Seismic Analysis by Generalised Force Method (GFM)
The Generalised Force Method (GFM) is a technique to estimate the dynamic behaviour of a building responding to the design seismic actions. The users need only to input some basic information of the building and an estimate of its deflection in static conditions. The version of GFM introduced herein for 2D analysis gives a quick estimate of the dynamic behaviour of a building to assist in decision making at any stages of the design and to verify results as reported by a commercial software. Although the code lateral force method forms part of GFM, it allows vertical irregularities in the building. For a building with less than 30 m in height above ground, we recommend using GFM based on the first mode of vibration only. Whereas, for a building with greater than or equal to 30 m in height above ground, we recommend using GFM based on the first three modes of vibration. The user has the opportunity to make the choice by the press of a button in below.
Select the preferred Generalised Force Method (GFM):
1. Input Parameters:
| I. No. of Lumped Masses: |
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.
| II. Probability Factor * Hazard Design Factor, kpZ: | |
| III. Ground Type: | |
| IV. Type of Structural System: |
V. Information about the Building:
Note:
- Please provide input for the roof on the first row and follow the descending pattern up to the base.
- The table can be filled manually or by performing a copy (CTRL+C) and paste (CTRL+V) of data from an Excel spreadsheet.
- The ‘Lateral Force’ can be user-defined or program calculated.
- For program calculated values of AS1170.4-2007(Amdt 2018) lateral forces, first provide the information: mass and height, and then press the button ‘PRESS TO GET LATERAL FORCE’. The lateral force will be calculated based on T1= 0.05 Kt H3/4 and will be generated into the table.
- The generated lateral forces can be used in any software to determine static storey deflections. The results of which are entered into the table.
| No. | Mass (tonnes) |
Height above Ground (m) |
(kN) |
User’s estimate of Static Deflection (mm) |
| 1 (top) | ||||
| 2 | ||||
| 3 | ||||
| 4 | ||||
| 5 | ||||
| 6 | ||||
| 7 | ||||
| 8 | ||||
| 9 | ||||
| 10 | ||||
| 11 | ||||
| 12 | ||||
| 13 | ||||
| 14 | ||||
| 15 | ||||
| 16 | ||||
| 17 | ||||
| 18 | ||||
| 19 | ||||
| 20 | ||||
| 21 | ||||
| 22 | ||||
| 23 | ||||
| 24 | ||||
| 25 |
(The input storey mass, storey height above ground, and static deflection are shown in Table 2.)
VI. Ratios of Natural Periods:
Note:
- The program requires the ratios of natural periods to determine the solution for higher modes of vibration. T1, T2, and T3 in the table are the natural periods of the first three modes of vibration.
- There are two methods for specifying the ratios of natural periods: user-defined and default.
- By pressing the ‘USER DEFINED’ button, we can specify the preferred ratios of the natural periods into the table.
- Similarly, by pressing the ‘DEFAULT’ button, the program will automatically consider the default ratios provided in the table.
| T2 / T1 | T3 / T1 |
| T2 / T1 | T3 / T1 |
| 0.25 | 0.125 |
VII. Modal Coefficients (Γ ϕ) for Second and Third Modes of Vibration:
Note:
- The modal coefficient is the displacement of each lumped mass of the structure under a specific mode of vibration divided by the response spectral displacement corresponding to the natural period of that mode of vibration.
- To determine the solution for higher modes of vibration, the modal coefficients for each normalised height are required.
- The ‘Normalised Height’ is the height of the lumped mass above ground divided by the total height of the building.
- There are two methods for specifying the modal coefficients: user-defined and default.
- By pressing the ‘USER DEFINED’ button, we can specify the preferred modal coefficients into the table.
- Similarly, by pressing the ‘DEFAULT’ button, the program will automatically consider the widely recognised modal coefficients provided in the table.
- The table can be filled manually or by performing a copy (CTRL+C) and paste (CTRL+V) of data from an Excel spreadsheet.
| Normalized Height | Modal Coefficients for Second Mode of Vibration | Modal Coefficients for Third Mode of Vibration |
| 1 | ||
| 0.9 | ||
| 0.8 | ||
| 0.7 | ||
| 0.6 | ||
| 0.5 | ||
| 0.4 | ||
| 0.3 | ||
| 0.2 | ||
| 0.1 | ||
| 0 |
| Normalized Height | Modal Coefficients for Second Mode of Vibration | Modal Coefficients for Third Mode of Vibration |
| 1 | 0.639 | 0.312 |
| 0.9 | 0.373 | 0.066 |
| 0.8 | 0.077 | -0.181 |
| 0.7 | -0.201 | -0.31 |
| 0.6 | -0.419 | -0.258 |
| 0.5 | -0.547 | -0.062 |
| 0.4 | -0.573 | 0.168 |
| 0.3 | -0.501 | 0.315 |
| 0.2 | -0.356 | 0.321 |
| 0.1 | -0.173 | 0.194 |
| 0 | 0 | 0 |
2. Results:
2.1 Seismic Design Response Spectrum:
Figure 1: Acceleration-Displacement Response Spectrum (ADRS) Diagram
2.2 Solution for First Mode of Vibration:
The effective static displacement (δeff), and effective mass (Meff,i), stiffness (Keff,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 δeff 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 1. $$ {δ_{eff} = \frac{\sum^n_{i=1}{m_i \times δ_i^2}}{ \sum^n_{i=1} {m_i \times δ_i}}}\tag{1}$$ $$ {M_{eff,1} = \frac{\left(\sum^n_{i=1}{m_i \times δ_i}\right)^2}{ \sum^n_{i=1} {m_i \times δ_i^2}}}\tag{2}$$ $${K_{eff,1} = \frac{\sum^n_{i=1}{F_{i}}}{δ_{eff}}}\tag{3}$$ $$ {T_{1} = 2\pi \sqrt{\frac{M_{eff,1}}{K_{eff,1}}}}\tag{4}$$ $${δ^*_{i} = \frac{δ_{performance\:point}}{δ_{eff}}\times δ_i}\tag{5}$$ $${F^*_{i} = \frac{δ_{performance\:point}}{δ_{eff}}\times F_i}\tag{6}$$
The base shear (Vb,1) and overturning moment (OM) for the first mode of vibration are calculated using equation 7 and 8.
$$ {V_{b,1} = \sum^n_{i=1}{F^*_{i}}}\tag{7}$$
$$ {OM = \sqrt{\left(\sum^n_{i=1}({{F^*_{i}} \times H_{i}})\right)^2}}\tag{8}$$
Where:
mi = mass of each lumped mass ‘i’,
δi = static deflection of each lumped mass ‘i’,
n = number of lumped masses,
Fi = inertia force determined based on AS1170.4.
The results for the first mode are given in Table 1 and Table 2.
Γ ϕ = modal coefficient for higher modes of vibration,
Mtotal = total mass of the building.
RSD2 = spectral displacement at the second modal natural period,
RSD3 = spectral displacement at the third modal natural period,
RSA2 = spectral acceleration at the second modal natural period,
RSA3 = spectral acceleration at the third modal natural period,
hi = height of storey ‘i’,
Hi = height of storey ‘i’ from ground.
2.3 Solution including Higher Modes of Vibration:
I. Mode Shape and Modal Displacement:
Figure 2: Normalised Mode Shape for the First Three Modes of Vibration
Figure 3: Modal Displacement for the First Three Modes of Vibration
II. Effective Modal Mass, Modal Participation Ratio, Effective Stiffness and Natural Period for Higher Modes of Vibration:
Effective modal mass (Meff,i), modal participation ratio (MPRi), and effective stiffness (Keff,i) for second and third modes of vibration are calculated using equation 8 to 12. Similarly, natural period for second and third modes of vibration are determined from first mode period based on the ratio defined in Section 1 (VII: ‘a’ and ‘b’).
$$ {M_{eff,2} = \sum^n_{i=1}({m_{i}}\:{{Γ_{2}}\:\:{ϕ_{i,2}}})}\tag{7}$$
$$ {M_{eff,3} = \sum^n_{i=1}({m_{i}}\:{{Γ_{3}}\:\:{ϕ_{i,3}}})}\tag{8}$$
$$ {MPR_{i} = \frac{M_{eff,i}}{M_{total}}}\tag{9}$$
$${K_{eff,2} = \left(\frac{2\pi}{T_{2}}\right)^2 \times {M_{eff,2}}}\tag{10}$$
$${K_{eff,3} = \left(\frac{2\pi}{T_{3}}\right)^2 \times {M_{eff,3}}}\tag{11}$$
Where:
Γ ϕ = modal coefficient for higher modes of vibration,
Mtotal = total mass of the building.
Table 1: Effective Modal Mass, Modal Participation Ratio, Effective Stiffness and Natural Period for the First Three Modes of Vibration:
III. Storey Drift, Inter-Storey Drift, Storey Shear, Base Shear, and Overturning Moment
The combination of storey drift (δ*i,3 modes), inter-storey drift (θ*i,3 modes), storey shear (Vi,3 modes), base shear (Vb,3 modes), and total overturning moment (OM) of the first three modes of vibration are calculated using equation 13 to 18.
$$ {δ^*_{i,3\:modes} = \sqrt{{δ^{*2}_{i}} + {\left({{Γ_{2}}\:\:{ϕ_{i,2}}\:\:{RSD_{2}}}\right)}^2 + {\left({{Γ_{3}}\:\:{ϕ_{i,3}}\:\:{RSD_{3}}}\right)}^2}}\tag{12}$$
$$ {θ^*_{i,3\:modes} = \sqrt{{\left(\frac{{{δ^{*}_{i}-δ^{*}_{i-1}}}}{h_{i}}\right)}^2 + {\left(\frac{{{Γ_{2}}\:\:{(ϕ_{i,2}-ϕ_{i-1,2})}\:\:{RSD_{2}}}}{h_{i}}\right)}^2 + {\left(\frac{{{Γ_{3}}\:\:{(ϕ_{i,3}-ϕ_{i-1,3})}\:\:{RSD_{3}}}}{h_{i}}\right)}^2}}\tag{13}$$
$$ {F^*_{i,3\:modes} = \sqrt{{F^{*2}_{i}} + {\left({m_{i}}\:{{Γ_{2}}\:\:{ϕ_{i,2}}\:\:{RSA_{2}}}\right)}^2 + {\left({m_{i}}\:{{Γ_{3}}\:\:{ϕ_{i,3}}\:\:{RSA_{3}}}\right)}^2}}\tag{14}$$
$$ {V_{i,3\:modes} = \sqrt{\left(\sum^n_{i}{F^*_{i}}\right)^2 + {\left(\sum^n_{i}{m_{i}}\:{{Γ_{2}}\:\:{ϕ_{i,2}}\:\:{RSA_{2}}}\right)}^2 + {\left(\sum^n_{i}{m_{i}}\:{{Γ_{3}}\:\:{ϕ_{i,3}}\:\:{RSA_{3}}}\right)}^2}}\tag{15}$$
$$ {V_{b,3\:modes} = \sqrt{{\left({V_{b,1}}\right)}^2+{\left({V_{b,2}}\right)}^2+{\left({V_{b,3}}\right)}^2}}\tag{16}$$
$$ {OM = \sqrt{\left(\sum^n_{i=1}({{F^*_{i}} \times H_{i}})\right)^2 + {\left(\sum^n_{i}({m_{i}}\:{{Γ_{2}}\:\:{ϕ_{i,2}}\:\:{RSA_{2}}}\times H_{i})\right)}^2 + {\left(\sum^n_{i}({m_{i}}\:{{Γ_{3}}\:\:{ϕ_{i,3}}\:\:{RSA_{3}}}\times H_{i})\right)}^2}}\tag{17}$$
$$ {V_{b,1} = \sum^n_{i=1}{F^*_{i}}}\tag{18}$$
$$ {V_{b,2} = \sum^n_{i=1}{m_{i}}\:{{Γ_{2}}\:\:{ϕ_{i,2}}\:\:{RSA_{2}}}}\tag{19}$$
$$ {V_{b,3} = \sum^n_{i=1}{m_{i}}\:{{Γ_{3}}\:\:{ϕ_{i,3}}\:\:{RSA_{3}}}}\tag{20}$$
Where:RSD2 = spectral displacement at the second modal natural period,
RSD3 = spectral displacement at the third modal natural period,
RSA2 = spectral acceleration at the second modal natural period,
RSA3 = spectral acceleration at the third modal natural period,
hi = height of storey ‘i’,
Hi = height of storey ‘i’ from ground.
Table 1: Effective Modal Mass, Modal Participation Ratio, Effective Stiffness and Natural Period:
Figure 4: GFM Storey Drift
Figure 2: GFM Storey Drift
Figure 5: GFM Inter-Storey Drift
Figure 3: GFM Inter-Storey Drift
Figure 6: GFM Storey Shear
Figure 4: GFM Storey Shear
| In Table 2, mi is storey mass, Hi is storey height (above ground), δi_static is storey static deflection, and Fi_code is storey inertia forces defined in the input section. Similarly, δi_GFM is storey drift, Fi_GFM is storey inertia forces, θi_GFM is inter-storey drift, and Vi_GFM is storey shear determined based on generalised force method. |
Table 2: Summary of input information of the Floor Mass, Height and Static Displacements (δi), and GFM result of the Floor Inertia Forces (Fi_GFM), Displacements (δi_GFM), Inter-Storey Drift (θi_GFM), Storey Shear (Vi_GFM), Total Base Shear, and Total Overturning Moment:
3. Estimation of the 3D to 2D Displacement Ratio and the 3D Displacement Profile
This estimation is achieved based on the input of the floorplan of the building (drawn to scale). Please upload floor plan (any image file format) below.
Upload Image File of the Building Floor Plan Here:
Note: Any information outside the boundary of the floorplan and any written text within the floorplan need to be removed before uploading the image. Click ‘More Info’ button to see an example.
| Select The Direction of Loading: | |
| Input Length of the Red Rectangle along X-axis (in meters): |
Table 3: Torsional parameters and 3D/2D displacement ratio.
Figure: Comparison of the 2D and 3D displacement profile (results are based on GFM Method).
4. References:
  1. Khatiwada, P.; Lumantarna, E.; Lam, N.; Looi, D. Fast Checking of Drift Demand in Multi-Storey Buildings with Asymmetry. Buildings 2021, 11, 13.
  2. Khatiwada, P.; Lumantarna, E. Simplified Method of Determining Torsional Stability of the Multi-Storey Reinforced Concrete Buildings. CivilEng 2021, 2(2), 290-308.
3. Standard, A. (2007). AS 1170.4-2007 (Amdt 2018).Structural Design Actions, Part 4: Earthquake Actions in Australia.
4. Lumantarna, E.; Mehdipanah, A.; Lam, N.T.K. and Wilson, J.K. (2018). Chapter 4: “Methods of Structural Analysis for Buildings in Regions of Low to Moderate Seismicity”. Guideline on Design of Buildings and Structures in Low-to-moderate Seismicity Countries. Professional Guide: PG-002, Chinese National Engineering Research Centre for Steel Construction (Hong Kong branch, The Hong Kong Polytechnic University), ISBN-13: 978-962-367-818-6
Disclaimer
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.