Torsion is defined as the application of a twisting moment on the longitudinal axis of a prismatic piece. The following image shows the application of the strut and tie method to the torsion. This model will give us an idea of the efforts/forces that appear in the concrete and what type of reinforcement will be needed.
Two types of stresses will occur in the concrete. On the one hand, there will be tangential stresses in the concrete in the plane perpendicular to the axis of the piece. On the other hand, longitudinal efforts will be present in the beam, see the image shown above. Additionally we must check the interaction of the shear with the torsion if both phenomena are present on the concrete at the same time. We must not forget that the concrete strut is also requested in compression at the same time by the shear and the torsion.
- Tangential efforts in the plane perpendicular to the axis of the piece: The EC-2 shows us a very intuitive image of what is happening in the concrete when we apply torsion in a prismatic piece:
As we can see, the tangential forces are produced only in a certain closed area that is determined by an effective width (tef) and not in the entire section, thus leaving the internal part of the piece free of these tangential stresses. The EC-2 tells us how to calculate the tangential stresses produced by the torsion in this concrete area delimited by tef (effective width). All the formulation used can be found below, in the section "Formulation of torsional resistance". These stresses must be compared with the concrete tensile strength resistance of concrete, fctd.
- Longitudinal efforts: As we see in the strut and tie model, these tangential stresses move along the piece in the form of an "spiral". It shows how tensile ties are generated that represent the additional longitudinal reinforcement to be added to the beam and the compressed struts in the concrete that form a theta angle with the ties. This angle must be the same as the one used in the shear calculation, with the same limitations of it. The calculated horizontal reinforcement must be evenly distributed in the perimeter of the concrete considered in the calculation (tef explained above).
- Torsion-shear interaction: As we have mentioned previously, the shear and the torsion are present on the concrete at the same time. Therefore, we must check that the tangential stresses in the concrete due to both phenomena do not exceed the tensile strength of the concrete and that the compression strut also resists the sum of the compressions due to both phenomena. The EC-2 gives us a formula to consider this interaction for both cases, the tangential stresses and for the struts in compression. It also recommends a minimum reinforcement to be considered in the cases when, after taking into account the interaction, the concrete resists the interaction of both phenomena. In case of not resisting, we need to calculate the required shear reinforcement as explained below. However, if the compressed strut does not resist, the only solution will be either to modify the geometry or to increase the compressive strength of our concrete.
- Torsion reinforcement: If the concrete resistance is less than the applied tangential forces (do not forget the interaction with the shear), we must add torsional reinforcement. To calculate the torsional reinforcement, we can calculate an equivalent shear force on each face of our piece produced by the torsion. As we know the tangential stresses in the concrete due to the torsion, we can calculate the shear force on each side multiplying these stresses by the effective width where they are produced (tef) and by zi (the width / height of the piece minus tef). With this calculated shear force we can apply the formula of the shear reinforcement of the EC-2 to calculate the reinforcement needed. We should not forget that this reinforcement (torsion one) must be added to that produced by the shear force. Do not forget either that this reinforcement for the torsion has been calculated for each face, and it must be placed in the 4 faces of our section.
The torsional reinforcement must be placed in the form of closed stirrups. It is important that they are closed to properly resist the torsion.
Geometric characteristics | ||
Height (h) (mm): | ||
Width (b) (mm): | ||
Distance from the bottom face to the center of the reinforcement (R1) (mm): |
d (mm): |
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Concrete characteristics | |||
Concrete compressive characteristic strength (fck) (MPa): | Concrete partial factor: |
Steel characteristics | |||
Steel yield limit (fyk) (MPa): | Steel partial factor: |
Loads | |||
Torsional moment (kN.m): | |||
Normal force(kN) (Compression +): |
Fill in only for the calculation of the compression strut resistance and for the calculation of the shear-torsion interaction | ||
Vertical shear force (Following h)(kN): | Fill in only for the interaction Torsion-Shear |
Longitudinal reinforcement (Only for the concrete shear resistance - Interaction torsion shear | |||
Lower reinforcement (A1) (mm^2): | Fill in only to calculate the interaction with the shear force, to calculae the concrete shear resistance |
Capacity of the concrete struts calculation factors and the additional longitudinal required reinforcement | |||
Cracked concrete reduction factor (V1) (The EC-2 recommended value is automatically proposed): | State of stress in the compression chord coefficient (α cw) (EC-2 calculated coefficient, it is conservative to consider it equal to 1): | ||
cot(θ) (1 ≤ cot(θ)≤ 2.5) : |
Formulas for torsion concrete resistance | |||
Torsion resistance and aditional longitudinal reinforcement: | |||
Where: |
Results: | |||
Ratio of torsion concrete resistance | Automatic calculation after introducing all the input data | ||
Additional longitudinal reinforcerment needed for the torsion | Automatic calculation after introducing all the input data | ||
Ratio of the strut in compression | Automatic calculation after introducing all the input data |
Formulas for shear concrete resistance for the torsion-shear interaction | |||
Concrete shear resistance, the maximum value between: | |||
Where: | |||
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Concrete shear resistance, the maximum value between: | |||
Where: |
Results of the interaction torsion-shear: | |||
Ratio of torsion concrete resistance | Automatic calculation after introducing all the input data | ||
Ratio of the strut in compression considering the shear interaction | Automatic calculation after introducing all the input data |
Formulas for torsion and shear reinforcement calculation | |||
Torsion equivalent shear force (calculated for one side) and torsion transversal and shear reinforcement calculation: | |||
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Where: |
Shear reinforcement required: | |||
Torsion transversal reinforcement required | Automatic calculation after introducing all the input data | ||
Shear transversal reinforcement required | Automatic calculation after introducing all the input data |