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Seismic Spectrum Calculator to EC8

This calculator allows the user to evaluate the impact of an earthquake on a structure, by generating the Elastic Response Spectrum, Design Spectrum and the resulting base shear (seismic force).
All calculations are performed in accordance with EN 1998-1 2004 Eurocode 8: Design of structures for earthquake resistance - Part 1: General rules, seismic actions and rules for buildings. This code is typically referred to as "EC8".

Calculation

List of parameters used in this calculator


Spectrum Type=Type 1 or 2 (High or Low Seismicty Level)\footnotesize{Spectrum\ Type= \text{Type\ 1\ or\ 2\ (High\ or\ Low\ Seismicty\ Level)}}

Ground Type=A, B, C, D or E \footnotesize{Ground\ Type= \text{A,\ B,\ C,\ D\ or\ E\ }}

ξ=Viscous damping ratio of the structure (%)\xi=\footnotesize{ \text{Viscous\ damping\ ratio\ of\ the\ structure\ (}}\%\text{)}

agR=Reference peak ground acceleration on type A ground(g)\footnotesize{a_{gR} = \text{Reference peak ground acceleration on type A ground}\hspace{0.1cm}(g)}

T1=fundamental period of the building in the horizontal direction of interest (seconds)\footnotesize{T_1 = \text{fundamental period of the building in the horizontal direction of interest }(seconds)}

Technical notes

  1. Spectrum Type and
    
    input cells should be taken from the National Annex.
  2. Viscous damping ratio, ξ is typically taken as 5%
  3. Ground type (A, B, C or D) is described by the stratigraphic profiles of your site. Refer to EC8 Table 3.1.
  4. Fundamental period of the building, T1 may be approximated by empirical formulae or determined by dynamic analysis.

Inputs

General



Spectrum Type
:Type 1​



Ground Type
:A​



Importance Class
:2​



agR
:0.25g​


Elastic Spectrum



ξ
:500%​


Design Spectrum



q
:3​


Base Shear



T1
:0.16sec​



m
:2000t​



Number of storeys of the building
:3​


Output

Base Shear



Fb
:4087.49999999999kN​


Horizontal Seismic Response Spectra




Explanation

Why are structures affected by earthquakes?

An earthquake event moves the ground, causing a ground acceleration (E).
Earthquake ground acceleration causing movement in a structure

A building with negligible mass subject to an earthquake motion at it's base, will move back and forth following the exact motion of the earthquake but no force will be generated in the building. A building with mass subject to an earthquake motion at it's base will not move in one unit. There will be a difference in motion between the top of the building and its base, this is called relative displacement (U). This is similar to what happens to a person on a train when the train moves suddenly. It is the relative displacement that causes forces to be generated in the building.
Absolute displacement (D) of a massless structure under an earthquake

Relative displacement (U) of a structure with mass under an earthquake



How do we quantify the affect of an earthquake on a structure?

If we multiply this relative displacement by the stiffness of the building, we get a shear force.

relative displacement (U)×stiffness (k)=shear force (V)\text{relative displacement }(U) \times \text{stiffness }(k)= \text{shear force }(V)
Shear force (V) is the lateral earthquake force used in the design of the building. If we divide this shear force by the mass of the building, we get acceleration.

shear force (V)mass (M)=acceleration (a)\dfrac{\text{shear force }(V) }{\text{mass }(M)} = \text{acceleration }(a)
Acceleration (a) is a quantity used to measure the response of the building to the earthquake. We prefer to use acceleration rather then relative displacement because we can directly obtain the design force by multiplying acceleration by the mass of the building.
In structural design we are concerned about the maximum force, and hence the maximum acceleration in the building due to an earthquake.
Acceleration in a structure vs time, under an earthquake load



What is a response spectrum?

All structures (in fact, all things!) vibrate and have a frequency that it naturally vibrates in. This is called it's fundamental frequency (f1) and the reciprocal is called it's fundamental period (T1), which depend on the mass and stiffness of the structure. Under an earthquake, how much a building accelerates depends mainly on it's fundamental period.
To formulate a trend, we can take many structures with different fundamental period's

, subject them to the same earthquake and plot the maximum accelerations (also referred to as spectral accelerations) that occur in each structure due to that earthquake. This is called a response spectrum, it is a plot of how structures respond to an earthquake. Each response spectrum is specific to a unique earthquake event.
National codes will envelope all response spectra from earthquakes that are expected to occur in that area. The exact shape of the enveloped response spectrum is based on location and soil type. In the case of EC8, these factors are represented by the parameters

,

,

and

.
Example of an enveloped response spectrum provided in a national code (adapted from EC8 Figure 3.1)

Engineer's use seismic response spectrum plots to read off the spectral acceleration (Se) for their specific building with fundamental period (T1). Multiplying this spectral acceleration by the mass of their building gives the design base shear force that their building must resist. In a building, this base shear can be resisted by ground friction of a foundation raft or by lateral shear in piles.
In summary, to evaluate the impact of an earthquake on a structure the engineer will produce a response spectrum as per the national code and then determine the base shear force to design the structure against.

💡Elastic Response Spectrum vs Design Spectrum

Ductility of a structural system is the capacity of the system to dissipate energy under an earthquake. Ductile behaviour is inelastic (nonlinear).
However, inelastic structural analysis is time-consuming. EC8 allows engineers to avoid explicit inelastic structural analysis by performing an elastic analysis to obtain the Elastic Response Spectrum and then reducing it by a factor, named the behaviour factor (q). This reduction allows the ductility of the structure to be accounted for, and is called the Design Spectrum.
The Design Spectrum shall be used to obtain the seismic shear force that the structure must be designed to withstand.


EC8 Parameters and Equations

Below are the equations as provided in EC8, that are used in this calculator.

Horizontal Elastic Response Spectrum

Horizontal elastic response spectrum,

in units of acceleration due to gravity

is calculated as per EC8 Section 3.2.2.2 Equations 3.2, 3.3, 3.4, and 3.5.

0 ≤ T ≤ TB: Se(T)=agS[1+TTB(η 2.5−1)]0\ \le\ T\ \le\ T_B:\ S_e(T) = a_gS[1+\frac{{T}}{T_B}(\eta\ 2.5-1)]

TB ≤ T ≤ TC: Se(T)=agSη 2.5T_B\ \le\ T\ \le\ T_C:\ S_e(T) = a_gS\eta\ 2.5

TC ≤ T ≤ TD: Se(T)=agSη 2.5TCTT_C\ \le\ T\ \le\ T_D:\ S_e(T) = a_gS\eta\ 2.5\frac{{T_C}}{T}

TD ≤ T ≤ 4s: Se(T)=agSη 2.5TCTDT2T_D\ \le\ T\ \le\ 4s:\ S_e(T) = a_gS\eta\ 2.5\frac{{T_CT_D}}{T^2}

η =(10/(5+ξ) ≥ 0.55 \eta\ = \sqrt(10/(5+\xi)\ \geq\ 0.55\
The value of the periods

,

and

and of the soil factor

describing the shape of the elastic response spectrum depend upon the ground type. For five ground types (A to E) the recommended values of the parameters are provided in Table 3.2 and 3.3.
EC8 Section 3.2.2.2 Table 3.2 - Spectrum Type 1 parameters

EC8 Section 3.2.2.2 Table 3.3 - Spectrum Type 2 parameters


Horizontal Design Spectrum

Horizontal design spectrum,

is calculated as per EC8 Section 3.2.2.5 Equations 3.13, 3.14, 3.15, and 3.16.

0 ≤ T ≤ TB: Sd(T)=agS[23+TTB(2.5q−23)]0\ \le\ T\ \le\ T_B:\ S_d(T) = a_gS[\frac{{2}}{3}+\frac{{T}}{T_B}(\frac{{2.5}}{q}-\frac{{2}}{3})]

TB ≤ T ≤ TC: Sd(T)=agS[2.5q]T_B\ \le\ T\ \le\ T_C:\ S_d(T) = a_gS[\frac{{2.5}}{q}]

TC ≤ T ≤ TD: Sd(T)={=agS2.5qTCT≥β agT_C\ \le\ T\ \le\ T_D:\ S_d(T) = \begin{cases} = a_gS\frac{2.5}{q}\frac{T_C}{T}&\\\geq \beta\ a_g \end{cases}

TD ≤ T : Sd(T)={=agS2.5qTCTDT2≥β ag T_D\ \le\ T\ :\ S_d(T) = \begin{cases} = a_gS\frac{2.5}{q}\frac{T_CT_D}{T^2}&\\\geq \beta\ a_g \end{cases}\

Seismic Base Shear Force

The seismic base shear force,

is a horizontal force at the base of a structure caused by an earthquake. Refer to EC8 Section 4.3.3.2.2 for relevant equations and parameters.

Fb=Sd(T1) mλ T1 =Fundamental period of the building for lateral motion in the direction consideredm =Total mass of the buildingλ =Correction factor If T1≤2TC and the building has more than two storeys, λ =1,otherwise λ= 0.85F_b = S_d(T_1)\ m\lambda\\\footnotesize\ T_1\ = \text{Fundamental\ period\ of\ the\ building\ for\ lateral\ motion\ in\ the\ direction\ considered}\\m\ = \text{Total\ mass\ of\ the\ building}\\\lambda\ = \text{Correction\ factor}\\\ If\ T_1\le2T_C\ \text{and\ the\ building\ has\ more\ than\ two\ storeys, }\lambda\ =1,\text{otherwise } \lambda =\ 0.85


Importance Factor

As per EC8 Section 4.2.5, buildings are classified in four importance classes. An importance factor,

is assigned to each importance class and relates to the consequences of a structural failure.
If a structure serves a critical function, the impact of seismic forces are given greater consideration than those that serve minor importance for public safety. The importance factor affects parameters and design assumptions in seismic design.
EC8 Section 4.2.5 Table 4.3 - Importance Classes for Buildings


Behaviour Factor

The behaviour factor relates to the ductility of the structural system, that is, the capacity of the system to dissipate energy under an earthquake. The behaviour factor reduces the seismic forces obtained from a linear analysis, in order to avoid inelastic structural analysis in design.
A structure is assigned a ductility class, which includes a high capacity to dissipate energy (DCH) and a medium capacity to dissipate energy (DCM). For DCH and DCM the behaviour factor can be found in EC8 Section 5.2.2.2 Table 5.1 based on different structural types.
EC8 Section 5.2.2.2 Table 5.1 - Basic value of the behaviour factor



Acknowledgements

This calculation was built in collaboration with Hakan Keskin Learn more.

Related Resources

  1. Design Guide: Steel Bracing Systems
  2. Eurocode calculators