# Stiffening of Structures

### Technical Article

001431 05/02/2017

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 01.

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
$$\begin{array}{l}{\mathrm I}_\mathrm y\;=\;10.668\;+\;1.600\;=\;12.268\;\mathrm m^4\\{\mathrm I}_\mathrm z\;=\;1.084\;+\;0.009\;=\;1.093\;\mathrm m^4\end{array}$$

The cross-section properties determined in SHAPE‑THIN 8 are displayed in Figure 02.

##### 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 03.

#### Reference

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