THREE-DIMENSIONAL SEISMIC ANALYSIS OF MASONRY COMBINED SYSTEMS

Transcription

1 THREE-DIMENSIONAL SEISMIC ANALYSIS OF MASONRY COMBINED SYSTEMS N. Augenti an F. Parisi Associate Professor Dept. of Structural Engineering Universit of Naples Feerico II Naples Ital ABSTRACT: Ph.D. Stuent Dept. of Structural Engineering Universit of Naples Feerico II Naples Ital Recent numerical analses on some tpes of planar Masonr Combine Sstems (MCSs) showe a great variabilit of the lateral stiffness along the height following to the mutual interaction between ifferent parallel structural elements. This ma inuce significant eccentricities between centre of mass an centre of rigiit in aition to the ones potentiall ue to U-shape walls place at the perimeter of the builing or to irregular in-plan frame-wall ual sstems. The mixe structure of the builing becomes ramaticall prone to amage ue to twist-inuce isplacements so a 3D seismic analsis is require to account for its torsional response. In this paper a matrix algorithm for seismic analsis of structural sstems forme b masonr or Reinforce Concrete (RC) shear walls (with or without openings) frames an structural cores is propose. Such lateral loa-resisting elements are consiere to be arbitraril arrange in plan an subjecte to a generic pattern of horiontal forces. The torsional stiffness matrix of the cores is built up b means of the non-uniform torsion theor an the stiffness matrix of the whole mixe structure is subsequentl efine. The presente analtical formulation provies a goo preiction of the seismic behaviour of torsionall eccentric builings with MCSs an allow to evaluate strength an isplacement emans. KEYWORDS: Masonr Combine Sstems 3D seismic analsis non-uniform torsion theor.. INTRODUCTION The seismic behaviour of the Masonr Combine Sstems (MCSs) uner lateral forces is characterise b a significant mutual interaction between the lateral loa-resisting elements ue to a great variabilit of the stiffness over the height. In this case the seismic coes at both national an international levels require a multi-moal namic analsis of the builing structure. Since the lateral stiffnesses change along the height the centre of rigiit ma rasticall var from one to another store. On the contrar the centre of mass change significantl onl in particular conitions (i.e. eccentric mass concentrations) so these aitional in-plan eccentricities ma increase the ones potentiall ue to cores place at the perimeter of the builing or to non-orthogonal bracing sstems. In such cases the seismic response of the structure ma have significant torsional components so D analses on separate planar moels along two ifferent irections of the builing plan shoul result in large errors an lac of safet. For this reasons a 3D seismic analsis is require. The present wor eals with this topic looing to provie an analtical tool accurate an simple at the same time that is able to escribe the seismic behaviour of torsionall eccentric builings with MCSs. In particular the istribution of the horiontal forces between frames masonr or Reinforce Concrete (RC) shear walls an structural cores is stuie. The 3D analsis is carrie out b constructing the stiffness matrix of the whole structure which inclues both the Saint-Venant an warping torsional rigiities of the cores. The torsional stiffness matrix is efine through the non-uniform torsion theor.. THE PROPOSED ALGORITHM The 3D seismic analsis of builings forme b frames shear walls an structural cores is performe b appling the irect stiffness metho (also calle isplacement or eformation metho) that is particularl suite for computer-automate analses of staticall ineterminate structures.

2 The algorithm is base on the following hpotheses: - lateral loa-resisting elements mutuall connecte b rigi iaphragms; - horiontal forces applie to the centre of mass at each level; - frames with inextensible beams an columns; - prismatic soli (masonr or RC) shear walls with both flexural an shear flexibilities; - opene masonr shear walls iscretise in macro-elements b using the RAN metho (Augenti 004); - prismatic RC cores having thin-walle open or multi-celle cross-section; - equal interstore heights. Both the out-of-plane lateral stiffness an the torsional rigiit can be neglecte for shear walls an frames; the structural cores can be moelle as cantilever beams with both lateral stiffness along an irection an torsional rigiit. P-Delta effects an axial eformations are also neglecte in this stu since the are significant for metallic an tall structures. In the following treatment E p is the generic static or inematic parameter referre to the p-th store of -th wall. Stores an lateral loa-resisting elements are progressivel numbere from the top to the base an from the left to the right respectivel. Let us consier the structural sstem showe in Figure compose b () an opene masonr wall () a soli shear wall (3) a moment-resisting frame an (4) a thin-walle open cross-section core. We efine a Global Coorinate Sstem (GCS) O(x ) having x-plane overlappe to the p-th floor an -axis with right-hane positive orientation. Given the generic element whose cross-section has the shear centre C at the p-th store of the builing we introuce a Local Coorinate Sstem (LCS) C ( x ) having x parallels to the principal inertia axes an oriente as the -axis. O x ω x C Figure. Arrangement of lateral loa-resisting elements In general x are rotate of an angle ω about the respective x axes of the GCS so a rotation matrix of the structural element have to be efine to transform the coorinates from an orthonormal basis to another one. Assuming the iaphragms to be rigi in their own plane the isplacement of a point ling on the p-th floor is univocall efine b: () the horiontal translation along the x-axis u p ; () the horiontal translation along the -axis v p ; an (3) the torsional rotation about the -axis θ p. If the centres of rigiit are not coincient at each level or not line up in the -irection the seismic response of the structure has non-ero torsional component an the lateral-torsional coupling of the bracing elements have to be evaluate b means of a 3D analsis. In this secon case enoting b s the number of stores it follows that: () the structure has 3s Degrees of Freeom (DOFs); () external loa vector Q an isplacement vector S have imensions 3s ; the stiffness matrix K has sie equal to 3s. In the GCS the istribution of the horiontal forces between the lateral loa-resisting elements is governe b the equilibrium matrix equation: KS Q (.)

3 which can be specialise to the -th element an written in the LCS. Both the loa an isplacement vectors ma be ecompose along the three axes so the problem reuces to the construction of the stiffness matrix of the structure. The 3s 3s rotation matrix of the -th element ma be expresse as follows: cosω senω 0 Ω senω cosω I (.) where ever s s sub-matrix is iagonal. The vector S is relate in turn to S b the compatibilit matrix: C c c cs (.3) compose b a number of s square sub-matrices relate to each store p of the element uner consieration. The global coorinates of the shear centre appear in the latter matrix an allow the representation of the compatibilit matrix through nine s s iagonal sub-matrices b grouping together the coorinates of all the floor cross-sections of the -th element in the vectors X an Y. Finall the transfer matrix T of such element can then be efine so the isplacement vector in the LCS is (Poati 977): S ΩCS T S (.4) Since T is orthogonal the stiffness matrix of the -th element is given b: an can be written also in terms of sub-matrices as: T K T K T (.5) Kuu Kuv K u θ K K vu K vv K vθ K θ u K θv K θθ (.6) being (.7): uu u v K cos ω sen ω K K cosω senω vu uv u v vv u v K sen ω cos ω θ θ K K cosω Xsenω Ycosω senω Xcosω Ysenω θu u u v K K senω Xsenω Ycosω cosω Xcosω Ysenω θv v u v K X senω Ycosω X cosω Ysenω θθ u v θ

4 Such equations contain the lateral stiffness matrices u an θ an allow to construct the following stiffness matrix for the -th element in the LCS: v as well as the torsional stiffness matrix u 0 0 K 0 v θ (.8) If this lateral loa-bearing element is a shear wall or a moment-resisting frame both the out-of-plane lateral stiffnesses an the torsional stiffnesses can be neglecte so the computational wor associate with 3D analsis of the mixe structure is rasticall reuce. Augenti an Parisi (008) propose some analtical expressions of the lateral stiffness matrices for both opene masonr shear walls an soli (masonr or RC) shear walls. The stiffness matrix of the whole mixe structure is obtaine b assembling the matrices K. Obviousl in the case of mutuall orthogonal planar loa-bearing elements K uv an K vu are ero matrices so the translational components of motion along the x an axes are uncouple. In such conitions the coupling of the lateral loa-resisting elements is onl ue to the torsional response. Therefore the global isplacements at each level ma be erive b the inversion of Eqn... Subsequentl isplacements an loa vectors can be estimate for ever loa-bearing element at each store an one can then assess their seismic performance. To achieve a better phsical unerstaning of the 3D seismic istribution of the horiontal actions between ifferent lateral loa-resisting elements istribution matrices are usefull efine. Given that: T S T S K Q an S K Q (.9) one can get: Q K T K Q Q (.0) The istribution matrix of the -th element is square as well but whilst its sie is equal to s for planar moels it becomes 3s for 3D moels. Also the istribution matrix can be written in the following form: uu uv u θ vu vv vθ θ u θv θθ (.) where the sub-matrices are expresse as follows (.): K cosω D senω D X senω Ycosω D K cosω D senω D X senω Ycosω D u K u cosω Du senω Dv Xsenω Ycosω D K senω D cosω D X cosω Ysenω D K senω D cosω D X cosω Ysenω D v K v senω Du cosω Dv Xcosω Ysenω D uu u uu vu θ u uv u uv vv θ v θ θ θ θθ vu v uu vu θ u vv v uv vv θ v θ θ θ θθ

5 K D θ u θ θu K D θ v v θ v K D θθ v θθ Flexibilit sub-matrices of the structure appear in such formulas an ma be clearl erive b the inversion of the stiffness matrix of the whole mixe structure. Eqn..0 allows the irect istribution of the applie actions between the lateral loa-resisting elements an can be expresse also b a set of matrix equations so each loa vector of the element is a linear combination of the external loa ones. When there are non-orthogonal lateral loa-resisting elements the first two equations are couple themselves. This occurs also if centre of mass an centre of stiffness are coincient at ever level: in such case the algebraic sstem is compose just b two matrix equations but the sub-matrices uv an vu have nonero entries. These last ones become ero matrices onl when the external horiontal actions are applie along the principal irections of the floor plan which still change along the height even if the lateral loa-resisting elements have constant cross-sections. 3. MODELLING OF THE STOREY TORQUES The propose algorithm provie goo results onl if the store torques are accuratel moelle. Such operation can be uniquel performe after eccentricit matrices are built up. These lower triangular matrices can be efine in turn onl after the coorinates of the centres of rigiit are etermine at each level. In general such algebraic problem can be solve via irect or iterative proceures but in this case trial an error algorithms ma result in high numerical approximations on the final solutions because () the stiffness matrices have large conition numbers (ill-conitione problems) an () an unacceptable step-b-step propagation of errors ma occur (Feriger et al. 998 Higham 00 Quarteroni et al. 007). The centres of rigiit can be estimate through a simple irect proceure base on the efinition of centre of vector fiel. After the stiffness matrices are constructe in the GCS the position of the centres can be erive b solving the following equation: w K ( r r c) 0 (3.) where r c is the unnown raius vector of the centres of rigiit of the whole structure an r is the raius vector of the -the lateral force-resisting element. Such 3s vectors ma be ivie into the sub-vectors X c Y c Z c an X Y Z respectivel which contain the global coorinates x so it turns out to be: w rc D K r (3.) where D is the flexibilit matrix of the whole structure. Obviousl Eqn. 3. ma be written also as: Kuu Kuv K uθ Xc Kuu Kuv K u θ X w Kvu Kvv K vθ Yc K vu K vv K vθ Y Kθu Kθv Kθθ Zc K θu K θv K θθ Z (3.3) or also as a sstem of three matrix equations in the unnown vectors X c Y c Z c. The exact solution of the problem is forme also b the vector Z c which oes not contain the coorinates of the iaphragms because each centre of rigiit oes not actuall lie in the mile plane of the floor but it is genericall place in the 3D

6 space. One coul then assume Z c equal to the vector Z p containing the coorinates of the floors but such constraint equation shoul reuce the number of unnowns an the algebraic problem shoul become ineterminate. Thus to estimate irectl the positions of the centres of rigiit this equalit relation have to be impose onl after Eqn. 3. is solve. This mous operani oes not result in significant errors since store torques epen just on the eccentricities along the x an axes. 4. TORSIONAL STIFFNESS MATRIX OF A STRUCTURAL CORE The last formula of Eqns..7 allows the efinition of the torsional stiffness matrix of a lateral force-resisting element in the GCS: it inclues both the Saint-Venant torsional stiffness matrices as well as the warping rigiit matrix. The first ones are erive from the translational stiffness matrices written in the LCS; the latter is presente below for a generic structural core b appling the non-uniform torsion theor (Vlasov 96 Kolbrunner an Basler 969). For a prismatic thin-walle beam the funamental equation of torsion is the following linear non-homogeneous Orinar Differential Equation (ODE) of orer 4 with constant coefficients: () () θ ( ) θ ( ) m ( ) (4.) being: m the applie torque per unit length; θ the angle of twist; () the Saint-Venant torsional stiffness efine as prouct of the shear moulus G b the torsion constant I * ; an () the warping rigiit efine as prouct of the Young s moulus E b the warping constant I λλ. Note that λ is the generalise warping function efine b Augenti (99) for multi-celle cross-sections. The general solution of Eqn. 4. is: θ ( ) A sh α B chα C D θ ( ) (4.) where: A B C D are the constants of integration; α is the inverse characteristic length; an θ () is the particular solution of the ODE. Let us consier a prismatic cantilever thin-walle beam having torsional restraint at the base an subjecte to a torque M at the top. In such case Eqn. 4. becomes homogeneous an has a particular solution set to ero so the following expression of the angle of twist is erive b imposing the bounar conitions: M θ ( ) shα thα () ( chα ) α α (4.3) Since at H one can get: H th α H θ ( M M H ) () ( α ) α H ( th α H α H ) (4.4) the torsional stiffness at this height is given b: () α θ (4.5) th α H α H To obtain a generalise analtical expression of the torsional stiffness matrix of a core with height sh consier a cantilever beam of height 3H subjecte to the torques M M M 3 at H H 3H an let us analse three separate schemes on which we appl the principle of effects superposition (see Figure ).

7 θ () θ () θ (3) H θ () θ () θ (3) H 3 θ 3 () θ 3 () θ 3 (3) 3 H scheme 0 scheme scheme scheme 3 Figure. Effects superposition for a structural core of height 3H The angle of twist θ p at the generic level of the cantilever beam uner the applie torques ma be expresse as sum of three rates: the first one cause b the actions applie to the levels below the one hel in consieration; the secon one inuce b the action applie to the level taen in consieration; the last one ue to the actions applie to the levels above the one hel in consieration. Thus one can write: () () (3) θ θ θ θ () () (3) θ θ θ θ () () (3) θ θ θ θ (4.6) where each superscript is referre to the generic scheme to be superpose. Taing into account that eforme configurations of schemes () an () are compose b an elastic part an a rigi part the twist rotation-torque relationships can be simpl obtaine so the torsional flexibilit matrix θ ma be constructe. This approach ma be generalise to a cantilever beam of height sh reaching to an s s torsional flexibilit matrix whose entries are: α thα sh sh α thα ( s ) H α ( s ) H α θ () θ () thαh αh α θ s () θ () θ () θ s () { } shα s H thα sh chα s H α s H α thα ( s ) H α ( s ) H α thαh αh α (4.7) (4.8)

8 θ s () shαh thα sh chαh αh α shα H thα ( s ) H ( chαh ) αh α θ s () thαh αh α θ ss () (4.9) Finall the torsional stiffness matrix θ is erive b the inversion of θ so one can get the matrix K θθ. CONCLUSIONS The matrix algorithm propose in this wor allows to carr out 3D seismic analses of torsionall eccentric masonr-rc mixe structures forme b frames (masonr or RC) shear walls an cores connecte themselves b rigi iaphragms. The analtical formulation is base on the irect stiffness metho so simple equations have been presente to efine both the translational an torsional stiffness matrices for each lateral loa-resisting element. Either the lateral force vectors or the store torque vectors ma be estimate for each structural element b etermining the stiffness-base istribution of the external horiontal actions applie to the centres of mass. To moel the store torques an to run the seismic analsis the centres of rigiit have to be etermine first. This problem ma be solve b means of a irect metho which provie little approximations on the final numerical solutions. Also the istribution matrices ma be efine so the lateral loa-resisting elements that mainl resist the external horiontal actions can be ientifie. These analtical tools are ver important especiall when the seismic upgraing or retrofit of an existing builing is neee. An essential step of the propose metho is the construction of the torsional stiffness matrices of the lateral loa-resisting elements. The were efine for RC cores b analsing the behaviour of a prismatic cantilever thin-walle beams torsionall fixe at the base an subjecte to a generic pattern of torques. A generalise analtical expression of such matrices was foun b using the non-uniform torsion theor taen into account that the istribution of the torsional moment in the Saint-Venant an warping parts epens not onl on the ratio between the respective rigiities but also on the height of the structural element. The propose algorithm is applicable not onl for linear equivalent static analses but also for pushover analses in which a step-b-step upate of the stiffness matrix ma be simpl performe. REFERENCES Augenti N. (99). Introuione al calcolo elle strutture in parete sottile Liguori Naples Ital (in Italian). Augenti N. (004). Il calcolo sismico egli eifici in muratura UTET Turin Ital (in Italian). Augenti N. an Parisi F. (008). Preliminar analsis on masonr RC combine sstems: structural assessment. Proceeings of 4 th International Bric & Bloc Masonr Conference Sne Feriger J.H. (998). Numerical Methos for Engineering Applications John Wile & Sons New Yor U.S.A. Higham N.J. (00). Accurac an Stabilit of Numerical Algorithms Societ of Inustrial an Applie Mathematics Philaelphia U.S.A. Kolbrunner C.F. an Basler K. (969). Torsion in structures Springer-Verlag Berlin German. Poati P. (977). Teoria e tecnica elle strutture UTET Turin Ital (in Italian). Quarteroni A. Sacco R. an Saleri F. (007). Numerical Mathematics Springer-Verlag New Yor U.S.A. Vlasov V.Z. (96) Thin-walle elastic beams National Science Founation Washington D.C. U.S.A. Olbourne Press Lonon U.K.