As we do not have a copy of this book in our office, here are some aspects to pay attention to when modeling such structures.

**1) Floor slabs**

To simulate the high in-plane stiffness contributed by floor slabs, many programs have a rigid diaphragm feature. In STAAD.Pro versions V8i SELECT series 3 and older, this is possible using the command SLAVE ZX MASTER aa JOINT slave-joint-list. When a frequency and mode shape extraction is involved, it is important to set the master node to a point that is at or very close to the center of mass of the weights that are assigned to the slave joints. The farther the master node is from the center of mass, the more approximate the results will be.

Depending upon the complexity of the floor plan, the center of mass can be tedious to calculate. So, this method of assigning a rigid diaphragm can be tedious. There are 2 workarounds.

a) Model the floor slab using plate elements, even a coarse mesh should be OK. Make sure that the elements connect to all joints (beam ends) of the floor. To ignore the bending stiffness (as in the case of rigid diaphragms), declare the elements as PLANE STRESS. As plane stress elements have no out-of-plane shear or bending capacity, make sure every plate node is attached to a beam or a column, else there will be instabilities along FY. To simulate the inplane rigid effect of the slab, assign a modulus of elasticity that is say, a 100 times that of concrete. In this method, one doesn't need to worry about the center of mass. The stiffness and mass distribution are automatically reflected in the respective matrices.

b) If you have STAAD.Pro V8i SELECT Series 4 (Build 20_07_09_11), there is a feature called the **RIGID FLOOR DIAPHRAM**. Using that, all the nodes of a floor can be declared as being part of a rigid diaphragm. During the analysis, the program finds the center of mass based on the weights defined at those nodes and creates a master node at the center of mass. The engineer doesn't have to manually calculate the correct position of the master node.

For complex floor plans - L-shaped, Z-shaped, etc., method (a) is still probably more accurate.

**2) Doubly symmetric structures**

If the stiffness and mass are identical along global X and Z, the frequencies and participation factors should reflect that. This will be evident in the form of pairs of modes with identical values for frequencies, and transposed values for the mass participation factor and base shear (in a response spectrum analysis). However, in those pairs, the mode shape and participation factors will not necessarily be entirely in one direction alone, but will be at an angle to (and hence have a component along) both plan directions. In the example below, modes 2 and 3 form a double root mode, and, modes 5 and 6 form another.

In each pair, for any given direction, some mass participation is from one of those double root modes and the remaining participation is from the second mode in that pair. So, both modes in a pair must be used together, which means, the participating mass, base shear, etc. must be added up from those two. Thus, in the example shown, the mass participation factor along X for the first double root mode is equal to 72.52 (contribution from mode 2) plus 1.43 (contribution from mode 3), = 73.95%.

Both modes of the double root mode should be used in the response spectrum or time history solution. For example, if the 8th and 9th mode are double root modes, the CUT OFF MODE SHAPE value should be at least 9, not 8 for a response spectrum analysis.

Whether a certain software reports double root modes or not depends upon the solver it uses for eigensolution. If the engineer wants the response to be entirely in one direction alone, he/she can

i) Make the seismic weights along the X direction to be marginally different from those in Z. This will cause the frequencies and participation factors for X to be slighty different from those for Z, but the variation should be small. Hopefully, each mode will be for a specific direction only.

ii) Apply the masses along one direction only. If you apply them only in X and none along Z, the participation for the lateral modes will be fully in X and zero in Z.

**3) PDELTA effect**

raycxx has pointed out in the following thread how the geometric stiffness can be considered during the eigensolution.

http://communities.bentley.com/forums/thread/51141.aspx

We will elaborate on this either through a new discussion in this forum or through a Q&A on the STAAD FAQ page.