Note: A building of up to 25 storeys may have one lumped mass to represent one storey (i.e. N = No. of storeys). A lumped mass in a taller building may represent two or more storeys.

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(The input storey mass, storey height above ground, and static deflection are shown in Table 2.)

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The effective static displacement (δ_eff), and effective mass (M_eff,i), stiffness (K_eff,i), and natural period (T₁) for the first mode of vibration are determined using equation 1 to 4. Similarly, the first mode displacement (δ*_i) and storey forces (F*_i) are calculated by scaling the static displacement values by the ratio of displacement at the performance point to δ_eff using equation 5 and 6. The performance point is the point of intersection of capacity curve with the acceleration displacement response spectrum (ADRS) diagram as shown in Figure 2. $$ {δ_{eff} = \frac{\sum^n_{i=1}{m_i \times δ_i^2}}{ \sum^n_{i=1} {m_i \times δ_i}}}\tag{1}$$ $$ {M_{eff,1} = \frac{\left(\sum^n_{i=1}{m_i \times δ_i}\right)^2}{ \sum^n_{i=1} {m_i \times δ_i^2}}}\tag{2}$$ $${K_{eff,1} = \frac{\sum^n_{i=1}{F_{i}}}{δ_{eff}}}\tag{3}$$ $$ {T_{1} = 2\pi \sqrt{\frac{M_{eff,1}}{K_{eff,1}}}}\tag{4}$$ $${δ^*_{i} = \frac{δ_{performance\:point}}{δ_{eff}}\times δ_i}\tag{5}$$ $${F^*_{i} = \frac{δ_{performance\:point}}{δ_{eff}}\times F_i}\tag{6}$$

Effective modal mass (M_eff,i), modal participation ratio (MPR_i), and effective stiffness (K_eff,i) for second and third modes of vibration are calculated using equation 7 to 11. Similarly, natural period for second and third modes of vibration are determined from first mode period based on the ratio defined in 'Input Parameter-V'. $$ {M_{eff,2} = \sum^n_{i=1}({m_{i}}\:{{Γ_{2}}\:\:{ϕ_{i,2}}})}\tag{7}$$ $$ {M_{eff,3} = \sum^n_{i=1}({m_{i}}\:{{Γ_{3}}\:\:{ϕ_{i,3}}})}\tag{8}$$ $$ {MPR_{i} = \frac{M_{eff,i}}{M_{total}}}\tag{9}$$ $${K_{eff,2} = \left(\frac{2\pi}{T_{2}}\right)^2 \times {M_{eff,2}}}\tag{10}$$ $${K_{eff,3} = \left(\frac{2\pi}{T_{3}}\right)^2 \times {M_{eff,3}}}\tag{11}$$ Where:
m_i = mass of each lumped mass 'i',
δ_i = static deflection of each lumped mass 'i',
n = number of lumped masses,
F_i = inertia force determined based on EC8.
Γ ϕ = modal coefficient for higher modes of vibration,
M_total = total mass of the building.

The combination of storey drift (δ*_{i,3 modes}), inter-storey drift (θ*_{i,3 modes}), storey shear (V_{i,3 modes}), base shear (V_{b,3 modes}), and total overturning moment (OM) of the first three modes of vibration are calculated using equation 12 to 16. $$ {δ^*_{i,3\:modes} = \sqrt{{δ^{*2}_{i}} + {\left({{Γ_{2}}\:\:{ϕ_{i,2}}\:\:{RSD_{2}}}\right)}^2 + {\left({{Γ_{3}}\:\:{ϕ_{i,3}}\:\:{RSD_{3}}}\right)}^2}}\tag{12}$$ $$ {θ^*_{i,3\:modes} = \sqrt{{\left(\frac{{{δ^{*}_{i}-δ^{*}_{i-1}}}}{h_{i}}\right)}^2 + {\left(\frac{{{Γ_{2}}\:\:{(ϕ_{i,2}-ϕ_{i-1,2})}\:\:{RSD_{2}}}}{h_{i}}\right)}^2 + {\left(\frac{{{Γ_{3}}\:\:{(ϕ_{i,3}-ϕ_{i-1,3})}\:\:{RSD_{3}}}}{h_{i}}\right)}^2}}\tag{13}$$ $$ {F^*_{i,3\:modes} = \sqrt{{F^{*2}_{i}} + {\left({m_{i}}\:{{Γ_{2}}\:\:{ϕ_{i,2}}\:\:{RSA_{2}}}\right)}^2 + {\left({m_{i}}\:{{Γ_{3}}\:\:{ϕ_{i,3}}\:\:{RSA_{3}}}\right)}^2}}\tag{14}$$ $$ {V_{i,3\:modes} = \sqrt{\left(\sum^n_{i}{F^*_{i}}\right)^2 + {\left(\sum^n_{i}{m_{i}}\:{{Γ_{2}}\:\:{ϕ_{i,2}}\:\:{RSA_{2}}}\right)}^2 + {\left(\sum^n_{i}{m_{i}}\:{{Γ_{3}}\:\:{ϕ_{i,3}}\:\:{RSA_{3}}}\right)}^2}}\tag{15}$$ $$ {V_{b,3\:modes} = \sqrt{{\left({V_{b,1}}\right)}^2+{\left({V_{b,2}}\right)}^2+{\left({V_{b,3}}\right)}^2}}\tag{16}$$ $$ {OM = \sqrt{\left(\sum^n_{i=1}({{F^*_{i}} \times H_{i}})\right)^2 + {\left(\sum^n_{i}({m_{i}}\:{{Γ_{2}}\:\:{ϕ_{i,2}}\:\:{RSA_{2}}}\times H_{i})\right)}^2 + {\left(\sum^n_{i}({m_{i}}\:{{Γ_{3}}\:\:{ϕ_{i,3}}\:\:{RSA_{3}}}\times H_{i})\right)}^2}}\tag{17}$$ Where:
RSD₂ and RSD₃ = spectral displacement at the second, and third modal natural period,
RSA₂ and RSA₃ = spectral acceleration at the second, and third modal natural period,
h_i and H_i = height of storey 'i', and storey height from ground, respectively.

Table 2: Summary of input information of the Floor Mass, Height and Static Displacement (δ_i), and GFM result of the Floor Inertia Forces (F_{i_GFM}), Displacement (δ_{i_GFM}), Inter-Storey Drift (θ_{i_GFM}), Storey Shear (V_{i_GFM}), Total Base Shear, and Total Overturning Moment:

1. Building and Construction Standards Committee (2013). Guidebook for Design of Buildings in Singapore to Requirements in SS EN 1998-1.Building and Construction Standards Committee, Singapore.
2. European Commitee for Standardization (2004). EN 1998-1:2004 Eurocode 8: Design of structures for earthquake resistance - Part 1 : General rules, seismic actions and rules for buildings.
3. Building and Construction Standards Committee (2013). EXAMPLE CALCULATION – Seismic Actions to BC3: 2013.Building and Construction Standards Committee, Singapore.
4. Khatiwada, P.; Lumantarna, E.; Lam, N.; Looi, D. Fast Checking of Drift Demand in Multi-Storey Buildings with Asymmetry. Buildings 2021, 11, 13. https://doi.org/10.3390/buildings11010013

The authors assume no responsibility for any injury, damage, liability, negligence and/or otherwise to any individual or property from the use or application of any of the methods, products, instructions, or ideas contained in the material herein.

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