When an seismic analysis of a building is carried out, these two ideas should, as a general rule, be considered:

- The design lateral force must be obtained considering the building as a whole.

- The design lateral force will be then distributed on the different storeys.

There are two procedures commonly used for the obtention of the earthquake design lateral forces:

1. An analysis of the equivalent static forces.

2. A dynamic analysis.

**1.- Analysis of the equivalent static forces.**

Conceptually it is a dynamic analysis divided in a dynamic part and another static part to obtain the maximum displacement. It is restricted to only one vibration mode of the structure.

The analysis of the equivalent static forces is based in the following hypotheses:

- It is assumed that it is rigid.

- It is assumed that there is a perfect embedding of the structure and the foundations.

- During the movement of the ground all the points of the structure experiment the same accelerations.

- The leading effect of the earthquake is equivalent to a force with a magnitude that varies with height.

- Approximately determining the total horizontal force (basic shear) in the structure.

This kind of analysis has a series of limitations and each norm establishes a series of assumptions to which it is applicable. Generally, the analysis of equivalent static forces will be applicable for regular buildings, that is, buildings that have uniformly distributed masses and stiffnesses. It is restricted to structural systems that can represent two flat models with a behaviour that is mainly determined by its fundamental periods of vibration and, thus, not influenced by the contribution of higher periods.

The total lateral force caused by the earthquake is determined for each principal direction using:

Where:

Sd: ordinate of the design spectrum for the fundamental period of vibration T1.

Gtot: total weight of the structure considering the dead loads and some of the overloads according to the codes.

λ: mass reduction factor applicable under some conditions (see codes).

The distribution of the total lateral force over the storeys is based on a own virtual mode and it is calculated with the following formula:

Fdi: lateral force on storey i

Fd: total lateral force

si, sj: weighing factors for storeys i,j

Gi,Gj: weight of the storeys i,j (Gtot = sum of all the Gi)

The weighing factors are the horizontal displacements of the own virtual mode at the height of each storey. The own virtual mode can have the following distributions:

1) Distribution of the fundamental mode of vibration.

2) Linear distribution (selected by default)

3) Constant displacement

For the linear distribution the previous formula would end up as:

Zi: height of the storeys

**2. Dynamic analysis**

The dynamic analysis is classified in two types:

- Time history method.

- Response spectrum method.

STATIK gives the option of carrying out the earthquake analysis with the response spectrum method. In the future Generation 8 the time history method will also be available.

To carry out the earthquake analysis available in STATIK it is necessary to define the design spectrum of the acceleration of the ground. There are two ways of entering this spectrum:

- Using the formulae for the standard spectrum of a specific code.

- Entering a general curve (defined by the user).

If we consider Eurocode 8 as the norm, the formulation is as follows:

**Zone ɑ factor: **basic horizontal maximum acceleration A=α*g (g = acceleration of gravity = 9.81 m/s2)

**Soil type ‘A’, ‘B’, ‘C’, ‘D’, ‘E’: **specifies the limit ranges TB, TC and TD and other parameters of the spectrum.

**Soil type ‘defined by the user’: **the parameters TB, TC and TD can be specified by the user.

**Horizontal seismic excitation **(values of the elastic spectrum):

Type I

Type II

With these parameters, the design spectrum for the horizontal excitations is defined as:

**Vertical seismic excitation:**

For the vertical components of the earthquake's actions, the design spectrum is defined by the expressions above with the vertical design base acceleration, Av replacing A and with the following parameters:

Once the response spectrum is defined the seismic weight of the building must be obtained to then establish the mass matrix [M] and the stiffness matrix [K] using a mass system concentrated at each of the storeys, with each of the masses having one degree of freedom.

Using the [M] and [K] matrices obtained and based on the principles of dynamics the modal frequencies {w} and the corresponding vibration modes will be obtained {Φ}.

|K-M*w^2| = 0 --> {w}

([K]-w^2*[M]){Φi} = {0} --> {Φi}

The natural periods will be Ti = 2π/wi

The modal masses Mk of each mode k will be calculated using the equation below, n being the number of considered modes:

The modal participation factors Pk of each mode k will be obtained with the following expression, n being the number of considered modes:

Now the lateral forces for each storey and on each vibration mode (storey i, mode k) will be obtained:

Qik=Sd*Φik*Pk*Wi

And the shear on each storey:

Finally, the shear due to all of the considered modes is obtained combining the shears obtained for each of the modes using the complete quadratic combination method or the square root of square sum method.

Once the theory of each of the methods has been seen, what will we obtain in STATIK with each of them?

To answer that question we will consider the simple example of a 4 storey building supported only on columns:

Apart from the dead loads caused by the self-weight of the members, we are going to consider an overload on storeys L1-L3 (all of them except the top one).

Before comparing the results obtained with each of the methods it is important to know that ** the values that are obtained for the shear applying the response spectrum method will always be smaller than the ones obtained with the equivalent force method. **This is due to the fact that the equivalent force method is an approximation that prioritises safety and, as we have mentioned before, it is not always applicable.

If we define an standard spectrum according to Eurocode 8:

- **Results using the equivalent force method:**

The response spectrum is:

And the results of the shear on each storey and on each direction X and Y (in this case they will be equal because the building is symmetric):

We see that the maximum shear is **887.41 kN **and it will be the corresponding one to the total shear at the base of the building.

- **Results using the response spectrum method:**

STATIK will give us the shear of each of the pillars so to obtain a value which can be compared to the one obtained before **we must add up the shears at the base of the pillars of the lower storey. **We also have to choose one of the directions:

If we look at the shear in the X-direction, Vx:

We can see that we must sum the shear at the base of the pillars of the ‘L1-’ series. In the results table, these values are the ones corresponding to the shear for the ‘Distance = 0 m’ (the equivalent to the ones indicated below for the 16 pillars, C1-C16):

The sum of these values gives a total shear of **682.54 kN **which, indeed, is lower than the shear obtained with the previous method (**887.41 kN**).

**NOTE: **in the manual of CEDRUS (E.4.3.4.) the differences between the values obtained for each method are clarified.

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