Stiffening of Structures

Technical Article

Buildings must be designed and dimensioned in such a way that both vertical and horizontal loads are conducted safely and without large deformations in the building. Examples of horizontal loads are wind, unintentional inclination, earthquakes, or a blast.

Finite element analysis programs such as RFEM allow you to determine internal forces and design stiffening structural elements. In this program, you can model a building including all structural components, openings, and other elements, and perform a calculation of the entire model.

Predimensioning of stiffening system can be performed using manual calculation according to the calculation method described in [1] or by using a program such as SHAPE-THIN. This software provides engineers with a better understanding of the load transfer in a structure as well as the resistance contribution of the individual structural components.

Distribution of Horizontal Forces

The horizontal load distribution for bending or torsional loading on the stiffening components can be calculated according to the following formulas.

Forces Caused by Bending
$$\begin{array}{l}{\mathrm V}_{\mathrm y,\mathrm i}\;=\;\frac{{\mathrm V}_\mathrm y\;\cdot\;({\mathrm I}_{\mathrm z,\mathrm i}\;\cdot\;{\mathrm I}_\mathrm y\;-\;{\mathrm I}_{\mathrm{yz},\mathrm i}\;\cdot\;{\mathrm I}_\mathrm{yz})\;-\;{\mathrm V}_\mathrm z\;\cdot\;({\mathrm I}_{\mathrm z,\mathrm i}\;\cdot\;{\mathrm I}_\mathrm{yz}\;-\;{\mathrm I}_{\mathrm{yz},\mathrm i}\;\cdot\;{\mathrm I}_\mathrm z)}{{\mathrm I}_\mathrm y\;\cdot\;{\mathrm I}_\mathrm z\;-\;{\mathrm I}_\mathrm{yz}²}\\{\mathrm V}_{\mathrm z,\mathrm i}\;=\;\frac{{\mathrm V}_\mathrm y\;\cdot\;({\mathrm I}_{\mathrm{yz},\mathrm i}\;\cdot\;{\mathrm I}_\mathrm y\;-\;{\mathrm I}_{\mathrm y,\mathrm i}\;\cdot\;{\mathrm I}_\mathrm{yz})\;-\;{\mathrm V}_\mathrm z\;\cdot\;({\mathrm I}_{\mathrm{yz},\mathrm i}\;\cdot\;{\mathrm I}_\mathrm{yz}\;-\;{\mathrm I}_{\mathrm y,\mathrm i}\;\cdot\;{\mathrm I}_\mathrm z)}{{\mathrm I}_\mathrm y\;\cdot\;{\mathrm I}_\mathrm z\;-\;{\mathrm I}_\mathrm{yz}²}\end{array}$$
where
Vy,i, Vz,i is the shear force in the y- or z-direction, which affects the partial cross-section i
Vy, Vz is the shear force in the y- or z-direction, which affects the gross cross-section
Iy,i, Iz,i, Iyz,i are the moments of inertia of the partial cross-section i relating to the parallel axes Y and Z by the partial cross-section centroid Si
Iy, Iz are the total second moments of area relating to the overall centroid S

Forces Caused by Torsion
$$\begin{array}{l}{\mathrm V}_{\mathrm y,\mathrm i}\;=\;\frac{{\mathrm M}_\mathrm{xs}\;\cdot\;\lbrack{\mathrm I}_{\mathrm{yz},\mathrm i}\;\cdot\;({\mathrm y}_{\mathrm M,\mathrm i}\;-\;{\mathrm y}_\mathrm M)\;-\;{\mathrm I}_{\mathrm z,\mathrm i}\;\cdot\;({\mathrm z}_{\mathrm M,\mathrm i}\;-\;{\mathrm z}_\mathrm M)\rbrack}{\mathrm\Sigma\;\lbrack{\mathrm I}_{\mathrm\omega,\mathrm i}\;+\;{\mathrm I}_{\mathrm y,\mathrm i}\;\cdot\;({\mathrm y}_{\mathrm M,\mathrm i}\;-\;{\mathrm y}_\mathrm M)²\;-\;2\;\cdot\;{\mathrm I}_{\mathrm{yz},\mathrm i}\;\cdot\;({\mathrm y}_{\mathrm M,\mathrm i}\;-\;{\mathrm y}_\mathrm M)\;\cdot\;({\mathrm z}_{\mathrm M,\mathrm i}\;-\;{\mathrm z}_\mathrm M)\;+\;{\mathrm I}_{\mathrm z,\mathrm i}\;\cdot\;({\mathrm z}_{\mathrm M,\mathrm i}\;-\;{\mathrm z}_\mathrm M)²\rbrack}\\{\mathrm V}_{\mathrm z,\mathrm i}\;=\;\frac{{\mathrm M}_\mathrm{xs}\;\cdot\;\lbrack{\mathrm I}_{\mathrm y,\mathrm i}\;\cdot\;({\mathrm y}_{\mathrm M,\mathrm i}\;-\;{\mathrm y}_\mathrm M)\;-\;{\mathrm I}_{\mathrm{yz},\mathrm i}\;\cdot\;({\mathrm z}_{\mathrm M,\mathrm i}\;-\;{\mathrm z}_\mathrm M)\rbrack}{\mathrm\Sigma\;\lbrack{\mathrm I}_{\mathrm\omega,\mathrm i}\;+\;{\mathrm I}_{\mathrm y,\mathrm i}\;\cdot\;({\mathrm y}_{\mathrm M,\mathrm i}\;-\;{\mathrm y}_\mathrm M)²\;-\;2\;\cdot\;{\mathrm I}_{\mathrm{yz},\mathrm i}\;\cdot\;({\mathrm y}_{\mathrm M,\mathrm i}\;-\;{\mathrm y}_\mathrm M)\;\cdot\;({\mathrm z}_{\mathrm M,\mathrm i}\;-\;{\mathrm z}_\mathrm M)\;+\;{\mathrm I}_{\mathrm z,\mathrm i}\;\cdot\;({\mathrm z}_{\mathrm M,\mathrm i}\;-\;{\mathrm z}_\mathrm M)²\rbrack}\end{array}$$
where
Vy,i, Vz,i is the shear force in the y- or z-direction, which affects the partial cross-section
Mxs is the secondary torsional moment, which affects the gross cross-section
Iy,i, Iz,i, Iyz,i are the moments of inertia of the partial cross-section i relating to the parallel axes Y and Z by the partial cross-section centroid Si
Iω,i is the warping constant relating to the shear center of the partial cross-section Mi
yM,i, zM,i is the coordinate of the shear center of the partial cross-section Mi
yM, zM is the coordinate of the overall shear center M

Example

The distribution of horizontal loads in the stiffening elements is explained on the system displayed in Figure 1.

Figure 01 - System

Wall thickness t = 30 cm

Cross-Section Properties

Partial Cross-Section 1
$$\begin{array}{l}{\mathrm z}_{\mathrm S,1}\;=\;\frac{\displaystyle\frac{2.15\;\cdot\;0.30\;\cdot\;0.30}2\;+\;4.70\;\cdot\;0.30\;\cdot\;(\frac{4.70}2\;+\;0.30)\;+\;2.15\;\cdot\;0.30\;\cdot\;(0.30\;+\;4.70\;+\;\frac{0.30}2)}{2.15\;\cdot\;0.30\;\cdot\;2\;+\;4.70\;\cdot\;0.30}\;=\;2.65\;\mathrm m\\{\mathrm y}_{\mathrm S,1}\;=\;\frac{2.15\;\cdot\;0.30\;\cdot\;{\displaystyle\frac{2.15}2}\;\cdot\;2\;+\;4.70\;\cdot\;0.30\;\cdot\;{\displaystyle\frac{0.30}2}}{2.15\;\cdot\;0.30\;\cdot\;2\;+\;4.70\;\cdot\;0.30}\;=\;0.59\;\mathrm m\\{\mathrm I}_{\mathrm y,1}\;=\;2.15\;\cdot\;\frac{0.303}{12}\;\cdot\;2\;+\;2.15\;\cdot\;0.30\;\cdot\;(\frac{2.65\;-\;0.30}2)²\;\cdot\;2\;+\;0.30\;\cdot\;\frac{4.703}{12}\;+\;4.70\;\cdot\;0.30\;\cdot\;(0.00)²\;=\;10.668\;\mathrm m^4\\{\mathrm I}_{\mathrm z,1}\;=\;0.30\;\cdot\;\frac{2.153}{12}\;\cdot\;2\;+\;2.15\;\cdot\;0.30\;\cdot\;(\frac{2.15}2\;-\;0.59)²\;\cdot\;2\;+\;4.70\;\cdot\;\frac{0.303}{12}\;+\;4.70\;\cdot\;0.30\;\cdot\;(0.59\;-\;\frac{0.30}2)²\;=\;1.084\;\mathrm m^4\end{array}$$

Partial Cross-Section 2
$$\begin{array}{l}{\mathrm I}_{\mathrm y,2}\;=\;\frac{0.30\;\cdot\;4.003}{12}\;=\;1.600\;\mathrm m^4\\{\mathrm I}_{\mathrm z,2}\;=\;\frac{4.00\;\cdot\;0.303}{12}\;=\;0.009\;\mathrm m^4\end{array}$$

Gross Cross-Section
Iy = 10.668 + 1.600 = 12.268 m4
Iz = 1.084 + 0.009 = 1.093 m4

The cross-section properties determined in SHAPE-THIN 8 are displayed in Figure 2.

Figure 02 - Cross-Sectional Properties

Shear Forces of Partial Cross-Section
$$\begin{array}{l}{\mathrm V}_{\mathrm y,1}\;=\;\frac{100\;\cdot\;(1.084\;\cdot\;12.268)}{12.268\;\cdot\;1.093}\;=\;99.18\;\mathrm{kN}\\{\mathrm V}_{\mathrm y,2}\;=\;\frac{100\;\cdot\;(0.009\;\cdot\;12.268)}{12.268\;\cdot\;1.093}\;=\;0.823\;\mathrm{kN}\end{array}$$

The shear forces of the partial cross-section determined in SHAPE-THIN 8 are displayed in Figure 3.

Figure 03 - Shear Forces of Partial Cross-Section

Reference

[1] Beck, H. & Schäfer, H. (1969). Die Berechnung von Hochhäusern durch Zusammenfassung aller aussteifenden Bauteile zu einem Balken. Der Bauingenieur, (Heft 3).

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