Development of finite element tools for assisted seismic design

Transcription

1 Istituto Universitario di Studi Superiori di Pavia Università degli Studi di Pavia ROSE School EUROPEAN SCHOOL FOR ADVANCED STUDIES IN REDUCTION OF SEISMIC RISK Development of finite element tools for assisted seismic design A Project Submitted in Partial Fulfilment of the Requirements for the PhD in EARTHQUAKE ENGINEERING By ALESSANDRO REALI Supervisors: Prof. FERDINANDO AURICCHIO Prof. ALBERTO PAVESE Pavia, February 2005

2 The project entitled Development of finite element tools for assisted seismic design, by Alessandro Reali, has been approved in partial fulfilment of the requirements for the PhD in Earthquake Engineering. Ferdinando Auricchio Alberto Pavese

3 Abstract ABSTRACT The latest developments in the building codes make necessary the implementation of tools able to help engineers in all the phases of structural seismic design. In this framework, it is rapidly arising the interest for building software packages for R.C. structural aided design that can guide engineers to design a structure starting from the geometric modeling and arriving to the (automatic) generation of the design tables in accordance with the prescriptions of the recent codes (as Eurocode and the Italian Seismic Code). This work focuses on a finite element code that can be used as structural engine for such seismic aided design packages. Starting from the research oriented general purpose finite element code FEAPpv, a tool for frame analysis is developed, including features very important for a correct design of R.C. frames, as for example the modal combination with response spectrum and the possibility of including rigid slabs. Keywords: finite element method, frame analysis, structural aided design, modal combination, response spectrum. i

4 Contents DEVELOPMENT OF FINITE ELEMENT TOOLS FOR ASSISTED SEISMIC DESIGN CONTENTS ABSTRACT CONTENTS LIST OF TABLES LIST OF FIGURES i ii v vi 1. FEAPPV Frame analysis using FEAppv General structure for an input file Details on mesh generation Details on batch commands Examples Static and eigenvalue analysis of a cantilever beam Eigenvalue analysis of a 1-bay 3-frame 2-storey structure NEW IMPLEMENTATIONS FEAPpv modifications ii

5 Contents Errors in FEAPpv Input-output modifications to FEAPpv New features Input of a response spectrum Modal participation factors and effective modal masses Modal combination (CQC) Lumped masses Springs Master-slave constraints NUMERICAL TESTS Self-weight analysis of a 1-bay 3-frame 2-storey structure Response spectrum analysis of a 1-bay 3-frame 2-storey structure Response spectrum analysis of a 1-bay 3-frame 2-storey structure with lumped masses and springs Response spectrum analysis of a 1-bay 3-frame 2-storey structure with rigid constraints Rigid top floor Rigid diaphragm top floor Response spectrum analysis of a 3-bay 3-frame 4-storey structure with rigid diaphragm and lumped masses and springs Response spectrum analysis of a L-shaped 4-storey structure CONCLUSIONS 41 BIBLIOGRAPHY 42 A. INPUT AND OUTPUT FILES 43 A.1 Self-weight analysis of a 1-bay 3-frame 2-storey structure A.2 Response spectrum analysis of a 1-bay 3-frame 2-storey structure A.3 Response spectrum analysis of a 1-bay 3-frame 2-storey structure with lumped masses and springs A.4 Response spectrum analysis of a 1-bay 3-frame 2-storey structure with rigid top floor A.5 Response spectrum analysis of a 1-bay 3-frame 2-storey structure with rigid diaphragm top floor A.6 Response spectrum analysis of a 3-bay 3-frame 4-storey structure with rigid diaphragm and lumped masses and springs iii

6 Contents A.7 Response spectrum analysis of a L-shaped 4-storey structure iv

7 LIST OF TABLES LIST OF TABLES 1.1 Cantilever beam, material and geometric properties Cantilever beam, reference nodal displacements Cantilever beam, reference eigenvalues Cantilever beam, reference first eigenmode bay 3-frame 2-storey structure, material and geometric properties bay 3-frame 2-storey structure, reference eigenvalues bay 3-frame 2-storey structure under self-weight, fem versus SAP2000 reference results bay 3-frame 2-storey structure: response spectrum analysis, fem versus SAP2000 reference results Stiffness components for springs added to first floor nodes bay 3-frame 2-storey structure with lumped masses and springs: response spectrum analysis, fem versus SAP2000 reference results bay 3-frame 2-storey structure with rigid top floor: response spectrum analysis, fem versus SAP2000 reference results bay 3-frame 2-storey structure with rigid diaphragm top floor: response spectrum analysis, fem versus SAP2000 reference results bay 3-frame 4-storey structure, material and geometric properties bay 3-frame 4-storey structure: response spectrum analysis, fem versus SAP2000 reference results L-shaped 4-storey structure, material and geometric properties L-shaped 4-storey structure: response spectrum analysis, fem versus SAP2000 reference results v

8 LIST OF FIGURES LIST OF FIGURES 1.1 Local frame defined through the reference node (R) Views of the 1-bay 3-frame 2-storey structure under investigation in the x-z (top) and in the x-y planes (bottom) Response spectrum used for the analyses Views of the 3-bay 3-frame 4-storey structure under investigation in the x-z (top) and in the x-y planes (bottom) Views of the L-shaped 4-storey structure under investigation in the x-z (top) and in the x-y planes (bottom) vi

9 Chapter 1. FEAPpv 1. FEAPPV FEAPpv (A Finite Element Analysis Program - Personal Version, see Taylor (2001)) is an open-source general purpose finite element program developed at the University of California at Berkeley by Professor R. L. Taylor. It can be defined a research-oriented tool, because it is very simple for a user to enter the Fortran code and modify it or add new elements and features in order to make the program able to solve any problem under investigation. For a complete description of FEAPpv (and FEAP, which is the extended version), an exhaustive documentation is available under Professor Taylor s web page ( rlt/). The aim of this Chapter is to show some FEAPpv features related to frame analysis, which are the starting point of our project that mainly consists in obtaining a reliable structural analysis tool that can be used as structural engine for seismic aided design packages. In particular, the work described in this report has been carried on in collaboration with the ProSA developing staff, who have chosen to implement the work results into the ProSA (which stands for PROgettazione Strutturale Assistita, meaning Structural Aided Design ) project, which is a project promoted by Eucentre and Protezione Civile consisting in the construction of a modern and complete software package for seismic aided design. 1.1 FRAME ANALYSIS USING FEAPPV In this section we show, with the help of simple examples, how to perform different kinds of frame analyses using FEAPpv as is when downloaded. Note that the structure of the input file is very simple and is divided into two parts: the first one, starting with a title line, is where the problem to be solved is defined, the second one is where the solution commands are specified. 1

10 Chapter 1. FEAPpv In the following we describe step by step how to prepare input files to perform linear static and eigenvalue frame analyses General structure for an input file FEAPpv title of the analysis numnp numel nummat ndm ndf nen COORdinate list of coordinates ELEMent list of elements BOUNdary list of boundary conditions FORCe list of nodal forces DISPlacement list of prescribed displacements MATErial,num FRAMe ELAStic ISOTropic E nu DENSity 0 rho BODY 0 q_x q_y q_z CROSs section A I_11 I_22 I_12 J_t REFErence NODE xx yy zz END BATCh macro1 macro2 etc. END 2

11 Chapter 1. FEAPpv STOP Details on mesh generation FEAP section FEAPpv title of the analysis numnp numel nummat ndm ndf nen where: numnp number of nodal points numel number of elements nummat number of material sets ndm number of dimensions ndf number of degrees of freedom at each node nen maximum number of nodes connected to each element coordinate section COORdinate 1 0 x_1 y_1 z_1 2 0 x_2 y_2 z_2 etc. where: i x i y i z i element section global node number global node coordinates ELEMent 1 0 mat_1 n1_1 n2_1 2 0 mat_2 n1_2 n2_2 etc. where: i global element number mat i number of element material n1 i n2 i nodes connected to the element boundary section BOUNdary 1 0 dof_1 dof_2 dof_ dof_1 dof_2 dof_3... etc. 3

12 Chapter 1. FEAPpv where: i dof j = 0 free, global constrained node number dof j = 1 constrained force section FORCe 1 0 f_1 f_2 f_ f_1 f_2 f_3... etc. where: i f j global loaded node number j th component of the generalized load at node i displacement section DISPlacement 1 0 d_1 d_2 d_ d_1 d_2 d_3... etc. where: i d j global node number j th component of the generalized prescribed displacement at node i material section MATErial,num FRAMe ELAStic ISOTropic E nu DENSity 0 rho BODY 0 q_1 q_2 q_3 CROSs section A I_22 I_33 I_23 J_t REFErence NODE xx yy zz (or REFErence VECTor xx yy zz) where: 4

13 Chapter 1. FEAPpv num material set number E Young s modulus nu Poisson s ratio A cross section area rho material density q 1 x-body load q 2 y-body load q 3 z-body load I 22 2-moment of inertia I 33 3-moment of inertia I 23 product of inertia J t torsional constant (default: I 22+I 33) xx x reference node coordinate (or x reference vector component) yy y reference node coordinate (or y reference vector component) zz z reference node coordinate (or z reference vector component) The local axis frame defined through the reference node is shown in figure 1.1; otherwise, if the reference vector is used, its direction directly defines the one for local axis 2. Figure 1.1: Local frame defined through the reference node (R) Details on batch commands We distinguish in the following discussion two cases depending on the type of desired structural analysis. 5

14 Chapter 1. FEAPpv Static analysis To perform a static analysis the following set of commands is needed: TANGent,,1! solve the linear problem DISPlacement,all! print all nodal displacements STREss,all! print all internal actions REACtion,all! print all nodal reactions Eigenvalues and eigenvectors To perform an eigenvalue analysis the following set of commands is needed: MASS,CONS! form consistent mass matrix (LUMP for a lumped mass formulation) TANGent! form tangent matrix SUBSpace,n1,n2! compute eigenvalues by subspace iterations EIGVector,all,1! print all nodal modal displacements for 1st eigenvector EIGVector,all,2! print all nodal modal displacements for 2nd eigenvector the subspace command is followed by two numbers whose meaning is: n1 = number of eigenpairs to compute n2 = number of vectors to expand the subspace and improve convergence (default: min{n1+8, 2 n1, maximum number of eigenvalues}) 1.2 EXAMPLES We report here a couple of very simple examples in order to show how to use FEAPpv to perform static and eigenvalue analyses. We remark that in these examples the results relative to internal actions and reactions are not reported because in FEAPpv there exists an error about the output of these quantities. Anyway such an error has been found and corrected. Moreover, the body load command in FEAPpv simply puts half of the resultant load on each of the element ends, but also this has been corrected considering its flexural effects on the element ends. 6

15 Chapter 1. FEAPpv Note: here and in all the tests performed in this work, the solutions taken as reference for the numerical results are the ones resulting from the finite element code SAP2000 (see SAP.htm) Static and eigenvalue analysis of a cantilever beam The first example consists of the static and eigenvalue analyses of a cantilever beam, whose material and geometric properties are reported in Table 1.1, loaded by a shear load p = 1 KN at the free end in the x direction. The cantilever beam has been modeled by four frame elements. For the eigenvalue analysis a lumped mass matrix formulation has been adopted (only the first eigenmode is here requested as output). Young modulus E = KN/m 2 Poisson ratio ν = 0.2 Density ρ = 2.5 t/m 3 Length L = 4 m Cross section base b = 0.3 m Cross section height h = 0.4 m Table 1.1: Cantilever beam, material and geometric properties. Below we report the input and the output files for such a problem, named respectively icant.txt and ocant.txt. Input file: icant.txt FEAPpv 4-element cantilever beam (static & modal analysis) [KN m] 5,4,1,3,6,3 coordinate element boundary force material,1 frame 7

16 Chapter 1. FEAPpv end elastic isotropic 3.E7.2 density cross section E-4 1.6E e-3 reference node batch tangent,,1 displacement,all mass,lump subspace eigvector,all,1 end stop Output file: ocant.txt FEAPpv 4-element cantilever beam (static & modal analysis) [KN m] Solution date: Fri Feb 18 18:38: UNIX/PC 1.2a - 01/07/02 - Input Data Filename: Icant.txt Number of Nodal Points : 5 Number of Elements : 4 Spatial Dimension of Mesh : 3 Degrees-of-Freedom/Node (Maximum) - : 6 Number Element Nodes (Maximum) - : 3 Number of Material Sets : 1 Number Parameters/Set (Program) - : 200 Number Parameters/Set (Users ) - : 50 FEAPpv 4-element cantilever beam (static & modal analysis) [KN m] Nodal Coordinates node 1 Coord 2 Coord 3 Coord E E E E E E E E E E E E E E E+00 FEAPpv 4-element cantilever beam (static & modal analysis) [KN m] E l e m e n t s Elmt Mat Reg 1 Node 2 Node 3 Node

17 Chapter 1. FEAPpv FEAPpv 4-element cantilever beam (static & modal analysis) [KN m] N o d a l B. C. Node 1-b.c. 2-b.c. 3-b.c. 4-b.c. 5-b.c. 6-b.c FEAPpv 4-element cantilever beam (static & modal analysis) [KN m] Nodal Forces node 1 Force 2 Force 3 Force 4 Force 5 Force 6 Force E E E E E E+00 FEAPpv 4-element cantilever beam (static & modal analysis) [KN m] M a t e r i a l P r o p e r t i e s Material Set 1 for Element Type: frame Degree of Freedom Assignments Local Global Number Number T h r e e D i m e n s i o n a l F r a m e M e c h a n i c a l P r o p e r t i e s Truss & Frame Analysis Modulus E E+07 Poisson ratio Density 1-Gravity Load 2-Gravity Load 3-Gravity Load E E E E+00 C r o s s S e c t i o n P a r a m e t e r s Area I_xx I_yy I_xy J_zz Kappa_x Kappa_y E E E E E E E-01 9

18 Chapter 1. FEAPpv Formulation : Small deformation. No shear deformation Mass type : Lumped. Rotational Mass Factor: E+00 R e f e r e n c e C o o r d i n a t e s X-1 = E+00 X-2 = E+00 X-3 = E+00 E q u a t i o n / P r o b l e m S u m m a r y: Space dimension (ndm) = 3 Number dof (ndf) = 6 Number of equations = 24 Number nodes = 5 Average col. height = 8 Number elements = 4 Number profile terms = 168 Number materials = 1 FEAPpv 4-element cantilever beam (static & modal analysis) [KN m] M a c r o I n s t r u c t i o n s Macro Statement Variable 1 Variable 2 Variable 3 tang E E E+00 disp all E E E+00 mass lump E E E+00 subs E E E+00 eigv all E E E+00 *Macro 1 * tang v: t= Residual norm = E E+00 t= Condition check: D-max E+07; D-min E+04; Ratio E+04 Maximum no. diagonal digits lost: 2 End Triangular Decomposition t= Energy convergence test Maximum = E-04 Current = E-04 Relative = E+00 Tolerance = E-16 *Macro 2 * disp all v: t= FEAPpv 4-element cantilever beam (static & modal analysis) [KN m] N o d a l D i s p l a c e m e n t s Time E+00 Prop. Ld E+00 Node 1 Coord 2 Coord 3 Coord 1 Displ 2 Displ 3 Displ 4 Displ 5 Displ 6 Displ E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E+00 10

19 Chapter 1. FEAPpv E E E E E E-04 *Macro 3 * mass lump v: t= *Macro 4 * subs v: t= There are 0 eigenvalues less than shift E+00 SUBSPACE: Current eigenvalues, iteration D D D D D D D D D+07 SUBSPACE: Current residuals, iteration D D D D D D D D D-02 SUBSPACE: Square root of eigenvalues (rad/t) D D D D D D D D D+03 SUBSPACE: Square root of eigenvalues (Hz.) D D D D D D D D D+02 SUBSPACE: Period in units of time (T) D D D D D D D D D-03 *Macro 5 * eigv all v: t= FEAPpv 4-element cantilever beam (static & modal analysis) [KN m] N o d a l D i s p l a c e m e n t s Time E+00 Eigenvalue E+03 Node 1 Coord 2 Coord 3 Coord 1 EigV. 2 EigV. 3 EigV. 4 EigV. 5 EigV. 6 EigV E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E-15 *End of Macro Execution* t= The reference results from SAP2000 are reported in Tables 1.2, 1.3 and 1.4. It is possible to see that they completely agree with FEAPpv results. 11

20 Chapter 1. FEAPpv node u 1 [m] u 2 [m] u 3 [m] θ 1 θ 2 θ Table 1.2: Cantilever beam, reference nodal displacements. mode period [s] frequency [s 1 ] circ. freq. [rad s 1 ] eigenvalue [rad 2 s 2 ] Table 1.3: Cantilever beam, reference eigenvalues. node u 1 [m] u 2 [m] u 3 [m] θ 1 θ 2 θ E E E E E E E E E E E E-16 Table 1.4: Cantilever beam, reference first eigenmode Eigenvalue analysis of a 1-bay 3-frame 2-storey structure The second example consists of the eigenvalue analysis of a 1-bay 3-frame 2-storey structure (see Figure 1.2) whose material and geometric properties are reported in Table 1.5. Each member has been modeled by one frame element. A lumped mass matrix formulation has been adopted. Young modulus E = KN/m 2 Poisson ratio ν = 0.2 Density ρ = 2.5 t/m 3 x-beam length L x = 4.5 m x-beam cross section base b x = 0.25 m x-beam cross section height h x = 0.35 m y-beam length L y = 6 m y-beam cross section base b y = 0.25 m y-beam cross section height h y = 0.40 m column height H = 3 m column edge (square) l = 0.25 m Table 1.5: 1-bay 3-frame 2-storey structure, material and geometric properties. Below we report the input and the output files for such a problem, named respectively iframe.txt and oframe.txt. 12

21 Chapter 1. FEAPpv Figure 1.2: Views of the 1-bay 3-frame 2-storey structure under investigation in the x-z (top) and in the x-y planes (bottom). Input file: iframe.txt FEAPpv 1 bay 3 frames 2 storeys [KN m] (modal analysis) 18,26,3,3,6,2 coordinate element

22 Chapter 1. FEAPpv boundary material,1! column.25x.25 frame elastic isotropic 3.E7.2 density cross section 6.25E E E E-4 reference vector material,2! x-beam.25x.35 frame elastic isotropic 3.E7.2 density cross section 8.75E E E E-3 reference vector material,3! y-beam.25x.40 frame elastic isotropic 3.E7.2 density cross section 1.0E E E E-3 reference vector end batch tangent mass,lump subspace end 14

23 Chapter 1. FEAPpv stop Output file: oframe.txt FEAPpv 1 bay 3 frames 2 storeys [KN m] (modal analysis) Solution date: Tue Feb 22 09:33: UNIX/PC 1.2a - 01/07/02 - Input Data Filename: Iframe.txt Number of Nodal Points : 18 Number of Elements : 26 Spatial Dimension of Mesh : 3 Degrees-of-Freedom/Node (Maximum) - : 6 Number Element Nodes (Maximum) - : 2 Number of Material Sets : 3 Number Parameters/Set (Program) - : 200 Number Parameters/Set (Users ) - : 50 FEAPpv 1 bay 3 frames 2 storeys [KN m] (modal analysis) Nodal Coordinates node 1 Coord 2 Coord 3 Coord E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E+00 FEAPpv 1 bay 3 frames 2 storeys [KN m] (modal analysis) E l e m e n t s Elmt Mat Reg 1 Node 2 Node

24 Chapter 1. FEAPpv FEAPpv 1 bay 3 frames 2 storeys [KN m] (modal analysis) N o d a l B. C. Node 1-b.c. 2-b.c. 3-b.c. 4-b.c. 5-b.c. 6-b.c FEAPpv 1 bay 3 frames 2 storeys [KN m] (modal analysis) M a t e r i a l P r o p e r t i e s Material Set 1 for Element Type: frame Degree of Freedom Assignments Local Global Number Number T h r e e D i m e n s i o n a l F r a m e M e c h a n i c a l P r o p e r t i e s Truss & Frame Analysis Modulus E E+07 Poisson ratio

25 Chapter 1. FEAPpv Density 1-Gravity Load 2-Gravity Load 3-Gravity Load E E E E+00 C r o s s S e c t i o n P a r a m e t e r s Area I_xx I_yy I_xy J_zz Kappa_x Kappa_y E E E E E E E-01 Formulation : Small deformation. No shear deformation Mass type : Lumped. Rotational Mass Factor: E+00 R e f e r e n c e V e c t o r v-1 = E+00 v-2 = E+00 v-3 = E+00 FEAPpv 1 bay 3 frames 2 storeys [KN m] (modal analysis) M a t e r i a l P r o p e r t i e s Material Set 2 for Element Type: frame Degree of Freedom Assignments Local Global Number Number T h r e e D i m e n s i o n a l F r a m e M e c h a n i c a l P r o p e r t i e s Truss & Frame Analysis Modulus E E+07 Poisson ratio Density 1-Gravity Load 2-Gravity Load 3-Gravity Load E E E E+00 17

26 Chapter 1. FEAPpv C r o s s S e c t i o n P a r a m e t e r s Area I_xx I_yy I_xy J_zz Kappa_x Kappa_y E E E E E E E-01 Formulation : Small deformation. No shear deformation Mass type : Lumped. Rotational Mass Factor: E+00 R e f e r e n c e V e c t o r v-1 = E+00 v-2 = E+00 v-3 = E+00 FEAPpv 1 bay 3 frames 2 storeys [KN m] (modal analysis) M a t e r i a l P r o p e r t i e s Material Set 3 for Element Type: frame Degree of Freedom Assignments Local Global Number Number T h r e e D i m e n s i o n a l F r a m e M e c h a n i c a l P r o p e r t i e s Truss & Frame Analysis Modulus E E+07 Poisson ratio Density 1-Gravity Load 2-Gravity Load 3-Gravity Load E E E E+00 C r o s s S e c t i o n P a r a m e t e r s Area I_xx E E-03 18

27 Chapter 1. FEAPpv I_yy I_xy J_zz Kappa_x Kappa_y E E E E E-01 Formulation : Small deformation. No shear deformation Mass type : Lumped. Rotational Mass Factor: E+00 R e f e r e n c e V e c t o r v-1 = E+00 v-2 = E+00 v-3 = E+00 E q u a t i o n / P r o b l e m S u m m a r y: Space dimension (ndm) = 3 Number dof (ndf) = 6 Number of equations = 72 Number nodes = 18 Average col. height = 22 Number elements = 26 Number profile terms = 1512 Number materials = 3 FEAPpv 1 bay 3 frames 2 storeys [KN m] (modal analysis) M a c r o I n s t r u c t i o n s Macro Statement Variable 1 Variable 2 Variable 3 tang E E E+00 mass lump E E E+00 subs E E E+00 *Macro 1 * tang v: t= t= Condition check: D-max E+07; D-min E+04; Ratio E+03 Maximum no. diagonal digits lost: 2 End Triangular Decomposition t= *Macro 2 * mass lump v: t= *Macro 3 * subs v: t= There are 0 eigenvalues less than shift E+00 SUBSPACE: Current eigenvalues, iteration D D D D D D D D D+03 SUBSPACE: Current residuals, iteration D D D D D D D D D-05 SUBSPACE: Square root of eigenvalues (rad/t) D D D D D D D D D+01 19

28 Chapter 1. FEAPpv SUBSPACE: Square root of eigenvalues (Hz.) D D D D D D D D D+01 SUBSPACE: Period in units of time (T) D D D D D D D D D-02 *End of Macro Execution* t= The reference results from SAP2000 are reported in Table 1.6. It is possible to see that they completely agree with FEAPpv results. mode period [s] frequency [s 1 ] circ. freq. [rad s 1 ] eigenvalue [rad 2 s 2 ] Table 1.6: 1-bay 3-frame 2-storey structure, reference eigenvalues. 20

29 Chapter 2. New implementations 2. NEW IMPLEMENTATIONS In this Chapter we show the modifications and the new features implemented into the original version of the finite element program FEAPpv. For the new features included in the code it is given a brief theoretical introduction of the problem. All the tests performed in order to check the results of the final version of the program are reported in the next Chapter. 2.1 FEAPPV MODIFICATIONS In this Section we present the modifications and the corrections we needed to perform on the code FEAPpv. The corrections consist just in fixing a couple of trivial (but important from the point of view of frame analysis) errors present in the original code. The modifications are applied to the input-output part of the code, in order to make simpler the connection between the structural engine of the project ProSA and its main parts which are the pre- and post-processor tools Errors in FEAPpv As already mentioned in the previous Chapter, a couple of trivial errors have been found in the original version of FEAPpv. These errors are about the computation (in the output phase) of internal actions and the application of a uniformly distributed body load. The former was simply an index problem inside a do-loop. Once detected, it has been fixed without any problem. The latter consisted in the lack of the contribution to the right-hand-side vector of the bending moments due to the body load. In this way the program simply loaded each element end with 21

30 Chapter 2. New implementations half of the resultant of the distributed load. Also this bug has been fixed, adding the suitable terms to the element right-hand-side vector Input-output modifications to FEAPpv Some modifications to the standard input-output system of FEAPpv were necessary in order to make as simple as possible the connection between the finite element code and the ProSA preand post-processors. The ProSA developing staff have decided that at the moment the simplest way to connect the different ProSA tools was through auxiliary files. Almost all the original input-output structure of FEAPpv has been eliminated. As there was no more need for interactive and graphical features, they have been removed from the program, which has been built as a simple Fortran Console Application instead of Fortran Standard Graphics or QuickWin Application. Moreover, as the input and output files need to be generated in a directory which is different in general from the one where is placed the executable program, it has been necessary for the program to be able to read paths and to work in different directories. At the end of this process, the syntax to call (to be done from a DOS prompt) the program fem.exe, make it read the input file file1 from the directory C:\input and make it write the output on the file file2 in the directory C:\output is the following: fem.exe -ic:\input\file1 -oc:\output\file2 We remark that, if the files present an extension (for instance.txt), it has to be specified in the call to the executable. Moreover, we have decided to preserve the original structure of the input file, but the output has been simplified and through the new format it has been made more rational the access to the output information by another program (i.e. the post-processor). At the moment it has been decided to keep (even if unnecessary) the first part of the output as an echo of the input for check reasons. The new format of the output can be seen in one of the test files reported in Appendix A. 2.2 NEW FEATURES A number of both important and minor new features has been added to FEAPpv in order to make it suitable for frame seismic analysis. 22

31 Chapter 2. New implementations In the following we show each new feature Input of a response spectrum As a very important new feature that has been added to FEAPpv is the modal analysis with response spectrum (we will talk about that in the following), we needed a new mesh command in order to input a response spectrum. So it has been created the new mesh command spectrum. The syntax to input a response spectrum is: spectrum ξ n T 1 Sa 1 T 2 Sa 2... T n Sa n where: ξ is the damping ratio, n is the number of spectrum pairs, T i is the i th value of period and Sa i is the corresponding value of spectral acceleration. The resulting response spectrum will be the linear interpolation between the n input spectrum pairs. We remark that the spectrum refers to an earthquake that in general can act in any direction in the x-y plane, but the angle θ that defines this direction has not to be specified at this level: it will be given by the user during the call to the modal combination Modal participation factors and effective modal masses Before performing a response spectrum analysis, it is necessary to compute for each mode i the so called modal participation factors (Γ i ) and effective modal masses (Mi ), which are respectively a measure of the degree of the mode participation in the total response and of the portion of the total mass effectively participating in the mode response. They are defined as follows: Γ i = L i M i and M i = L2 i M i where L i = φ T i mi and M i = φ T i mφ i, being φ i the i th eigenmode vector, m the mass matrix and i the influence vector. The participation factors Γ i are necessary to compute each mode response (see next Subsection). The effective masses have to be known by the user, because they give information on how many modes need to be included in the modal combination, depending on what is prescribed by the reference seismic code (for instance, a widely used rule is to include a number of modes such 23

32 Chapter 2. New implementations that the total effective mass is at least 90% of the total mass). We remark that the sum of all the effective modal masses is equal to the total mass of the system. The computation of the quantities just described have been inserted in the macro command subspace and their values are printed to the output file immediately after the frequencies of the system Modal combination (CQC) Of course a fundamental tool for earthquake engineering analysis is the dynamic analysis with response spectrum. This method consists in estimating the peak seismic response for a structure by combining the peak responses associated to single vibration modes and computed by means of a response spectrum. The uncoupled equation of motion for the i th mode is: q i (t) + 2ξ i ω i q i (t) + ωi 2 q i (t) = Γ i ü g (t) where q i (t) is the modal coordinate (such that the contribution of the i th mode to the nodal displacement u(t) is u i (t) = φ i q i (t), with φ i the i th eigenmode vector), ξ i is the damping ratio, ω i is the (circular) frequency, Γ i is the modal participation factor and u g (t) is the ground motion. Comparing this equation with the one for the single degree of freedom system (whose displacement is indicated with D i ), we get: q i (t) = Γ i D i (t). So: u i (t) = Γ i φ i D i (t). Now, given the earthquake response spectrum, for each mode i we can enter it with the period T i getting the spectral acceleration S a,i and then the spectral displacement S d,i = S a,i /ω 2 i. From the spectral displacement S d,i we obtain the peak modal nodal displacement as: U i = Γ i φ i S d,i Finally, to get an estimate of the peak displacement, we have to combine the peak modal displacements. Many combination rules have been proposed in the literature, the most famous of which are the absolute sum (ABSSUM), the square-root-of-sum-of-squares (SRSS) and the complete quadratic combination (CQC), see Der Kiureghian (1981) and Wilson et al. (1981). Since the complete quadratic combination seems to be the rule giving in general the best estimate for peak response quantities, it is the one we have chosen in order to perform the modal combination in our code. In this way, according to CQC estimate, the k th component of the peak nodal displacement vector U CQC is: 24

33 Chapter 2. New implementations U CQC,k = N N ρ ij U i,k U j,k i=1 j=1 where N is the number of modes included in the combination and ρ ij [0, 1] is the correlation coefficient between modes i and j. Different expressions for the correlation coefficient ρ ij have been published in the years. We have used the one proposed first in Der Kiureghian (1981), which is the most widely used. The expression, supposing the same damping ratio for all modes (ξ i = ξ i), is the following: where β ij = ω i /ω j. ρ ij = 8ξ 2 (1 + β ij )β 3/2 ij (1 β 2 ij )2 + 4ξ 2 β ij (1 + β ij ) 2 Besides displacements, other output quantities, as internal actions and nodal reactions, need to be computed for each mode and then combined by a CQC rule. In order to do that, we simply have to pass, for each mode and for each element, the corresponding nodal displacements to the subroutine that computes, element-wise, the internal actions and we have to pass, for each mode, the global displacement vector to the subroutine that computes the reaction forces. Finally we just combine each modal quantity according to the CQC rule used for displacements. This analysis procedure has been implemented in the macro command mods, which performs the complete quadratic combination for displacements, internal forces and reactions. The results are directly printed in the output file (printing is automatic and no more commands are required). The syntax to perform the complete quadratic modal combination (to be obviously performed only after a call to the macro command subspace ) is: mods,,n,θ where n is the number of modes to be included in the combination and θ is the angle (in degrees, equal to zero by default) that defines the direction of the ground motion in the x-y plane (it is the counter-clock-wise angle with the positive direction of the x-axis) Lumped masses Another feature which was not present in FEAPpv but is useful in Earthquake Engineering is the possibility to append to a node a lumped mass. In order to do that we have to add the values of the lumped masses to the diagonal of the mass matrix in the suitable positions. Lumped masses can now be introduced in the system through the mesh command cmas with the following syntax: 25

34 Chapter 2. New implementations cmas n i m i j m j... l m l where n is the number of lumped masses to be added, i, j,..., l are the global number of the nodes to which the masses are appended and m i is the value of the lumped mass added to node i Springs Also the possibility to add a spring to a node was not originally present in FEAPpv. In order to do that, we have to add the stiffness components of the spring to the diagonal of the global stiffness matrix in the suitable positions. The mesh command prepared to introduce springs in the structure is nspr and the syntax is as follows: nspr n i k i,x k i,y k i,z k i,θx k i,θy k i,θz j k j,x k j,y k j,z k j,θx k j,θy k j,θz... l k l,x k l,y k l,z k l,θx k l,θy k l,θz where n is the number of springs to be added, i, j,..., l are the global number of the nodes to which the springs are appended and k i,x k i,y k i,z k i,θx k i,θy k i,θz are the values of the translational and rotational stiffness of the spring added to node i in the three cartesian components Master-slave constraints It is very common in structural analysis to make use of constraints in order to approximate the behaviour of real frames including the effects of nearly rigid elements such as slabs or diaphragms. In order to construct a general framework for rigid constraints, we allow to rigidly connect the displacements and the rotations of a node to the ones of another one through a master-slave option. This means that a node s can be defined as the slave of a master node m, in the sense 26

35 Chapter 2. New implementations that the degrees of freedom of the former are rigidly connected to the ones of the latter, through the following relations: u (s) x u (s) y u (s) z = u (m) x = u (m) y = u (m) z u (s) θ x = u (m) θ x u (s) θ y u (s) θ z = u (m) θ y = u (m) θ z + (z (s) z (m) )u (m) θ y (z (s) z (m) )u (m) θ x + (y (s) y (m) )u (m) θ z (y (s) y (m) )u (m) θ z + (x (s) x (m) )u (m) θ z (x (s) x (m) )u (m) θ y As some kinds of constraint (as a rigid diaphragm, for example) do not involve all the degrees of freedom of the slave node, it is possible to constrain only a subset of the degrees of freedom of a node. The master-slave option has to be called just after the end command closing the FEAPpv mesh input section using the following syntax: master slave x m,1 y m,1 z m,1 x 1 y 1 z 1 flag 1,x flag 1,y flag 1,z flag 1,θx flag 1,θy flag 1,θz slave x m,2 y m,2 z m,2 x 2 y 2 z 2 flag 2,x flag 2,y flag 2,z flag 2,θx flag 2,θy flag 2,θz... slave x m,n y m,n z m,n x n y n z n flag n,x flag n,y flag n,z flag n,θx flag n,θy flag n,θz where x i, y i, z i are the coordinates of the i th slave node and x m,i, y m,i, z m,i are the coordinates of the corresponding master node; the following flags are equal to 1 if the corresponding degree of freedom is free and 0 if it is constrained. 27

36 Chapter 3. Numerical tests 3. NUMERICAL TESTS In this Chapter, we report some examples performed in order to test the new features implemented in the program. The results obtained are compared with the ones from SAP2000 and all of them show the reliability of the analyses performed. To begin with, we test all the different features on the simple three-dimensional 1-bay 3-frame 2-storey structure presented in Chapter 1 (see Figure 1.2 and Table 1.5). Moreover, in the last two examples, we test also more complicated structures. The complete input and output files for all the analyses reported in this Chapter are reported in Appendix A. 3.1 SELF-WEIGHT ANALYSIS OF A 1-BAY 3-FRAME 2-STOREY STRUC- TURE As a first test, the structure sketched in Figure 1.2 has been tested under its self-weight assumed to be a uniform load of intensity ρg = KN/m 3 in the negative direction of the z-axis. We take as reference values to check our results the displacements, the internal forces and the reactions at some specific points, in particular we check: generalized displacements ([m] and rad) at node 18, of coordinates (4.5, 12, 6); internal actions ([KN] and [KNm]) at the ends of element 5 connecting nodes 3 (I) and 4 (J), of coordinates, respectively (0, 0, 3) and (4.5, 0, 3); reactions ([KN] and [KNm]) at node 1, of coordinates (0, 0, 0). In Table 3.1 we report the results of the analysis using our code (named simply fem in the following) and SAP

37 Chapter 3. Numerical tests It is possible to see that the results from both the programs completely agree. quantity fem results SAP2000 results u 18 u 18 u 18 θ 18 θ 18 θ 18 x y z x y z 1.265E-19 0 N5 I N5 J V22,5 I 4.440E E-17 V J V I V J 22, E E-17 33, , Mt,5 I E E-16 Mt,5 J E E-16 M22,5 I M22,5 J M33,5 I 9.642E E-16 M33,5 J E E-16 Fx Fy Fz Mx My Mz E E-17 Table 3.1: 1-bay 3-frame 2-storey structure under self-weight, fem versus SAP2000 reference results. 29

38 Chapter 3. Numerical tests 3.2 RESPONSE SPECTRUM ANALYSIS OF A 1-BAY 3-FRAME 2-STOREY STRUCTURE The same structure of the previous Section has been tested also for response spectrum analysis. The response spectrum selected is a piece-wise linear interpolation of the Eurocode 8 spectrum for subsoil class B and for a value of peak ground acceleration (PGA) of 3m/s 2 (see Figure 3.1), and it is directed along a direction that forms a 30 degree angle with the positive direction of the x-axis. We remark that the first ten modes have been included in the combination. In Table 3.2 we report the results of the analysis using fem and SAP2000. It is possible to see that the results from both the programs completely agree. quantity fem results SAP2000 results u 18 u 18 u 18 θ 18 θ 18 θ 18 x y z x y z N5 I 3.168E N5 J 3.168E V22,5 I V J V I V J 22, , , Mt,5 I Mt,5 J M22,5 I M22,5 J M33,5 I M33,5 J Fx Fy Fz Mx My Mz Table 3.2: 1-bay 3-frame 2-storey structure: response spectrum analysis, fem versus SAP2000 reference results. 30

39 Chapter 3. Numerical tests S a [m/s 2 ] T [s] Figure 3.1: Response spectrum used for the analyses. 3.3 RESPONSE SPECTRUM ANALYSIS OF A 1-BAY 3-FRAME 2-STOREY STRUCTURE WITH LUMPED MASSES AND SPRINGS Here we repeat the analysis of the previous Section adding lumped masses on the top floor and springs to the nodes of the first floor. In particular we add a 10 t lumped mass to each node of the top floor and a spring with stiffness components as indicated in Table 3.3 to each node of the first floor. k x [KN/m] k y [KN/m] k z [KN/m] k θx [KNm] k θy [KNm] k θz [KNm] Table 3.3: Stiffness components for springs added to first floor nodes. In Table 3.2 we report the results of the analysis using fem and SAP2000. It is possible to see that the results from both the programs completely agree. 31

40 Chapter 3. Numerical tests quantity fem results SAP2000 results u 18 u 18 u 18 θ 18 θ 18 θ 18 x y z x y z N5 I E N5 J E V22,5 I V J V I V J 22, , , Mt,5 I Mt,5 J M22,5 I M22,5 J M33,5 I M33,5 J Fx Fy Fz Mx My Mz Table 3.4: 1-bay 3-frame 2-storey structure with lumped masses and springs: response spectrum analysis, fem versus SAP2000 reference results. 32

41 Chapter 3. Numerical tests 3.4 RESPONSE SPECTRUM ANALYSIS OF A 1-BAY 3-FRAME 2-STOREY STRUCTURE WITH RIGID CONSTRAINTS In this Section we repeat the analysis described in Section 3.2 using rigid constraints on the nodes of the top floor Rigid top floor In this first case the constraint on the nodes of the top floor consists of a rigid body constraint. In Table 3.5 we report the results of the analysis using fem and SAP2000. It is possible to see that the results from both the programs completely agree. quantity fem results SAP2000 results u 18 u 18 u 18 θ 18 θ 18 θ 18 x y z 2.152E x y z 1.191E E-16 N5 I 3.088E E-12 N5 J 3.088E E-12 V22,5 I V J V I V J 22, , , Mt,5 I Mt,5 J M22,5 I M22,5 J M33,5 I M33,5 J Fx Fy Fz Mx My Mz Table 3.5: 1-bay 3-frame 2-storey structure with rigid top floor: response spectrum analysis, fem versus SAP2000 reference results. 33