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Design of RC members in biaxial bending

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

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\(N_{Ed}\)\(\)750kN
Bending moment X\(M_{Ed,X}\)\(\)225kNm
Bending moment Z\(M_{Ed,Z}\)\(\)315kNm

The characteristics of the cross section are listed below.

Overall width of a cross-section\(b\)\(\)500mm
concrete cover\(d_1\)\(\)70mm
concrete cover\(d_2\)\(\)70mm
ratio for interaction
diagram choice

And finally, the material parameters

Characteristic compressive cylinder
strength of concrete at 28 days
Characteristic yield strength
of reinforcement
Coefficient taking account
of long term effects
Partial factor for concrete\(\gamma_c\)\(\)1,50
Partial factor for reinforcing
Design value of concrete
compressive strength
\(f_{cd}\)\(\alpha_{cc} f_{ck}/\gamma_c\)16.666,67kPa
Design value for yield
strength of reinforcement

When the model is ready, you can start the analysis, and you will obtain the following diagrams.

RC design biaxial bending
Bending moment Y
RC design biaxial bending
Bending moment Z
Normal force

Focusing on the RC member design results, you can see that the total reinforcement area in the cross section is \(A_{s,tot}=46,17cm^2\).

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
Fictitious eccentricity\(e’_y\)\(e_y + \beta \cdot e_z \cdot b / h\)0,733m
Effective uniaxial moment\(M’_z\)\(N_{Ed}\cdot e’_y\)549,84kNm
Parameterized bending moment\(\mu\)\(M’_z/b\cdot h^2 \cdot f_{cd}\)0,33
RC design interaction diagrams
Coefficient from
interaction diagram
\(A_s / b \cdot h\)\(\)0,02
Total reinforcement area\(A_{s,tot}\)\(\)48,00cm^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