### Accidental eccentricity

The discussion in this topic assumes that Y is vertical, however if Z is vertical then Y and Z should be interchanged below.

Most earthquake loading codes require extra torsional effects in buildings to be taken into account to allow for uncertainty in geometric and material properties that could cause extra torsion in the structure during a seismic event. This means that the spectral response analysis must allow for the masses being moved by the accidental eccentricity in each of the +ve and -ve directions normal to the earthquake direction.

A common approach for modelling accidental eccentricity is to define rigid or semi-rigid diaphragms at the floor levels and then create extra "eccentric" mass cases that have the masses moved in each of the +X, -X, +Z and -Z eccentric directions. The method of modelling the eccentric mass cases depends on the type of diaphragm used as follows.

1. Rigid diaphragms are modelled by using horizontal master-slave constraints to link the nodes throughout the floor to a master node placed at the centre of mass of the floor. This causes the floor to behave as a rigid element in the horizontal plane while still allowing independent movement in the vertical direction. The accidental eccentricity is taken into account by connecting four stiff and weightless members to the centre of mass node of the floor and extending them out along the orthogonal plan directions with a length equal to the accidental eccentricity. The total mass for the floor is then lumped at the end of each of those members, one for each "eccentric" mass case.

2. Semi-rigid diaphragms require the floors to be modelled using plate/shell elements, and rely on those elements to provide the horizontal in-plane rigidity instead of needing master-slave constraints. Because the plate/shell elements have some in-plane flexibility, the diaphragms are not totally rigid and are therefore classed as "semi-rigid". The accidental eccentricity is modelled by keeping the floor masses in their original positions but factoring them up on one side of the centre of mass and factoring them down on the other side. This has the effect of maintaining the same total mass but moving its centre of mass by the desired accidental eccentricity. This is done for each of the four eccentric directions, resulting in four "eccentric" mass cases. Accidental eccentricity in semi-rigid diaphragms only involves adding the extra eccentric mass cases and does not require any changes to the structural model.

Note that if you have multiple source mass cases (or combinations) that require a separate spectral analysis for each one then you would probably have a different center of mass in each mass case and this makes the rigid diaphragm approach impractical. In this case the semi-rigid approach would still work except that each source mass case would require its own set of eccentric mass cases.

So with either diaphragm type, a single initial mass case would spawn four extra "eccentric" mass cases. It would also commonly require an extra "concentric" mass case that has the same centre of mass and total mass as the initial mass case but with self mass (if included) converted to lumped masses and combined with the applied lumped masses. For a rigid diaphragm this concentric mass case would have all of its mass lumped at the master node located at the centre of mass, whereas for a semi-rigid diaphragm it would be distributed around the floor nodes.

These five mass cases would then be used to create six spectral load cases as shown below:

1.  The concentric mass case for the earthquake in the X direction.
2.  The concentric mass case for the earthquake in the Z direction.
3.  The +X eccentric mass case for the earthquake in the Z direction.
4.  The -X eccentric mass case for the earthquake in the Z direction.
5.  The +Z eccentric mass case for the earthquake in the X direction.
6.  The -Z eccentric mass case for the earthquake in the X direction.

Combination load cases would also be required to allow for the reversal of each primary spectral load case, plus to combine the effect of the earthquake acting in one horizontal direction with a proportion (usually 30%) of the earthquake acting in the other horizontal direction. This could result in a total of 78 combination load cases.

So based on an initial mass case and two orthogonal earthquake directions, if accidental eccentricity is considered then you would finish up with 5 extra mass cases, 6 spectral load cases and 78 combination load cases. Without accidental eccentricity you would have just 1 mass case, 2 spectral load cases and 10 combination load cases.

Creating all of these mass, spectral and combination load cases can be a tedious process and prone to error if done manually. SPACE GASS therefore has some useful tools to automate this process for you.

1. The "Diaphragm generation" tool not only generates the rigid or semi-rigid diaphragms, but also has options to create all of the required extra mass, spectral and combination load cases.

2. The "Generate spectral load cases and combinations" tool can also be used to generate all of the required extra mass, spectral and combination load cases if the diaphragms have already been created.

Refer to these tools for details on how to operate them and for lists of the primary and combination load cases that would be created.

If you want to combine the spectral analysis results with static analysis results then you should create further combination load cases that combine the static and spectral load cases and combinations.

The structure should be designed to resist the envelope of all of these load cases.