|
||
|
|
||
Knowing the position of the shear center of each storey of a building is a commonly requested information. Unfortunately, SCIA Engineer does not currently provide that result in an automated way. This document described a method to determine the position of the shear center of each storey, using the available tools.
The proposed method is based on the following definition:
A horizontal force applied at any point above the shear center of a storey will cause zero inter-storey rotation in the considered storey.
Based on that definition, in principle, 3 load cases are necessary to provide information about the structure, applying respectively unit forces in X and Y directions and unit moments around the Z axis. The rotation of each storey around the Z axis will be obtained from FEM analysis. Based on those results, the position of the shear center can be determined.
After FEM analysis, the rotation of each storey around Z-axis it obtained for each load case
= rotation of storey i around Z-axis for load case FX
= rotation of storey i around Z-axis for load case FY
= rotation of storey i around Z-axis for load case MZThe position of the shear center is deducted by determining the coefficient which, applied to the MZ load case and then adding the FX or FY load case, produces a zero rotation. The corresponding eccentricity of the unit force gives the position of the shear center.
The position of the shear center of storey i is given by
where
is the point of application of unit forces Fx,i, Fy,i and Mz,i above the i-th storey (typically, the point of application of the resultant of storey forces above the current storey, applied to the mass center of each storey)
is the inter-storey rotation around the Z-axis due to a unit force applied in the X-direction above the i-th storey
is the inter-storey rotation around the Z-axis due to a unit force applied in the Y-direction above the i-th storey
is the inter-storey rotation around the Z-axis due to a unit moment applied around the Z-axis above the i-th storey
The 3 basic load cases described above, with unit forces and moments applied to each storey, cannot be defined simply in SCIA Engineer. However, the method can be adapted easily to use directly results of standard ELF seismic load cases.
Create 2 seismic load cases with the following settings:
Please note, that 2 additional accidental eccentricity load cases are automatically created for each seismic load case.
After FEM analysis, basic results can be extracted.
Storey forces Fx,i and Fy,i can be obtained from Summary storey results for both seismic load cases:
Values can be copied to a spreadsheet from Table results.
Storey moments Mz(Fx),i and Mz(Fy),i, corresponding to accidental eccentricity load cases, are not directly available in results. They can, however, be easily calculated from storey forces.
where
(resp. 
) is the accidental eccentricity parameter for seismic load case in X-direction (resp. Y) (default = 0.05)
(resp. 
) is the width of storey i perpendicularly to the direction of the seismic load case X (resp. Y)
(resp. 
) is the storey force for storey i for seismic load case in X-direction (resp. Y)
Please note, that there are here 2 sets of values of storey moments instead of one, as described in the theoretical explanation above. This is because the storey forces used here are not unit forces. They are real seismic forces, which can be different in X and Y direction. In the theoretical explanation above, unit forces were used, which are identical in X and Y directions.
The coordinates xG,i and yG,i of the center of gravity of each storey can be obtained from Summary storey results for the mass combination used for the seismic load cases:
Values can be copied to a spreadsheet from Table results.
Storey rotations can be obtained from standard results Displacement of nodes or 2D members > Displacement of nodes.
In all cases, the value of interest is Fiz.
If a storey uses a rigid diaphragm, the rotation of any node of that storey can be used directly.
If not, a simple method consists of extracting the displacement of all nodes of the considered storey, copying them to a spreadsheet and calculating the average value of Fiz for that storey.
is obtained from the seismic load case in X-direction
is obtained from the seismic load case in Y-direction
is obtained from the accidental eccentricity load case linked to the seismic load case in X-direction
is obtained from the accidental eccentricity load case linked to the seismic load case in Y-directionThat operation must be repeated for each storey separately.
Calculate resultant forces and moments per storey
Calculate the position of the resultant forces per storey
Calculate inter-storey rotations
Calculate inter-storey unit rotations
In principle, 
, so it should not be necessary to compute storey moments nor extract storey rotations for both directions. Processing only one accidental eccentricity load case is sufficient for the entire process.
Calculate the position of the shear center
