Eccentric Seismic and Response Spectra Load Cases on Semirigid Diaphragms

After a semirigid diaphragm is meshed, the program processes all mass loads over the diaphragm. At the end of this process, all the mass are converted into point (nodal) mass loads and they are directly applied at mesh nodes. In order to explain the current method used for seismic and response spectra load cases, an example is provided in this section (in the following example, a diaphragm loaded in (+) Y direction with (+) eccentricity). The following steps are pursued for this purpose:

Procedure for Eccentric Seismic Loads for Semirigid Diaphragms

The program calculates mass center for the diaphragm. This is shown in (a) in the preceding figure. Mass center line is determined (i.e., a line passing through the mass center and parallel to global Y-axis)
For the load case in Y-direction, the diaphragm is divided into two zones: one zone at the left side and one zone of right side of the mass center line ( (b) in the preceding figure). For each zone, mass centers are calculated according to the following equations:
$c m_{1} = \frac{\sum_{i = 1}^{n} m_{i} c_{i}}{\sum_{i = 1}^{n} m_{i}} = \frac{\sum_{i = 1}^{n} m_{i} c_{i}}{m_{1}}$ (6.22)

$c m_{2} = \frac{\sum_{j = 1}^{m} m_{j} c_{j}}{\sum_{j = 1}^{n} m_{j}} = \frac{\sum_{j 1}^{m} m_{j} c_{j}}{m_{2}}$ (6.23)

Where cm₁ and cm₂ are the mass center coordinates for the zone 1 and 2, respectively ( (b) in the preceding figure). Note that the index " i " is reserved for nodes located inside Zone 1, and the index " j " is reserved for nodes in Zone 2.
If it is a rigid diaphragm, calculated seismic story loads are concentrated at a single point " cm_e ", (i.e., cm_e = cm + eccentricity for (+) eccentricity in Y-direction). In other words, the mass center is shifted to its new location, "cm_e" to account for eccentric loading. Once the concentrated load is applied at this location, additional moment (i.e., accidental torsion) is created at " cm ".
For a semirigid diaphragm, two new terms are introduced, " ${\bar{m}}_{1}$ " and " ${\bar{m}}_{2}$ ", and they are obtained from solving the following two equations:
$c m_{e} = \frac{{\bar{m}}_{1} c m_{1} + {\bar{m}}_{2} c m_{2}}{{\bar{m}}_{1} + {\bar{m}}_{2}}$ (6.24)

${\bar{m}}_{1} + {\bar{m}}_{2} = \sum_{p = 1}^{k} m_{p} = m = Total mass of the diaphragm$ (6.25)

Then,

${\bar{m}}_{1} = - (\sum_{p = 1}^{k} m_{p}) (\frac{c m_{2} - c m_{e}}{c m_{1} - c m_{2}})$ (6.26)

${\bar{m}}_{2} = + (\sum_{p = 1}^{k} m_{p}) (\frac{c m_{1} - c m_{e}}{c m_{1} - c m_{2}})$ (6.27)

The terms " ${\bar{m}}_{1}$ " and " ${\bar{m}}_{2}$ " can be interpreted as the amount of lumped masses at each zone so that the final system has a combined mass center at "cm_e". Then, the program calculates effective mass coefficients for each zone as follows:

$a_{1} = \frac{{\bar{m}}_{1}}{\sum_{j = 1}^{n} m_{j}} = \frac{{\bar{m}}_{1}}{m_{1}}$ (6.28)

$a_{2} = \frac{{\bar{m}}_{2}}{\sum_{j = 1}^{n} m_{j}} = \frac{{\bar{m}}_{2}}{m_{2}}$ (6.29)
Finally, each point mass (or calculated seismic point load at each node) in zones 1 and 2 are multiplied with α₁ and α₂, respectively. Once, each point mass (or point seismic load) is modified as explained above, the analysis is carried out with this modified configuration. Note that it is for seismic load cases when each seismic point loads are modified with α₁ and α₂. Similarly, it is response spectra load cases when each point mass is modified with α₁ and α₂.

The following is summarized based on the above procedure:

a_{1} = \frac{{\bar{m}}_{1}}{m_{1}}, a_{2} = \frac{{\bar{m}}_{2}}{m_{2}}

(6.30)

And also note that diaphragm mass is always conserved:

m_{1} + m_{2} = {\bar{m}}_{1} + {\bar{m}}_{2} = m

(6.31)

α₁m₁ + α₂m₂ = m

(6.32)

In the following figure, (A) shows an example of eccentrically loaded diaphragm (i.e., Y+E). Note that the arrows of the right side of the diagram are relatively longer than those on left side, which indicates that these seismic point loads are amplified with α₂ whereas those on left side are reduced with α₁. In addition, the deflected shape shown in (B) indicates presence of eccentric loading.

(A) Eccentric seismic loads, (B) deflected shape under eccentric seismic loads

If it is a seismic load case, the seismic point loads are modified as explained above. If it is a dynamic load case (Response Spectra Analysis), nodal masses are modified according to the same method explained.