# Stiffening of Structures

## Technical Article on the Topic Structural Analysis Using Dlubal Software

### 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

Formula 1

$$Vy,i = Vy · (Iz,i · Iy - Iyz,i · Iyz) - Vz · (Iz,i · Iyz - Iyz,i · Iz)Iy · Iz - Iyz²Vz,i = Vy · (Iyz,i · Iy - Iy,i · Iyz) - Vz · (Iyz,i · Iyz - Iy,i · Iz)Iy · Iz - Iyz²$$

where
Vy,i, Vz,i: shear force in the y- or z‑direction, which affects the partial cross‑section i
Vy, Vz: shear force in the y- or z‑direction, which affects the gross cross‑section
Iy,i, Iz,i, Iyz,i: 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: total second moments of area relating to the overall centroid S

Forces Caused by Torsion

Formula 2

$$Vy,i = Mxs · [Iyz,i · (yM,i - yM) - Iz,i · (zM,i - zM)]Σ [Iω,i + Iy,i · (yM,i - yM)² - 2 · Iyz,i · (yM,i - yM) · (zM,i - zM) + Iz,i · (zM,i - zM)²]Vz,i = Mxs · [Iy,i · (yM,i - yM) - Iyz,i · (zM,i - zM)]Σ [Iω,i + Iy,i · (yM,i - yM)² - 2 · Iyz,i · (yM,i - yM) · (zM,i - zM) + Iz,i · (zM,i - zM)²]$$

where
Vy,i, Vz,i: shear force in the y- or z‑direction, which affects the partial cross‑section
Mxs: secondary torsional moment, which affects the gross cross‑section
Iy,i, Iz,i, Iyz,i:  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: warping constant relating to the shear center of the partial cross‑section Mi
yM,i, zM,i: coordinate of the shear center of the partial cross‑section Mi
yM, zM: coordinate of the overall shear center M

#### Example

The distribution of horizontal loads on the stiffening elements is explained on the structural system shown in Figure 01.

Wall thickness t = 30 cm

Cross-Section Properties

Partial cross-section 1

Formula 3

$$zS,1 = 2,15 · 0,30 · 0,302 + 4,70 · 0,30 · (4,702 + 0,30) + 2,15 · 0,30 · (0,30 + 4,70 + 0,302)2,15 · 0,30 · 2 + 4,70 · 0,30 = 2,65 myS,1 = 2,15 · 0,30 · 2,152 · 2 + 4,70 · 0,30 · 0,3022,15 · 0,30 · 2 + 4,70 · 0,30 = 0,59 mIy,1 = 2,15 · 0,30312 · 2 + 2,15 · 0,30 · (2,65 - 0,302)² · 2 + 0,30 · 4,70312 + 4,70 · 0,30 · (0,00)² = 10,668 m4Iz,1 = 0,30 · 2,15312 · 2 + 2,15 · 0,30 · (2,152 - 0,59)² · 2 + 4,70 · 0,30312 + 4,70 · 0,30 · (0,59 - 0,302)² = 1,084 m4$$

Partial cross-section 2

Formula 4

$$Iy,2 = 0.30 · 4.00312 = 1.600 m4Iz,2 = 4.00 · 0.30312 = 0.009 m4$$

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 02.

Shear Forces of Partial Cross-Section

Formula 5

$$Vy,1 = 100 · (1.084 · 12.268)12.268 · 1.093 = 99.18 kNVy,2 = 100 · (0.009 · 12.268)12.268 · 1.093 = 0.823 kN$$

The shear forces of the partial cross‑section determined in SHAPE‑THIN 8 are displayed in Figure 03.

#### Reference

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

#### Sonja von Bloh, M.Sc.

Product Engineering & Customer Support

Ms. von Bloh provides technical support for our customer and is responsible for the development of the SHAPE‑THIN program.

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• Updated 06/21/2021

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