# 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

The lateral force method called also Equivalent static load 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

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**

## 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:

ΣGk, j"" + ""ΣψE, iQk, 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.

**Calculations parameters and calculations steps**

Calculation input parameters can be set in the Calculation dialog in Analysis/ Seismic analysis in the Setup as can be seen below.

## Calculation methods selection

National codes always provides, which Seismic calculation method to be performed for different structure, where and when it should be performed and what other effects to be considered (torsional effect, P-Δ effect).

As an example in Norwegian code NS3491-12, seismic calculation is not necessary if the acceleration from the design spectrum is S_{d}(T_{1}) ≤ 0,5 m/s^{2} where T_{1} is the base vibration period. In EC8 3.2.1 some criteria can be found.

FEM-Design provides three types of calculation methods in harmony with EC8 and NS3491-12.

These three methods really cover two basic concepts:

- Lateral force method, where the base shear force can be distributed in two ways (Static linear/mode shape),
- Modal response spectrum analysis (Modal analysis).

### Lateral force method

In some codes called equivalent static analysis.EC8 as well NS3491-12 uses this method. The user may not use this method in other codes.

This method can be used to calculate the seismic effect in horizontal plan, x' and/or y' direction. The main point of this method is to calculate base shear force taking into account the base vibration period and design spectrum in x' or y' direction which is distributed into those nodes of the structure where there are nodal masses. The base shear force formula is taken from EC8 4.3.3.2.2(1)P:

F_{b} = S_{d}(T_{1}) m λ

where:

- S
_{d}(T_{1}) is the value of design spectrum at T1 (means the acceleration of the structure), - T1 is the fundamental period of vibration of the building for lateral motion in the direction considered,
- m is the total mass of the building, above the foundation or above the top of a rigid basement. Remark: the FEM-Design always ta- kes into account the total mass of the structure including the base- ment,
- λ is the correction factor, the value of which is equal to: 0,85 if
- T
^{1}≤ 2 TC and the building has more than two storeys, or λ = 1,0 otherwise.

From this formula it can be seen that the base shear force is nothing else than the total seismic force of inertia (from second Newton’s law) which acts between the ground and the structure.

Distribution of the base shear force can occur in two ways which is described below.

**Linear distribution of horizontal seismic forces (Static, linear shape)**

In this method the distribution of base shear force happens according to a simplified fundamental mode shape which is approximated by horizontal displacements that increased linearly along the height (see EC8 4.3.3.2.3(3)). The seismic action effects shall be determined by applying to the x' or y' direction. The horizontal forces are:

where:

- F
_{b}is the seismic base shear force, - F
_{i}is the horizontal force acting on the place of mi, - z
_{i}, z_{j}are the heights of the masses m_{i}, m_{j }above the foundation level.

According to NS3491-12 the distribution formula is:

where:

- k = 1 for T
_{1}≤ 0,5 sec - k = 2 for T
_{1}≥ 2,5 sec

In the 0,5-2,5 interval the value of the k is interpolated linearly.

As a matter of fact eigenfrequency calculation is not necessary for this method, because giving the base period time in x' and y' direction is enough for the calculation. Practically, eigenfrequency calculation is performed before setting this data, but these data can be defined using experimental formulas as well. Investigation can be done in x' or y' direction, or both together.

The user may set the calculation direction to be performed by selecting the desi- red direction. To set the desired x'-y' direction user should give the α angle (α is the angle between the global x and x'). α = 0,0 means x'-y' directions coincide with global x-y directions. More details can be found in Horizontal direction setting for seismic calculation to set the correct seismic effect direction (α).

If any of the above mentioned situations happen, the static, mode shape or modal analysis should be used.

**Distribution of seismic forces according to fundamental mode shapes (Static, mode shape)**

In this method the distribution of base shear force happens according to the base vibration shape (see EC8 4.3.3.2.3(2)P). The horizontal forces acting on the place of mi are:

where:

- s
_{i}, s_{j}are the horizontal displacements of masses - m
_{i}, m_{j}in the fundamental mode shape.

The following table shows how to select the base vibration shape. The table contains all mode shapes (No.), the vibration time (T(s)) and effective masses of the mode shapes in x' and y' directions (mx(%) and my(%)). As you can see the effective masses are given in a relative form to the total or reduced mass of the structure. The reduced mass means the total mass above the foundation or above the rigid basement. The value of the effective mass is referred to how the mode shape respond to a ground motion direction, so the effective mass shows the participation weight of the mode shape.

It is recommended to select that mode shape which gives the largest effective mass as the fundamental mode shape. The method allows to Select one mode shape in x´ or/and y´ direction(s).

### Modal response spectrum analysis

This method can be used in all national codes.

The essence of the method is the calculation of the structural response for different ground motions by the sufficient summation of more vibration shapes. Method gives possibility to take into account full x, y and z direction investigation. In the table below, more vibration mode shape could be selected in x', y' and z' directions if necessary. The last row of the table shows that in a given ground motion direction how large is the sum of the considered effective masses compared to the total or reduced mass of the structure.

According to EC8 4.3.3.3.1(3) and NS3491-12 sum of the effective mass of the chosen mode shapes - at least in horizontal direction - should reach 90% of total mass. Additionally every mode shape has to be taken into account which effective mass is larger than 5%.

According to the EC8 and NS3491-12 the summation rule in the individual directions can be selected in the lower part of the seismic analysis setup dialog. In all other codes there is no possibility to choose, the SRSS rule is used for summation. According to EC8 4.3.3.3.2, the summation rule possibilities are the following:

where:

- E
_{E}is the seismic action effect under consideration (force, displacement, etc.), - E
_{Ei}is the value of this seismic action effect due to the vibration mode i, - r
_{ij}is the interaction between two vibration periods taking into ac- count the declining ratio:

The **CQC **(Complete Quadratic Combination) summation rule might be adopted when individual direction, two vibration modes are dependent to each other if they satisfy the following condition:

T_{j }/ T_{i }> 0,9 with T_{j }≤ T_{i}

FEM-Design always applies the selected rule for the summation except if the **Automatic **is highlighted. If the **Automatic **is selected then the rule selection procedure is as follows:

Always three directions (if there were more than one mode shape selected in that column) is investigated weather all mode shapes are independent from each other or not.

If at least one dependent situation exists in a direction, the program automatically uses the CQC rule for all mode shapes in that direction, otherwise SRSS rule is used.

## Other setting possibilities

### Horizontal direction setting

Generally codes speak about the seismic calculation in X-Y directions. However results in these directions give the maximum effect if the mass and elastic properties of the structure ensure that the calculated mode shapes lay in X-Z or Y-Z plane. Nevertheless it is not always achieved in practice. To achieve the unfavorable direction, where the results from a ground motion are maximum, the user can Set the Alpha angle or may get the program suggestion by using Auto but- ton.

The most unfavorable direction can be found when any of the mx', my' is zero and the other is maximum in a row. Using Auto button, program gives the most unfavorable directions, but there are certain restrictions: this directions can be ensured only for one mode shape. The program selects the row where the effective mass is the maximum.

As an example, on the left hand side figure you can see a badly adjusted x'-y' direction. Appling Auto button, program arranges the direction for the 73,8% effective mass and correct it to 98,3%.

Of course this also can be reached if the user rotates the whole geometry with a specified angle.

### Effective mass setting

FEM-Design always takes into account the entire mass of the structure in the calculation of base shear force which was mentioned in Lateral force method. It was also mentioned, EC8 defines the total mass without the basement, this is called Reduced mass in this manual. The effective masses are generally compared to the Reduced mass, but this is not valid for the massive basement with elastic foundation.

If the above mentioned situation is the case, it might happen that the sum of the effective masses of a column is larger than 100%. The user may compare the modal effective masses to the total mass or reduced mass by pushing the Eff. mass button.

In FEM-Design Reduced mass means the difference between the total mass of the structure and the basement mass. The basement mass is the sum of all masses which lay on the foundation level which can be set in the Others tab of seismic load.

It is uninteresting from the calculation point of view that effective masses are compared to the total or the reduced mass because these values are given in percentage and only gives information about which mode shape is the fundamental or which shapes are dominant in a given direction.

## Combination rule, rotation and second order effects

According to EC8 4.3.3.5, the combination rule of x', y' and maybe Z direction effects, namely the seismic calculation of final results (Seismic max.), can be selected from the following two possibilities:

The first rule which is called SRSS is implemented to all the other codes than EC8 and NS3491-12 and there is no possibility for rule selection.

### Torsional effect

According to EC8 4.3.2 the program gives possibility to take into account the accidental mass distribution of the structure by the calculation of the torsional effect. This means that from the horizontal seismic forces a Z directional torsional moment can be calculated according to EC8 4.3.3.3.3 (EC8 4.17 equation) as follows:

M_{ai }= e_{ai} F_{i}

where:

- M
_{ai}is the torsional moment applied at the mi point about the vertical axis, - e
_{ai}is the accidental eccentricity of mass i in accordance with expression (EC8 4.3 formulas) for all relevant directions:

e_{ai} = ± 0,05 L_{i}

- L
_{i}is the floor-dimension perpendicular to the direction of seismic action (Lx',i or Ly',i), - F
_{i}is the horizontal force acting on the place of mi in x' or y' direction, when static method is used. In the modal analysis, this force is calculated, selecting the mode shape which gives the largest effective mass (fundamental shape). Using this mode shape this force is calculated according to static, mode shape. So, the total mass and not the effective mass of the structure is taken into account which belongs to this fundamental mode shape.

The explanation of the floor-dimension (L_{x',i }and L_{y',i}) on the ith storey:

It was seen that the influence of uncertainties of mass position was modeled by the rotation effect. According to our experiment using the FE method, when a plate, a wall and beams are divided into several elements the accidental torsional effect is not reasonable.

### Second-order effects (P-∆ effects)

Only EC8 gives a possibility to calculate the second order effect which is done according to 4.4.2.2(2). The second order effect is ignored if the following condition is fulfilled in all storeys and all horizontal directions:

where:

- θ is the interstorey drift sensitivity coefficient,
- P
_{tot}is the total gravity load at and above the storey considered in the seismic design situation. Remark: this total gravity load is calculated back from the nodal masses. - d
_{r}is the design interstorey drift, evaluated as the difference of the average lateral displacements ds (see Displacement calculation) at the top and bottom of the storey under consideration and calculated in accordance with EC8 4.3.4, - V
_{tot}is the total seismic storey shear force, - h is the interstorey height.

If 0,1 < θ ≤ 0,2, the second order effect is taken into account by multiplying the relevant seismic action effects (the internal and reaction forces) by a factor equal to 1/(1-θ).

According to EC8 4.4.2.2(4)P the θ coefficient shall not exceed 0,3. When θ >0.3, FEM-Design sends a warning message and continues the calculation using θ = 0,0.

The 0,2-0,3 interval is missing in EC8. In this case FEM-Design sends a warning message and continues the calculation using calculated θ.

## Displacement calculation

The displacement calculation is made according to EC8 4.3.4 using the following formula:

d_{s }= q_{d} d_{e}

where:

- d
_{s}is the displacement of a point of the structural system induced by the design seismic action, - qd is the displacement behavior factor, assumed equal to q unless otherwise specified,
- d
_{e}is the displacement of the same point of the structural system, as determined by a linear analysis based on the design response spec- trum.

FEM-Design uses the above formula only to calculate the summarized and combined the so called final results displacements. The displacements obtained from the single shapes and torsional effects won’t be modified.

**The results of seismic calculation**

The seismic results are very similar to statical results with some more items as follow: equivalent seismic forces and base shear force. The results shown separately from mode shapes, torsional effects, sum of the directions (Sum, x'…) and the final results (Seismic max.).

Desired results can be selected from the result dialog as it is shown below. Among the equivalent load results not only the nodal forces can be seen but also the base shear force, and in case of torsional effect the total torsional moment as well.

**Summation of static and seismic effects**

The seismic effect’s results can be considered together with static effects in two ways:

- seismic forces applied as real static forces in load cases,
- the final results can be combined with a static load combination or taking into account in load groups.

## Seismic loads in static load cases

Horizontal seismic forces and torsional effects which were calculated using the static method according to EC8 or NS3491-12 additionally can be added to the static load cases. However it is recommended to have the seismic forces in separate load case(s) in order not to mix up them with the normal static loads.

In the static calculations, load cases which contain seismic forces behave like the other normal static forces. Consequently they can be inserted in load combinations and load groups. If they are inserted in the load combinations then they can take part in the imperfection and stability analysis. Of course there is no possibility to convert the masses from these load cases. As it can be seen in the table above, this effect can be taken into account by positive or negative sign as well, because the seismic effect means vibration between +/- extreme values, but the results are shown only in positive direction for the sake of simplicity.

As it is shown above all the seismic possible cases can be found in the list but only those cases are valid which were calculated in seismic calculation. The calculated static loads from seismic effect can be found among the seismic results in the **Equivalent loads**.

## Final results of seismic effect in load combination

Final results of seismic effect (Seismic max.) always obtained from the total summation of all components. These results which actually means extreme +/- values, can be added to a load combination as a special load case with arbitrary factor and it can be applied in all codes.

The combination of the Seismic max. and the other static loads results is calculated in a special way.

## Final results of seismic effect in load groups

Final results of seismic effect (Seismic max.) can be applied in load groups in all codes. Using EC8 and NS3491-12 program give possibility to have Seismic load type beside Permanent, Temporary and Accidental load type. In all other codes it is recommended to apply the Seismic load type in the Accidental load type.

The final results of seismic effect take part with +/- sign automatically in the load group combination.

**Useful tips, which method to use?**

It is hard to answer this question, even for experienced engineers. However, some basic concept can be formulated: