Do you need to design a reinforced concrete element under biaxial bending? In this example you can see how we validate WeStatiX’s accuracy in the calculation of the reinforcement cross-sectional area for a beam subjected to biaxial bending.
![RC design for biaxial bending](https://docs.westatix.com/wp-content/uploads/2019/11/image-144-1024x477.png)
In WeStatiX you can find the model we utilized for this verification: it’s a cantilever beam subjected to biaxial bending and compressive axial force on its free end.
Axial force | \(N_{Ed}\) | \(\) | 750 | kN |
Bending moment X | \(M_{Ed,X}\) | \(\) | 225 | kNm |
Bending moment Z | \(M_{Ed,Z}\) | \(\) | 315 | kNm |
The characteristics of the cross section are listed below.
Description | Symbol | Value | UM | |
---|---|---|---|---|
Overall width of a cross-section | \(b\) | \(\) | 500 | mm |
Height | \(h\) | \(\) | 400 | mm |
concrete cover | \(d_1\) | \(\) | 70 | mm |
concrete cover | \(d_2\) | \(\) | 70 | mm |
\(d’\) | \(h-d_1\) | 330 | mm | |
– | \(b’\) | \(b-d_2\) | 430 | mm |
ratio for interaction diagram choice | \(d’/h\) | \(\) | 0,18 | – |
And finally, the material parameters
Description | Symbol | value | UM | |
---|---|---|---|---|
Characteristic compressive cylinder strength of concrete at 28 days | \(f_{ck}\) | \(\) | 25.000,00 | kPa |
Characteristic yield strength of reinforcement | \(f_{yk}\) | \(\) | 550.000,00 | kPa |
Coefficient taking account of long term effects | \(\alpha_{cc}\) | \(\) | 1,00 | – |
Partial factor for concrete | \(\gamma_c\) | \(\) | 1,50 | – |
Partial factor for reinforcing steel | \(\gamma_s\) | \(\) | 1,15 | – |
Design value of concrete compressive strength | \(f_{cd}\) | \(\alpha_{cc} f_{ck}/\gamma_c\) | 16.666,67 | kPa |
Design value for yield strength of reinforcement | \(f_{yd}\) | \(f_{yk}/\gamma_{s}\) | 478.260,87 | kPa |
When the model is ready, you can start the analysis, and you will obtain the following diagrams.
![RC design biaxial bending](https://docs.westatix.com/wp-content/uploads/2019/11/image-99-1024x348.png)
![RC design biaxial bending](https://docs.westatix.com/wp-content/uploads/2019/11/image-101-1024x342.png)
![](https://docs.westatix.com/wp-content/uploads/2019/11/image-145-1024x315.png)
Focusing on the RC member design results, you can see that the total reinforcement area in the cross section is \(A_{s,tot}=46,17cm^2\).
![](https://docs.westatix.com/wp-content/uploads/2019/11/image-146-1024x488.png)
You can verify it with briefly with the interaction diagrams for the reinforced concrete design of a cross-section under biaxial bending. [1]
Parameterized axial force | \(\nu\) | \(N_d/b \cdot h \cdot f_{cd}\) | 0,225 | – |
\(\beta\) | \(0,6+\nu\) | 0,825 | – | |
Fictitious eccentricity | \(e’_y\) | \(e_y + \beta \cdot e_z \cdot b / h\) | 0,733 | m |
Effective uniaxial moment | \(M’_z\) | \(N_{Ed}\cdot e’_y\) | 549,84 | kNm |
Parameterized bending moment | \(\mu\) | \(M’_z/b\cdot h^2 \cdot f_{cd}\) | 0,33 | – |
![RC design interaction diagrams](https://docs.westatix.com/wp-content/uploads/2021/03/biaxialbendinginteraction.png)
Coefficient from interaction diagram | \(A_s / b \cdot h\) | \(\) | 0,02 | – |
Total reinforcement area | \(A_{s,tot}\) | \(\) | 48,00 | cm^2 |
So the error is
\( \epsilon = 1-\frac{46,17}{48,00} = 3,81\% \)Which is acceptable since the interaction diagram method is approximate. WeStatiX’s solution is therefore verified.
[1] Scriptum zur Vorlesung BETONBAU 1 nach EC 1992-1-1, Technische Universität Wien, Institut für Tragkonstruktionen – Herausgegeben von Prof. Dr.-Ing. Johann KOLLEGER