Nonlinear beam unit analysis method considering prestressed tendon bonding slippage

Abstract

The invention discloses a nonlinear beam unit analysis method considering prestressed tendon bonding slippage, which comprises the following steps: establishing an overall structure analysis model according to the geometric dimension of a prestressed concrete beam, dividing calculation units, and assembling unit stiffness matrixes into a structural overall stiffness matrix; applying load and boundary conditions to the overall structure analysis model, and solving the overall analysis model by adopting a Newton iteration method; and extracting calculation results of structural deformation, stress, section internal force, prestressed tendon relative slippage and the like, drawing a load-deformation curve, and calculating structural rigidity and bearing capacity according to the slope and peak value of the curve. The calculation unit provided by the invention is an 8-degree-of-freedom beam unit, not only is the stress characteristic of the beam structure considered, but also the bond slip characteristic of the prestressed tendon is considered, the problem that a traditional beam unit analysis model cannot consider the bond slip effect of the prestressed tendon is solved, and compared with an entity unit analysis model, the convergence is better, and the calculation efficiency is higher.

Description

TECHNICAL FIELD

[0001] The present application relates to the field of bridge deterioration analysis, and particularly relates to a nonlinear beam element analysis method considering prestressed tendon bond slip. BACKGROUND

[0002] In prestressed concrete beams, the bond effect between the prestressed tendon and the concrete beam is the key to ensure the cooperative stress between them. However, due to factors such as grouting construction defects, steel corrosion, and fatigue effects, the bond effect between the prestressed tendon and the concrete beam will deteriorate, and the bond slip effect will inevitably occur. In addition, to improve the ductility and self-centering performance of the structure, the engineering application of unbonded prestressed concrete beams is increasing, and the unbonded prestressed tendon and the concrete beam body are in an unbonded connection state with vertical deformation coordination and only a small axial friction effect between them. For such prestressed concrete beams, the bond slip effect of the prestressed tendon makes the traditional beam theory no longer satisfy the plane section assumption, and a reasonable analysis model needs to be established to consider the influence of the bond slip effect of the prestressed tendon on the mechanical properties of the structure.

[0003] Currently, there are two main types of analysis models considering the bond slip effect of the prestressed tendon. One is the solid element analysis model method: the concrete beam and the prestressed tendon are modeled by solid elements, the prestressed duct is deducted from the geometric model of the concrete beam, and the bond slip effect is considered by setting the bond contact relationship between the outer surface of the prestressed tendon and the inner surface of the prestressed duct in the concrete beam. However, the solid element analysis model has many problems such as many element degrees of freedom, large amount of calculation, poor convergence of bond contact, and low calculation efficiency, which is not convenient for application in actual engineering. The second type is the beam-bar combined element method: the traditional beam element model cannot consider the bond slip effect of the prestressed tendon; to consider the bond slip, the concrete beam is modeled by a beam element, and the prestressed tendon is modeled by a bar element, and a connecting spring is set between the nodes of the concrete beam element and the nodes of the prestressed tendon bar element to consider the bond slip effect of the prestressed tendon by adjusting the stiffness of the connecting spring. This method needs to use different element types to model the concrete beam and the prestressed tendon respectively, and the node correspondence relationship of the beam element and the bar element and the setting process of the connecting spring are very complex, which is not convenient for modeling and analysis of large-scale engineering structures. Therefore, a new type of element analysis model that is simple to model and efficient to calculate is needed to meet the mechanical property analysis requirements of prestressed concrete beams considering the bond slip effect. SUMMARY

[0004] The application provides a nonlinear beam element analysis method and device considering prestressed tendon bond slip, and solves the above problems.

[0005] The application provides a nonlinear beam element analysis method considering prestressed tendon bond slip, which comprises the following steps:

[0006] S1, an overall structure analysis model is established according to the geometric size of a prestressed concrete beam, calculation elements are divided, and an element stiffness matrix is assembled into an overall stiffness matrix of the structure;

[0007] S2, a load and a boundary condition are applied to the overall structure analysis model, and the overall analysis model is solved by using a Newton iteration method;

[0008] S3, calculation results such as structure deformation, stress, sectional internal force and relative slip of the prestressed tendon are extracted, a load-deformation curve is drawn, and the structure stiffness and bearing capacity are calculated according to the slope and peak value of the curve.

[0009] The nonlinear beam element analysis method considering prestressed tendon bond slip provided by the application, as a preferred mode, the specific acquisition method of the element stiffness matrix in step S1 is:

[0010] S11, a beam element degree of freedom and a deformation mode are defined;

[0011] S12, a beam element section is defined, a sectional stiffness matrix and a sectional resistance vector are derived;

[0012] S13, the sectional stiffness matrix and the resistance vector are integrated, and the element stiffness matrix and the resistance vector are derived.

[0013] The nonlinear beam element analysis method considering prestressed tendon bond slip provided by the application, as a preferred mode, step S11 specifically comprises:

[0014] The length of the beam element is Le, including two nodes i and j, and there are 8 degrees of freedom, and the element displacement vector u is as follows:

[0015] u=[u i v i θ i Δ i u j v j θ j Δj ] T ;

[0016] The unit resistance vector F is as follows:

[0017] F = [N i V i M i T i N j V j M j T j ] T ;

[0018] In the formula, u i , v i , θ i , δ i respectively represent the axial direction, vertical direction, rotation angle and relative slip deformation of the prestressed tendon of the i node, u j , v j , θ j , δ j respectively represent the axial direction, vertical direction, rotation angle and relative slip deformation of the prestressed tendon of the j node, N i , V i , M i , T i respectively represent the axial force, shear force, bending moment and slip additional force of the i node, N j , V j , M j , T j respectively represent the axial force, shear force, bending moment and slip additional force of the j node.

[0019] The unit basic deformation vector u b of the beam is as follows:

[0020] u b = [Δu θ' θ' Δ i Δ j ] T ;

[0021] The unit displacement vector u is calculated as follows:

[0022] u b = Pu;

[0023]

[0024] Wherein, P is a conversion matrix.

[0025] Three deformation modes of the beam unit are set, including: axial deformation mode, bending deformation mode, and prestressed tendon bonding slip deformation mode.

[0026] The displacement field in the beam unit is interpolated by Hermite polynomial according to three deformation modes, and the calculation method is as follows:

[0027]

[0028] Wherein, x is the section position coordinate in the unit (0≤x≤L e ), is the displacement field vector in the unit, N(x) is the shape function, du(x), θ(x) and Δ(x) respectively represent the axial deformation, the rotation deformation and the relative slip deformation of the x position in the unit, ξ=x / L e is the relative position coordinate parameter;

[0029] The section deformation of the x position in the unit is calculated, and the formula is as follows:

[0030] E(x)=[ε b (x) ε t (x) φ(x) Δ(x)] T =B(x)u b ;

[0031]

[0032] Wherein, E(x) is the section deformation vector of the x position, ε b (x) and φ(x) are respectively the axial average strain and the curvature of the x position section, ε t (x) represents the prestressed steel bar strain of the x position section, and B(x) is the x position section geometry matrix.

[0033] The nonlinear beam unit analysis method considering the prestressed steel bar bond slip provided by the application specifically comprises the following steps:

[0034] The unit section is established according to the geometric size of the concrete beam, and the section is meshed;

[0035] The ordinary steel bar and the prestressed steel bar mesh are divided into one mesh that can only axially deform according to the actual cross-sectional area and the centroid position of each steel bar;

[0036] The storage data of each mesh are recorded as [y k , z k , A k , ε k , σ k , E k , Ω k ],

[0037] Wherein, y k , z k , A k , ε k , σk , E k , Ω k respectively represent the vertical coordinate, the horizontal coordinate, the area, the axial strain, the axial stress, the tangent modulus and the grid type identifier of the kth grid; for the prestressed tendon grid, Ω k is 1 or 0;

[0038] The axial strain ε k of each grid is calculated, and the formula is as follows:

[0039] ε k = [1- Ω k Ω k -y 0] E(x) ;

[0040] The grid axial stress σ k and the tangent modulus E k are obtained by substituting the grid strain into the material constitutive relationship,

[0041] The concrete constitutive relationship adopts the Concrete02 model in OpenSees, the steel bar constitutive relationship adopts the Steel01 model, and the prestressed tendon material constitutive relationship adopts the Steel02 model;

[0042] The prestressed tendon slip displacement of the x-position section is substituted into the bond slip constitutive relationship to obtain the bond force f s (x) and the bond stiffness ρ s (x) ;

[0043] The bond slip constitutive relationship is linear and nonlinear;

[0044] The linear bond slip constitutive relationship is as follows:

[0045] f s (x) = ρ s (x) · Δ(x) ;

[0046] The nonlinear bond slip constitutive relationship is as follows:

[0047]

[0048] Wherein, ρ s (x) = 0, τ0 is the bond strength, s0 is the slip displacement corresponding to the maximum bond force, τ1 is the residual bond strength, and s1 is the slip displacement corresponding to the bond failure;

[0049] The cross-sectional stiffness matrix k s (x) is calculated, and the formula is as follows:

[0050]

[0051] The cross-sectional resistance vector rs (x), the formula is as follows:

[0052]

[0053] The nonlinear beam element analysis method considering prestressed tendon bond slip, as a preferred mode, step S13 specifically comprises:

[0054] The calculation unit stiffness matrix K is calculated, and the formula is as follows:

[0055]

[0056] The calculation unit resistance vector F is calculated, and the formula is as follows:

[0057]

[0058] The nonlinear beam element analysis method considering prestressed tendon bond slip, as a preferred mode, step S2 specifically comprises:

[0059] The support freedom of the corresponding position unit of the support is constrained; in particular, the relative slip freedom of the node at the anchoring end of the prestressed tendon is constrained;

[0060] First, the self-weight and prestressed tendon tension of the structure are applied, and the prestressed tendon tension is applied by the initial stress mode of the Steel02 constitutive model; after reaching the equilibrium state, the live load is applied to analyze the stress of the structure.

[0061] The present application has the following beneficial effects:

[0062] 1) The calculation unit proposed in the present application adopts the mode of additional freedom, and a new type of beam element analysis model considering prestressed tendon bond slip is derived, which can realize the overall consideration of the concrete beam, the prestressed tendon and the bond action between them in one unit, without the need for separate modeling of concrete and prestressed tendon; the modeling efficiency is improved;

[0063] 2) The calculation unit proposed in the present application is an 8-degree-of-freedom beam element, which considers the stress characteristics of the beam structure and the bond slip characteristics of the prestressed tendon, solves the problem that the traditional beam element analysis model cannot consider the bond slip effect of the prestressed tendon, and has better convergence and higher calculation efficiency compared with the solid element analysis model. BRIEF DESCRIPTION OF DRAWINGS

[0064] Figure 1 It is a flow chart of a nonlinear beam element analysis method considering prestressed tendon bond slip;

[0065] Figure 2 It is a beam element diagram of a nonlinear beam element analysis method considering prestressed tendon bond slip;

[0066] Figure 3 A nonlinear beam element analysis method considering prestressed tendon bond slip. DETAILED DESCRIPTION

[0067] The technical solutions in the embodiments of the present application will be clearly and completely described below with reference to the drawings in the embodiments of the present application. Obviously, the described embodiments are only some of the embodiments of the present application, but not all the embodiments of the present application.

[0068] Embodiment 1

[0069] As shown in the drawings, a nonlinear beam element analysis method considering prestressed tendon bond slip, Figure 1

[0070] Basic assumptions are made

[0071] 1) It is assumed that, except for the prestressed tendon, the deformation of the concrete beam conforms to the plane section assumption, and the shear deformation can be ignored;

[0072] 2) The vertical and rotational deformations between the prestressed tendon and the concrete beam are consistent, and the axial relative deformation considers the bond slip relationship;

[0073] Definition of beam element freedom and deformation mode

[0074] 1) The basic composition of the prestressed concrete beam considering the prestressed tendon bond slip effect; the length of the beam element is defined as Le, including i and j two nodes, and there are 8 degrees of freedom, the element displacement vector u and the element resistance vector F, and the formulas are as follows,

[0075] u = [u i v i θ i Δ i u j v j θ j Δ j ] T

[0076] F = [N i V i M i Τ i N j V j M j Τ j ] T

[0077] In the formula, u i , v i , θ i , Δ i respectively represent the axial, vertical, rotational and prestressed tendon relative slip deformation of i node, u​j j j j i i i i j j j j

[0078] 2) Define the basic deformation vector u of beam element b , and calculate the element displacement vector u,

[0079] u b = [Δu θ' θ' Δ i Δ j ] T

[0080] u b = Pu

[0081]

[0082] In the formula, P is the conversion matrix.

[0083] 3) Define three deformation modes of beam element, including: axial deformation mode, bending deformation mode, and prestressed tendon bond slip deformation mode; the displacement field in the element is interpolated by Hermite polynomial according to the three deformation modes,

[0084]

[0085] In the formula, x is the section position coordinate in the element (0≤x≤L e ), is the displacement field vector in the element, N(x) is the shape function, du(x), θ(x), Δ(x) respectively represent the axial deformation, the rotation deformation and the relative slip deformation at x position in the element, and ξ=x / L e is the relative position coordinate parameter.

[0086] 4) Calculate the section deformation at x position in the element,

[0087] E(x) = [ε b (x) ε t (x) φ(x) Δ(x)] T = B(x)u b

[0088] ​​​​​​​​​​​​

[0089] where E(x) is the cross-section deformation vector at x position, ε b (x) and φ(x) are the axial mean strain and curvature of the cross-section at x position, ε t (x) represents the strain of the prestressed tendon at x position, and B(x) is the geometry matrix of the cross-section at x position.

[0090] Definition of beam element cross-section, derivation of cross-section stiffness matrix and cross-section resistance vector

[0091] 1) Establish the element cross-section according to the geometric size of the concrete beam, and divide the cross-section into grids; the grids of ordinary steel bars and prestressed tendons are divided according to the actual cross-sectional area and the position of the centroid, and each steel bar is divided into one grid; each grid can only deform axially.

[0092] 2) Define the storage data of each grid as [y k , z k , A k , ε k , σ k , E k , Ω k ], where y k , z k , A k , ε k , σ k , E k , Ω k represent the vertical coordinate, horizontal coordinate, area, axial strain, axial stress, tangent modulus and grid type identifier of the kth grid, respectively; for the prestressed tendon grid, Ω k = 1, otherwise Ω k = 0.

[0093] 3) Calculate the axial strain of each grid,

[0094] ε k = [1-Ω k Ω k -y 0]E(x)

[0095] 4) Define the material constitutive relationship of concrete, steel and prestressed tendon, and substitute the grid strain into the material constitutive relationship to obtain the axial stress σ k and tangent modulus E k of the grid. The constitutive relationship of concrete adopts the Concrete02 model in OpenSees, the constitutive relationship of steel adopts the Steel01 model, and the constitutive relationship of prestressed tendon adopts the Steel02 model.

[0096] 5) Define the bond-slip constitutive relation of the prestressed tendon, and substitute the slip displacement of the x-section of the prestressed tendon into the bond-slip constitutive relation to obtain the bond force f s (x) and the bond stiffness p s (x). The bond-slip constitutive relation can be linear or nonlinear as needed; the linear bond-slip constitutive relation is as follows: f s (x) = 0; the nonlinear bond-slip constitutive relation is as follows.

[0097] f s (x) = p s (x) · Δ(x)

[0098]

[0099] In the formula, τ0 is the bond strength, s0 is the slip displacement corresponding to the maximum bond force, τ1 is the residual bond strength, and s1 is the slip displacement corresponding to the bond failure.

[0100] 6) The sectional stiffness matrix and the sectional resistance vector are calculated according to the following formula:

[0101]

[0102]

[0103] In the formula, k s (x) and r s (x) respectively represent the stiffness matrix and the resistance vector of the x-section.

[0104] Integrate the sectional stiffness matrix and the resistance vector to derive the element stiffness matrix and the resistance vector

[0105] 1) Numerical integration is performed using the obtained sectional stiffness matrix k s (x) to obtain the element stiffness matrix;

[0106]

[0107] 2) Numerical integration is performed using the obtained sectional resistance vector r s (x) to obtain the element resistance vector.

[0108]

[0109] According to the geometric size of the prestressed concrete beam, an overall structural analysis model is established, and the overall analysis conditions are as follows:

[0110] The implementation case beam is a continuous beam with a span of 4.8+4.8m, a rectangular cross section, and a cross section size of 150*300mm; four steel bars with a diameter of 12mm are arranged at the bottom of the beam, and two steel bars with a diameter of 12mm are arranged at the top of the beam; one A s 15.2 prestressed reinforcement, the free sliding between the prestressed reinforcement and the beam body, the bonding strength is 0; the compressive strength of the concrete is 36.7MPa, the nominal yield strength of the prestressed reinforcement is 1860MPa, and the prestressed reinforcement tension force is 1196MPa; the load is symmetrically applied at the three-point position, and the deformation and bearing capacity of the prestressed concrete beam under the bonding and sliding conditions are tested and analyzed.

[0111] The overall structure analysis model is divided into calculation units according to a length of 100mm, and the unit stiffness matrix is assembled into the overall stiffness matrix of the structure.

[0112] The load and boundary conditions are applied to the overall structure analysis model, and the Newton iteration method is used to solve the overall analysis model.

[0113] 1) The support freedom of the element corresponding to the position of the support is constrained; in particular, the relative sliding freedom of the node at the prestressed reinforcement anchoring end needs to be constrained.

[0114] 2) The load is applied in two analysis steps: first, the self-weight and the prestressed reinforcement tension force are applied, and the prestressed reinforcement tension force is applied through the initial stress of the Steel02 constitutive model; after reaching the balanced state, the live load is applied to analyze the stress of the structure.

[0115] The calculation results of the structure deformation, stress, cross section internal force and prestressed reinforcement relative sliding are extracted. The relative sliding of the prestressed reinforcement can be obtained by extracting the deformation of the 4th freedom of the node. The load-deformation curve is drawn, and the structure stiffness and bearing capacity are calculated according to the curve slope and peak value.

[0116] 1) The calculation results of the new nonlinear beam element analysis model proposed in the application are in good agreement with the test results, while the residual displacement calculated by the traditional beam element analysis model has obvious deviation from the test results due to the fact that the bonding and sliding effect is not considered.

[0117] 2) The new nonlinear beam element model of the whole beam proposed by the method of the application has only 384 degrees of freedom, and does not involve complex contact relationship, so the calculation efficiency is higher than that of the entity element model.

[0118] The above is only the preferred specific embodiment of the application, but the protection scope of the application is not limited to this, any person skilled in the art can make equivalent replacement or change according to the technical scheme and the inventive concept of the application within the technical range disclosed by the application, which should be covered in the protection scope of the application.

Claims

1. A nonlinear beam element analysis method considering prestressed tendon bond slip, characterized by: The following steps are involved: S1. Establish an overall structural analysis model based on the geometric dimensions of the prestressed concrete beam, divide the calculation units, and assemble the unit stiffness matrices into the overall structural stiffness matrix; S2. Apply loads and boundary conditions to the overall structural analysis model and use the Newton iteration method to solve the overall analysis model; S3. Extract the calculation results of structural deformation, stress, cross-sectional internal force, and relative slip of prestressed tendons, draw the load-deformation curve, and calculate the structural stiffness and bearing capacity based on the slope and peak value of the curve.

2. The nonlinear beam element analysis method considering prestressed tendon bond slip according to claim 1, characterized in that: The specific method for obtaining the element stiffness matrix in step S1 is: S11. Define the degrees of freedom and deformation modes of the beam element; S12. Define the beam element section and derive the section stiffness matrix and section resistance vector; S13. Integrate the section stiffness matrix and resistance vector to derive the unit stiffness matrix and resistance vector.

3. The nonlinear beam element analysis method considering prestressed tendon bond slip according to claim 2 is characterized by: The step S11 specifically includes: Assume that the length of the beam element is Le, including two nodes i and j, with a total of 8 degrees of freedom. The formula for the element displacement vector u is as follows: u=[u i v i i i D i you j v j i j D j ] T ; The formula for the unit resistance vector F is as follows: F=[N i V i M i T i N j V j M j T j ] T ; Where u i 、v i ,θ i , δ i They represent the axial, vertical, rotational and relative slip deformation of prestressed tendons at node i, u j 、v j ,θ j , δ j They represent the axial, vertical, rotational and relative slip deformation of the prestressed tendon at node j, respectively. N i 、V i 、M i 、T i They represent the axial force, shear force, bending moment and additional slip force of node i, N j 、V j 、M j 、T j They represent the axial force, shear force, bending moment and additional slip force of node j respectively; The element basic deformation vector u of the beam b , the formula is as follows: you b =[Δuθ′θ′Δ i D j ] T ; Calculate the unit displacement vector u, the formula is as follows: u b =Punch; Where P is the transformation matrix; There are three deformation modes of beam elements, including axial deformation mode, bending deformation mode, and prestressed tendon bond slip deformation mode; The displacement field in the beam element is calculated using Hermite polynomial interpolation according to the three deformation modes. The calculation method is as follows: Where x is the coordinate of the cross-section position in the unit (0≤x≤L e ), is the displacement field vector within the unit, N(x) is the shape function, du(x), θ(x), and Δ(x) represent the axial deformation, angular deformation, and relative slip deformation at the x position within the unit, respectively, ξ = x / L e is the relative position coordinate parameter; Calculate the cross-sectional deformation at the x position within the element using the following formula: E(x)=[e b (x)e t (x)φ(x)Δ(x)] T =B(x)u b ; Among them, E(x) is the deformation vector of the section at position x, ε b (x) and φ(x) are the axial average strain and curvature of the section at position x, respectively, and ε t (x) represents the strain of the prestressed tendon at the section at position x, and B(x) is the geometric matrix of the section at position x.

4. The nonlinear beam element analysis method considering prestressed tendon bond slip according to claim 3 is characterized by: The step S12 specifically includes: Establish unit sections according to the geometric dimensions of the concrete beam and divide the sections into grids; The mesh of ordinary steel bars and prestressed steel bars is divided into a mesh that can only deform in the axial direction according to the actual cross-sectional area and centroid position; The storage data of each grid is recorded as [y k ,z k ,A k ,ε k ,σ k ,E k ,Ω k ], Among them, y k 、z k 、A k , ε k , σ k 、E k ,Ω k Represent the vertical coordinate, transverse coordinate, area, axial strain, axial stress, tangent modulus and mesh type identifier of the kth mesh respectively; for the prestressed tendon mesh, Ω k is 1 or 0; Calculate the axial strain ε for each grid k , the formula is as follows: e k =[1-Ω k Oh k -y 0]E(x); Substitute the mesh strain into the material constitutive relationship to obtain the mesh axial stress σ k and tangent modulus E k , The concrete constitutive relationship adopts the Concrete02 model in OpenSees, the steel bar constitutive relationship adopts the Steel01 model, and the prestressed tendon material constitutive relationship adopts the Steel02 model; Substitute the slip displacement of the prestressed tendon at the x-section into the bond-slip constitutive relation to obtain the bond force f s (x) and bond stiffness ρ s (x); The bond-slip constitutive relations are linear and nonlinear; The linear bond-slip constitutive relation is as follows: f s (x)=ρ s (x)·Δ(x); The nonlinear bond-slip constitutive relation is as follows: Among them, ρ s (x) = 0, τ0 is the bond strength, s0 is the slip displacement corresponding to the maximum bond force, τ1 is the residual bond strength, and s1 is the slip displacement corresponding to bond failure. Calculate the section stiffness matrix k s (x), the formula is as follows: Calculate the cross-sectional resistance vector r s (x), the formula is as follows:

5. The nonlinear beam element analysis method considering prestressed tendon bond slip according to claim 4, characterized in that: The step S13 specifically includes: Calculate the element stiffness matrix K, the formula is as follows: Calculate the unit resistance vector F using the following formula:

6. The nonlinear beam element analysis method considering prestressed tendon bond slip according to claim 1, characterized in that: The step S2 specifically includes: Apply constraints to the support degrees of freedom of the units at the corresponding positions of the supports; in particular, the relative sliding degrees of freedom of the nodes at the anchorage ends of the prestressed tendons need to be constrained; First, the self-weight of the structure and the tension of the prestressed tendons are applied. The tension of the prestressed tendons is applied through the initial stress method of the Steel02 constitutive model. After reaching the equilibrium state, the live load is applied to analyze the structural stress.

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