Graphic dynamic prediction of polarized earthquake incidence response for plan-irregular single story buildings

M. Faggella; R. Gigliotti; G. Mezzacapo; E. Spacone

doi:10.1007/s10518-018-0357-1

Bulletin of Earthquake Engineering

pp 1–31 | Cite as

Graphic dynamic prediction of polarized earthquake incidence response for plan-irregular single story buildings

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Authors and affiliations

M. FaggellaEmail author
R. Gigliotti
G. Mezzacapo
E. Spacone

Original Research Paper

First Online: 24 March 2018

Received: 22 June 2017

Accepted: 20 March 2018

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Abstract

A graphical dynamic model is presented to predict the directional earthquake response of two-ways plan-asymmetric buildings. The theoretical principles inherent to torsional dynamics and vibrations are investigated and the dynamic directional response is rationally explained based on modal rotational kinematics about modal torsional pivots. Seismic forces and response decomposition are handled through geometric modal torsional trends and the earthquake incidence response envelopes are described through directional modal participation radii and graphic spectrum-based ‘8-shaped’ directional influence circles. The graphic approach provides good predictions of the maximum response and of the critical angle computed through directional combination methods. A nonlinear 3D frame model with eccentric infills is analyzed through linear and nonlinear response history analyses (RHA, NLRHA) changing the earthquake incidence angle. Upon confirmation of weak modal coupling, graphic-dynamic modal torsional trends and directional inelastic response envelopes are used to predict the nonlinear response. Directional incremental dynamic analyses and uncoupled modal response history analyses confirm the prediction of polarization.

Keywords

Plan asymmetry Irregular buildings Torsional behavior Directional earthquake incidence Response spectrum analysis Incremental dynamic analysis

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Appendix: Symbols and glossary of terms

G: Center of mass
K: Center of stiffness
e _x: Eccentricity in the X direction
e _y: Eccentricity in the Y direction
m: Total floor mass
I _p: Floor polar moment of inertia about center of mass G
\( \rho = \sqrt {{{I_{p} } \mathord{\left/ {\vphantom {{I_{p} } m}} \right. \kern-0pt} m}} \): Radius of gyration
k _x: Principal stiffness in the x direction
k _y: Principal stiffness in the y direction
\( \rho_{x} = \sqrt {{{k_{\theta } } \mathord{\left/ {\vphantom {{k_{\theta } } {k_{x} }}} \right. \kern-0pt} {k_{x} }}} \): Semi-axis of the ellipse of elasticity in the Y direction
\( \rho_{y} = \sqrt {{{k_{\theta } } \mathord{\left/ {\vphantom {{k_{\theta } } {k_{y} }}} \right. \kern-0pt} {k_{y} }}} \): Semi-axis of the ellipse of elasticity in the X direction
\( \alpha = \frac{{\rho_{y} }}{{\rho_{x} }} = \sqrt {\frac{{k_{x} }}{{k_{y} }}} \): Aspect ratio of the ellipse of elasticity
\( \varepsilon_{x} = \frac{{e_{x} }}{\rho } \): Normalized eccentricity in the X direction
\( \varepsilon_{y} = \frac{{e_{y} }}{\rho } \): Normalized eccentricity in the Y direction
\( d_{xy} = \sqrt {e_{x}^{2} + \rho_{y}^{2} + (\alpha e_{y} )^{2} } \): Coupled stiffness gyrator
\( q_{xy} = \frac{{d_{xy} }}{\rho } = \sqrt {q_{x}^{2} + (\alpha \varepsilon_{y} )^{2} } \): Translationality
\( {\mathbf{u}} = \left\{ {\begin{array}{*{20}c} {u_{x} (t)} \\ {u_{y} (t)} \\ {\theta (t)} \\ \end{array} } \right\} = \sum\limits_{n = 1}^{3} {\left\{ {\begin{array}{*{20}c} {y_{c,n} } \\ { - x_{c,n} } \\ 1 \\ \end{array} } \right\}\theta_{n} (t)} \): Degrees of freedom, mode shapes, time functions
\( {\varvec{\Upphi}} = \left\{ {\begin{array}{*{20}c} {y_{c,n} } \\ { - x_{c,n} } \\ 1 \\ \end{array} } \right\},\quad \theta_{n} (t) \): Mode shape n time-dependent rotation function
\( C\left( t \right) = \left\{ {\begin{array}{*{20}c} {x_{c} (t)} \\ {y_{c} (t)} \\ \end{array} } \right\} = \left\{ {\begin{array}{*{20}c} {{{ - u_{y} (t)} \mathord{\left/ {\vphantom {{ - u_{y} (t)} {\theta (t)}}} \right. \kern-0pt} {\theta (t)}}} \\ {{{u_{x} (t)} \mathord{\left/ {\vphantom {{u_{x} (t)} {\theta (t)}}} \right. \kern-0pt} {\theta (t)}}} \\ \end{array} } \right\} \): Instant center of rotation
\( C_{n} = \left\{ {\begin{array}{*{20}c} {x_{c,n} } \\ {y_{c,n} } \\ \end{array} } \right\} \): Fixed modal node/pivot, center of rotation of mode n (graphic mode)
\( C_{1} ,C_{2} ,C_{3} \): Vertices of the modal triangle of pivots/nodes
\( {\mathbf{k}} = k_{y} \left[ {\begin{array}{*{20}c} {\alpha^{2} } & 0 & { - \alpha^{2} e_{y} } \\ 0 & 1 & {e_{x} } \\ { - \alpha^{2} e_{y} } & {e_{x} } & {d_{xy}^{2} } \\ \end{array} } \right] \): Stiffness matrix associated to u
\( {\mathbf{u}}_{\rho } = \left\{ {\begin{array}{*{20}c} {u_{x} (t)} \\ {u_{y} (t)} \\ {\rho \theta (t)} \\ \end{array} } \right\} = \sum\limits_{n = 1}^{3} {\left\{ {\begin{array}{*{20}c} {y_{c,n} } \\ { - x_{c,n} } \\ \rho \\ \end{array} } \right\}\theta_{n} (t)} \): Equivalent translational degrees of freedom
\( {\mathbf{k}}_{\rho } = k_{y} \left[ {\begin{array}{*{20}c} {\alpha^{2} } & 0 & { - \alpha^{2} \varepsilon_{y} } \\ 0 & 1 & {\varepsilon_{x} } \\ { - \alpha^{2} \varepsilon_{y} } & {\varepsilon_{x} } & {q_{xy}^{2} } \\ \end{array} } \right] \): Stiffness matrix associated to u_ρ
\( l_{n,r} = \frac{1}{{{\varvec{\Upgamma}}_{n,r} }} = \frac{{{\varvec{\Upphi}}_{n}^{t} {\mathbf{m}}{\varvec{\Upphi}}_{n} }}{{{\varvec{\Upphi}}_{n}^{t} {\mathbf{mr}}_{r} }} \): Directional modal participation radius of mode n along r
\( {\mathbf{s}}_{n,r} = \frac{1}{{l_{n,r} }}{\mathbf{m}}{\varvec{\Upphi}}_{n} \): Mode n seismic force vector (side of triangle of modal torsional pivots)
\( \gamma_{n,r} \): Orientation angle of earthquake direction r with respect to mode n force/side
D_n(t): Duhamel function of mode n
P _n,r: Directional sampling point of mode n Duhamel function along direction r
e _n: Eccentricity of C_n with respect to G
h _n: Height of the triangle of pivots with respect to C_n
\( \Delta_{n,r} \): Distance of mode n pivot C_n from the seismic force direction r
\( m_{eff,n,r} = m\frac{{\Delta_{n,r} }}{{l_{n,r} }} \): Directional effective modal mass of mode n for earthquake along r
\( s_{n,r} = \frac{{\Delta_{n,r} }}{{h_{n} }}m \): Directional intensity of mode n seismic force for earthquake along r
\( m_{ineff,n,r} = \frac{{s_{n,r} }}{{\sin \gamma_{n,r} }} \): Directional ineffective modal mass of mode n for earthquake along r

Modal nodes (or modal torsional pivots) The fixed centers of rotation C_i of the mode shapes of a rigid-diaphragm single-story system. For in-height uniform multi story systems the modal nodes become modal pivotal axes (Athanatopoulou et al. 2006).

Ellipse of elasticity The ellipse centered in the center of stiffness K, with semi-axes ρ_x and ρ_y respectively along the Y and X directions (Belluzzi 1942).

Graphic modal analysis (GMA) A graphic procedure to determine the mode shapes and frequencies of a single-story rigid diaphragm system, and to identify other parameters relevant to dynamic seismic analysis, such as modal participation factors, modal participating masses, etc.

Ellipse of elasticity modal analysis (EEMA) Graphic GMA method based on recursive antipolarities with respect to the ellipse of elasticity and to the mass gyrator circle, which determines the mode shapes through their modal nodes C_i and/or the lines of action of the modal forces, equivalent to the numerical Ritz method for modal analysis.

Mohr circle modal analysis (MCMA) Graphic non-recursive GMA method, which determines the modal pivots C_i of the mode shapes and the lines of action of modal forces of one-way asymmetric rigid single story-systems through the construction of the Mohr Circle of the system of equations of torsional dynamics.

Triangle of modal torsional pivots/forces Graphic triangle with edges given by the modal nodes C_i and sides along the lines of action of the modal forces.

Graphic response history analysis (GRHA) Graphic method, based on EEMA or MCMA, which visualizes and determines the modal response of a rotational single-story system at a generic time t, based on the points whose modal rotation represents the Duhamel functions D_i(t). Each modal response function is later linearly superimposed via kinematic combination rules.

Directional modal participation radii Distances determined based on the GMA and on the direction of the seismic action, which locate the ‘Duhamel sampling points’, where the functions D_i(t) are read.

Directional effective modal masses Components/projections of the graphic modal earthquake forces decomposed along the line of action of the seismic action, which sum up to the total mass.

Directional ineffective modal masses Components/projections of the graphic modal earthquake forces decomposed orthogonal to the line of action of the seismic force, which sum up to zero.

Graphic response spectrum analysis (GRSA) Graphic method, based on GRHA and on the Displacement Response Spectrum function S_d(T), which determines the spectrum-based rotational response of each mode, based on the envelope of the Duhamel function and on the directional modal participation radii.

Directional spectral influence circles Graphic envelope of the modal directional response at a given point of the floor, based on GRSA construction and on modal participation radii. For each mode the circles are two (‘8-shaped’) and are symmetric with respect to the heights of the triangle of torsional pivots.

References

Anastassiadis K (1985) Caracteristiques elastiques des batiments a etages. In: Annales de l’ITBTP no. 435 (in French) Google Scholar
Athanatopoulou A (2005) Critical orientation of three correlated seismic components. Eng Struct 27(1):301–312CrossRefGoogle Scholar
Athanatopoulou AM, Makarios T, Anastassiadis K (2006) Earthquake analysis of isotropic asymmetric multistory buildings. Struct Des Tall Spec Build 15(4):417–443CrossRefGoogle Scholar
Beghini LL, Carrion J, Beghini A, Mazurek A, Baker WF (2014) Structural optimization using graphic statics. Struct Multidiscip Optim 49(3):351–366CrossRefGoogle Scholar
Belluzzi O (1942) Scienza delle Costruzioni. Zanichelli, Bologna (in Italian) Google Scholar
Benvenuto E (2006) La Scienza delle Costruzioni e il suo sviluppo storico. Edizioni di Storia e Letteratura, RomaGoogle Scholar
Braga F, Faggella M, Gigliotti R, Laterza M (2005) Nonlinear dynamic response of HDRB and hybrid HDRB-friction sliders base isolation systems. Bull Earthq Eng 3(3):333–353. https://doi.org/10.1007/s10518-005-1242-2 CrossRefGoogle Scholar
Chopra AK (2012) Dynamics of structures: theory and applications to earthquake engineering. Prentice Hall, Englewood CliffsGoogle Scholar
Chopra AK, Goel RK (2004) A modal pushover analysis procedure to estimate seismic demands for unsymmetric-plan buildings. Earthq Eng Struct Dynam 33(8):903–927CrossRefGoogle Scholar
Computers and Structures Inc (CSI) (2008) SAP2000 Integrated finite element analysis and design of structures. CSI, BerkeleyGoogle Scholar
Cremona L (1890) Graphical statics: two treatises on the graphical calculus and reciprocal figures in graphical statics (trans.: Thomas Hudson Beare). Clarendon Press, OxfordGoogle Scholar
Culmann K (1865) Die Graphische Statik (in German) Google Scholar
Dempsey KM, Irvine HM (1979) Envelopes of maximum seismic response for a partially symmetric single storey building model. Earthq Eng Struct Dyn 7(2):161–180CrossRefGoogle Scholar
Dolsek M, Fajfar P (2001) Soft story effects in uniformly infilled reinforced concrete frames. J Earthq Eng 5(1):1–12Google Scholar
Faggella M (2013) Graphical modal analysis and earthquake statics of linear one-way asymmetric single-story structure. In: COMPDYN 2013 4th ECCOMAS thematic conference on computational methods in structural dynamics and earthquake engineering, Kos Island, GreeceGoogle Scholar
Faggella M (2014a) The ellipse of elasticity and Mohr circle-based graphic dynamic modal analysis of torsionally coupled systems. In: Proceedings of the 9th international conference on structural dynamics, EURODYN 2014, 30th June–2nd July, Porto, PortugalGoogle Scholar
Faggella M (2014b) Graphic dynamic earthquake response analysis of linear torsionally coupled 2DOF systems. In: Proceedings of the 9th international conference on structural dynamics, EURODYN 2014, 30th June–2nd July, Porto, PortugalGoogle Scholar
Faggella M (2014c) Graphical dynamic earthquake response analysis of one-way asymmetric systems. In: Proceedings of the 10th national conference in earthquake engineering, earthquake engineering research institute, AnchorageGoogle Scholar
Faggella M (2014d) Graphical dynamic earthquake response of two-ways asymmetric systems based on directional modal participation radii. In: CST2014, the twelfth international conference, on computational structures technology, Naples, Italy, 2–5 SeptemberGoogle Scholar
Faggella M, Gigliotti R (2015) Graphical earthquake response analysis of torsional systems based on Mohr circle. In: XVI conference ANIDIS L’Ingegneria Sismica in Italia, L’Aquila 13–17 SeptemberGoogle Scholar
Faggella M, Barbosa AR, Conte JP, Spacone E, Restrepo JI (2013) Probabilistic seismic response analysis of a 3D reinforced concrete building. Struct Saf 44:11–27. https://doi.org/10.1016/j.strusafe.2013.04.002 CrossRefGoogle Scholar
Faggella M, Gigliotti R, Mezzacapo G, Spacone E (2015a) Graphical dynamic trends for earthquake incidence response of plan-asymmetric systems. In: COMPDYN 2015, 5th ECCOMAS thematic conference on computational methods in structural dynamics and earthquake engineering, Crete Island, Greece, 25–27 May 2015Google Scholar
Faggella M, Mezzacapo G, Gigliotti R, Spacone E (2015b) Significance of earthquake incidence on response of plan-irregular infilled R/C buildings. In: COMPDYN 2015, 5th ECCOMAS thematic conference on computational methods in structural dynamics and earthquake engineering, Crete Island, Greece, 25–27 May 2015Google Scholar
Faggella M, Gigliotti R, Morrone C, Spacone E (2017) Mohr circle-based graphical vibration analysis and earthquake response of asymmetric systems. In: X international conference on structural dynamics, EURODYN 2017, Rome 10–17 September 2017 (Submitted)Google Scholar
Fajfar P, Marusic D, Perus I (2005) Torsional effects in the pushover-based seismic analysis of buildings. J Earthq Eng 9(6):831–854Google Scholar
Fivet C, Zastavni D (2013) Constraint-based graphic statics: new paradigms of computer-aided structural equilibrium design. J Int Assoc Shell Spat Struct 54(4):271–280Google Scholar
Fontara I-KM, Kostinakis KG, Manoukas GE, Athanatopoulou AM (2015) Parameters affecting the seismic response of buildings under bi-directional excitation. Struct Eng Mech 53(5):957–979CrossRefGoogle Scholar
Fujii K (2014) Prediction of the largest peak nonlinear seismic response of asymmetric buildings under bi-directional excitation using pushover analyses. Bull Earthq Eng 12:909–938. https://doi.org/10.1007/s10518-013-9557-x CrossRefGoogle Scholar
Hejal R, Chopra AK (1989) Earthquake analysis of a class of torsionally-coupled buildings. Earthq Eng Struct Dyn 18:305–323CrossRefGoogle Scholar
Iori T, Poretti S (2015) SIXXI 2. Storia dell’ingegneria strutturale in Italia, Gangemi (in Italian)Google Scholar
Italian Code NTC08 (2008) Norme tecniche per le costruzioni. D.M. 14.01.08, G.U. No.9 – 04.02.08, 2008 (in Italian) Google Scholar
Kalkan E, Reyes JC (2015) Significance of rotating ground motions on behavior of symmetric-and asymmetric-plan structures: Part II. multi-story structures. Earthq Spectra 31(3):1613–1628CrossRefGoogle Scholar
Lin J, Tsai K (2007) Simplified seismic analysis of asymmetric building systems. Earthq Eng Struct Dyn 36:459–479CrossRefGoogle Scholar
Maxwell JC (1864) On reciprocal figures and diagrams of forces. Philos Mag J Ser 4(27):250–261CrossRefGoogle Scholar
McKenna F, Fenves GL (2001) OpenSees manual. Pacific Earthquake Engineering Research (PEER) Center. http://opensees.berkeley.edu/OpenSees. Accessed 2017
Mehrabi AB, Shing PB (1997) Finite element modeling of masonry-infilled RC frames. J Struct Eng ASCE 123(5):604–613CrossRefGoogle Scholar
Menun C, Der Kiureghian A (1998) A replacement for the 30%, 40% and SRSS rules for multicomponent seismic analysis. Earthq Spectra 14(1):153–163CrossRefGoogle Scholar
Mohammad AF, Faggella M, Gigliotti R, Spacone E (2014) Influence of bond-slip effect and shear deficient column in the seismic assessment of older infilled frame R/C structures. In: Proceedings of the 9th international conference on structural dynamics, EURODYN 2014, 30th June—2nd July, Porto, PortugalGoogle Scholar
Mohammad AF, Faggella M, Gigliotti R, Spacone E (2016) Seismic performance of older R/C frame structures accounting for infills-induced shear failure of columns. Eng Struct. https://doi.org/10.1016/j.engstruct.2016.05.010 Google Scholar
Mohr O (1900) Welch Umsttinde bedingen die Elastizitltsgrenze und den Bruch eines Materials? Zeitschrift Verein Deutsch Iwenieur 44(1524–1530):1572–1577 (in German) Google Scholar
Mohr O (1914) Abhandlugen aus dem Gebiete der technische Mechanik, 2nd edn. Ernst und Sohn, Berlin (in German) Google Scholar
Palermo M, Sammarco E, Silvestri S, Gasparini G, Trombetti T (2014) Nonlinear seismic response of planar asymmetric systems. In: Second European conference on earthquake engineering and seismology, Istanbul, Aug. 25–29Google Scholar
Reyes J, Chopra AK (2011) Evaluation of three-dimensional modal pushover analysis for unsymmetric-plan buildings subjected to two components of ground motion. Earthq Eng Struct Dyn 40(13):1475–1494CrossRefGoogle Scholar
Reyes J, Kalkan E (2015) Significance of rotating ground motions on behaviour of symmetric-and asymmetric-plan structures: Part 1. Single-story structures. Earthq Spectra 31(3):1591–1612CrossRefGoogle Scholar
Rigato AB, Medina RA (2007) Influence of angle of incidence on seismic demands for single-storey structures subjected to bi-directional ground motions. Eng Struct 29:2593–2601CrossRefGoogle Scholar
Ritter W (1865) Anwendungen der Graphischen Statik, 3, (Appendix), Zurich (in German)Google Scholar
Skinner RI, Robinson WH, Mc Verry GH (1993) An introduction to seismic isolation. Wiley, New YorkGoogle Scholar
Trombetti TL, Conte JP (2005) New insight into and simplified approach to seismic analysis of torsionally coupled one-story, elastic systems. J Sound Vib 286:265–312CrossRefGoogle Scholar
Vamvatsikos D, Cornell CA (2002) Incremental dynamic analysis. Earthq Eng Struct Dyn 3:491–514CrossRefGoogle Scholar
Van Mele T, Block P (2014) Algebraic graph statics. Comput Aided Des 53:104–116CrossRefGoogle Scholar

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Authors and Affiliations

M. Faggella
- 1
Email author
R. Gigliotti
- 1
G. Mezzacapo
- 1
E. Spacone
- 2

1.Sapienza University of RomeRomeItaly
2.University of Chieti-PescaraPescaraItaly

Graphic dynamic prediction of polarized earthquake incidence response for plan-irregular single story buildings

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