Vibration analysis method for building structure with eccentric rigidity and mass
Through the Lagrangian equation and vibration mode decomposition reaction spectrum method, the accuracy of vibration analysis of rigid mass eccentric building structures is solved, providing a theoretical basis for design, and improving the safety of the building and material use efficiency.
Patent Information
- Application Number
- CN202411331315.4
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- Filing Date
- 2024-09-24
- Publication Date
- 2024-12-27
AI Technical Summary
The prior art lacks accurate vibration analysis methods for eccentric building structures with stiffness mass, resulting in a significant increase in building vibration response under wind and seismic loads, and the load of structural components may exceed the design value.
The Lagrangian equation is used to derive the equation of motion, and the linear combination of the non-eccentric structure vibration mode of the stiffness mass is used to reflect the vibration mode, and the three-dimensional vibration response is calculated by the vibration mode decomposition reaction spectrum method to accurately calculate the displacement and acceleration response of the building structure.
It provides an accurate vibration analysis method for eccentric building structures with rigidity quality, ensures the theoretical basis of the design, improves the reliability and safety of the design, reduces the cost of post-reinforcement or transformation, and reasonably controls the use of building materials.
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Figure CN119203564A_ABST
Abstract
Description
Technical Field
[0001] The present application relates to the field of force analysis, and in particular to a vibration analysis method for a building structure with eccentric stiffness and mass. Background Art
[0002] Steel structure houses have the advantages of light weight, high strength, short construction period, good wind and earthquake resistance, and recyclability. Promoting the development of steel structure buildings meets the requirements of intelligent construction and building industrialization. The steel structure design standard requires: "The building plan layout should be simple and regular, the structural plan layout should be symmetrical, the line of action of the combined force of the horizontal load should be close to the stiffness center of the lateral force resisting structure; the dynamic characteristics of the two main axis directions of the high-rise steel structure should be similar."
[0003] However, in actual steel structure construction projects, due to architectural modeling needs and lighting and ventilation requirements, the mass and stiffness centers of buildings often do not coincide with the geometric centers. Under the dynamic effects of wind loads, earthquake loads, etc., the vibration response of such steel structure buildings with eccentric stiffness and mass will increase significantly. In extreme cases, the eccentricity of mass and stiffness may cause the loads borne by some structural components to exceed the design value. At present, there is no accurate vibration analysis method for building structures with eccentric stiffness and mass to guide actual construction. Summary of the invention
[0004] The purpose of this application is to provide a vibration analysis method for a building structure with eccentric stiffness and mass, which can accurately calculate the vibration of the building structure with eccentric stiffness and mass, thereby providing a theoretical basis for the design of the building structure with eccentric stiffness and mass.
[0005] The embodiment of the present application is implemented as follows:
[0006] The present application provides a method for analyzing vibration of a building structure with eccentric stiffness and mass, comprising the following steps:
[0007] The equations of motion of the building structure with eccentric stiffness mass are derived using Lagrange equations;
[0008] The linear combination of vibration modes of the building structure with non-eccentric stiffness and mass is used to reflect the vibration modes of the building structure with eccentric stiffness and mass.
[0009] The three-dimensional vibration response of building structures with eccentric stiffness and mass is calculated using the mode decomposition response spectrum method.
[0010] In some optional embodiments, the motion equation of the building structure with eccentric stiffness mass is derived using the Lagrange equation, including the following steps:
[0011] Assume that the building structure with eccentric stiffness and mass has n floors, and the width is B, the length is D, and the height is H. The stiffness center of each floor is on the same elastic axis, and the mass center is on the same inertia axis. The coordinates of the mass center of the i-th floor are (X m ,Y m ,Z i ), the mass of the i-th layer is m i , the radius of gyration relative to the center of mass is r i , the coordinates of the stiffness center of the i-th layer are (X r ,Y r ,Z i ), the lateral stiffness of the building structure relative to the stiffness center in the x direction is k xi The lateral stiffness of the building structure relative to the stiffness center in the y direction is k yi , the torsional stiffness of the building structure is k zi ;
[0012] Define the counterclockwise direction as the positive direction of torsion. Under the action of dynamic load, the displacement of the geometric center of the building structure with eccentric stiffness mass at time t in the x direction is u(z,t), the displacement in the y direction is v(z,t), the torsion angle in the z direction is θ(z,t), and the displacement of the geometric center of the i-th floor is recorded as (u i ,v i ,θ i ), the displacement of the mass center is recorded as (u mi ,v mi ,θ mi ), the displacement of the stiffness center is recorded as (u mi ,v mi ,θ mi ), the displacement of the geometric center, the displacement of the mass center, and the displacement of the stiffness center of the i-th layer have the following relationship:
[0013] u mi =u i -Y m θ i ,v mi =v i +X m θ i ,θ mi =θ i i=1,2,…n;
[0014] u ri =u i -Y r θ i ,v ri =v i +X r θ i ,θ ri =θ i i=1,2,…n;
[0015] Then the kinetic energy T of the building structure with eccentric stiffness mass is:
[0016]
[0017] The kinetic energy of a building structure with eccentric stiffness and mass can be rewritten as an equation with the displacement of the geometric center as a variable:
[0018]
[0019] Where: r i is the radius of gyration of the i-th floor structure of the building structure with eccentric stiffness and mass relative to the mass center, is the radius of gyration relative to the geometric center;
[0020] Then the strain energy U of the building structure with eccentric stiffness mass is:
[0021]
[0022] Where: k xi , k yi and k zi are the stiffness of the i-th layer structure relative to the stiffness center in the x, y and z directions respectively. The strain energy of the building structure with eccentric stiffness mass is rewritten as an equation with the geometric center displacement as a variable:
[0023]
[0024] The dynamic load acting on the i-th floor of the building structure with eccentric stiffness and mass is F xi , F yi and F zi , then the work W done by the external force is:
[0025]
[0026] The calculation formula of the total potential energy V of the building structure with eccentric stiffness mass is: V = UW; the calculation formula of the Lagrangian quantity L of the building structure with eccentric stiffness mass is: L = TV; in rectangular coordinates, the Lagrangian equation is:
[0027]
[0028] According to the above formula, the motion equation of the i-th layer of the building structure with eccentric stiffness and mass is calculated:
[0029]
[0030] Where: It is the torsional stiffness of the i-th floor structure of the building structure with eccentric stiffness mass relative to the geometric center.
[0031] In some optional embodiments, using a linear combination of vibration modes of a building structure with non-eccentric stiffness and mass to reflect the vibration mode of a building structure with eccentric stiffness and mass includes the following steps:
[0032] The motion equations of the i-th layer of the building structure with eccentric stiffness and mass are summarized and rewritten into matrix form:
[0033]
[0034] In the formula, the mass sub-matrix and stiffness sub-matrix are both of n×n order, and the displacement sub-vector and load sub-vector are both of n×1 order. The specific expressions are:
[0035]
[0036] M xx =M yy =M,M xz =M zx =-Y m M,M yz =M zy =X m M;
[0037]
[0038] K xz =K zx =-Y r K xx ,K yz =K zy =X r K yy ;
[0039]
[0040] If the non-diagonal elements of the mass matrix and the stiffness matrix are not zero, then the i-th order vibration mode of the building structure with eccentric stiffness and mass is as follows:
[0041]
[0042] Where: Ψ represents the vibration mode of the building structure with eccentric stiffness and mass; Represents the vibration mode of the building structure with non-eccentric stiffness and mass. The superscript (i) represents the i-th order vibration mode, and the subscripts x, y and z represent the components of the vibration mode in different directions. The i-th order vibration mode refers to the vibration mode that includes the three directions of x, y and z.
[0043] The displacement of the vibration mode of the building structure with eccentric stiffness and mass can be expressed as the generalized coordinate η i (t) Combination of vector and vibration mode:
[0044]
[0045] Where: η i (t) is a 3×1 subvector. Substitute the above formula into the matrix form of the motion equation of the i-th floor structure of the building structure with eccentric stiffness and mass and multiply it on the left. get:
[0046]
[0047] In the formula, the mass sub-matrix and stiffness sub-matrix are both of 3×3 order, the displacement sub-vector and load sub-vector are both of 3×1 order, and when the non-diagonal elements are ignored, the i-th order equation of the building structure with eccentric stiffness and mass can be written as follows:
[0048]
[0049] In the formula, the eigenvalue is the approximate solution of the natural frequency of the i-th modal of the building structure with eccentric stiffness and mass, and the eigenvector is the mode combination coefficient of the building structure with eccentric stiffness and mass. The specific expressions of the elements in the formula are as follows:
[0050]
[0051] In some optional implementation schemes, when using the mode decomposition response spectrum method to calculate the three-dimensional vibration response of a building structure with an eccentric stiffness and mass, the mode decomposition response spectrum method is first used to calculate the force integral of the building structure with an eccentric stiffness and mass, and then the mutual power spectrum of the forces of the building structure with an eccentric stiffness and mass is calculated based on the force integral of the building structure with an eccentric stiffness and mass, and finally the displacement of the coordinates of the building structure with an eccentric stiffness and mass in different directions is calculated based on the mutual power spectrum of the forces of the building structure with an eccentric stiffness and mass.
[0052] In some optional implementation schemes, the force integral of the building structure with eccentric stiffness and mass is calculated using the mode decomposition response spectrum method as follows:
[0053]
[0054] Where: is the i-th order force integral of the building structure with non-eccentric stiffness mass in the s direction; F is the amplitude of the i-th order s-direction vibration mode of the building structure with non-eccentric stiffness mass at the z height; s (z,t) is the dynamic force in the s direction at the z height of the building structure with non-eccentric stiffness mass.
[0055] In some optional implementation schemes, when calculating the mutual power spectrum of the force of the building structure with eccentric stiffness and mass according to the force integral of the building structure with eccentric stiffness and mass, the mutual power spectrum of the i-order force in different directions of the building structure with eccentric stiffness and mass is obtained according to the force integral of the building structure with eccentric stiffness and mass as follows:
[0056]
[0057] Then the mutual power spectrum of the i-th order generalized force of the building structure with eccentric stiffness and mass is:
[0058]
[0059] In the formula, It is the amplitude of the mth vibration mode of the i-order motion equation of the building structure with eccentric stiffness and mass in the s direction, and it is also the combination coefficient of the mth component of the i-order vibration mode of the building structure with eccentric stiffness and mass in the s direction.
[0060] In some optional implementation schemes, when calculating the displacement of coordinates in different directions on the building structure with eccentric stiffness and mass according to the mutual power spectrum of the force of the building structure with eccentric stiffness and mass:
[0061] According to the random vibration theory of mode decomposition and the mutual power spectrum of the i-th order generalized force of the building structure with eccentric stiffness and mass, the i-th order response spectrum of the building structure with eccentric stiffness and mass in different directions is obtained:
[0062]
[0063] Where: H m (-if) and H n (if) is the mechanical admittance of the i-th order motion equation, expressed as follows:
[0064]
[0065] Where: is the mth-order generalized mass of the i-th-order motion equation of the building structure with eccentric stiffness mass, f m is the mth order natural frequency of the i-th order motion equation, ζ m is the mth-order modal damping ratio of the i-th-order motion equation; f m The eigenvalue of the i-th order equation of the building structure with eccentric stiffness and mass is obtained. The expression is as follows:
[0066]
[0067] Where: The mass in different directions of the i-th order motion equation of the building structure with eccentric stiffness mass;
[0068] According to the i-order response spectrum of the building structure with eccentric stiffness and mass in different directions, the i-order displacement and acceleration response spectrum of the building structure with eccentric stiffness and mass is obtained:
[0069]
[0070] The variance of the total displacement and acceleration response of a building structure with eccentric stiffness and mass is:
[0071]
[0072] Where: s or t is x, y and z, the diagonal terms represent the variance of the displacement or acceleration in that direction, and the non-diagonal terms represent the covariance of the displacement or acceleration in different directions;
[0073] Assume that the stiffness mass of the building structure is eccentric i The displacement response of the geometric center at the height is (u, v, θ), and the coordinates of the building structure with eccentric stiffness and mass (x i ,y i ,z i ) is located at (uy i θ,v+x i θ,θ), so the coordinates of the building structure with eccentric stiffness mass (x i ,y i ,z i ) in different directions of displacement and acceleration response variance is:
[0074]
[0075] The beneficial effects of the present application are as follows: the vibration analysis method for a building structure with eccentric stiffness and mass provided in the present embodiment comprises the following steps: using the Lagrange equation to derive the motion equation of the building structure with eccentric stiffness and mass; using the linear combination of the vibration modes of the building structure with non-eccentric stiffness and mass to reflect the vibration mode of the building structure with eccentric stiffness and mass; using the vibration mode decomposition response spectrum method to calculate the three-dimensional vibration response of the building structure with eccentric stiffness and mass. The vibration analysis method for a building structure with eccentric stiffness and mass provided in the present embodiment can accurately calculate the vibration condition of the building structure with eccentric stiffness and mass, thereby providing a theoretical basis for the design of the building structure with eccentric stiffness and mass. BRIEF DESCRIPTION OF THE DRAWINGS
[0076] In order to more clearly illustrate the technical solutions of the embodiments of the present application, the drawings required for use in the embodiments will be briefly introduced below. It should be understood that the following drawings only show certain embodiments of the present application and therefore should not be regarded as limiting the scope. For ordinary technicians in this field, other related drawings can be obtained based on these drawings without paying creative work.
[0077] Figure 1 A schematic flow chart of a method for analyzing vibration of a building structure with eccentric stiffness and mass provided in an embodiment of the present application;
[0078] Figure 2 A schematic diagram of a model of a building structure with eccentric stiffness and mass in a vibration analysis method for a building structure with eccentric stiffness and mass provided in an embodiment of the present application;
[0079] Figure 3 A diagram for defining the direction of stiffness and mass eccentricity in the stiffness and mass eccentricity building structure vibration analysis method provided in an embodiment of the present application;
[0080] Figure 4 The top floor acceleration response diagram when the building mass is eccentric provided in the embodiment of the present application;
[0081] Figure 5 This is a top floor acceleration response diagram when the building stiffness is eccentric provided in an embodiment of the present application. DETAILED DESCRIPTION
[0082] In order to make the purpose, technical solution and advantages of the embodiments of the present application clearer, the technical solution in the embodiments of the present application will be clearly and completely described below in conjunction with the drawings in the embodiments of the present application. Obviously, the described embodiments are part of the embodiments of the present application, rather than all the embodiments. The components of the embodiments of the present application described and shown in the drawings here can be arranged and designed in various different configurations.
[0083] Therefore, the following detailed description of the embodiments of the present application provided in the accompanying drawings is not intended to limit the scope of the present application for which protection is sought, but merely represents selected embodiments of the present application. Based on the embodiments in the present application, all other embodiments obtained by ordinary technicians in the field without creative work are within the scope of protection of the present application.
[0084] It should be noted that similar reference numerals and letters denote similar items in the following drawings, and therefore, once an item is defined in one drawing, further definition and explanation thereof is not required in subsequent drawings.
[0085] In the description of the present application, it should be noted that the terms "center", "upper", "lower", "left", "right", "vertical", "horizontal", "inner", "outer", etc. indicate positions or positional relationships based on the positions or positional relationships shown in the accompanying drawings, or the positions or positional relationships in which the product of the application is usually placed when in use. They are only for the convenience of describing the present application and simplifying the description, and do not indicate or imply that the device or element referred to must have a specific position, be constructed and operated in a specific position, and therefore cannot be understood as a limitation on the present application. In addition, the terms "first", "second", "third", etc. are only used to distinguish the description, and cannot be understood as indicating or implying relative importance.
[0086] In addition, the terms "horizontal", "vertical", "overhanging" and the like do not mean that the components are required to be absolutely horizontal or overhanging, but can be slightly tilted. For example, "horizontal" only means that its direction is more horizontal than "vertical", and does not mean that the structure must be completely horizontal, but can be slightly tilted.
[0087] In the description of this application, it should also be noted that, unless otherwise clearly specified and limited, the terms "set", "install", "connect", and "connect" should be understood in a broad sense, for example, it can be a fixed connection, a detachable connection, or an integral connection; it can be a mechanical connection or an electrical connection; it can be a direct connection, or it can be indirectly connected through an intermediate medium, or it can be the internal communication of two elements. For ordinary technicians in this field, the specific meanings of the above terms in this application can be understood according to specific circumstances.
[0088] In the present application, unless otherwise clearly specified and limited, a first feature being "above" or "below" a second feature may include that the first and second features are in direct contact, or may include that the first and second features are not in direct contact but are in contact through another feature between them. Moreover, a first feature being "above", "above" and "above" a second feature includes that the first feature is directly above and obliquely above the second feature, or simply indicates that the first feature is higher in level than the second feature. A first feature being "below", "below" and "below" a second feature includes that the first feature is directly below and obliquely below the second feature, or simply indicates that the first feature is lower in level than the second feature.
[0089] The features and performance of the vibration analysis method for building structures with eccentric stiffness and mass of the present application are further described in detail below in conjunction with the embodiments.
[0090] like Figure 1 and Figure 2 As shown, the embodiment of the present application provides a vibration analysis method for a building structure with eccentric stiffness and mass, comprising the following steps:
[0091] Step 1: Use Lagrange's equation to derive the motion equation of the building structure with eccentric stiffness and mass;
[0092] like Figure 2 As shown, suppose that the building structure with eccentric stiffness and mass has n floors, and the width is B, the length is D, and the height is H. The stiffness center of each floor is on the same elastic axis, and the mass center is on the same inertia axis. The coordinates of the mass center of the i-th floor are (X m ,Y m ,Z i ), the mass of the i-th layer is m i The radius of gyration relative to the center of mass is r i , the coordinates of the stiffness center of the i-th layer are (X r ,Y r ,Z i ), the lateral stiffness of the building structure relative to the stiffness center in the x direction is k xi The lateral stiffness of the building structure relative to the stiffness center in the y direction is k yi , the torsional stiffness of the building structure is k zi ;
[0093] Define the counterclockwise direction as the positive direction of torsion. Under the action of dynamic load, the displacement of the geometric center of the building structure with eccentric stiffness mass at time t in the x direction is u(z,t), the displacement in the y direction is v(z,t), the torsion angle in the z direction is θ(z,t), and the displacement of the geometric center of the i-th floor is recorded as (u i ,v i ,θ i ), the displacement of the mass center is recorded as (u mi ,v mi ,θ mi ), the displacement of the stiffness center is recorded as (u mi ,v mi ,θ mi ), the displacement of the geometric center, the displacement of the mass center, and the displacement of the stiffness center of the i-th layer have the following relationship:
[0094] u mi =u i -Y m θ i ,v mi =v i +X m θ i ,θ mi =θ i i=1,2,…n;
[0095] u ri =u i -Y r θ i ,v ri =vi +X r θ i ,θ ri =θ i i=1,2,…n;
[0096] Then the kinetic energy T of the building structure with eccentric stiffness mass is:
[0097]
[0098] The kinetic energy of a building structure with eccentric stiffness and mass can be rewritten as an equation with the displacement of the geometric center as a variable:
[0099]
[0100] Where: r i is the radius of gyration of the i-th structure relative to the mass center, is the radius of gyration relative to the geometric center;
[0101] Then the strain energy U of the building structure with eccentric stiffness mass is:
[0102]
[0103] Where: k xi , k yi and k zi are the stiffness of the i-th layer structure relative to the stiffness center in the x, y and z directions respectively. The strain energy of the building structure with eccentric stiffness mass is rewritten as an equation with the geometric center displacement as a variable:
[0104]
[0105] The dynamic load acting on the i-th layer structure is F xi , F yi and F zi , then the work W done by the external force is:
[0106]
[0107] The calculation formula of the total potential energy V of the building structure with eccentric stiffness mass is: V = UW; the calculation formula of the Lagrangian quantity L of the building structure with eccentric stiffness mass is: L = TV; in rectangular coordinates, the Lagrangian equation is:
[0108]
[0109] According to the above formula, the motion equation of the i-th layer structure is calculated:
[0110]
[0111]
[0112] Where: is the torsional stiffness of the i-th layer structure relative to the geometric center.
[0113] Step 2: Use the linear combination of vibration modes of the building structure with non-eccentric stiffness and mass to reflect the vibration mode of the building structure with eccentric stiffness and mass;
[0114] The motion equations of the i-th structure are summarized and rewritten in matrix form:
[0115]
[0116] In the formula, the mass sub-matrix and stiffness sub-matrix are both of n×n order, and the displacement sub-vector and load sub-vector are both of n×1 order. The specific expressions are:
[0117]
[0118] M xx =M yy =M,M xz =M zx =-Y m M,M yz =M zy =X m M;
[0119]
[0120]
[0121]
[0122]
[0123] If the non-diagonal elements of the mass matrix and the stiffness matrix are not zero, then the i-th order vibration mode of the building structure with eccentric stiffness and mass is as follows:
[0124]
[0125] Where: Ψ represents the vibration mode of the building structure with eccentric stiffness and mass; Represents the vibration mode of the building structure with non-eccentric stiffness and mass. The superscript (i) represents the i-th order vibration mode, and the subscripts x, y and z represent the components of the vibration mode in different directions. The i-th order vibration mode refers to the vibration mode that includes the three directions of x, y and z.
[0126] The displacement of the vibration mode of the building structure with eccentric stiffness and mass can be expressed as the generalized coordinate η i (t) Combination of vector and vibration mode:
[0127]
[0128] Where: η i (t) is a 3×1 subvector. Substitute the above formula into the matrix form of the motion equation of the i-th structure and multiply it on the left. get:
[0129]
[0130] In the formula, the mass sub-matrix and stiffness sub-matrix are both of 3×3 order, the displacement sub-vector and load sub-vector are both of 3×1 order, and when the non-diagonal elements are ignored, the i-th order equation of the building structure with eccentric stiffness and mass can be written as follows:
[0131]
[0132] In the formula, the eigenvalue is the approximate solution of the natural frequency of the i-th modal of the building structure with eccentric stiffness and mass, and the eigenvector is the mode combination coefficient of the building structure with eccentric stiffness and mass. The specific expressions of the elements in the formula are as follows:
[0133]
[0134] Step 3: Use the mode decomposition response spectrum method to calculate the three-dimensional vibration response of the building structure with eccentric stiffness and mass.
[0135] First, the modal decomposition response spectrum method is used to calculate the force integral of the building structure with eccentric stiffness and mass. Then, the mutual power spectrum of the force of the building structure with eccentric stiffness and mass is calculated based on the force integral of the building structure with eccentric stiffness and mass. Finally, the displacement of the coordinates of the building structure with eccentric stiffness and mass in different directions is calculated based on the mutual power spectrum of the force of the building structure with eccentric stiffness and mass. Specifically:
[0136] The force integral of the building structure with eccentric stiffness and mass calculated by the modal decomposition response spectrum method is as follows:
[0137]
[0138] Where: is the i-th order force integral of the building structure with non-eccentric stiffness mass in the s direction; F is the amplitude of the i-th order s-direction vibration mode of the building structure with non-eccentric stiffness mass at the z height; s (z,t) is the dynamic force in the s direction at the z height of the building structure with non-eccentric stiffness mass.
[0139] When calculating the mutual power spectrum of the force of the building structure with eccentric stiffness and mass according to the force integral of the building structure with eccentric stiffness and mass, the mutual power spectrum of the i-order force in different directions of the building structure with eccentric stiffness and mass is obtained according to the force integral of the building structure with eccentric stiffness and mass as follows:
[0140]
[0141] Then the mutual power spectrum of the i-th order generalized force of the building structure with eccentric stiffness and mass is:
[0142]
[0143] In the formula, It is the amplitude of the mth vibration mode of the i-order motion equation of the building structure with eccentric stiffness and mass in the s direction, and it is also the combination coefficient of the mth component of the i-order vibration mode of the building structure with eccentric stiffness and mass in the s direction.
[0144] When calculating the displacement of the coordinates of the building structure with eccentric stiffness and mass in different directions according to the mutual power spectrum of the force of the building structure with eccentric stiffness and mass, the i-order response spectrum of the displacement of the building structure with eccentric stiffness and mass in different directions is obtained according to the random vibration theory of mode decomposition and the mutual power spectrum of the i-th order generalized force of the building structure with eccentric stiffness and mass:
[0145]
[0146] Where: H m (-if) and H n (if) is the mechanical admittance of the i-th order motion equation, expressed as follows:
[0147]
[0148] Where: is the mth-order generalized mass of the i-th-order motion equation of the building structure with eccentric stiffness mass, f m is the mth order natural frequency of the i-th order motion equation, ζ m is the mth-order modal damping ratio of the i-th-order motion equation; f m The eigenvalue of the i-th order equation of the building structure with eccentric stiffness and mass is obtained. The expression is as follows:
[0149]
[0150] Where: The mass in different directions of the i-th order motion equation of the building structure with eccentric stiffness mass;
[0151] According to the i-order response spectrum of the building structure with eccentric stiffness and mass in different directions, the i-order displacement and acceleration response spectrum of the building structure with eccentric stiffness and mass is obtained:
[0152]
[0153] The variance of the total displacement and acceleration response of a building structure with eccentric stiffness and mass is:
[0154]
[0155] Where: s or t is x, y and z, the diagonal terms represent the variance of the displacement or acceleration in that direction, and the non-diagonal terms represent the covariance of the displacement or acceleration in different directions;
[0156] Assume that the stiffness mass of the building structure is eccentric i The displacement response of the geometric center at the height is (u, v, θ), and the coordinates of the building structure with eccentric stiffness and mass (x i ,y i ,z i ) is located at (uy i θ,v+x i θ,θ), so the coordinates of the building structure with eccentric stiffness mass (x i ,y i ,z i ) in different directions of displacement and acceleration response variance is:
[0157]
[0158] The vibration analysis method of the building structure with eccentric rigidity and mass provided in the embodiment of the present application is to derive the motion equation of the building structure with eccentric rigidity and mass by using the Lagrange equation, and then use the linear combination of the vibration mode of the building structure with non-eccentric rigidity and mass to reflect the vibration mode of the building structure with eccentric rigidity and mass, and then use the vibration mode decomposition response spectrum method to calculate the three-dimensional vibration response of the building structure with eccentric rigidity and mass, so as to accurately calculate the vibration of the building structure with eccentric rigidity and mass, thereby providing a theoretical basis for the design of the building structure with eccentric rigidity and mass. The present application improves the reliability and safety of the design of the building structure with common mass rigidity eccentricity, and ensures that the building can resist the influence of various dynamic loads (such as earthquakes, wind loads, etc.) in long-term use. Based on the calculation method of the present invention, the rigidity distribution and mass distribution of the structure are accurately predicted and adjusted in the early stage of structural design to achieve the optimal vibration performance. This pre-optimization design strategy can not only effectively reduce the later reinforcement or transformation costs caused by structural vibration problems, but also reasonably control the use of building materials while maintaining structural safety, and reduce construction costs.
[0159] Example 1
[0160] The plane size of a steel structure high-rise building is 12m×12m and the height is 100m. The basic wind pressure for 10 years is 0.3kN / m 2 , the average wind speed at 100m height is 32.73m / s. The damping ratio is 2%, and the gyration radius r is calculated according to r 2 =1 / 12(B 2 +D2 ) calculation. Assume that the unit volume mass of a steel structure high-rise building is 180kg / m 3 The first three modal coefficients of non-eccentric steel structure high-rise buildings are determined according to the following table.
[0161]
[0162] The first three natural frequencies of non-eccentric steel structure high-rise buildings are:
[0163] f x1 =f y1 =0.253,f z1 =0.347; f x2 =f y2 =0.764,f z2 =0.988; f x3 =f y3 =1.376,f z3 =1.702;
[0164] Figure 3 The definition of the stiffness or mass eccentricity direction of a steel structure high-rise building is given: the x-axis is in the same direction as the dynamic load, the y-axis is perpendicular to the dynamic load, the coordinate origin is at the geometric center, when the eccentricity direction is in the same direction as the dynamic load direction, it is called positive eccentricity, and when the eccentricity direction is opposite to the dynamic load direction, it is called negative downwind eccentricity. The structural vibration analysis method of the stiffness and mass eccentricity provided in the embodiment of the present application can calculate the vibration response results of the steel structure high-rise building under the action of wind load under the condition of stiffness and mass eccentricity, and draw the following diagram based on the vibration response results: Figure 4 The top floor acceleration response diagram when the building mass is eccentric is shown in Figure 5 The top floor acceleration response diagram when the building stiffness is eccentric, Figure 4 and Figure 5 Medium A is the top corner acceleration of the building, σ xx is the downwind acceleration of the geometric center of the top floor of the building, σ yy is the crosswind acceleration of the geometric center of the top floor of the building, σ zz is the torsional acceleration of the geometric center of the top floor of the building; X r represents the eccentricity of the stiffness in the x direction, D represents the depth of the building's side wind surface, and the ratio of the eccentricity to the building size in the corresponding direction is the eccentricity.
[0165] The embodiments described above are part of the embodiments of the present application, rather than all of the embodiments. The detailed description of the embodiments of the present application is not intended to limit the scope of the present application for protection, but merely represents the selected embodiments of the present application. Based on the embodiments in the present application, all other embodiments obtained by ordinary technicians in this field without creative work are within the scope of protection of the present application.
Claims
1. A vibration analysis method for building structures with eccentric stiffness and mass, characterized in that: The following steps are involved: The equations of motion of the building structure with eccentric stiffness mass are derived using Lagrange equations; The linear combination of vibration modes of the building structure with non-eccentric stiffness and mass is used to reflect the vibration modes of the building structure with eccentric stiffness and mass. The three-dimensional vibration response of building structures with eccentric stiffness and mass is calculated using the mode decomposition response spectrum method.
2. The method for analyzing vibration of building structures with eccentric stiffness and mass according to claim 1, characterized in that: The following steps are used to derive the equations of motion for a building structure with eccentric stiffness and mass using the Lagrange equations: Assume that the building structure with eccentric stiffness and mass has n floors, and the width is B, the length is D, and the height is H. The stiffness center of each floor is on the same elastic axis, and the mass center is on the same inertia axis. The coordinates of the mass center of the i-th floor are (X m ,Y m ,Z i ), the mass of the i-th layer is m i , the radius of gyration relative to the center of mass is r i , the coordinates of the stiffness center of the i-th layer are (X r ,Y r ,Z i ), the lateral stiffness of the building structure relative to the stiffness center in the x direction is k xi The lateral stiffness of the building structure relative to the stiffness center in the y direction is k yi , the torsional stiffness of the building structure is k zi ; Define the counterclockwise direction as the positive direction of torsion. Under the action of dynamic load, the displacement of the geometric center of the building structure with eccentric stiffness mass at time t in the x direction is u(z,t), the displacement in the y direction is v(z,t), the torsion angle in the z direction is θ(z,t), and the displacement of the geometric center of the i-th floor is recorded as (u i ,v i ,θ i ), the displacement of the mass center is recorded as (u mi ,v mi ,θ mi ), the displacement of the stiffness center is recorded as (u mi ,v mi ,θ mi ), the displacement of the geometric center, the displacement of the mass center, and the displacement of the stiffness center of the i-th layer have the following relationship: you mi =u i -Y m i i ,v mi =v i +X m i i ,i mi =θ i i=1,2,…n; you ri =u i -Y r i i ,v ri =v i +X r i i ,i ri =θ i i=1,2,…n; Then the kinetic energy T of the building structure with eccentric stiffness mass is: The kinetic energy of a building structure with eccentric stiffness and mass can be rewritten as an equation with the displacement of the geometric center as a variable: Where: r i is the radius of gyration of the i-th floor structure of the building structure with eccentric stiffness and mass relative to the mass center, is the radius of gyration relative to the geometric center; Then the strain energy U of the building structure with eccentric stiffness mass is: Where: k xi , k yi and k zi are the stiffness of the i-th floor structure of the building structure with eccentric stiffness and mass relative to the stiffness center in the x, y and z directions respectively. The strain energy of the building structure with eccentric stiffness and mass is rewritten as an equation with the geometric center displacement as a variable: The dynamic load acting on the i-th floor of the building structure with eccentric stiffness and mass is F xi , F yi and F zi , then the work W done by the external force is: The calculation formula of the total potential energy V of the building structure with eccentric stiffness mass is: V = UW; the calculation formula of the Lagrangian quantity L of the building structure with eccentric stiffness mass is: L = TV; in rectangular coordinates, the Lagrangian equation is: According to the above formula, the motion equation of the i-th layer of the building structure with eccentric stiffness and mass is calculated: Where: It is the torsional stiffness of the i-th floor structure of the building structure with eccentric stiffness mass relative to the geometric center.
3. The vibration analysis method for building structures with eccentric stiffness and mass according to claim 2 is characterized in that: Using the linear combination of the vibration modes of the building structure with non-eccentric stiffness and mass to reflect the vibration modes of the building structure with eccentric stiffness and mass includes the following steps: The motion equations of the i-th layer of the building structure with eccentric stiffness and mass are summarized and rewritten into matrix form: In the formula, the mass sub-matrix and stiffness sub-matrix are both of n×n order, and the displacement sub-vector and load sub-vector are both of n×1 order. The specific expressions are: M xx =M yy =M,M xz =M zx =-Y m M,M yz =M zy =X m M; K xz =K zx =-Y r K xx ,K yz =K zy =X r K yy ; If the non-diagonal elements of the mass matrix and the stiffness matrix are not zero, then the i-th order vibration mode of the building structure with eccentric stiffness and mass is as follows: Where: Ψ represents the vibration mode of the building structure with eccentric stiffness and mass; Represents the vibration mode of the building structure with non-eccentric stiffness and mass. The superscript (i) represents the i-th order vibration mode, and the subscripts x, y and z represent the components of the vibration mode in different directions. The i-th order vibration mode refers to the vibration mode that includes the three directions of x, y and z. The displacement of the vibration mode of the building structure with eccentric stiffness and mass can be expressed as the generalized coordinate η i (t) Combination of vector and vibration mode: Where: η i (t) is a 3×1 subvector. Substitute the above formula into the matrix form of the motion equation of the i-th floor structure of the building structure with eccentric stiffness and mass and multiply it on the left. get: In the formula, the mass sub-matrix and stiffness sub-matrix are both of 3×3 order, the displacement sub-vector and load sub-vector are both of 3×1 order, and when the non-diagonal elements are ignored, the i-th order equation of the building structure with eccentric stiffness and mass can be written as follows: In the formula, the eigenvalue is the approximate solution of the natural frequency of the i-th modal of the building structure with eccentric stiffness and mass, and the eigenvector is the mode combination coefficient of the building structure with eccentric stiffness and mass. The specific expressions of the elements in the formula are as follows:
4. The method for analyzing vibration of building structures with eccentric stiffness and mass according to claim 3 is characterized in that: When using the mode decomposition response spectrum method to calculate the three-dimensional vibration response of a building structure with eccentric stiffness and mass, the mode decomposition response spectrum method is first used to calculate the force integral of the building structure with eccentric stiffness and mass, and then the mutual power spectrum of the force of the building structure with eccentric stiffness and mass is calculated based on the force integral of the building structure with eccentric stiffness and mass. Finally, the displacement of the coordinates of the building structure with eccentric stiffness and mass in different directions is calculated based on the mutual power spectrum of the force of the building structure with eccentric stiffness and mass.
5. The method for analyzing vibration of building structures with eccentric stiffness and mass according to claim 4 is characterized in that: The force integral of the building structure with eccentric stiffness and mass calculated by the modal decomposition response spectrum method is as follows: Where: is the i-th order force integral of the building structure with non-eccentric stiffness mass in the s direction; F is the amplitude of the i-th order s-direction vibration mode of the building structure with non-eccentric stiffness mass at the z height; s (z,t) is the dynamic force in the s direction at the z height of the building structure with non-eccentric stiffness mass.
6. The method for analyzing vibration of building structures with eccentric stiffness and mass according to claim 5, characterized in that: When calculating the mutual power spectrum of the force of the building structure with eccentric stiffness and mass according to the force integral of the building structure with eccentric stiffness and mass, the mutual power spectrum of the i-order force in different directions of the building structure with eccentric stiffness and mass is obtained according to the force integral of the building structure with eccentric stiffness and mass as follows: Then the mutual power spectrum of the i-th order generalized force of the building structure with eccentric stiffness and mass is: In the formula, It is the amplitude of the mth vibration mode of the i-order motion equation of the building structure with eccentric stiffness and mass in the s direction, and it is also the combination coefficient of the mth component of the i-order vibration mode of the building structure with eccentric stiffness and mass in the s direction.
7. The method for analyzing vibration of building structures with eccentric stiffness and mass according to claim 6, characterized in that: When calculating the displacement of the coordinates in different directions of the building structure with eccentric stiffness and mass according to the mutual power spectrum of the force of the building structure with eccentric stiffness and mass: According to the random vibration theory of mode decomposition and the mutual power spectrum of the i-th order generalized force of the building structure with eccentric stiffness and mass, the i-th order response spectrum of the building structure with eccentric stiffness and mass in different directions is obtained: Where: H m (-if) and H n (if) is the mechanical admittance of the i-th order motion equation, expressed as follows: Where: is the mth-order generalized mass of the i-th-order motion equation of the building structure with eccentric stiffness mass, f m is the mth order natural frequency of the i-th order motion equation, ζ m is the mth-order modal damping ratio of the i-th-order motion equation; f m The eigenvalue of the i-th order equation of the building structure with eccentric stiffness and mass is obtained. The expression is as follows: Where: The mass in different directions of the i-th order motion equation of the building structure with eccentric stiffness mass; According to the i-order response spectrum of the building structure with eccentric stiffness and mass in different directions, the i-order displacement and acceleration response spectrum of the building structure with eccentric stiffness and mass is obtained: The variance of the total displacement and acceleration response of a building structure with eccentric stiffness and mass is: Where: s or t is x, y and z, the diagonal terms represent the variance of the displacement or acceleration in that direction, and the non-diagonal terms represent the covariance of the displacement or acceleration in different directions; Assume that the stiffness mass of the building structure is eccentric i The displacement response of the geometric center at the height is (u, v, θ), and the coordinates of the building structure with eccentric stiffness and mass (x i ,y i ,z i ) is located at (uy i θ,v+x i θ,θ), so the coordinates of the building structure with eccentric stiffness mass (x i ,y i ,z i ) in different directions of displacement and acceleration response variance is: