To determine the deformation developed by external moment M, applied at the centre of stiffness C_{T}, the initial system X0Y is transferred (by parallel translation) to the principal system xC_{T}y. The centre of mass is transferred to the principal system along the structural eccentricities e_{ox}, e_{oy} in accordance with the following expressions:
Principal coordinate system
, , , (6’)
The displacement of the diaphragm consists essentially of a rotation θ_{z} about the C_{T}, inducing a displacement δ_{i} at each column top i with coordinates x_{i},y_{i} in respect to the coordinate system with origin the C_{T}. If the distance between the point i and the C_{T} is r_{i}, the two components of the (infinitesimal) deformation δ_{i} are equal to δ_{xi}=-θ_{z}·y_{i} and δ_{yi}=θ_{z}·x_{i}.
The shear forces V_{xi} and V_{yi} in each column developed from the displacements δ_{xi}, δ_{yi} are:
V_{xi}=K_{xi}·δ_{xi}=K_{xi}·(-θ_{z}·y_{i}) → V_{xi}=-θ_{z}·K_{xi}·y_{i} and V_{yi}=K_{yi}·δ_{yi}=K_{yi}·(θ_{z}·x_{i}) → V_{yi}=θ_{z}·k_{yi}·x_{i}
The resultant moment of all shear forces V_{xi}, V_{yi} about the centre of stiffness is equal to the external moment M_{CT}, i.e.
M_{CT}=Σ(-V_{xi}·y_{i}+V_{yi}·x_{i}+K_{zi}) → M_{CT}= θ_{z}·Σ(K_{xi}·y_{i}^{2}+K_{yi}·x_{i}^{2}+K_{zi})
Torsional stiffness K_{zi} of column i
Columns resist the rotation of the diaphragm by their flexural stiffness expressed in terms K_{xi}·y_{i}^{2 }, K_{yi}·x_{i}^{2} (in N·m), and their torsional stiffness K_{zi}, which is measured in units of moment e.g. N·m.
The torsional stiffness of a column is given by the expression K_{z}=0.5E·I_{d}/h, where 0.5Ε is the material shear modulus G, usually taken equal to 0.5^{ }of the elasticity modulus, h is the height of the column and I_{d} is the torsional moment of inertia of the column’s cross-section, taken from the following table.
Torsional stiffness of the floor diaphragm
, where (7’)
The quantity Κ_{θ}_{ }is the torsional stiffness of the diaphragm and is measured in N·m. The quantities K_{x}=Σ(K_{xi}), K_{y}=Σ(K_{yi}), measured in N/m, imply the lateral stiffnesses of the diaphragm in x and y direction respectively.
Definitions:
Lateral stiffness Κ_{j} of diaphragm denotes the force in direction j required to cause a relative parallel displacement of the diaphragm by one unit in considered direction.
Torsional stiffness K_{θ} of diaphragm denotes the moment required to cause relative rotation of the diaphragm by one unit.
Note
The torsional stiffness of columns K_{z} is very small and is usually omitted.