Methods of overlapping vibration modes and combining results from different excitation directions in STATIK

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Implementing a dynamic calculation in STATIK is simply carried out using a very intuitive dialog box. However, we think it is important to thoroughly explain the way this program overlaps the vibration modes and how the results of different earthquake excitation directions are combined when designing a structure.

The definition box for ‘Calculation of the response spectrum’ looks like this:

 

In this article we will focus on the functions shown on the lower part:

 

On the left side of the calculation specifications definition we see that there is one last column called ‘Overlapping’. It is here where we will choose the overlapping method of the vibration modes. On the right part we have the necessary options to define the combination of results of the different calculation specifications shown on the left. We can also see a drop-down menu in which we will choose the combination method we want.

- METHODS OF OVERLAPPING VIBRATION MODES

Because the maximum amplitudes of the vibration modes considered do not happen at the same time, the problem is reduced to knowing how the results derived from this modes must be combined to obtain an appropriate result with the purpose of creating a design since a simple sum would be too conservative.

To a more conceptual level we would say that the shapes of the vibration modes cannot be overlapped, but rather that only the scalar results calculated for each mode can, as in the external forces or the reaction components of an specific point of a structure, for example.

To explain the different overlapping methods that STATIK offers we will use the following nomenclature:

E: the result of interest, for example, the normal force on a point of the structure.

Ei: the value of E for Ai,max (maximum displacements of mode i).

Emax: the maximum value of E (all modes considered, combined)

1) Sum of the absolute value of the contributions

This mode has no importance from a practical point of view since, as we have previously explained, it gives results that are too conservative.

If we take as an example the simple structure below, made of a pillar built-in its base and a small lintel:

Which is subjected to an earthquake excitation in the 3 directions (X,Y,Z) on which we carry out an overlapping of the modes with the summation method for all of them:

 

For example, the moment diagram My obtained for the first calculation specification AS1_AX is:

 

The value of the moment of 4.86 kNm at the base of the support is obtained as the sum of the absolute values of the moments obtained for each of the vibration modes analysed.

The calculation that it makes with moments obtained for the modes analysed (5 in this case) is:

My = E1+E2+E3+E4+E5 = 4.86 kN

2) SRSS Method (Square-Root-of-Sum-of-Squares).

This method assumes that all the modal responses can be considered independent from one another. In the opposite case, this method would not be conservative. According to Eurocode 8, two modes can be considered independent from one another if their periods Ti and Tj (Ti<=Tj) satisfy the condition Ti<=0.9*Tj.

Let’s see what happens if we obtain the same value for the moment of the previous example using this method:

 

The value obtained is now 4.79 kNm, slightly lower than the one obtained with the previous method. In this case, the result is obtained doing the square root of the sum of the squares of the obtained moments for each of the vibration modes being analysed.

The calculation that it makes with moments obtained for the modes analysed (5 in this case) is:

My = (E1^2+E2^2+E3^2+E4^2+E5^2)^1/2 = 4.79 kN

3) Complete quadratic combination (C.Q.C)

If the condition of indepence of the modes is not fulfilled to apply the SRSS, then the C.Q.C. must be applied.

If the second term of the expression in the square root is being reduced, the method ends up converging to the SRSS method. If we look at the expression for the correlation coefficients:

 

That convergence of the method to the SRSS will happen if the damping is reduced, if the ratio r between the two frequencies is too big or if the product Ei*Ej of the modes with dependent frequencies is too small.

With this method, the value of the moment that we obtain in our example is:

 

We see that the value is identical to the one obtained with the SRSS mode. This is because the explained convergence has occured because of the specific characteristics of this model, but it will not always be like this.

- COMBINATING THE RESULTS OF THE DIFFERENT EXCITATION DIRECTIONS

An analysis of the response spectrum is always carried out for a determined excitation direction, for example, direction X. For a complete analysis, at least one excitation for direction Y must be considered (perpendicular to X) and we can even take into account some vertical excitation on the Z axis.

The effects of the excitation in the different directions are named E_Ex, E_Ey, E_Ez and can be combined using one of the following methods:

a) Sum of the absolute values of the results

Very conservative.

b) SRSS Method (Square-Root-of-Sum-of-Squares)

Standard method.

c) Eurocode 8 method

Let’s see, from the example we are using for this explanation, the difference in the results when each of the combination methods is applied. We will focus in this case in the results of My in the intersection point between the support and the lintel.

Method A) Sum of the results obtained for each direction:

The moment resultant must be My = 0+0.13 + 0.22 = 0.35 kNm

 

We see in the table that it is, indeed, that way.

Method B) SRSS:

 

If we select now the combination method SRSS, the moment resultant would be obtained as My = (0^2+0.13^2+0.22^2)^1/2 = 0.25 kNm.

 

We see in the table that, indeed, it is that way.

Method C) Eurocode 8 method

 

The moment resultant would be obtained as:

My = max (E1, E2, E3) = 0.25 kNm

E1 = 1.0*0+0.3*0.13+0.3*0.22 = 0.11 kNm

E2 = 0.3*0+1.0*0.13+0.3*0.22 = 0.20 kNm

E3 = 0.3*0+0.3*0.13+1.0*0.22 = 0.25 kNm

 

We see in the table that, indeed, it is that way.

Finally, we would like to pay attention to the option in the calculation definition box highlighted in the following figure:

 

If we check this option, the program will compare the results obtained for each of the calculation specifications or excitation directions (provided that we check the box of the ‘Max.+’ column of that specification), giving the maximum one. In addition, it is possible to compare it with the results of the combination as long as the combination (for example, K3) is checked (and defined) and it will give the maximum results. Evidently, if we compare the values obtained for the specifications with the combination values, the latter ones will always be greater. For this reason, STATIK only allows us to use the combination values to design cross sections interacting with FAGUS. The option of obtaining the maximum values if we want to compare the values of the different excitation directions with each other.

If we do so in our example:

 

We can see how the value of My, for the point we have taken as reference, for example, is the one obtained for the AS1_Az specification, since it is the greatest:

 

We also confirm what we meant regarding this value being lower that the one obtained by combination (0.25 kNm),  not being valid for designing purposes.