# ZPA Method in Response Spectrum Analysis

### Technical Article

In a multi-modal response spectrum analysis, it is important to determine a sufficient number of eigenvalues of the structure and to consider their dynamic responses. Regulations such as EN 1998-1 [1] and other international standards require to activate 90 % of the structural mass. This means: to determine as many eigenvalues that the sum of the effective modal mass factor is greater 0.9.

For very large models with many degrees of freedom or for structures that have fundamental eigenvalues in the high-frequency range (for example, piping systems), these requirements may be difficult to meet. In such cases, the ZPA method (Zero-Period-Acceleration) becomes more important. This method is also able to consider masses in the supports themselves, which has a decisive influence on the support forces.

There are three frequency ranges that can be used to categorize the system responses differently (see Figure 01): (1) the low frequency range, (2) the medium frequency range, and (3) the high frequency range.

The low-frequency and medium-frequency range (f <f _{ZPA} ) is the range usually considered in structural dynamics. Structures have dominant natural frequencies in this range and in many cases more than 90% of the structural mass is activated with these eigenvalues. The system responses in the low-frequency range are periodic and results of different eigenvalues are out of phase. The superposition of responses from individual eigenvalues is quadratic either with the SRSS rule or better with the CQC rule.

In the high-frequency range (f> f _{ZPA} ), the system responses are pseudo-static, the responses from individual eigenvalues are in-phase. The modal superposition can thus be performed as an algebraic sum. Instead of taking these high frequencies into account dynamically, it is common to determine the missing activated masses and to add the responses of the entire high-frequency range pseudo-statically by means of the ZPA value. The ZPA value corresponds to the value from the response spectrum for the period T = 0 sec; ZPA = S _{a} (T = 0). However, it is also conceivable to define a self-defined value for the ZPA value. This method is called the ZPA method, the missing mass method, or also the static correction [2, 3, 4] .

Frequencies in the middle range (f _{SP} <f <f _{ZPA} ) provide system responses that are periodic to parts and pseudo-static to parts. These frequencies can be combined with special superposition rules such as the Gupta method [2] to consider the algebraic summation of the pseudo-static components. However, a superposition by means of the usual quadratic rules such as the CQC rule is also common.

The frequency f _{SP} (sp = spectral peak) corresponds to the maximum value of the spectral acceleration. The ZPA frequency f _{ZPA} (ZPA = Zero-Period-Acceleration) is the minimum frequency with which the acceleration approximately reaches the ZPA value.

#### Calculation of ZPA components

The proportions of the activated masses at each individual node in the system can be determined as follows:

$${\mathbf m}_\mathrm j\boldsymbol\;=\;\sum_{\mathrm i=1}^\mathrm p\;{\mathrm\Gamma}_\mathrm{ij}\;{\mathbf u}_\mathrm i$$

where

i = 1 ... p = Number of eigenvalues that are considered in the response spectrum analysis

j = direction of earthquake excitation**m** _{j} = (m _{X, j} , m _{Y, j} , m _{Z, j} ) = parts of the activated masses on each node in the excitation direction j

Γij = participation factors for the eigenvalue i and the excitation direction j**u** _{i} = (u _{X} , u _{Y} , u _{Z} ) ^{T} = eigenmode of the eigenvalue i on a single node, mass normalized with M _{i} = **u** _{i} ^{T} ∙ **M** ∙ **u** _{i} = 1 kg

The proportion of missing, unactivated masses at each individual node is the difference to the total structural mass and is determined as follows:**m** _{j, missing} = **1** - **m** _{j}

The equivalent loads on each node and consequently deformations and internal forces for the proportion of non-activated masses are determined as follows:**F** _{j} = **m** _{j, missing} ∙ ZPA _{j} ∙ **M**

where**F** _{j} = (F _{X, j} , F _{Y, j} , F _{Z, j} ) = equivalent loads on each node for the proportion of unactivated masses resulting from the excitation direction j

ZPA _{j} = spectral acceleration S _{a, j} (T = 0) in the direction of the excitation j**M** = (M _{X} , M _{Y} , M _{Z} ) = mass at the individual nodes in the system

The results of the ZPA components determined in this way are treated as another eigenvalue in the modal superposition. The superposition with the results of dynamically considered eigenvalues can be performed by means of the SRSS rule or as an absolute sum. The absolute sum provides results that are on the safe side.

#### Implementation in DYNAM Pro - Forced Vibrations

In DYNAM Pro - Forced Vibrations, the ZPA method is applied when the "Apply Static Correction" check box is selected. The setting is shown in Figure 02.

Figure 02 - Activation of ZPA Analysis in DYNAM Pro - Forced Vibrations

In DYNAM Pro, the determination of the non-activated masses and the resulting equivalent loads is performed internally. The ZPA value is defined with the value from the response spectrum for the period T = 0 sec; ZPA = S _{a} (T = 0). The results of the ZPA component are superimposed as an absolute sum with the results of dynamically considered eigenvalues.

R _{t} = | R _{SRSS/CQC} | + | R _{missing} |

where

R _{t} = results after modal and directional superposition including the ZPA component

R _{SRSS/CQC} = Results of the dynamically considered eigenvalues, modally superimposed with the SRSS or CQC rule

R _{missing} = Results of the ZPA component

The results after superimposition are exported as result combinations in the main program RSTAB.

#### Example

A cantilever with five degrees of freedom shows how the ZPA method is implemented in DYNAM Pro - Forced Vibrations. A very simple system is considered to enable the traceability of the results. A rigid cross-section RO 508.0x10.0 with I _{y} = 48.520 cm ^{4} made of steel S 235 is selected in order to achieve frequencies above the value f _{ZPA} with relevant mass contribution. The self-weight of the beam of 612.3 kg is distributed evenly over the six nodes (including support nodes). In addition, a mass of 1 t is defined at node 5. The masses as well as the excitation of the system act in the X-direction. The structure with mass distribution, resulting natural frequencies and effective modal masses is shown together with the user-defined response spectrum in Figure 03.

In this example, the ZPA value is S _{a} = 2.00 m/s ^{2} . This is the acceleration value for the period T = 0 sec. If the response spectrum is plotted against the frequencies, the limit value results in the frequency f _{ZPA} with 100 Hz. The first two frequencies f _{1} = 19.8 Hz and f _{2} = 92.8 Hz are thus in the middle frequency range (see Figure 01) and are considered dynamically. The remaining three natural frequencies are high frequency and can be taken into account by the ZPA method.

As a default setting, the masses of fixed supports are not taken into account in the eigenvalue calculation in DYNAM Pro. These masses have no influence on the determined natural frequencies and only in this way it is possible to achieve effective modal mass factors of 100%.

If, however, you want to explicitly consider the influence of the masses in the supports in the ZPA method, you have to activate them in DYNAM Pro with the setting shown in Figure 04. In this example, the masses in the supports are considered.

Figure 04 - Detail Setting in DYNAM Pro

By activating "Neglect Masses", the default setting of the considered masses is changed. If the table of "nodal supports" is left empty, masses are also considered on supports.

The following table determines the participation factors Γ _{X} , the proportions of the activated masses **m** _{X} , the proportions of the non-activated masses **m** _{X, missing} and the resulting equivalent loads at the six nodes in the system. The calculation basis of the ZPA method was discussed in the previous section.

Node | MassM _{x} | Participation factor Γ _{X} | 4 - Mode Shape u _{X} | Proportions of activated Masses m _{x} | Proportions of missing Masses m _{x, missing} | Equivalent Loads F _{X} [N] | ||

Shape 1 | Shape 2 | Shape 1 | Shape 2 | |||||

1 | 61.23 | 0.078350 | -0.056290 | 0.3220 | 0.6780 | 83.03 | ||

2 | 122.46 | 0.056790 | -0.008520 | 1.1325 | -0.1325 | -32.44 | ||

3 | 122.46 | 24.12 | 27.85 | 0.036140 | 0.027190 | 1.6290 | -0.6290 | -154.05 |

4 | 122.46 | 0.018110 | 0.038290 | 1.5033 | -0.,5033 | -123.26 | ||

5 | 1,122.46 | 0,005100 | 0.021670 | 0.7266 | 0.2734 | 613.82 | ||

6 | 61.23 | 0.000000 | 0.000000 | 0.0000 | 1.0000 | 122.46 |

The shear forces, moments, and support forces resulting from these equivalent loads are shown in Figure 05.

Figure 05 - Results of ZPA Method

The final results of the multi-modal response spectrum analysis taking into account the ZPA method result from the results of the first two eigenvalues (here modally superimposed with the SRSS rule) and the results of the ZPA component (see Figure 05).

In Figure 06, the results of the response spectrum analysis taking into account the first two eigenvalues (DLF2 and RC2 in the model) are compared with the final results including the ZPA component (DLF3 and RC3 in the model). If you look at Figure 05 and Figure 06, you can clearly see the absolute summation that is used in DYNAM Pro.

Figure 07 shows the results of the response spectrum analysis for all five eigenvalues for comparison. The ZPA method takes into account the masses in the supports. This results in a greater support force P _{X} = 2.57 kN. The internal forces are on the safe side due to the superposition as an absolute sum (compare Figure 07 with Figure 06).

Figure 07 - Results of Multi-Modal Response Spectrum Analysis Considering Five Eigenvalues

#### Summary

This example showed how the ZPA method is implemented in DYNAM Pro and that the results are verifiable and verifiable. This method is useful and recommended if high-frequency frequencies of the structure activate relevant mass components and if larger support masses are available in the structure.

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