A mechanical model and calculation method considering the rotation of the top and bottom plates of laminated rubber isolation bearings

Abstract

This invention discloses a mechanical model and calculation method for considering the effects of top and bottom plate rotation on laminated rubber isolation bearings. This model incorporates a new "top plate rotation spring" at the top of the existing "double-spring" mechanical model. This spring rotates by an angle β under the action of the top plate bending moment M, accounting for the effects of top plate rotation. This mechanical model accurately simulates top plate rotation on laminated rubber isolation bearings and is expected to provide a more precise analysis method for the design and computational analysis of seismic isolation structures.

Description

Technical Field

[0001] The present invention belongs to the field of civil engineering, and specifically relates to a mechanical model that takes into account the influence of the rotation of the top and bottom plates of a laminated rubber seismic isolation bearing, and also relates to a calculation method for the mechanical model that takes into account the influence of the rotation of the top and bottom plates of a laminated rubber seismic isolation bearing. The method can accurately simulate the rotation of the top plate of the laminated rubber seismic isolation bearing, and provide a more accurate analysis method for the design and calculation analysis of the seismic isolation structure. Background Art

[0002] Laminated rubber isolation bearings are widely used in the seismic isolation design of buildings and bridge structures. By alternately stacking thin steel plates and thin rubber sheets, the rubber sheets can withstand large vertical loads due to the restraining effect of the thin steel plates. At the same time, they have the ability to deform in horizontal shear, so they can dissipate earthquake energy through horizontal deformation, ensure structural safety, and achieve the purpose of seismic isolation. Traditional seismic isolation structure design usually assumes that the internal rubber layer of the bearing rotates during the horizontal deformation process, but its top and bottom plates do not rotate. The mechanical model used to simulate its mechanical properties also assumes that the top and bottom plates do not rotate, and only uses a rotation spring to simulate the rotational stiffness of its internal rubber layer (such as Figure 4 (as shown); however, in actual application, due to the overturning moment of the superstructure or the eccentric load borne by the bearing, the top plate of the bearing may rotate during the horizontal deformation process; especially in bridge engineering, rubber seismic isolation bearings are often placed at the top of the pier. The high pier will make the boundary conditions of the bearing flexible, so the top and bottom plates of the bearing may rotate under the action of an earthquake; the rotation of the top plate of the bearing will significantly affect the mechanical properties of the bearing, such as reducing the vertical critical load of the bearing in the direction of top plate rotation, reducing the horizontal stiffness of the bearing in the direction of rotation, and causing the bearing to be damaged prematurely in the direction of top plate rotation. Summary of the Invention

[0003] In order to solve the problems existing in the above-mentioned prior art, the purpose of the present invention is to provide a mechanical model that takes into account the influence of the rotation of the top and bottom plates of the laminated rubber seismic isolation bearing, which can effectively simulate the mechanical behavior of the laminated rubber seismic isolation bearing under the rotation of the top and bottom plates. It is particularly suitable for simulating the mechanical behavior of the bearing under the rotation of the top and bottom plates caused by flexible boundary conditions when the bearing is placed at the top of the pier in bridge engineering. This mechanical model can be used for the design and computational analysis of seismic isolation structures.

[0004] Another object of the present invention is to provide a calculation method for a mechanical model that takes into account the influence of the rotation of the top and bottom plates of the laminated rubber seismic isolation bearing, re-establish the force balance equation and deformation coordination equation of the model, and re-establish the calculation method of the model, so that the mechanical behavior of the bearing under the condition of rotation of the top and bottom plates of the bearing can be simulated. MATLAB is used for programming, and the model calculation results are compared with the experimental results to verify the effectiveness of the mechanical model.

[0005] To further achieve the above objectives, the present invention adopts the following technical solutions:

[0006] A mechanical model that considers the influence of the rotation of the top and bottom plates of a laminated rubber seismic isolation bearing includes: adding a "top plate rotation spring" on the top of the "double spring" mechanical model. Under the action of the top plate bending moment M, the top plate rotation spring will produce an angle β, which is used to consider the influence of the rotation of the bearing top plate.

[0007] Accordingly, the present invention also claims protection for a calculation method of a mechanical model that considers the influence of the rotation of the top and bottom plates of the laminated rubber seismic isolation bearing, using the aforementioned mechanical model that considers the influence of the rotation of the top and bottom plates of the laminated rubber seismic isolation bearing, comprising the following steps:

[0008] (1) The force balance equation and deformation coordination equation of the mechanical model are as follows:

[0009] M s =P(s+h b θ)+Fh b +M (6)

[0010] Q=Pθ+F (7)

[0011] M=k r β (8)

[0012] P=k z v (9)

[0013] u x =h b θ+s (10)

[0014]

[0015] α=β (12)

[0016] Where, P is the vertical pressure of the support top plate; F is the horizontal force of the support; M is the local bending moment of the support top plate; M s is the internal rotation spring bending moment; Q is the horizontal spring restoring force; u x is the horizontal deformation of the support; u zis the vertical deformation of the support; α is the rotation angle of the support top plate; s is the deformation of the horizontal shear spring; v is the vertical spring deformation; θ is the rotation angle of the internal rotation spring; β is the rotation angle of the top plate rotation spring; k z is the vertical spring stiffness; k r is the spring stiffness of the top plate, h b It is the height of the support excluding the top and bottom plates;

[0017] (2) The constitutive model of the horizontal spring in the mechanical model is as follows:

[0018] Q=Q l +Q h (13)

[0019]

[0020]

[0021]

[0022]

[0023] Where Q is the horizontal spring restoring force; Q l and Q h are the linear part and hysteresis part of the horizontal restoring force of the support respectively; G is the shear modulus of the rubber material; A is the cross-sectional area of ​​the support; u x is the horizontal deformation of the support; T r is the total thickness of the bearing rubber layer; n is the empirical parameter of the hysteresis model; Y and Y0 are the yield strengths of the bearing in the deformed and undeformed states, respectively; P is the vertical pressure of the bearing top plate; λ is an empirical parameter; P cr and P cr0 are the vertical critical loads of the support under deformation and non-deformation states respectively; A r is the overlapping area of ​​the support top and bottom plates under horizontal deformation;

[0024] (3) The constitutive model of the internal rotation spring in the mechanical model is as follows:

[0025]

[0026]

[0027]

[0028]

[0029] Where ω and r are empirical parameters; M s is the internal rotation spring bending moment; M y is the yield moment of the support; Z is the section modulus of the support (πD3 / 32, D is the cross-sectional diameter of the support); A is the cross-sectional area of ​​the support; P is the vertical pressure of the support top plate; σ y is the tensile yield strength of the rubber material; θ y k is the yield angle of the internal rotation spring; θ0 is the initial rotation stiffness of the internal rotation spring; T r is the total thickness of the bearing rubber layer; K is the bulk modulus of the rubber material; I is the section moment of inertia; I1 and I2 are the first-order and second-order modified Bessel functions of the first kind, respectively; G is the shear modulus of the rubber material; S1 is the first shape coefficient of the bearing;

[0030] (4) The stiffness of the vertical spring in the mechanical model is defined as follows:

[0031]

[0032]

[0033] Where k z is the vertical spring stiffness; E c is the compression modulus of the rubber layer; A is the cross-sectional area of ​​the support; T r is the total thickness of the bearing rubber layer; G is the shear modulus of the rubber material; S1 is the first shape coefficient of the bearing.

[0034] Optionally, the aforementioned calculation method of the mechanical model considering the influence of the rotation of the top and bottom plates of the laminated rubber isolation bearing is used to analyze the mechanical performance of the bearing, including two working conditions: 1) pseudo-static bearing test condition; 2) earthquake motion loading condition;

[0035] 1) Under the pseudo-static bearing test condition, the vertical pressure P of the bearing top plate, the bearing top plate rotation angle α and the bearing horizontal deformation u x , controlled by the hydraulic system, the model control equation (6-12) is rewritten into the following root-finding form:

[0036]

[0037] Where, P is the vertical pressure of the support top plate; F is the horizontal force of the support; M is the local bending moment of the support top plate; M s is the internal rotation spring bending moment; Q is the horizontal spring restoring force; u x is the horizontal deformation of the support; u z is the vertical deformation of the support; α is the rotation angle of the support top plate; s is the deformation of the horizontal spring; v is the deformation of the vertical spring; θ is the rotation angle of the internal rotation spring; β is the rotation angle of the top plate rotation spring; k z is the vertical spring stiffness; k r is the spring stiffness of the top plate; h b It is the height of the support excluding the top and bottom plates;

[0038] Where x = [F, M, u z ,s,θ,v,β] T is an unknown vector. At each global calculation step, the Newton iteration method can be used to calculate the converged x value at this time. The expression is as follows:

[0039] x (k) =x (k-1) -[J(x (k-1) )] -1 g(x (k-1) ) (25)

[0040] Where k = 1, 2, 3, ..., represents the number of iterations; the initial value of the iteration x (0) The convergence value of the previous global iteration step can be used. is the Jacobian matrix, which is expressed as follows:

[0041]

[0042] Where h b is the height of the support excluding the top and bottom plates (i.e., the sum of the thickness of all rubber layers and thin steel plates); P is the vertical pressure of the support top plate; Q is the horizontal spring restoring force; M s is the internal rotation spring bending moment; s is the deformation of the horizontal spring; θ is the internal rotation spring angle; k r k is the spring stiffness of the top plate; z is the vertical spring stiffness;

[0043] 2) For earthquake loading conditions, the Newmark-β time-stepping method is used for calculation:

[0044] At each global calculation step, the force increment vector [dF, dP, dM] T and displacement increment vector [du x ,du z ,dα] T The relationship between , that is, the tangent stiffness matrix, is derived by following the steps below:

[0045] First, formula (6-9) can be rewritten into the following differential form:

[0046]

[0047]

[0048] dM=k r dβ (29)

[0049] dP=k z dv (30)

[0050] Where h b is the height of the support excluding the top and bottom plates (i.e., the sum of the thickness of all rubber layers and thin steel plates); F is the horizontal force increment of the support; P is the vertical pressure of the support top plate; M is the local bending moment increment of the support top plate; Q is the horizontal spring restoring force; M s is the internal rotation spring bending moment; s is the deformation of the horizontal spring; θ is the internal rotation spring angle; v is the vertical spring deformation; β is the top plate rotation spring angle; k r k is the spring stiffness of the top plate; z is the vertical spring stiffness;

[0051] Rewrite formula (27-30) into the following matrix form:

[0052]

[0053] Where h b is the height of the support excluding the top and bottom plates (i.e., the sum of the thickness of all rubber layers and thin steel plates); F is the horizontal force of the support; P is the vertical pressure of the support top plate; M is the local bending moment of the support top plate; Q is the horizontal spring restoring force; M s is the internal rotation spring bending moment; s is the deformation of the horizontal spring; θ is the internal rotation spring angle; v is the vertical spring deformation; β is the top plate rotation spring angle; k r k is the spring stiffness of the top plate; z is the vertical spring stiffness; secondly, formula (10-12) is rewritten into the following differential form:

[0054] du x =ds+h b dθ (32)

[0055] du z =θds+(h b θ+s)dθ+dv (33)

[0056] dα=dβ (34)

[0057] Where h b is the height of the support excluding the top and bottom plates; s is the horizontal spring deformation; θ is the internal rotation spring angle; v is the vertical spring deformation; α is the support top plate angle; β is the top plate rotation spring angle; u x is the horizontal deformation of the support; u z is the vertical deformation of the support;

[0058] Rewrite formula (32-34) into the following matrix form:

[0059]

[0060] Where h b is the height of the support excluding the top and bottom plates; s is the horizontal spring deformation; θ is the internal rotation spring angle; v is the vertical spring deformation; α is the support top plate angle; β is the top plate rotation spring angle; u x is the horizontal deformation of the support; u z is the vertical deformation of the support;

[0061] Rewrite equations (31) and (35) into the following matrix form:

[0062]

[0063] Where h b is the height of the support excluding the top and bottom plates; s is the horizontal spring deformation; θ is the internal rotation spring angle; v is the vertical spring deformation; α is the support top plate angle; u x is the horizontal deformation of the support; u z is the vertical deformation of the support; P is the vertical pressure of the support top plate; M is the local bending moment of the support top plate; Q is the horizontal spring restoring force; F is the horizontal force of the support; k r k is the spring stiffness of the top plate; z is the vertical spring stiffness; M s is the internal rotation spring bending moment;

[0064] The tangent stiffness matrix Kt is obtained from formula (36) as follows:

[0065]

[0066] Where h b is the height of the support excluding the top and bottom plates (i.e., the sum of the thickness of all rubber layers and thin steel plates); s is the horizontal spring deformation; θ is the internal rotation spring angle; P is the vertical pressure of the support top plate; Q is the horizontal spring restoring force; M s is the internal rotation spring bending moment; k r k is the spring stiffness of the top plate; z is the vertical spring stiffness.

[0067] From the above, since the existing "double spring" model does not have the degree of freedom of the top and bottom plate rotation angle, it only has an internal rotation spring to simulate the rotation of the rubber sheet inside the support, and therefore cannot analyze the influence of the rotation of the top and bottom plates of the support on the mechanical behavior of the support; the present invention adds a "top plate rotation spring" to the top plate of the original "double spring" model, increases the degree of freedom of "top plate rotation", and re-establishes the force balance equation and deformation coordination equation of the model, and re-establishes the calculation method of the model, so that the mechanical behavior of the support under the condition of rotation of the top and bottom plates of the support can be simulated. By comparing the calculation results of the mechanical model proposed in the present invention with the existing experimental results, the results show that the model proposed in the present invention can effectively simulate the influence of the rotation of the top and bottom plates of the support on the mechanical behavior of the support, verifying the effectiveness of the mechanical model proposed in the present invention.

[0068] Compared with the prior art, the present invention has at least the following beneficial effects: the mechanical model proposed in the present invention can effectively simulate the mechanical behavior of the laminated rubber seismic isolation bearing when the top and bottom plates rotate, and is particularly suitable for simulating the mechanical behavior of the bearing when the bearing is placed at the top of the pier and the top and bottom plates rotate due to flexible boundary conditions in bridge engineering; the mechanical model can be used for the design and computational analysis of seismic isolation structures, providing a more accurate analysis method for the design and computational analysis of seismic isolation structures. BRIEF DESCRIPTION OF THE DRAWINGS

[0069] Figure 1 Schematic diagram of the mechanical model proposed in this invention considering the rotation of the top and bottom plates of the support, where (a) is the deformation diagram of the support; (b) is the initial state and deformation state diagram of the mechanical model;

[0070] Figure 2 This is a flow chart of the calculation method of the model of the present invention under the pseudo-static support test condition;

[0071] Figure 3 Comparison of the calculated and experimental results of the mechanical model proposed in this invention under vertical loads p = 6, 7, 8 MPa and top plate rotation angles α = 0, 0.03 rad; the blue solid line is the experimental result; the red dotted line is the calculated result using the model of this patent;

[0072] Figure 4 This is the existing support deformation mode and corresponding support mechanical model diagram without considering the rotation of the support top and bottom plates. DETAILED DESCRIPTION

[0073] The technical solutions in the embodiments of the present invention will be clearly and completely described below in conjunction with the drawings in the embodiments of the present invention; it is obvious that the described embodiments are only part of the embodiments of the present invention, rather than all the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without making creative work are within the scope of protection of the present invention.

[0074] like Figure 4 As shown in the existing "double spring" mechanical model (Koh and Kelly (1988)), it is composed of an internal rotation spring, a horizontal spring, a vertical spring and a hinge axis. Under the combined action of the horizontal force F and the vertical force P, the support undergoes a horizontal displacement u x and vertical displacement u z , the internal rotation spring rotates θ, the vertical spring deforms v, and the horizontal spring deforms s; Figure 4 It can be seen that when the original "double-spring" mechanical model deforms, the top and bottom plates always remain horizontal and do not rotate. Therefore, the influence of the rotation of the top and bottom plates of the support on its mechanical properties cannot be considered.

[0075] The force balance equation and deformation coordination equation of the existing "double spring" mechanical model are as follows:

[0076] M s =P(s+h b θ)+Fh b +M (1)

[0077] Q=Pθ+F (2)

[0078] P=k z v (3)

[0079] u x =h b θ+s (4)

[0080]

[0081] Where, P is the vertical pressure of the support top plate; F is the horizontal force of the support; M s is the internal rotation spring bending moment; Q is the horizontal spring restoring force; u x is the horizontal deformation of the support; u z is the vertical deformation of the support; s is the deformation of the horizontal spring, v is the deformation of the vertical spring; θ is the rotation angle of the internal rotation spring; k z is the vertical spring stiffness; h b The height of the support excluding the top and bottom plates (i.e. the sum of the thickness of all rubber layers and thin steel plates);

[0082] Example 1. Figure 1As shown, the present invention provides a mechanical model that considers the influence of the rotation of the top and bottom plates of the laminated rubber seismic isolation bearing. On the basis of the existing "double spring" mechanical model, a new "top plate rotation spring" is added to the top. Under the action of the top plate bending moment M, the spring will rotate at an angle β, which is used to consider the influence of the rotation of the bearing top plate.

[0083] Example 2. The present invention provides a method for calculating a mechanical model that takes into account the influence of the rotation of the top and bottom plates of a laminated rubber isolation bearing, comprising the following steps:

[0084] (1) The force balance equation and deformation coordination equation of the mechanical model proposed in the present invention are as follows:

[0085] M s =P(s+h b θ)+Fh b +M (6)

[0086] Q=Pθ+F (7)

[0087] M=k r β (8)

[0088] P=k z v (9)

[0089] u x =h b θ+s (10)

[0090]

[0091] α=β (12)

[0092] Where, P is the vertical pressure of the support top plate; F is the horizontal force of the support; M is the local bending moment of the support top plate; M s is the internal rotation spring bending moment; Q is the horizontal spring restoring force; u x is the horizontal deformation of the support; u z is the vertical deformation of the support; α is the rotation angle of the support top plate; s is the deformation of the horizontal shear spring; v is the vertical spring deformation; θ is the rotation angle of the internal rotation spring; β is the rotation angle of the top plate rotation spring; k z is the vertical spring stiffness; k r is the spring stiffness of the top plate, h b The height of the support excluding the top and bottom plates (i.e. the sum of the thickness of all rubber layers and thin steel plates);

[0093] Comparing formula (1-5) and formula (6-12), we can see that the difference between the new model proposed in this invention and the original "double spring" mechanical model is that the "top plate rotation spring" is added, thereby increasing the local degree of freedom β ( Figure 1In the model, the top plate rotation spring angle) and the global degree of freedom α (support top plate rotation angle) are added to the control equations. Formulas (8) and (12) are added.

[0094] (2) The constitutive model of the horizontal spring in this mechanical model is as follows:

[0095] Q=Q l +Q h (13)

[0096]

[0097]

[0098]

[0099]

[0100] Where Q is the horizontal spring restoring force; Q l and Q h are the linear part and hysteresis part of the horizontal restoring force of the support respectively; G is the shear modulus of the rubber material; A is the cross-sectional area of ​​the support; u x is the horizontal deformation of the support; T r is the total thickness of the bearing rubber layer; n is the empirical parameter of the hysteresis model; Y and Y0 are the yield strengths of the bearing in the deformed and undeformed states, respectively; P is the vertical pressure of the bearing top plate; λ is an empirical parameter; P cr and P cr0 are the vertical critical loads of the support under deformation and non-deformation states respectively; A r is the overlapping area of ​​the top and bottom plates of the support under horizontal deformation; the dot on the symbol represents the "first-order derivative".

[0101] (3) In order to accurately simulate the nonlinear mechanical characteristics of the support, the constitutive model of the internal rotation spring in the mechanical model is as follows:

[0102]

[0103]

[0104]

[0105]

[0106] Where ω and r are empirical parameters; M s is the internal rotation spring bending moment; M y is the yield moment of the support; Z is the section modulus of the support (πD 3 / 32, D is the cross-sectional diameter of the support); A is the cross-sectional area of ​​the support; P is the vertical pressure of the support top plate; σy is the tensile yield strength of the rubber material; θ y k is the yield angle of the internal rotation spring; θ0 is the initial rotation stiffness of the internal rotation spring; T r is the total thickness of the bearing rubber layer; K is the bulk modulus of the rubber material; I is the section moment of inertia; I1 and I2 are the first-order and second-order modified first-kind Bessel functions, respectively (calculated using the fixed built-in function besseli in MATLAB); G is the shear modulus of the rubber material; S1 is the first shape coefficient of the bearing;

[0107] (4) The stiffness of the vertical spring in this mechanical model is defined as follows:

[0108]

[0109]

[0110] Where k z is the vertical spring stiffness; E c is the compression modulus of the rubber layer; A is the cross-sectional area of ​​the support; T r is the total thickness of the bearing rubber layer; G is the shear modulus of the rubber material; S1 is the first shape coefficient of the bearing.

[0111] (5) In practical applications, it is necessary to use the mechanical model of the present invention to perform calculations and analyze the mechanical properties of the bearing, specifically involving two working conditions: 1) quasi-static bearing test conditions; 2) seismic loading conditions;

[0112] 1) Under the pseudo-static bearing test condition, the vertical pressure P of the bearing top plate, the bearing top plate rotation angle α and the bearing horizontal deformation u x , controlled by the hydraulic system, the model control equation (6-12) can be rewritten into the following root-finding form:

[0113]

[0114] Where, P is the vertical pressure of the support top plate; F is the horizontal force of the support; M is the local bending moment of the support top plate; M s is the internal rotation spring bending moment; Q is the horizontal spring restoring force; u x is the horizontal deformation of the support; u z is the vertical deformation of the support; α is the rotation angle of the support top plate; s is the deformation of the horizontal spring; v is the deformation of the vertical spring; θ is the rotation angle of the internal rotation spring; β is the rotation angle of the top plate rotation spring; k z is the vertical spring stiffness; k r is the spring stiffness of the top plate; h b The height of the support excluding the top and bottom plates (i.e. the sum of the thickness of all rubber layers and thin steel plates);

[0115] Where x = [F, M, u z ,s,θ,v,β] T is an unknown vector. At each global calculation step, the Newton iteration method can be used to calculate the converged x value at this time. The expression is as follows:

[0116] x (k) =x (k-1) -[J(x (k-1) )] -1 g(x (k-1) ) (25)

[0117] Where k = 1, 2, 3, ..., represents the number of iterations; the initial value of the iteration x (0) The convergence value of the previous global iteration step can be used. is the Jacobian matrix, which is expressed as follows:

[0118]

[0119] Where h b is the height of the support excluding the top and bottom plates (i.e., the sum of the thickness of all rubber layers and thin steel plates); P is the vertical pressure of the support top plate; Q is the horizontal spring restoring force; M s is the internal rotation spring bending moment; s is the deformation of the horizontal spring; θ is the internal rotation spring angle; k r k is the spring stiffness of the top plate; z is the vertical spring stiffness; symbol represents partial differential;

[0120] Under the pseudo-static support test condition, the calculation method (programming calculation) of the model proposed by the present invention is as follows: Figure 2 shown.

[0121] 2) For seismic loading conditions, the Newmark-β time-stepping method is used for calculation. The Newmark-β method is a general calculation method for structural dynamics.

[0122] At each global calculation step, the force increment vector [dF, dP, dM] T , and displacement increment vector [du x ,du z ,dα] T The relationship between , that is, the tangent stiffness matrix, can be derived by following the steps below:

[0123] First, formula (6-9) can be rewritten into the following differential form:

[0124]

[0125]

[0126] dM=k r dβ (29)

[0127] dP=k z dv (30)

[0128] Where h b is the height of the support excluding the top and bottom plates (i.e., the sum of the thickness of all rubber layers and thin steel plates); F is the horizontal force increment of the support; P is the vertical pressure of the support top plate; M is the local bending moment increment of the support top plate; Q is the horizontal spring restoring force; M s is the internal rotation spring bending moment; s is the deformation of the horizontal spring; θ is the internal rotation spring angle; v is the vertical spring deformation; β is the top plate rotation spring angle; k r k is the spring stiffness of the top plate; z is the vertical spring stiffness; the symbol d represents the increment, and the symbol represents partial differential;

[0129] Rewrite formula (27-30) into the following matrix form:

[0130]

[0131] Where h b is the height of the support excluding the top and bottom plates (i.e., the sum of the thickness of all rubber layers and thin steel plates); F is the horizontal force of the support; P is the vertical pressure of the support top plate; M is the local bending moment of the support top plate; Q is the horizontal spring restoring force; M s is the internal rotation spring bending moment; s is the deformation of the horizontal spring; θ is the internal rotation spring angle; v is the vertical spring deformation; β is the top plate rotation spring angle; k r k is the spring stiffness of the top plate; z is the vertical spring stiffness; the symbol d represents the increment, and the symbol represents partial differential;

[0132] Secondly, formula (10-12) can be rewritten into the following differential form:

[0133] du x =ds+h b dθ (32)

[0134] du z =θds+(h b θ+s)dθ+dv (33)

[0135] dα=dβ (34)

[0136] Where h bis the height of the support excluding the top and bottom plates (i.e., the sum of the thickness of all rubber layers and thin steel plates); s is the horizontal spring deformation; θ is the internal rotation spring angle; v is the vertical spring deformation; α is the support top plate angle; β is the top plate rotation spring angle; u x is the horizontal deformation of the support; u z is the vertical deformation of the support; the symbol d represents the increment;

[0137] Rewrite formula (32-34) into the following matrix form:

[0138]

[0139] Where h b is the height of the support excluding the top and bottom plates (i.e., the sum of the thickness of all rubber layers and thin steel plates); s is the horizontal spring deformation; θ is the internal rotation spring angle; v is the vertical spring deformation; α is the support top plate angle; β is the top plate rotation spring angle; u x is the horizontal deformation of the support; u z is the vertical deformation of the support; the symbol d represents the increment;

[0140] Rewrite equations (31) and (35) into the following matrix form:

[0141]

[0142]

[0143] Where h b is the height of the support excluding the top and bottom plates (i.e., the sum of the thickness of all rubber layers and thin steel plates); s is the horizontal spring deformation; θ is the internal rotation spring angle; v is the vertical spring deformation; α is the support top plate angle; u x is the horizontal deformation of the support; u z is the vertical deformation of the support; P is the vertical pressure of the support top plate; M is the local bending moment of the support top plate; Q is the horizontal spring restoring force; F is the horizontal force of the support; k r k is the spring stiffness of the top plate; z is the vertical spring stiffness; M s is the internal rotation spring bending moment; the symbol d represents the increment; the symbol represents partial differential;

[0144] The tangent stiffness matrix Kt can be obtained from formula (36) as follows:

[0145]

[0146] Where h bis the height of the support excluding the top and bottom plates (i.e., the sum of the thickness of all rubber layers and thin steel plates); s is the horizontal spring deformation; θ is the internal rotation spring angle; P is the vertical pressure of the support top plate; Q is the horizontal spring restoring force; M s is the internal rotation spring bending moment; k r k is the spring stiffness of the top plate; z is the vertical spring stiffness; the symbol d represents the increment; the symbol represents partial differential.

[0147] Application examples:

[0148] The mechanical model proposed by the present invention (see Appendix Figure 1 The model parameters are shown in Table 1. Matlab numerical calculation software is used for programming calculation (code is attached). The vertical load p of the support is set as 6, 7, and 8 MPa respectively, and the top plate rotation angle α of the support is set as 0 and 0.03 rad respectively. The model calculation results are shown in the attached Figure 3 As shown;

[0149] Table 1 Model parameter values

[0150]

[0151]

[0152] From the attached Figure 3 It can be seen that the hysteresis curves obtained from the computational simulation using the model of the present invention are highly consistent with the experimental results. This mechanical model has a high degree of accuracy in simulating the horizontal mechanical properties of the bearing, thus demonstrating the effectiveness of the mechanical model of the present invention. Because the mechanical simulation of the present invention can accurately simulate experimental phenomena, it is expected to provide a more accurate analysis method for the design and computational analysis of seismic isolation structures.

[0153] It will be easily understood by those skilled in the art that the above description is only a preferred embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions and improvements made within the spirit and principles of the present invention should be included in the scope of protection of the present invention.

Claims

1. A calculation method for a mechanical model that considers the influence of the rotation of the top and bottom plates of a laminated rubber seismic isolation bearing, wherein the mechanical model that considers the influence of the rotation of the top and bottom plates of a laminated rubber seismic isolation bearing is adopted, characterized in that: The mechanical model includes adding a "top plate rotation spring" on top of the "double spring" mechanical model. Under the action of the top plate bending moment M, the top plate rotation spring will generate an angle β to consider the influence of the support top plate rotation. The calculation method includes the following steps: (1) The force balance equation and deformation coordination equation of the mechanical model are as follows: , , , , Where, P is the vertical pressure of the support top plate; F is the horizontal force of the support; M is the local bending moment of the support top plate; M s is the internal rotation spring bending moment; Q is the horizontal spring restoring force; u x is the horizontal deformation of the support; u z is the vertical deformation of the support; α is the rotation angle of the support top plate; s is the deformation of the horizontal shear spring; v is the vertical spring deformation; θ is the rotation angle of the internal rotation spring; β is the rotation angle of the top plate rotation spring; k z is the vertical spring stiffness; k r is the spring stiffness of the top plate, h b It is the height of the support excluding the top and bottom plates; (2) The constitutive model of the horizontal spring in the mechanical model is as follows: , , , , , Where Q is the horizontal spring restoring force; Q l and Q h are the linear part and hysteresis part of the horizontal restoring force of the support respectively; G is the shear modulus of the rubber material; A is the cross-sectional area of ​​the support; u x is the horizontal deformation of the support; T r is the total thickness of the bearing rubber layer; n is the empirical parameter of the hysteresis model; Y and Y0 are the yield strengths of the bearing in the deformed and undeformed states, respectively; P is the vertical pressure of the bearing top plate; λ is an empirical parameter; P cr and P cr0 are the vertical critical loads of the support under deformation and non-deformation states respectively; A r is the overlapping area of ​​the support top and bottom plates under horizontal deformation; (3) The constitutive model of the internal rotation spring in the mechanical model is as follows: , , , , Where ω and r are empirical parameters; M s is the internal rotation spring bending moment; M y is the yield moment of the support; Z is the section modulus of the support; A is the cross-sectional area of ​​the support; P is the vertical pressure on the top plate of the support; σ y is the tensile yield strength of the rubber material; θ y k is the yield angle of the internal rotation spring; θ0 is the initial rotation stiffness of the internal rotation spring; T r is the total thickness of the bearing rubber layer; K is the bulk modulus of the rubber material; I is the section moment of inertia; I1 and I2 are the first-order and second-order modified Bessel functions of the first kind, respectively; G is the shear modulus of the rubber material; S1 is the first shape coefficient of the bearing; (4) The stiffness of the vertical spring in the mechanical model is defined as follows: , , Where k z is the vertical spring stiffness; E c is the compression modulus of the rubber layer; A is the cross-sectional area of ​​the support; T r is the total thickness of the bearing rubber layer; G is the shear modulus of the rubber material; S1 is the first shape coefficient of the bearing.

2. The calculation method of the mechanical model considering the rotation effect of the top and bottom plates of the laminated rubber isolation bearing according to claim 1 is characterized in that: The mechanical properties of the bearing include two working conditions: 1) pseudo-static bearing test condition; 2) Earthquake loading conditions; 1) Under the pseudo-static bearing test condition, the vertical pressure P of the bearing top plate, the bearing top plate rotation angle α and the horizontal deformation u of the bearing x , controlled by the hydraulic system, the model control equation (6-12) is rewritten into the following root-finding form: , Where, P is the vertical pressure of the support top plate; F is the horizontal force of the support; M is the local bending moment of the support top plate; M s is the internal rotation spring bending moment; Q is the horizontal spring restoring force; u x is the horizontal deformation of the support; u z is the vertical deformation of the support; α is the rotation angle of the support top plate; s is the deformation of the horizontal spring; v is the deformation of the vertical spring; θ is the rotation angle of the internal rotation spring; β is the rotation angle of the top plate rotation spring; k z is the vertical spring stiffness; k r is the spring stiffness of the top plate; h b It is the height of the support excluding the top and bottom plates; Where x = [ F, M, u z , s, θ, v, β ] T is the unknown vector. At each global calculation step, the Newton iteration method is used to calculate the converged x value at this time. The expression is as follows: , Where k = 1,2,3,…, represents the number of iterations; the initial value of the iteration x (0) Using the convergence value of the last global iteration step, J(x) (J ij = ∂g i / ∂x j ) is the Jacobian matrix, which is expressed as follows: , Where h b is the height of the support excluding the top and bottom plates; P is the vertical pressure of the support top plate; Q is the horizontal spring restoring force; M s is the internal rotation spring bending moment; s is the deformation of the horizontal spring; θ is the internal rotation spring angle; k r k is the spring stiffness of the top plate; z is the vertical spring stiffness; 2) For earthquake loading conditions, the Newmark-β time-stepping method is used for calculation: At each global calculation step, the force increment vector [dF, dP, dM] T and displacement increment vector [ du x , du z , dα] T The relationship between , that is, the tangent stiffness matrix, is derived by following the steps below: First, rewrite formula (6-9) into the following differential form: , , , Where h b is the height of the support excluding the top and bottom plates; F is the horizontal force increment of the support; P is the vertical pressure of the support top plate; M is the local bending moment increment of the support top plate; Q is the horizontal spring restoring force; M s is the internal rotation spring bending moment; s is the deformation of the horizontal spring; θ is the internal rotation spring angle; v is the vertical spring deformation; β is the top plate rotation spring angle; k r k is the spring stiffness of the top plate; z is the vertical spring stiffness; Rewrite formula (27-30) into the following matrix form: , Where h b is the height of the support excluding the top and bottom plates (i.e., the sum of the thickness of all rubber layers and thin steel plates); F is the horizontal force of the support; P is the vertical pressure of the support top plate; M is the local bending moment of the support top plate; Q is the horizontal spring restoring force; M s is the internal rotation spring bending moment; s is the deformation of the horizontal spring; θ is the internal rotation spring angle; v is the vertical spring deformation; β is the top plate rotation spring angle; k r k is the spring stiffness of the top plate; z is the vertical spring stiffness; Next, formula (10-12) is rewritten into the following differential form: , Where h b is the height of the support excluding the top and bottom plates; s is the horizontal spring deformation; θ is the internal rotation spring angle; v is the vertical spring deformation; α is the support top plate angle; β is the top plate rotation spring angle; u x is the horizontal deformation of the support; u z is the vertical deformation of the support; Rewrite formula (32-34) into the following matrix form: , Where h b is the height of the support excluding the top and bottom plates; s is the horizontal spring deformation; θ is the internal rotation spring angle; v is the vertical spring deformation; α is the support top plate angle; β is the top plate rotation spring angle; u x is the horizontal deformation of the support; u z is the vertical deformation of the support; Rewrite formulas (31) and (35) into the following matrix form: , Where h b is the height of the support excluding the top and bottom plates; s is the horizontal spring deformation; θ is the internal rotation spring angle; v is the vertical spring deformation; α is the support top plate angle; u x is the horizontal deformation of the support; u z is the vertical deformation of the support; P is the vertical pressure of the support top plate; M is the local bending moment of the support top plate; Q is the horizontal spring restoring force; F is the horizontal force of the support; k r k is the spring stiffness of the top plate; z is the vertical spring stiffness; M s is the internal rotation spring bending moment; The tangent stiffness matrix Kt is obtained from formula (36) as follows: , Where h b is the height of the support excluding the top and bottom plates; s is the horizontal spring deformation; θ is the internal rotation spring angle; P is the vertical pressure of the support top plate; Q is the horizontal spring restoring force; M s is the internal rotation spring bending moment; k r k is the spring stiffness of the top plate; z is the vertical spring stiffness.

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