Centre of stiffness and elastic displacements of the diaphragm
The current paragraph examines the special case of orthogonal columns in parallel arrangement. The general case is examined in Appendix C.
Figure 5.4.3.11: Simple onestorey structure comprising four columns, whose tops are connected by a rigid slabdiaphragm. Figure 5.4.3.11: Simple onestorey structure comprising four columns, whose tops are connected by a rigid slabdiaphragm.
Figure 5.4.3.12: Parallel translation of the diaphragm in both directions and rotation ,due to a force H applied to the centre of mass C_{M}.(Χ0Υ initial coordinate system, xC_{Ty} main coordinate system) Figure 5.4.3.12: Parallel translation of the diaphragm in both directions and rotation ,due to a force H applied to the centre of mass C_{M}.(Χ0Υ initial coordinate system, xC_{Ty} main coordinate system)
The diaphragmatic behaviour may be considered as a superposition of three cases: (a) parallel translation of the diaphragm along the X direction due to horizontal force component H_{X}, (b) parallel translation of the diaphragm along the Y direction due to horizontal force component H_{Y}, (c) rotation of the diaphragm due to moment M_{CT} applied at the centre of stiffness C_{T}. The horizontal seismic forces are applied at each mass point, while the resultant force is applied at the centre of mass C_{M}. In case the direction of the force H passes through the point C_{T}_{ }as well as C_{M} the moment has zero value and therefore the diaphragm develops zero rotation. Translation of centre of stiffness C_{T} along x direction
Figure 5.4.3.2: Parallel translation along the x direction due to force H_{x }applied at C_{T} Figure 5.4.3.2: Parallel translation along the x direction due to force H_{x }applied at C_{T}
In case a horizontal force H_{x} is applied at C_{T} in x direction, the following 2 equilibrium equations apply:  The sum of forces in x direction is equal to H_{x}, i.e. H_{x}=Σ(V_{xoi} ) (i).
 The sum of moments V_{xoi} about the point C_{T} is equal to zero, i.e. Σ(V_{xoi} ·y_{i})=0 (ii).
Each column i carries a shear force V_{xoi}=δ_{xo}·K_{xi}. Σ (V_{xoi})= Σ ( δ _{xo} · K_{xi})= δ _{xo} · Σ (K_{xi}), expression (i) gives H_{x}=δ_{xo}·Σ(K_{xi}) → H_{x}=K_{x} · δ _{xo} where K_{x}= Σ (K_{xi}). Expression (ii) gives Σ(V_{xoi}·[Y_{i}Y_{CT}])=0 → Σ(V_{xoi}·Y_{i} )Σ(V_{xoi}·Y_{CT})=0 → Y_{CT}·Σ(V_{xoi})= Σ(V_{xoi} ·Y_{CT}) → Y_{CT}= Σ ( δ _{xo} · K_{xi} · Y_{i})/ Σ ( δ _{xo} · K_{xi}) → Y_{CT}= Σ (K_{xi} · Y_{i})/ Σ (K_{xi}) Translation of centre of stiffness C_{T} along y direction
Figure 5.4.3.3: Parallel translation in y direction due to force H_{y }applied at C_{T} Figure 5.4.3.3: Parallel translation in y direction due to force H_{y }applied at C_{T}
Accordingly, the corresponding expressions are derived for direction y. H_{y}=K_{y} · δ _{yo} where K_{y}=Σ(K_{yi}) andX_{CT}=Σ(K_{yi}·X_{i})/Σ (K_{yi}) Summarising, the centre of stiffness and the lateral stiffnesses are defined by the following expressions: Centre of stiffness and lateral stiffnesses: Rotation of the diaphragm by an angle θ_{z} about C_{T}
Figure 5.4.3.4: Displacements due to rotation developed from moment M applied at CT Figure 5.4.3.4: Displacements due to rotation developed from moment M applied at CT
Principal coordinate system 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 twocomponents 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 crosssection, taken from the following table.  
 
 
 where n is taken from the following expression  The torsional stiffness of columns K_{z} is very small and is usually omitted. Torsional stiffness of the floor diaphragm 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 directionrespectively . Torsional stiffness K _{θ} of diaphragm denotes the moment required to cause relative rotation of the diaphragm by one unit. Torsional stiffness ellipse, torsional radii and equivalent system Question: Create a simple idealised equivalent structural system with the same seismic behaviour as the actual structural system. Solution: Four idealised columns E_{1}, E_{2} and E_{3}, E_{4 }are placed_{ }symmetrically with respect to the centre C _{T} and the axes x and y, i.e. all four idealised columns have the same absolute value of coordinates x and y. Each idealised column isassumed to have stiffnesses K_{x}=1/4·Σ(K_{xi}) and K_{y}=1/4 ·Σ(K_{yi}).
Figure 5.4.3.51 Figure 5.4.3.51
This system satisfies the two conditions of the actual system, concerning the translations of all diaphragm columns. Stiffness by x: 4·1/4·Σ(K_{xi)}=Σ(K_{xi)} and stiffness by y: 4·(1/4)·Σ (K_{yi)}=Σ(K_{yi}) To satisfy the third condition, the torsional stiffness of the idealised system should be K _{θ,} _{eq} =[4 · (1/4) · Σ(K_{yi}_{)}·y^{2}+4·(1/4)·Σ(K_{xi}_{)}·x^{2}]=Σ(K_{xi})·y^{2}+Σ(K_{yi})·x ^{2} equal to the torsional stiffness of the actual system K _{θ} _{,re} = Κ_{θ} = Σ (K_{xi} · y_{i}^{2}+K_{yi} · x_{i}^{2}+K_{zi}) . i.e. K_{θ}_{,eq}=K_{θ}_{,re} → Σ(K_{yi})·x^{2}+Σ(K_{xi})· y^{2}=K_{θ} → Torsional radii of the diaphragm: The curve equation (8') is an ellipse with centre C_{T}, direction that of the principal axes (in this case the direction of the initial system) and semi axes r_{x}, r_{y}_{, }(torsional radii of the diaphragm). The torsional behaviour of a floor can be described by the torsional stiffness ellipse (C_{T}, r_{x}, r_{y}) which represents the equivalent distribution of the diaphragm stiffness. The radii r_{x}, r_{y}, of the ellipse are called torsional radii. There are infinite solutions of idealised dual system sets, whose most characteristic is the one with the four idealised columns placed in the four ellipse ends. Generally, there are infinite solutions with ntuple diametrically opposed systems, where each idealised column stiffness is equal to 1/n of the total system stiffness. Superposition of the three cases: All the previous calculations depend on the structure geometry and are not affected by the magnitude of the external loading. For instance the centre of stiffness, the structural eccentricities, and the torsional radii, are independent of the seismic force magnitude. Next, the displacements and stress resultants of the structure, due to external seismic loading H, will be calculated. The relevant seismic force H is applied at the centre of mass C_{M} of the diaphragm. This force is resolved in the two forces H_{x} and H_{y} parallel to the two axes of the principal system. In order to perform the previous analysis, theforces H_{x}, H_{y} are transferred to the centre of stiffness C_{T} together with the moment M_{CT} according to the expression: Seismic moment at the centre of stiffness: The following quantities are calculated using the external magnitudes H_{x}, H_{y}, M_{CT}:  the displacements δ _{xo} , δ _{yo} and θ _{z} of the diaphragm's centre of stiffness
Displacements of the centre of stiffness:  the displacements δ _{xi} , δ _{yi} of each column top
 shear forces and bending moments of each column in its local system
Seismic shear forces and bending moments:
Figure 5.4.3.52 Figure 5.4.3.52
M_{ji,1}M_{ji,2}=V_{ji} ·h  The distribution of shear forces depends only on the geometry of the structural system. This means that if the modulus of elasticity of column material changes, column displacements will also change but shear forces will not.
 The rotation effect increases with the distance of a column from the centre of stiffness, e.g. a column placed at the storey perimeter bears a bigger rotational charge.
 The deformations due to rotation, develop biaxial bending in columns, i.e. simultaneous bending in both directions. Therefore rotation should be limited to the extent possible.
 The biaxial bending of the columns is constantly changing due to the change of directions of seismic forces and displacements. To this end, in contrast to the nonearthquake resistant structures, the exact position of increased reinforcement cannot be predicted, thus it is placed relatively uniformly in the column perimeter.
 Even in geometrically symmetric structures, the accidental eccentricities [*]NoteSee Chapter 6. can develop intense biaxial stresses. To this end, high torsional stiffness is extremely useful. High torsional stiffness is mainly obtained from placing strong columns, usually walls, along the building perimeter.
 The high torsional stiffness is not necessary only in earthquake resistant buildings but also in every building possibly subjected to horizontal actions such as the wind or asymmetrical snow loading or to other unpredictable factors.
 The torsional stiffness of a floor diaphragm depends mainly on the position of the centre of stiffness in relation to the centre of mass, and on the magnitude of the torsional radii. Both factors depend only on the geometry of the structure and not on the magnitude of the seismic loadings.
