# Seismic analysis

**Contents**

# Introduction

Seismic calculation is a special case of forced vibration calculation, when the exciting effect is the ground acceleration which is time dependent and of course not periodical. The response of the structure to the ground acceleration will be a vibration like motion. The structure gets forces of inertia which is calculated ac- cording to the Newton’s law (F = m a) and it is proportional to the mass and acceleration. These equivalent forces of course cause internal forces, stresses and if they are larger than the limit value the structure may collapse.

From the above explanation we found that originally this is a dynamic phenomenon when the acceleration and so the inertia forces change in each second. The interaction of the ground and structure is complicated so in a given time the acceleration of the structure depends on several components:

- ground acceleration (the seismic magnitude and its development on time),
- the elasticity of the structure,
- the mass and mass distribution of the structure,
- the connection between the structure and ground, namely soil type.

Another complicated problem is to define exact direction of the ground motion in the seismic investigation. Generally the ground movement is assumed as an arbitrary horizontal motion but the vertical motion also may cause problem to the structure. Fundamentally the calculation process can be divided into three methods:

## Time history

This calculation is carried out as an ordinary forced vibration when the excitation is a time dependent acceleration function. These functions can be registered or simulated seismograms. Mathematically we always solve the differential equation system of the vibration by a suitable method (e.g. step-by-step method). From the results of the equation system (means the displacement of the structure) the internal forces can be calculated and the design can be performed.

Theoretically the method is exact, but several circumstances strongly constrain the usage:

- statistically the number of seismograms are insufficient,
- the calculation is very complicated and the runtime is long.

Because of the above mentioned difficulties, this method is not widespread and is not implemented in FEM-Design.

## Modal analysis

As was mentioned in the above method, the vibrations arising from the seismic effect are difficult to predict. So the modal analysis assumption starts from the investigation of the most unfavorable ground motion directions and the period time.

Expected value of the maximal accelerations belongs to the individual periods which are prescribed in the national codes and named as acceleration response spectrum. The horizontal axis shows the frequency or vibration time of a single degree mass-spring system and the vertical axis shows the maximum corresponding acceleration. (In the Civil Engineering practice vibration period is used instead of frequency.)

The results which belong to the different ground motion directions and structural eigenfrequencies are summarized on the basis of the probability theory, which assumes that not all the effects appear in the same time. Most frequently used summation rule is the SRSS (Square Root of Sum of Squares).

Although the modal analysis is the most accepted method all over the world (as well as in EC8), it has some disadvantages. Some of them are listed as follows:

- the results which are calculated using the SRSS summation rule are not simultaneous. For example for a bending moment in a point of the structure we can’t show the simultaneous normal force in the same point, because the summation is carried out from component to component separately. Consequence of the summation rule, other calculations (second order application, stability analysis) are not interpreted,
- mainly from the application of the statistical method, the graphical results weakly can be followed compare to the results of statical calculation,
- in a lot of cases great number of vibration shapes should be calculated to reach reasonable results which require long calculation time.

Despite of all disadvantages of this method, we can expect most trustable results if the code requirements are fulfilled.

## Lateral force method (Equivalent static load method)

The lateral force method partly eliminates the disadvantages of the previous method with simplification in certain cases. The method postulates that the dis- placement response of the structure for ground motion can be described with one (or both x', y' directions) mode shape. While this means generally a simplification or approximation, this method is suitable for a part of the structure (EC8 prescribes the condition of application). In this method the mode shape of the structure is a linear deviation or it is equivalent to the calculated fundamental vibration shape. In the case of linear deviation or mode shape the period also can be calculated by approximate formula.

The application of this method gives possibility to transform the seismic lateral forces to simple static loads and it is applicable as follows:

- these loads (seismic load cases) can be combined with other static loads,
- second order and stability analysis can be performed,
- it is also possible to use these loads for hand calculation, so the results can be checked easily.

This method is usable in FEM-Design with two options if the code permits:

- assumption of linear deviation shape when the period also can be defined by the user (Static, linear shape),
- application of the calculated fundamental vibration shape as the deformed shape of the structure and its period (Static, mode shape).

## National codes

Remarks in application of national codes:

- before releasing the current version of FEM-Design, only the Eurocode and Norwegian national code contained special description for seismic calculation. In the other codes FEM-Design supports only the general mo- dal analysis,
- most of the countries did not prepare the National Application Document (NAD) for the universal Eurocode, so the program uses the general pres-cription.

Supported national codes and methods:

British | Modal analysis |

Code independent | Modal analysis |

Danish | Modal analysis |

Eurocode (NA: - ) | EC8-2005 (No NAD, static method, modal analysis) |

Eurocode (NA: British ) | EC8-2005 (No NAD, static method, modal analysis) |

Eurocode (NA: German ) | EC8-2005 (No NAD, static method, modal analysis) |

Eurocode (NA: Italian ) | EC8-2005 (No NAD, static method, modal analysis) |

Finnish (B4:2001) | Modal analysis |

Finnish (By50:2005) | Modal analysis |

German | Modal analysis |

Hungarian | Modal analysis |

Norwegian | NS3491-12 (static method, modal analysis) |

Swedish | Modal analysis |

Norwegian code differs from Eurocode in a few places, so they are reviewed together and the differences are marked separately.

# Input data

Lorem ipsum dolor sit amet, consectetur adipiscing elit, sed do eiusmod tempor incididunt ut labore et dolore magna aliqua. Ut enim ad minim veniam, quis nostrud exercitation ullamco laboris nisi ut aliquip ex ea commodo consequat. Duis aute irure dolor in reprehenderit in voluptate velit esse cillum dolore eu fugiat nulla pariatur. Excepteur sint occaecat cupidatat non proident, sunt in culpa qui officia deserunt mollit anim id est laborum.

## Dynamic calculations and Mass definitions

To calculate the seismic effect it is necessary to know the vibration shapes and corresponding periods, except the static method (lateral force method: linear force distribution). Therefore a dynamic calculation should be done before performing seismic calculation, which gives sufficient vibration shapes of the structure. To perform the dynamic calculation, it is necessary to define mass distribution which can be defined in Load tab as concentrated mass or load case-mass conversion.

According to EC8 3.2.4(2), mass distribution should be made in the following way:

ΣG_{k}, _{j}"" + ""Σψ_{E, i}Q_{k, i }

where:

- ψ
_{E, i}is the combination coefficient for variable action i (see EC8 4.2.4), it shall be computed from the following expression:

ψ_{E, i }= ϕ ψ^{2, }i

The recommended values for ϕ are listed in EC8 Table 4.2.

The above formula means that mass conversation is made from all dead load without any factor, also masses in gravity direction temporary loads with reduced value.

## Design spectrum

The program contains EC8 and NS3491-12 predefined design spectra or the user can define its own spectra if necessary. The vertical spectrum is necessary when the vertical affect taken into account.

**EC8 design spectrum**

The code gives the horizontal and vertical spectra and although the value of variables is prescribed, they can be modified if necessary.

**Horizontal spectra**

Data of horizontal design spectra:

- Type type of spectra, which there are two in the code,
- Ground ground type, which can be A, B, C, D and E,

The above two data specify the values of S, TB, TC and TD, which can be found in EC8, table 3.2 and 3.3.

- ag is the design ground acceleration on type A ground (ag = γI ag R),
- S is the soil factor,
- q is the behavior factor, which depends on material and type of the structure,
- beta (β) is the lower bound factor for the horizontal design spectrum.

The Sd(T) horizontal design spectrum is based on EC8 3.2.2.5 as follow:

**Vertical spectra**

The built-in vertical design spectrum is derived from the horizontal spectrum using the aυg / ag multiplicator which can be found in EC8 table 3.4 and described in 3.2.2.5(5)-(7).

**Other input parameters (Others tab)**

In the Others tab, the user should set some parameters that effect the calculation and results.

- Ksi(ξ) is the viscous damping ratio, expressed as a percentage, gene- rally 5%. This data is used in modal analysis when the sum- mation of the effect of the same direction vibration shapes is carried out by the CQC (Complete Quadratic Combination), see later.
- qd is the displacement behavior factor, assumed equal to q unless otherwise specified.
- Foundation level when Static-linear shape is used, the program assumes that the foundation level is defined on that height. It means the pro- gram calculates the mass height from that level. In the other two calculation methods (Static-mode shape and Modal analysis) base shear force is drawn in that level and it is taken into consideration in the so called reduced mass calculation (details in Effective mass setting).

**NS3491-12 design spectrum**

**Horizontal spectra**

The built-in horizontal design spectrum is based on the following formula:

S_{d}(T_{i}) = k_{Q} k_{S} γ_{1} a_{g} S_{e}(T_{i}) k_{f, spiss}

where:

- Ksi(ξ) is the declining ratio for the structure, given in %. Usually 5%,
- k
_{Q}is a structure factor, dependent on the type of structure, - k
_{S}is a soil factor, dependent on the type of ground, - Gamma 1(γ
_{1}) is a seismic factor, dependent on the seismic class, - a
_{g}is the maximum ground acceleration, dependent on location and reference period, - S
_{e}(T_{i}) is the acceleration for the period Ti in the normalized response spectra, see below, - k
_{f,spiss}is a factor dependent on the reference period used.

**Vertical spectra**

S_{νd}(T_{ν,i}) = k_{ν} γ_{1} a_{g} S_{e}(T_{ν,i}) k_{f, spiss}

where

- k
_{ν}is the ratio between horizontal and vertical response spectra, mostly set to 0,7.

The normalized response spectrum in Norwegian code is based on four different formulas, each covering a part of the possible periods from 0 to 4 seconds. Periods over 4 seconds has to be treated in a different way anyhow, and can therefore be based on a manually written response spectrum.

In FEM-Design, we assume, the spectrum is constant for periods over 4 seconds and equal to the value of S_{d}(T = 4).

where:

- T is the vibration period,
- T
_{B}= 0,1sec, - T
_{C}= 0,25sec - T
_{D}= 1,5sec - η is a factor describing how the swaying declines, calculated as:

**Other input parameters (Others tab)**

In the NS3491-12 code only foundation level should be set.

**Design spectra in the other national codes**

Except for the above mentioned two codes, the user has in all cases to define the spectra in table or in a graphical way. In the Others tab only the foundation level should be set.

# 3. Calculations parameters and calculations steps

## Calculation methods selection

## Other setting possibilities

## Combination rule, rotation and second order effects

## Displacement calculation

# 4. The results of seismic calculation

# 5. Summation of static and seismic effects

## Seismic loads in static load cases

## Final results of seismic effect (Seismic max.) in load combination

## Final results of seismic effect (Seismic max.) in load groups

# 6. Useful tips, which method to use?