the self-weight of the pier. η is the maximum displacement of the pier, and s is the length of the pier curve.

[0040] The implementation of formula (2) is to calculate the exact solution of the y-direction deformation of the pier 3 using the power series method, but this solution process is very cumbersome and not convenient for engineering applications. Therefore, the present invention combines engineering practical experience and mechanical principles to simplify the axial uniformly distributed load of the pier into a concentrated load at the top of the column with a magnitude of mql. The numerical value of m is related to the boundary conditions of the pier: when the top of the pier is directionally constrained, m≈0.358; when the top of the pier is free, m≈0.390. Therefore, for general bearing constraints, m = 0.39 can be taken conservatively. Then formula (2) can be approximately simplified to formula (3):

[0041] (3)

[0042] Among them, is the calculation length coefficient of the pier; Pcr refers to the buckling load of the pier.

[0043] According to formula (3), it can be seen that the calculation length coefficient of the pier is only related to the boundary conditions and has nothing to do with the load. After obtaining the stability coefficient λ of the pier through buckling analysis, the following formula (4) is established:

[0044] (4)

[0045] After arranging formula (4), it can be obtained that the calculation length coefficient μ of the pier, the axial pressure at the imaginary fixed point position, and the section stiffness satisfy formula (5):

[0046] (5)

[0047] The present invention proposes an analogy method for determining the calculation length coefficient of the pier, that is, for the same pressure-bearing member, when only the boundary conditions are different, it can be known from formula (3):

[0048] (6)

[0049] Where P cr1 and P cr 2 are the buckling load magnitudes of the same pier under two different boundary conditions.

[0050] After arranging formula (6), formula (7) can be obtained:

[0051] (7)

[0052] Further substituting formula (5) into and arranging formula (7), formula (8) can be obtained,

[0053] (8)

[0054] Formula (8) shows that for the same pier, under different boundary conditions, the calculation length coefficient μ of the pier has the relationship shown in formula (8).

[0055] Since those skilled in the art know that for the same pier, if the upper end is free and unrestrained; the lower end is based on the actual soil layer parameter information, and the "pile-soil" constraint between the target pier and the pile foundation is established according to the "m" method; only considering the single vertical load P applied at the top of the pier, without including the self-weight load G of the target pier and its upper main beam, the calculation length coefficient μ of this pier is 2.

[0056] Then, as Figure 3 shown, the method for determining the calculation length coefficient of the pier described in the present invention:

[0057] S1. Based on the actual dimensions of the bridge structure, establish an integral bridge model 1 that includes the upper main girder structure and the lower pier structure. The boundary conditions of this model 1 are as follows: between the main girder and the target pier, establish actual bearing constraints according to the bearing design parameters; between the target pier and the foundation pile, establish "pile-soil" constraints according to the actual soil layer parameter information by the "m" method. The loads of this model 1 include: the self-weight load G of the target pier and its upper main girder, the secondary dead load, and the vehicle load converted into a uniform load. The vehicle load is a uniform line load applied based on the influence line of the target pier.

[0058] S2. Conduct a buckling analysis on model 1, calculate the stability coefficient λ1 of the target pier, and extract the axial force P at the top of the column of the target pier. Among them, the buckling analysis is an ideal linear elastic bifurcation point stability analysis.

[0059] S3. Based on the actual dimensions of the target pier, establish a model 2 that only includes the target pier. The boundary conditions of this model 2 are as follows: the upper end of the target pier is free and unconstrained; the lower end establishes "pile-soil" constraints between the target pier and the foundation pile according to the actual soil layer parameter information by the "m" method. The loads of model 2 include: a single vertical load P applied at the top of the target pier, excluding the self-weight load G of the target pier and its upper main girder.

[0060] S4. Conduct a buckling analysis on model 2, and calculate the stability coefficient of the target pier λ 2 ;

[0061] S5. Determine the calculated length coefficient of the target pier according to formula (9) ;

[0062] (9)

[0063] Example 2. Verification of the effectiveness of the method of the present invention

[0064] As Figure 4 shown, the target pier is a three-column bent pier with a pier column diameter of 1.8 m and C40 concrete is used; the foundation pile diameter is 2.0 m and C30 concrete is used. The total height of the pier column is 30 m, the length of the foundation pile is 36 m, and the center distance of the piles is 5.7 m. According to the formula in Appendix L of the "Code for Design of Highway Bridge Foundations" (JTG 3363-2019), determine the theoretical fixed point position of the foundation pile. Among them, the deformation coefficient of the foundation pile

[0065]

[0066] Then the theoretical fixed point position is .

[0067] According to the method of the present invention, in the first step, a model 1 including the target pier is established and a buckling analysis is carried out. The boundary conditions of model 1 are as follows: displacement constraints in the longitudinal and transverse bridge directions (DX = 0, DY = 0) are applied to the upper end of the pier; the lower section of the pier is on the pile foundation, and according to the actual soil layer parameter information, a "pile-soil" spring constraint between the pier and the pile foundation is established by the "m" method. For the specific method, refer to Appendix L of the "Code for Design of Highway Bridge and Culvert Foundations" (JTG 3363-2019). Table 1 is the table of pile foundation node constraint stiffness.

[0068] Table 1

[0069]

[0070] The load of model 1 is to apply a vertical load N = 1000 to the top node of the pier, and the buckling analysis is calculated respectively when considering and not considering the self-weight load of the structure.

[0071] Perform a buckling analysis on model 1 and calculate the buckling load P cr1 ; after the buckling analysis is completed, the stability coefficient of the pier is obtained λ 1 and extract the axial force N1 at the position where the distance from the pile top to the target pier is h = 2.0 / a where the buckling load P cr1 = λ 1 N1. The following Table 2 shows the buckling analysis results considering the self-weight load of model 1 and not considering the self-weight load of model 1.

[0072] Table 2

[0073]

[0074] In the second step, a model 2 including only the target pier is established and a buckling analysis is carried out.

[0075] The boundary conditions of model 2 are as follows: the upper end of the pier is free and unconstrained, being a cantilever member; the lower part of the pier is at the pile foundation position, and according to the actual soil layer parameter information, a "pile-soil" spring constraint is established by the "m" method. For the specific method, refer to Appendix L of the "Code for Design of Highway Bridge and Culvert Foundations" (JTG 3363-2019).

[0076] The load of model 2 is to apply a vertical load N = 1000 to the top node of the pier, and the buckling analysis is calculated respectively when considering and not considering the self-weight load of the structure.

[0077] Perform a buckling analysis on model 2 and calculate the buckling load P cr2 ; after the buckling analysis is completed, the stability coefficient of the pier is obtained λ 2 and extract the axial force at the position where the distance from the pile top to the target pier is h = 2 / aAxial force N2 at the position (calculating the buckling load P cr2 = λ 2 N2. The following Table 3 shows the buckling analysis results considering the self-weight load of Model 2 and not considering the self-weight load of Model 2.

[0078] Table 3

[0079]

[0080] For the same pier, for Models 1 and 2 with different boundary conditions, their buckling loads P cr1 and P cr2 are:

[0081]

[0082] When neither the self-weight load of Model 1 nor that of Model 2 is considered, P cr1 is 266402 and P cr2 is 33018, then

[0083]

[0084] And the theoretical calculated length coefficient

[0085]

[0086] It can be seen that the theoretical solution of 0.35 has a high degree of coincidence with the calculated value of 0.352.

[0087] When considering the self-weight load of Model 1 and Model 2,

[0088]

[0089] And the theoretical calculated length coefficient

[0090]

[0091] It can be seen that the theoretical solution of 0.375 has a high degree of coincidence with the calculated value of 0.371.

[0092] In summary, it can be known that the method provided by the present invention is highly consistent with the calculation results of Euler theory, fully indicating that the method described in the present invention is correct, feasible and effective.

[0093] Example 3, Application example of the method of the present invention

[0094] As Figure 5 shown, a 4x50m beam bridge, with a precast T-beam structure on the upper part and a three-column bent pier on the lower part. Its design parameters are for the three-column bent pier (as Figure 4As shown in the figure, the pier column has a diameter of 1.8 m and is made of C40 concrete; the foundation pile has a diameter of 2.0 m and is made of C30 concrete. The total height of the pier column is 30 m, the length of the foundation pile is 36 m, and the distance between the pile centers is 5.7 m.

[0095] The piers are numbered 1 to 5 from left to right, and the bearings adopt friction pendulum isolation bearings. The design parameters and layout of the bearings are as Figure 6 shown. The target pier is pier No. 2, and the calculation length coefficient of the target pier is μ .

[0096] First step, according to the actual dimensions of the bridge structure, establish Model 1 of a continuous bridge including the upper main girder structure and the lower pier structure

[0097] Boundary conditions of Model 1: Based on the actual bearing design parameters between the main girder and the pier, establish actual bearing constraints; at the position of the foundation pile of the lower pier, according to the actual soil layer parameter information, establish "pile-soil" spring constraints according to the "m" method, and the specific values are shown in Table 1.

[0098] Loads of Model 1: According to the actual stress, the loads applied to the bridge structure include self-weight loads, secondary dead loads, vehicle loads (applying uniform line loads based on the influence line of the target pier), etc.

[0099] Second step, conduct buckling analysis on Model 1 to calculate the buckling stability coefficient of the target pier λ 1 = 10.469, and extract the axial force P = 8714.5 kN at the top of the target pier column.

[0100] Third step, establish Model 2 that only includes the target pier, as Figure 7 shown. The boundary conditions of this Model 2 include: the upper end of the target pier is free and unconstrained; the lower end is based on the actual soil layer parameter information, and the "pile-soil" constraint between the target pier and the foundation pile is established according to the "m" method; the loads include: a single vertical load P applied to the top of the target pier, excluding the self-weight load G of the target pier and its upper main girder.

[0101] Fourth step, apply a concentrated load P = 8714.5 kN to the top of the pier column in Model 2, and the self-weight load of the pier = 2646.5 kN, conduct buckling analysis, and calculate the stability coefficient λ 2 = 3.7915;

[0102] Fifth step, calculate the calculation length coefficient of the target pier μ

[0103]

[0104] As can be seen from the above calculation process and results, the method for determining the calculated length coefficient of bridge piers using the method of the present invention μ has a clear load value, a simple calculation process, is applicable to bridge piers with complex cross-sections, can also include multiple interference factors such as external loads and complex boundaries, and has high calculation accuracy.

Claims

1. A method for determining the computational length coefficient of a pier, characterized in that: It includes the following steps: S1. According to the actual dimensions of the bridge structure, establish an integral bridge model 1 including the upper main girder structure and the lower pier structure; The boundary conditions of the model 1 include: establishing a bearing constraint between the main girder and the target pier; establishing a "pile - soil" constraint between the target pier and the pile foundation; The loads of the model 1 include: self - weight load G, secondary dead load and vehicle load; S2. Conduct a buckling analysis on the model 1, calculate the stability coefficient λ1 of the target pier, and extract the single vertical load P applied to the top of the target pier; S3. According to the actual dimensions of the target pier, establish a model 2 including only the target pier; The boundary conditions of the model 2 include: no constraint at the upper end of the target pier; establishing a "pile - soil" constraint at the lower end with the pile foundation; The loads of the model 2 include: the single vertical load P applied to the top of the target pier, without considering the self - weight load G of the target pier; S4. Perform a buckling analysis on Model 2 to calculate the stability coefficient of the target pier λ 2 ; S5. Determine the calculated length coefficient of the target pier according to the formula ; 。 2. The method for determining the calculation length coefficient of a bridge pier according to claim 1, characterized in that: The buckling analysis is an ideal linear elastic bifurcation point stability analysis.

3. The method for determining the calculation length coefficient of a bridge pier according to claim 1, wherein: The "pile - soil" constraint means establishing the "pile - soil" constraint according to the actual soil layer parameter information by the "m" method.

4. The method for determining the calculated length coefficient of a bridge pier according to claim 1, wherein: The vehicle load is a uniform line load applied based on the influence line of the target pier.

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