Center of mass and rigidity calculations

The purpose of the Center of Mass (CoM) and Center of Rigidity (CoR) calculations is to provide the user with a feel for the structure as to the eccentricity between mass and rigidity occurring naturally within the structure as a result of the positioning of loads and the stiffness of the lateral force resisting systems. Minimizing this eccentricity is the key to effective seismic design in earthquake-prone regions.

Tekla Structural Designer provides two alternative approaches for these calculations: an analytical approach, which is only performed for "modal mass" and "base shear" combinations; and an approximate approach, which is available for loadcases and other combinations, but not "modal mass" and "base shear" combinations.

While the CoM and CoR results are displayed, they are not currently used in any further calculations within Tekla Structural Designer.

Analytical approach

Using the analytical approach, calculations are performed for all "Modal Mass" and "Base Shear" type combinations defined in the model. Provided the structure is stable (without mechanism warnings or excessive deflection) this approach is applicable in all situations, including models with rigid diaphragms, semi-rigid diaphragms, or no diaphragms at a given level.

The CoM and CoR are calculated as follows:

CoM: The CoM is determined based on the loads present in the relevant combination.
CoR: The CoR is determined using analytical methods allowing for the mass distribution when there is no rigid diaphragm present.

For meaningful results, each modal mass combination should accurately represent the structural state for which CoM and CoR are required (e.g., the seismic weight combination for seismic situations).

Provided that at least one modal mass combination (either "modal mass" or "base shear" type) has been defined, the analytical approach is initiated from the Analyze ribbon by clicking Calculate in the Centre of Mass and Rigidity group.

CoM and CoR calculations are then performed for every defined modal mass combination.

Graphical results for the first modal mass combination will be displayed in the active window provided that in the Scene Content window, Analytical has been selected under Center of Mass & Rigidity.

As shown below, you then have three analytical display options.

Slabs/Roofs/Diaphragms - a CoM and CoR is shown for each slab/roof/diaphragm
Floors - a single CoM and CoR is shown for each floor
Model - a single CoM and CoR is shown for the entire model

If you have defined more than one modal mass combination, you can click Show Results on the Analyze ribbon to cycle through the results for the second, and each additional modal mass combination.

Example:

Consider the following model in which each slab is set to be a rigid diaphragm. The top three floors are formed by discrete towers.

Display CoM and CoR for Floors

When the Scene Content settings are configured as below, a single CoM and CoR result is displayed at each level.

While this calculation is appropriate for the lower levels, it is not as useful at the top of the model where the towers can move independently of each other.

Hovering over a CoM or CoR in the graphical view will display a tooltip indicating CoM location (X and Y coordinates), CoR location (X and Y coordinates), and the offset distance between each CoM and CoR pair.

Selecting a CoM or CoR in the graphical view will display their coordinates in the Properties window.

Display CoM and CoR for Slabs/Roofs/Diaphragms

For this particular model it will be more informative to display results for diaphragms rather than floors. This can be done by changing the settings in the Scene Content window as follows:

Using this option separate results are displayed for each diaphragm, hence two results are shown at each of the upper levels.

Note that if a slab is not designated as a rigid or semi-rigid diaphragm and is not set to be meshed, no results will be provided for that slab. Slabs must be meshed if slab results are required.

Display CoM and CoR for Model

The last option in the Scene Content window enables you to display a single result for the entire model.

Scene Content entity information controls

The entity information controls in the Scene Content window allow you to select what is displayed at each location.

The controls are:

"Geometry" - a red arrow for CoR and a blue arrow for CoM.

"Text" - the offset distance between CoR and CoM.

"Origin" - if required, a green arrow can be displayed to indicate the origin at each level from which the CoR and CoM coordinates are measured.

Analytical approach - tabular results

To view tabular CoM and CoR results:

On the Analyze ribbon, click Model & Results
In the View Type group on the Analyze ribbon select CoM/CoR
In the View Type group on the Analyze ribbon select Table Options
Select the results you want to display and click OK

The analytical calculation explained

The analytical calculation of the CoR considers both the mass distribution and the stiffness of the structure below the level under consideration, with the following all being taken into account:

Interconnected core wall systems
Coupling of beam and frame systems
Independent diaphragms on floors
The true in-plane stiffness of semi-rigid diaphragms

Mass distribution is critical as it indicates where inertia forces will act. These inertia forces counterbalance the moment created by the eccentricity between the CoM and CoR.

For floors with a single rigid diaphragm, mass distribution is irrelevant, and results can be verified using a single point load. However, for flexible floors with multiple rigid diaphragms, semi-rigid diaphragms, or no diaphragms, the results are sensitive to the mass distribution within the modal mass combination. To test these cases, multiple point loads are necessary to accurately replicate the mass distribution in the floor within the modal mass combination for verification purposes.

A note for those attempting manual verification: ensure that Analyze > Settings > Modification Factors for "Building Analysis" and "Modal Analysis (and RSA)" are set to the same values.

Approximate approach

The approximate approach can be used for any loadcase or any combination that isn't a "Modal Mass" or "Base Shear" class of combination.

To use this approach you must first ensure the Approximate center of mass & rigidity is selected for display in the Scene Content window.

The approximate centers of mass and rigidity can then be displayed by selecting the required loadcase or combination in the Loading list.

For the selected loadcase or combination, arrows indicate the centers of mass and rigidity for each individual rigid or semi-rigid diaphragm. (Unlike the analytical approach, FE meshed slab areas not designated as rigid or semi-rigid diaphragms are not included.)

Hovering over a CoM or CoR in the graphical view will highlight the diaphragm to which the calculation relates. A tooltip will also be displayed indicating CoM location (X and Y coordinates), CoR location (X and Y coordinates), and the offset distance between each CoM and CoR pair.

Selecting a CoM or CoR in the graphical view will display their coordinates in the Properties window.

The center of mass coordinates that are reported for the approximate method are constantly updated as you develop the model, with the calculations only considering loading applied in the plane of the diaphragm. Loads applied between diaphragms have no impact - most notably in this regard will be the self weight of columns and walls.