12_动力弹塑性分析(英文)

1 A n a l y s i s M a n u a l Inelastic Time History Analysis 2 1. Boundary Conditions 1.1 General Link Element kry krz joint i krx kdx kdy kdz x z y local coordinate axis cyiL cyjL L cziL czjL joint j Fig 1.1 Composition of General Link Element The General Link element is used to model dampers, base isolators, compression-only element, tension-only element, plastic hinges, soil springs, etc. The 6 springs individually represent 1 axial deformation spring, 2 shear deformation springs, 1 torsional deformation spring and 2 bending deformation springs as per Figure 1.1. Among the 6 springs, only selective springs may be partially used, and linear and nonlinear properties can be assigned. The general link can be thus used as linear and nonlinear elements. The General Link element can be largely classified into Element type and Force type depending on the method of applying it to analysis. The Element type general link element directly reflects the nonlinear behavior of the element by renewing the element stiffness matrix. The Force type on the other hand, does not renew the stiffness matrix, but rather reflects the nonlinearity indirectly by converting the member forces calculated based on the nonlinear properties into external forces. 3 First, the Element type general link element provides three types, Spring, Dashpot and Spring and Dashpot. The Spring retains linear elastic stiffness for each of 6 components, and the Dashpot retains linear viscous damping for each of 6 components. The Spring and Dashpot is a type, which combines Spring and Dashpot. All of the three types are analyzed as linear elements. However, the Spring type general link element can be assigned inelastic hinge properties and used as a nonlinear element. This can be mainly used to model plastic hinges, which exist in parts in a structure or nonlinearity of soils. However, this can be used as a nonlinear element only in the process of nonlinear time history analysis by direct integration. Also, viscous damping is reflected in linear and nonlinear time history analyses only if “Group Damping” is selected for damping for the structure. The Force type general link element can be used for dampers such as Viscoelastic Damper and Hysteretic System, seismic isolators such as Lead Rubber Bearing Isolator and Friction Pendulum System Isolator, Gap (compression-only element) and Hook (tension-only element). Each of the components retains effective stiffness and effective damping. You may specify nonlinear properties for selective components. The Force type general link element is applied in analysis as below. First, it is analyzed as a linear element based on the effective stiffness while ignoring the effective damping in static and response spectrum analyses. In linear time history analysis, it is analyzed as a linear element based on the effective stiffness, and the effective damping is considered only when the damping selection is set as “Group Damping”. In nonlinear time history analysis, the effective stiffness acts as virtual linear stiffness, and as indicated before, the stiffness matrix does not become renewed even if it has nonlinear properties. Also, because the nonlinear properties of the element are considered in analysis, the effective damping is not used. This is because the role of effective damping indirectly reflects energy dissipation due to the nonlinear behavior of the Force type general link element in linear analysis. The rules for applying the general link element noted above are summarized in Table 1.1. When the damping selection is set as “Group Damping”, the damping of the Element type general link element and the effective damping of the Force type general link element are reflected in analysis as below. First, when linear and nonlinear analyses are carried out based on modal superposition, they are reflected in the analyses through modal damping ratios based on strain energy. On the other hand, when linear and nonlinear analyses are carried out by direct integration, they are reflected through formulating the element damping matrix. If element stiffness or element-mass-proportional damping is specified for the general link element, the analysis is carried out by adding the damping or effective damping specified for the properties of the general link element. 4 Table 1.1 Rules for applying general link element (Damping and effective damping are considered only when the damping option is set to “Group Damping”.) General Link Element Element Type Force Type Properties Elastic Damping Effective Stiffness Effective Damping Static analysis Elastic X Elastic X Response spectrum analysis Elastic X Elastic X Modal Superposition Elastic Linear Elastic Linear Linear Time History Analysis Direct Integration Elastic Linear Elastic Linear Modal Superposition Elastic Linear Elastic (virtual) X Nonlinear Time History Analysis Direct Integration Elastic & Inelastic Linear Elastic (virtual) X The locations of the 2 shear springs may be separately specified on the member. The locations are defined in ratios by the distances from the first node relative to the total length of the member. If the locations of the shear springs are specified and shear forces are acting on the nonlinear link element, the bending moments at the ends of the member are different. The rotational deformations also vary depending on the locations of the shear springs. Conversely, if the locations of the shear springs are unspecified, the end bending moments always remain equal regardless of the presence of shear forces. The degrees of freedom for each element are composed of 3 translational displacement components and 3 rotational displacement components regardless of the element or global coordinate system. The element coordinate system follows the convention of the truss element. Internal forces produced for each node of the element consist of 1 axial force, 2 shear forces, 1 torsional moment and 2 bending moments. The sign convention is identical to that of the beam element. In calculating the nodal forces of the element, the nodal forces due to damping or effective damping of the general link element are found based on Table 1.1. However, the nodal forces due to the element mass or element-stiffness-proportional damping are ignored. 2. Consideration for Damping 2.1 Modal damping based on strain energy In real structures, damping properties are different for different materials, and sometimes damping devices are locally installed. The MIDAS programs thus enable us to specify different damping 5 properties by elements. However, the damping matrices of such structures generally are of non- classical damping, and their modes can not be decomposed. Accordingly, modal damping ratios are calculated on the basis of a concept of strain energy in order to reflect different damping properties by elements in modal superposition in a dynamic analysis. The damping ratio of a single degree of vibration system having viscous damping can be defined by the ratio of dissipated energy in a harmonic motion to the strain energy of the structure. 4 D S E E ξ π= where, ED : Dissipated energy ES : Strain energy In a structure with multi-degrees of freedom, the dynamic behavior of a particular mode can be investigated by the dynamic behavior of the single degree of freedom system having the corresponding natural frequency. For this, two assumptions are made to calculate the dissipated energy and strain energy pertaining to particular elements. First, the deformation of the structure is assumed to be proportional to the mode shapes. The element nodal displacement and velocity vectors of the structure in a harmonic motion based only on the i-th mode with the corresponding natural frequency can be written as: ( ) ( ) , , , , sin cos i n i n i i i n i i n i i t t ω θ ω ω θ = + = + u φ u φ� where, ,i nu : Nodal displacement of n-th element due to i-th mode of vibration ,i nu� : Nodal velocity of n-th element due to i-th mode of vibration ϕi,n : i-th Mode shape corresponding to n-th element’s degree of freedom ωi : Natural frequency of i-th mode θi : Phase angle of i-th mode Secondly, the element’s damping is assumed to be viscous damping, which is proportional to the element’s stiffness. 2 n n n i h ω=C K where, 6 Cn : Damping matrix of n-th element Kn : Stiffness matrix of n-th element hn : Damping ratio of n-th element The dissipated energy and strain energy can be expressed as below under the above assumptions. ( ) ( ) , , , , , , , , , 2 1 1, 2 2 T T D i n n i n n i n n i n T T S i n n i n i n n i n E i n h E i n π π= = = = u C u φ K φ u K u φ K φ � where, ED (i, n) : Dissipated energy of n-th element due to i-th mode of vibration ES (i, n) : Strain energy of n-th element due to i-th mode of vibration The damping ratio of i-th mode for the entire structure can be calculated by summing the energy for all the elements corresponding to the i-th mode. ( ) ( ) , , 1 1 , , 1 1 , 4 , N N T D n n i n n i n n i N N T S n i n n i n n E i n h E i n ξ π = = = = = = ⋅ ∑ ∑ ∑ ∑ φ K φ φ K φ 7 3. Boundary Nonlinear Time History Analysis 3.1 Overview Boundary nonlinear time history analysis, being one of nonlinear time history analyses, can be applied to a structure, which has limited nonlinearity. The nonlinearity of the structure is modeled through General Link of Force Type, and the remainder of the structure is modeled linear elastically. Out of convenience, the former is referred to as a nonlinear system, and the latter is referred to as a linear system. Boundary nonlinear time history analysis is analyzed by converting the member forces of the nonlinear system into loads acting in the linear system. Because a linear system is analyzed through modal superposition, this approach has an advantage of fast analysis speed compared to the method of direct integration, which solves equilibrium equations for the entire structure at every time step. The equation of motion for a structure, which contains General Link elements of Force Type, is as follows: ( ) ( )( ) ( ) ( ) ( ) ( ) - ( )S N P N L NMu t Cu t K K u t B p t B f t f t+ + + = +�� � where, M : Mass matrix C : Damping matrix SK : Elastic stiffness without General Link elements of Force Type NK : Effective stiffness of General Link elements of Force Type PB , NB : Transformation matrices ( )u t , ( )u t� , ( )u t�� : Nodal displacement, velocity & acceleration ( )p t : Dynamic load ( )Lf t : Internal forces due to the effective stiffness of nonlinear components contained in General Link elements of Force Type ( )Nf t : True internal forces of nonlinear components contained in General Link elements of Force Type The term fL (t) on the right hand side is cancelled by the nodal forces produced by KN on the left hand side, which correspond to the nonlinear components of General Link of Force Type. Only the true internal forces of the nonlinear components fN (t) will affect the dynamic behavior. The reason for using the effective stiffness matrix KN is that the stiffness matrix of KS alone can become unstable depending on the connection locations of the general link elements of the force type. Mode shapes and natural frequencies on the basis of mass and stiffness matrices can be calculated through Eigenvalue Analysis or Ritz Vector Analysis. The damping is considered by modal damping ratios. Using the orthogonality of the modes, the above equation is transformed into the equation of Modal 8 Coordinates as follows: 2 ( ) ( ) ( )( ) 2 ( ) ( ) T T T i P i N L i N N i i i i i T T T i i i i i i B p t B f t B f t q t q t q t M M M φ φ φξ ω ω φ φ φ φ φ φ+ = + ++�� � where, iφ : Mode shape vector of the i-th mode iξ : Damping ratio of the i-th mode iω : Natural frequency of the i-th mode ( )iq t , ( )iq t� , ( )iq t�� : Generalized displacement, velocity & acceleration at the i-th mode The fN (t) and fL (t) on the right hand side are determined by the true deformations and the rates of changes in deformations in the local coordinate systems of the corresponding general link elements of the force type. However, the true deformations of the elements contain the components of all the modes without being specific to any particular modes. The above modal coordinate kinetic equation thus cannot be said to be independent by individual modes. In order to fully take the advantage of modal analysis, we assume fN (t) and fL (t) at each analysis time step so that it becomes a kinetic equation in the independent modal coordinate system. First, using the analysis results of the immediately preceding step, the generalized modal displacement and velocity of the present step are assumed; based on these, fN (t) and fL (t) for the present step are calculated. Again from these, the generalized modal displacement and velocity of the present stage are calculated. And the deformations and the rates of changes in deformations of the general link elements of the force type are calculated through a combination process. The entire calculation process is repeated until the following convergence errors fall within the permitted tolerance. ( 1) ( ) ( 1) ( ) ( ) max ( ) j j i i q ji i q n∆t q n∆t q n∆t ε + + −=     ( 1) ( ) ( 1) ( ) ( ) max ( ) j j i i q ji i q n∆t q n∆t q n∆t ε + + −=    � � � � ( 1) ( ) , , ( 1) , ( ) ( ) max ( )M j j M i M i f ji M i f n∆t f n∆t f n∆t ε + + −=     where, ( ) ( ) , ( ∆ )( ∆ ) T i N T i i j j N M i B f n t M f n t φ φ φ= ∆t : Magnitude of time step n : Time step j : Repeated calculation step i : Mode number The above process is repeated for each analysis time step. The user directly specifies the maximum number of repetitions and the convergence tolerance in Time History Load Cases. If convergence is not reached, the program automatically subdivides the analysis time interval ∆t and begins reanalyzing. The nonlinear properties of the general link elements of the force type are expressed in terms of differential equations. Solutions to the numerical analysis of the differential equations are required to 9 calculate and correct the internal forces corresponding to nonlinear components in the process of each repetition. MIDAS programs use the Runge-Kutta Fehlberg numerical analysis method, which is widely used for that purpose and known to provide analysis speed and accuracy. 3.2 Cautions for Eigenvalue Analysis The boundary nonlinear time history analysis in MIDAS is based on modal analysis, and as such a sufficient number of modes are required to represent the structural response. A sufficient number of modes are especially required to represent the deformations of general link elements of the force type. A representative example may be the case of the seismic response analysis of a friction pendulum system isolator. In this type of isolator, the internal force of the element’s axial direction component is an important factor for determining the behavior of the shear direction components. Accordingly, unlike other typical seismic response analyses, the vertical modes play an important role. The number of modes must be sufficiently enough so that the sum of the modal masses in the vertical direction is close to the total mass. When the eigenvalue analysis is used to achieve such objective, a very large number of modes may be required. This may lead to a very long analysis time. If Ritz Vector Analysis is used, the mode shapes and natural frequencies can be found reflecting the distribution of dynamic loads with respect to each degree of freedom. This allows us to include the effects of higher modes with a relatively small number of modes. For example, in the case of a friction pendulum system isolator, we can select the ground acceleration in the Z-direction and static Load Case Names related to the self weight of the structure in the input dialog box of Ritz Vector analysis. Natural frequencies and mode shapes related mainly to the vertical movement can be obtained. In general, the Ritz Vector analysis is known to provide more accurate analysis results with a fewer number of modes compared to Eigenvalue analysis. 3.3 Combining Static and Dynamic Loads Unlike linear time history analysis, the principle of superposition cannot be applied to nonlinear time history analysis. The analysis results of static loads and dynamic loads cannot be simply combined as if they could occur concurrently. In order to account for the effects of static and dynamic loads simultaneously, the static loads must be applied in the form of dynamic loads, and then a boundary nonlinear time history analysis is carried out. MIDAS/Civil provides the Time Varying Static Loads functionality, which enables us to input static loads in the form of dynamic loads. First, we enter the Ramp function of Normal Data Type in Dynamic Forcing Function. Next, we can enter the Static Load Cases pertaining to the vertical direction and previously defined Function Names in Time Varying Static Load. The shape of the Ramp function should be such that the converted static loading is completely loaded and the resulting vibration is sufficiently dampened before the Arrival Time of the ground acceleration. In order to reduce the time that takes to dampen the vibration resulting from the loading of the converted static loads, we may select the option of the 99% initial damping ratio in Time History Load Case. In addition, the static loads are maintained while the ground acceleration is acting. 10 3.4 Effective Stiffness In a boundary nonlinear time history analysis, the entire structure is divided into linear and nonlinear systems. Nonlinear member forces stemming from the nonlinear system are converted into external dynamic loads acting on the linear system for the analysis. At this point, the linear system alone may become unstable depending on the locations of the general link elements of the force type comprising the nonlinear system. Therefore, modal analysis is carried out after stabilizing the structure using the effective stiffness. If the structure becomes unstable after removing the general link elements of the force type, appropriate effective stiffness need to be entered to induce the natural frequencies and mode shapes, which closely represent the true nonlinear behavior. The appropriate effective stiffness in this case is generally greater than 0, and a smaller than or equal to the value of the initial stiffness of nonlinear properties is used. The initial stiffness corresponds to the dynamic properties of different element types that will be covered in the latter section; namely, kb for Viscoelastic Damper, k for Gap, Hook and Hysteretic System, and ky & kz for Lead Rubber Bearing Isolator and Friction Pendulum System. The initial stiffness is entered as effective stiffness to carry out linear static analysis or linear dynamic analysis and obtain the response prior to enacting nonlinear behavior. In order to approximate linear dynamic analysis, appropriate Secant Stiffness is entered as effective stiffness on the basis of the anticipated maximum deformation. This is an attempt to closely resemble the behavior of n