❓ What is Torsion?

Torsion is a force that causes a structural member to twist about its longitudinal axis. It occurs when a load acts at a distance from the centre of a member.

In reinforced concrete beams, if the applied torsion is too large, spiral diagonal cracks may form around the beam.

❓ What is Torsion Resistance?

Torsion resistance is the ability of a structural member to resist an applied twisting moment without without exceeding its strength.

Common symbol in eurocode:

T_Rd

Torsional Analysis

Torsional behaviour is commonly evaluated using the thin-walled hollow section analogy. A solid rectangular section is treated as an equivalent thin-walled hollow section, where the following properties need to be considered:

Effective wall thickness, t_ef,i
Perimeter of the centreline of the effective wall, U_k
Area enclosed by the centreline of the effective wall, A_k

These equivalent section properties are then used to determine the torsional shear flow and the torsional resistance of the member.

Torsional Resistance

In reinforced concrete design, torsion resistance is usually provided by:

Concrete, before it crack, which can resist a limited amount of torsion through its inherent tensile strength
Shear reinforcement, consisting of closed stirrups (links) and longitudinal reinforcement, which forms a space truss mechanism to carry torsional forces after cracking. The design of torsional reinforcement is discussed below.

After torsional cracking occurs, the concrete contributes little to the torsional resistance, and the majority of the torsion is resisted by the reinforcement system.

🧩 Dependence of Torsional Resistance from Reinforcement

From Eurocode 2, T_Rd mainly depends on: Cl. 6.3.2, BSEN 1992-1-1:2004

Inclined angle of compression strut, θ
Cross-sectional area of torsional reinforcement, A_sw
Spacing of the link, s
Design yield strength of shear reinforcement, f_ywd
Section height, h
Section width, b_w

🔍 Details of Formula of TRd,c and TRd,max

The torsion resistance provided by torsional links can be calculated in below formula Eq. 6.26, 6.30 & 6.31, BSEN 1992-1-1:2004.

Torsion Resistance w/o vertical reinforcement

\begin{aligned}T_{Rd,c} &= 2A_{k}f_{ctd}t_{ef,i}\end{aligned}

Eq. 6.26 & 6.31, BSEN 1992-1-1:2004

where
A_k = area enclosed by the centre-lines of connecting walls, including inner hollow area
t_ef,i = effective wall thickness
f_ctd = design tension yield strength of concrete

Maximum Torsion Resistance

\begin{aligned}T_{Rd,max} &= 2v_{1}\alpha_{cw}f_{cd}A_{k}t_{ef,i}sin(\theta)cos(\theta)\end{aligned}

Eq. 6.30, BSEN 1992-1-1:2004

where
A_k = area enclosed by the centre-lines of connecting walls, including inner hollow area
α_cw= coeff. for stress in compression chord, take 1 for non-prestressing case
b_w =

\frac{A_{sl}}{b_{w}d}≤ 0.02

f_cd = design yield strength of concrete
v₁ = v =

0.6[1 - \frac{f_{ck}}{250}]

, strength reduction factor for concrete cracked in shear
t_ef,i = effective wall thickness
θ= inclined degree of compression strut

Given a section with Torsion T_Ed, find the required Torsion links

Calculate Equivalent Hollow Thin-wall Section

t_ef,i = (b⋅h)/(2⋅(b+h))

A_k = (b - t_ef,i)(h - t_ef,i)

U_k = 2 ⋅ ((b - t_ef,i)+(h - t_ef,i))

Determine if concrete is sufficient for compression

T_Rd,max = 2v₁α_cwf_cdA_kt_ef,isinθcosθ

V_Rd,max = α_cwbzf_cdv₁/(cotθ + tanθ)

Combine Shear and Torsion Check for compression = (T_Ed/T_Rd,max) + (V_Ed/V_Rd,max)

If (T_Ed/T_Rd,max) + (V_Ed/V_Rd,max) > 1, larger section is required

If (T_Ed/T_Rd,max) + (V_Ed/V_Rd,max) < 1, section is sufficient for compression, continue for step 3

Determine if concrete is sufficient for tension

T_Rd,c = 2A_kf_ctdt_ef,i

V_Rd,c = C_Rd,ck(100 ρ₁ f_ck) ^1/3 b d

Combine Shear and Torsion Check for tension = (T_Ed/T_Rd,c) + (V_Ed/V_Rd,c)

If (T_Ed/T_Rd,c) + (V_Ed/V_Rd,c) > 1, additional torsional links is required, see step 4

If (T_Ed/T_Rd,x) + (V_Ed/V_Rd,c) < 1, only minimum shear links is required

Determine required torsional links

A_sw,req/s = T_Ed/(2A_kf_ydcotθ)

Determine required longitudinal links

A_sl,req = (T_EdU_kcotθ)/(2A_kf_yd)

📝Summary & Key Takeaways

Torsion is a force that causes a structural member to twist about its longitudinal axis. Torsion failure is commonly associated with the formation of spiral diagonal cracks.
Torsion resistance is ability of a structural member to resist an applied twisting moment without without exceeding its strength.
Common torsion calculation procedures are demonstrated in this article, such as calculating the required torsional links area for a given torsional force.

#rc design #torsion design #formula #structural

CivilSimple Team

The CivilSimple Team writes practical engineering guides for the profession and the curious. All articles are reviewed for technical accuracy before publication.