Theoretical Background > ULS Torsion

A full design for torsion shall be made only when the static design of a structure depends on torsional resistance of elements. In statically indeterminate structures where torsion arises from compatibility only, it will be normally unnecessary to consider torsion at ultimate limit states.

The program applies the method given in § 6.3 EC2 to calculate torsional resistance. Such method refers to thin walled closed sections in which equilibrium is satisfied by a closed shear flow. Solid sections may be modelled by equivalent thin-walled sections. Complex shapes, such T or flanged sections may be divided into more sub-sections, each of which is modelled by equivalent thin-walled sections, and the total resistance taken as the sum of the capacities of individual elements.

The method of calculation assumes an annular flow of shear stresses and, consequently, one can assume for resistance a tubular truss in which tensile stresses are balanced by the longitudinal and transverse reinforcement therein and the oblique compression forces by the struts of outer concrete.

Are explicitly excluded from this discussion (and from the program) sections that consist in open thin-walled profiles.

SLU check

EC2 (§6.3.2) define tef = A/u as the effective wall thickness given by the ratio between the concrete area (including inner hollow area) and the outer perimeter u. The thickness tef need not be taken as less than twice the distance between edge and centre of longitudinal bars. For hollow sections the real thickness is an upper limit.

Torsion check in a polygonal equivalent thin-walled section is satisfied if the acting torsion force TEd is not greater of the following three torsional resistances provided by concrete struts and by the transverse (hoops) and longitudinal reinforcement:

a) Concrete: TRd,max = 2 Ak ⋅ tef ⋅ f’cd ⋅ cot ϑ / (1+ cot2ϑ)

b) Hoops: TRsd = 2 ⋅ Ak⋅ As/s ⋅ fywd ⋅ cot ϑ

c) Longitudinal bars: TRld = 2 ⋅ Ak ⋅ ΣAsl / uk ⋅ fyd / cot ϑ

where:

Ak area enclosed by the centre-lines of the connecting walls, including inner hollow areas

uk perimeter of area Ak

As area of cross-section of the hoop bar (one bar)

s pitch of hoops

ΣAsl cross-sectional area of longitudinal reinforcement for torsion

f’cd = v ⋅ fcd [see (6.9)EC2]

fywd design yield stress of the hoops

fywd design yield stress of the longitudinal bars

cot ϑ = cotangent of the angle ϑ of compression struts

The maximum resistance of a member subjected to torsion and shear is limited by the capacity of compression struts. In order not to exceed this capacity the following safety condition should be satisfied:

TEd/TRd,max + VEd/VRd,max ≤ 1.0 (1)

This safety condition is named "safety factor" in the program output of predefined section.

When torsion and shear are simultaneously present the angle ϑ assumed in the checks must be the same.

For light action effects the absence of cracking is expressed by:

TEd/TRd,c + VEd/VRd,c ≤ 1.0 (2)

where:

TRd,c = 2 fctd ⋅ tef ⋅ Ak

Predefined sections

For this typology the program performs (with the above formulas) both the calculation of verification (check) that the design of the reinforcement. If combinations with torsion and shear force acting simultaneously interaction check is carried out with the relationship (1).

As default value of cotϑ is assumed that assigned in Code and reinforcement options window. In presence of bending moment a portion of the upper and lower area of longitudinal bars for bending is reserved to torsion resistance (proportionally to the length of sides of the section). The area of web bars completes the longitudinal reinforcement for torsion. All the longitudinal area of bars resisting to torsion are non included in the resisting are for bending.

Polygonal sections in torsion

Because of their generality for General section typology is not expected torsion check in interaction with bending and shear. So for this section typology program performs only the torsion verification for the assigned reinforcement; also shown in output are the areas of stirrups and longitudinal bars strictly necessary to balance torsion force. Torsional concrete struts resistance TRd,max is also calculated in order to manually control the above relationship (1). The torsional reinforcement areas may be, at last, added to those already calculated for bending and shear and so to get the final design of the section. You must assign for Cot ϑ the same value assumed for the shear force acting in the same combination.

The program proposes as results a disposition of the reinforcement that respect the detailing of standards (§9.2.3 EC2):

- The longitudinal bars spacing of the torsion link should not exceed u/8 or the requirement in 9.2.2(6) or the lesser dimension of the beam corss-section.

- The longitudinal bars should be so arranged that there is at least one bar at each corner, the others being distributed uniformly around the inner periphery of the links, with a spacing not greater than 350 mm.