=============== Capacity Models =============== Introduction ------------ The nonlinear behaviour of each building class is characterised by a capacity curve, which defines its global lateral (shear) strength and deformation capacity. Capacity curves are obtained from nonlinear static "pushover" analysis and are commonly expressed in the acceleration-displacement response spectrum (ADRS) domain, i.e., spectral displacement (SD) versus spectral acceleration (SA). Idealised Capacity Curve Shapes ------------------------------- To reflect differences in structural nonlinear behaviour, three idealised curve shapes are considered: **Bilinear curves** For buildings that experience negligible elastic stiffness degradation and no significant post-yield loss of lateral strength (e.g., steel and timber frames). **Trilinear curves** For buildings that exhibit an initial reduction in elastic stiffness due to cracking while maintaining a relatively stable post-yield lateral strength (e.g., reinforced concrete bare frames, dual systems, unreinforced masonry). **Quadrilinear curves** For structures that undergo substantial reduction in lateral strength at large deformation levels (e.g., reinforced concrete infilled frames). The mathematical relationship between SA, SD, and T used to define the capacity curve is: .. math:: SA = SD \left(\frac{2\pi}{T}\right)^2 where :math:`SD_y` and :math:`SA_y` denote the spectral displacement and acceleration at yielding, :math:`SD_u` is the ultimate spectral displacement, :math:`T_1` is the first-mode vibration period, and :math:`T_y` is the vibration period at yielding. SDOF to MDOF Calibration ------------------------ Unlike simplified single-degree-of-freedom (SDOF) oscillators, the vulnerability model adopts stick-and-mass multi-degree-of-freedom (MDOF) systems with masses lumped at floor levels and connected by nonlinear springs defining the storey-shear force versus drift behaviour. These models explicitly capture storey-level demands like peak storey drift (PSD) and peak floor acceleration (PFA). Conversion from Capacity Curve to Pushover Curve ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ The capacity curves are converted to pushover curves relating roof displacement (:math:`u_{roof}`) to base shear (:math:`V_{base}`). The conversion follows structural dynamics principles: .. math:: u_{roof} = SD \cdot \Gamma_1 .. math:: V_{base} = SA \cdot \Gamma_1 \cdot m_{SDOF} where :math:`\Gamma_1` is the first-mode participation factor and :math:`m_{SDOF}` is the effective mass of the SDOF oscillator (taken equal to 1.0 ton). MDOF Storey Mass Calculation ~~~~~~~~~~~~~~~~~~~~~~~~~~~~ The storey mass of the MDOF model (:math:`m_{MDOF}`) is computed as: .. math:: m_{MDOF} = \frac{\phi_1^T [I] \phi_1}{(\phi_1^T [I] \bar{J})^2} where: * :math:`[I]` is the identity matrix of floor masses with diagonal entries specifying the proportion of :math:`m_{MDOF}` assigned to each floor * :math:`\bar{J}` is a unity vector of length n (the number of storeys) * :math:`\phi_1` is the first-mode shape of the building * :math:`\phi_1^T` is its transpose Storey Force-Drift Relationships ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ The shear force-drift relationship is proportioned for individual storeys based on :math:`\phi_1` following Lu et al. (2014): .. math:: V_i = \frac{\sum_{j=i}^{n} \phi_{1,i} m_i}{\sum_{k=1}^{n} \phi_{1,i} m_i} V_{base} .. math:: \Delta_i = (\phi_{1,i} - \phi_{1,i-1}) u_{roof} where: * :math:`V_i` and :math:`\Delta_i` are the shear force and drift values associated with storey i * :math:`\phi_{1,i}` is the first-mode ordinate at that storey Mode Shape Assumptions ~~~~~~~~~~~~~~~~~~~~~~ The first-mode shape :math:`\phi_1` is obtained through eigenvalue analysis, considering uniform mass and stiffness across all storeys, except at the roof where mass is decreased by 25%. For buildings exhibiting soft-storey behaviour, the stiffness of the first storey is reduced by 80%. For bare-frame structures without walls/infills, Eurocode 1-Part 4 recommends: .. math:: \phi_{1,i} = \left(\frac{i}{n}\right)^{0.6} where i is the storey index. Capacity Model References ------------------------- The capacity models are derived from peer-reviewed analytical studies and empirical data. Table 2 below summarises the primary material types and corresponding references used for calibrating the capacity curves. .. list-table:: Table 2: Capacity Model References by Primary Material :header-rows: 1 :widths: 25 75 * - Primary Material - Literature References * - Adobe and Rammed Earth - Tarque et al. (2012); Villar-Vega et al. (2017); Lang et al. (2018); Karanikoloudis & Lourenço (2018); Silveira et al. (2018); Battaglia et al. (2021) * - Unreinforced Masonry - Barbat et al. (2006); Borzi et al. (2008); Villar-Vega et al. (2017); Lang et al. (2018) * - Reinforced Masonry - Voon & Ingham (2006); Shedid et al. (2008); Haach et al. (2010); Murcia-Delso & Shing (2012); Lotfy et al. (2022); FEMA (2024a) * - Confined Masonry - Magenes & Calvi (1992); Nucera et al. (2012); Erkoseoglu et al. (2014); Perez Gavilan et al. (2015); Villar-Vega et al. (2017); Ahmed et al. (2018) * - Reinforced Concrete - Crowley et al. (2008); Bal et al. (2010); Ricci et al. (2011); Tischer et al. (2012); Villar-Vega et al. (2017); Lang et al. (2018); Tarquini et al. (2019); Parra et al. (2019); Ugalde et al. (2019); Aljawhari et al. (2020); Suzuki & Iervolino (2020); Ugalde & Lopez-Garcia (2020); Farsangi et al. (2021); Gondaliya et al. (2023); FEMA (2024b); Nafeh & O'Reilly (2024) * - Steel Structures - Sabelli et al. (2003); McCormick et al. (2007); Tasnimi & Mohebkhah (2011); Ramirez et al. (2012); Radnić et al. (2013); Markulak et al. (2013); Abou-Elfath et al. (2017); Eladly (2017); Del Carpio R. et al. (2019); Jouneghani & Haghollahi (2020); FEMA (2024a) * - Timber Structures - Goda & Atkinson (2011); Loss et al. (2013); Villar-Vega et al. (2017); Lang et al. (2018); FEMA (2024a) Interactive Viewer ------------------ Use the interactive viewer below to explore capacity curves for different building classes. .. raw:: html
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References ---------- The capacity models are derived from the following peer-reviewed literature, organised by primary construction material: **Adobe and Rammed Earth** Tarque et al. (2012); Villar-Vega et al. (2017); Lang et al. (2018); Karanikoloudis & Lourenço (2018); Silveira et al. (2018); Battaglia et al. (2021) **Unreinforced Masonry** Barbat et al. (2006); Borzi et al. (2008); Villar-Vega et al. (2017); Lang et al. (2018) **Reinforced Masonry** Voon & Ingham (2006); Shedid et al. (2008); Haach et al. (2010); Murcia-Delso & Shing (2012); Lotfy et al. (2022); FEMA (2024a) **Confined Masonry** Magenes & Calvi (1992); Nucera et al. (2012); Erkoseoglu et al. (2014); Perez Gavilan et al. (2015); Villar-Vega et al. (2017); Ahmed et al. (2018) **Reinforced Concrete** Crowley et al. (2008); Bal et al. (2010); Ricci et al. (2011); Tischer et al. (2012); Villar-Vega et al. (2017); Lang et al. (2018); Tarquini et al. (2019); Parra et al. (2019); Ugalde et al. (2019); Aljawhari et al. (2020); Suzuki & Iervolino (2020); Ugalde & Lopez-Garcia (2020); Farsangi et al. (2021); Gondaliya et al. (2023); FEMA (2024b); Nafeh & O'Reilly (2024) **Steel Structures** Sabelli et al. (2003); McCormick et al. (2007); Tasnimi & Mohebkhah (2011); Ramirez et al. (2012); Radnić et al. (2013); Markulak et al. (2013); Abou-Elfath et al. (2017); Eladly (2017); Del Carpio R. et al. (2019); Jouneghani & Haghollahi (2020); FEMA (2024a) **Timber Structures** Goda & Atkinson (2011); Loss et al. (2013); Villar-Vega et al. (2017); Lang et al. (2018); FEMA (2024a) Full Citations ~~~~~~~~~~~~~~ * Abou-Elfath H., Ramadan M., Alkanai F.O. (2017). Upgrading the seismic capacity of existing RC buildings using buckling restrained braces. *Alexandria Engineering Journal*, 56(2), 251-262. * Ahmed S., Shahzada K., Khan S.W. (2018). Seismic vulnerability assessment of confined masonry structures by macro-modeling approach. *Structures*, 14, 282-293. * Aljawhari K., Gentile R., Galasso C. (2020). A fragility-based framework to assess the seismic risk of an RC building portfolio. *Earthquake Spectra*, 36(4), 1797-1822. * Bal I.E., Crowley H., Pinho R., Gülay F.G. (2010). Structural characteristics of Turkish RC building stock in Northern Marmara region for loss assessment applications. *ISET Journal of Earthquake Technology*, 47(2-4), 35-69. * Barbat A.H., Pujades L.G., Lantada N. (2006). Performance of buildings under earthquakes in Barcelona, Spain. *Computer-Aided Civil and Infrastructure Engineering*, 21(8), 573-593. * Battaglia L., Ferreira T.M., Lourenço P.B. (2021). Seismic fragility assessment of masonry building aggregates: A case study in the old city centre of Seixal, Portugal. *Earthquake Engineering & Structural Dynamics*, 50(5), 1358-1377. * Borzi B., Pinho R., Crowley H. (2008). Simplified pushover-based vulnerability analysis for large-scale assessment of RC buildings. *Engineering Structures*, 30(3), 804-820. * Crowley H., Pinho R., Bommer J.J., Bird J.F. (2008). Development of a displacement-based method for earthquake loss assessment. IUSS Press, Pavia. * Del Carpio R.M., Mosqueda G., Lignos D.G. (2019). Experimental investigation of steel building gravity framing systems under strong earthquake shaking. *Soil Dynamics and Earthquake Engineering*, 116, 230-241. * Eladly M.M. (2017). Cyclic testing of steel frames with buckling-restrained braces. *Journal of Constructional Steel Research*, 138, 325-338. * Erkoseoglu G., Altunel E., Beyhan G. (2014). Seismic fragility assessment of confined masonry construction. *Bulletin of Earthquake Engineering*, 12(6), 2831-2855. * Farsangi E.N., Tasnimi A.A., Yang T.Y., Takewaki I. (2021). Seismic performance of RC buildings designed based on INBC. *Engineering Structures*, 232, 111806. * FEMA (2024a). FEMA P-58: Seismic Performance Assessment of Buildings. Federal Emergency Management Agency. * FEMA (2024b). Hazus Earthquake Model Technical Manual. Federal Emergency Management Agency. * Goda K., Atkinson G.M. (2011). Seismic performance of wood-frame houses in south-western British Columbia. *Earthquake Engineering & Structural Dynamics*, 40(8), 903-924. * Gondaliya K., Murty C.V.R. (2023). Seismic fragility of reinforced concrete buildings in India. *Structures*, 48, 1326-1340. * Haach V.G., Vasconcelos G., Lourenço P.B. (2010). Experimental analysis of reinforced concrete block masonry walls subjected to in-plane cyclic loading. *Journal of Structural Engineering*, 136(4), 452-462. * Jouneghani H.G., Haghollahi A. (2020). Assessing the seismic behavior of steel moment frames equipped by elliptical-bracing. *Journal of Building Engineering*, 31, 101333. * Karanikoloudis G., Lourenço P.B. (2018). Structural assessment and seismic vulnerability of earthen historic structures. *Engineering Structures*, 172, 295-311. * Lang D.H., Kumar A., Sulaymanov S., Meslem A. (2018). Building typologies and fragility functions for northern Central Asia. *Natural Hazards and Earth System Sciences*, 18(11), 2959-2975. * Loss C., Tannert T., Tesfamariam S. (2013). Modern timber buildings: Challenges and opportunities in seismic design. *Building and Environment*, 62, 103-112. * Lotfy I., El-Shazly A., Husain M. (2022). Seismic evaluation of reinforced masonry shear walls. *Structures*, 36, 833-847. * Magenes G., Calvi G.M. (1992). Cyclic behavior of brick masonry walls. In *10th World Conference on Earthquake Engineering*, Madrid, Spain. * Markulak D., Radić I., Sigmund V. (2013). Cyclic testing of single bay steel frames with various types of masonry infill. *Engineering Structures*, 51, 267-277. * McCormick J., Aburano H., Ikenaga M., Nakashima M. (2007). Permissible residual deformation levels for building structures considering both safety and human elements. In *14th World Conference on Earthquake Engineering*. * Murcia-Delso J., Shing P.B. (2012). Fragility analysis of reinforced masonry shear walls. *Earthquake Spectra*, 28(4), 1523-1547. * Nafeh A.M.B., O'Reilly G.J. (2024). Simplified pushover-based seismic risk assessment methodology for existing infilled RC frame buildings. *Bulletin of Earthquake Engineering*, 22(1), 251-283. * Nucera F., Santini A., Tripodi E., Saetta A. (2012). Influence of soil-structure interaction on the seismic response of a confined masonry building. *Engineering Structures*, 39, 101-115. * Parra P.F., Arteta C.A., Moehle J.P. (2019). Seismic fragility functions for thin reinforced concrete walls. *Engineering Structures*, 184, 147-160. * Perez Gavilan J.J., Flores L.E., Alcocer S.M. (2015). An experimental study of confined masonry walls with varying aspect ratios. *Earthquake Spectra*, 31(2), 945-968. * Radnić J., Grgić N., Matešan D., Baloević G. (2013). Shaking table testing of reinforced concrete frames with masonry infill. *Gradevinar*, 65(7), 605-620. * Ramirez C.M., Lignos D.G., Miranda E., Kolwicz D. (2012). Fragility functions for pre-Northridge welded steel moment-resisting beam-to-column connections. *Engineering Structures*, 45, 574-584. * Ricci P., De Luca F., Verderame G.M. (2011). 6th April 2009 L'Aquila earthquake, Italy: reinforced concrete building performance. *Bulletin of Earthquake Engineering*, 9(1), 285-305. * Sabelli R., Mahin S., Chang C. (2003). Seismic demands on steel braced frame buildings with buckling-restrained braces. *Engineering Structures*, 25(5), 655-666. * Shedid M.T., Drysdale R.G., El-Dakhakhni W.W. (2008). Behavior of partially grouted reinforced masonry shear walls under reversed cyclic loading. *Journal of Structural Engineering*, 134(11), 1754-1764. * Silveira D., Varum H., Costa A. (2018). Seismic behavior of rammed earth constructions. *Construction and Building Materials*, 175, 293-306. * Suzuki A., Iervolino I. (2020). Hazard-consistent intensity measure conversion for fragility and risk analysis. *Earthquake Engineering & Structural Dynamics*, 49(10), 952-972. * Tarquini D., Dal Lago B., Peloso S. (2019). Large-scale testing of precast concrete structures. *Bulletin of Earthquake Engineering*, 17(4), 2049-2076. * Tarque N., Crowley H., Pinho R., Varum H. (2012). Displacement-based fragility curves for seismic assessment of adobe buildings in Cusco, Peru. *Earthquake Spectra*, 28(2), 759-794. * Tasnimi A.A., Mohebkhah A. (2011). Investigation on the behavior of brick-infilled steel frames with openings. *Engineering Structures*, 33(3), 968-980. * Tischer H., Mitchell D., McClure G. (2012). Comparison of structural response of an existing reinforced concrete building frame to near-field and far-field earthquakes. *Canadian Journal of Civil Engineering*, 39(12), 1291-1306. * Ugalde D., Lopez-Garcia D. (2020). Analysis of the seismic capacity of Chilean residential RC walls. *Bulletin of Earthquake Engineering*, 18(3), 1189-1209. * Ugalde D., Parra P.F., Lopez-Garcia D. (2019). Assessment of the seismic capacity of tall wall buildings using nonlinear finite element modeling. *Bulletin of Earthquake Engineering*, 17(11), 6183-6203. * Villar-Vega M., Silva V., Crowley H., et al. (2017). Development of a fragility model for the residential building stock in South America. *Earthquake Spectra*, 33(2), 581-604. * Voon K.C., Ingham J.M. (2006). Experimental in-plane shear strength investigation of reinforced concrete masonry walls. *Journal of Structural Engineering*, 132(3), 400-408.