Torsion

Abstract

This chapter presents the design methods of RC elements subjected to torsion. After an introductory note on the stress distribution in beam elements as deduced from the basic structural mechanics, the peripheral truss model is described, with its more recent improvements, for the torsional resistance calculations of RC beams. The interaction of torsion with the other internal force components of bending moment, shear and axial action is treated for the actual design applications. In the final section, with reference to the overall stability of the building examined in the previous chapters, the calculation of the corewall system is developed under the pertinent horizontal actions.

The original version of this chapter was revised: For detailed information please see Erratum. The erratum to this chapter is available at 10.1007/978-3-319-52033-9_11

Appendix: Torsion

8.1.1 Table 8.1: Torsion: Elastic Design—Formulas

Reinforced concrete elements subject to circulatory torsion.

Symbols

T :

Torsional moment

W :

Resisting torsional module of the section

J :

Torsional moment of inertia of the section

τ :

Maximum shear stress

G :

Elastic shear modulus

χ :

Torsional curvature

h _s :

Depth of resisting section

b _s :

Width of resisting section

A = h _s b _s :

Area enclosed by the resisting perimeter reinforcement

A _s :

Sectional area of a closed stirrup

s :

Spacing of stirrups

a _s  = A _s/s :

Unit area of stirrups

A _l :

Total area of longitudinal bars

u = 2(h _s + b _s):

Perimeter of resisting peripheral reinforcement

a _l  = A _l /u :

Unit area of longitudinal reinforcement

Uncracked Section

• Circular section (r = radius of the section)

  $$ \begin{aligned} J & = \frac{{\pi r^{4} }}{2}\quad \chi = \frac{T}{GJ} \\ W & = \frac{{\pi r^{3} }}{2}\quad \tau = \frac{T}{W} \\ \end{aligned} $$

• Circluar hollow section (r _e, r _i  = external and internal radii)

  $$ \begin{array}{*{20}c} {J = \frac{\pi }{2}(r_{\text{e}}^{4} - r_{i}^{4} )} & {\chi = \frac{T}{GJ}} \\ {W = \frac{J}{{r_{\text{e}} }}} & {\tau = \frac{T}{W}} \\ \end{array} $$

• Rectangular section (h, b = longer and shorter sides)

  $$ \begin{array}{*{20}c} {J = k_{2} hb^{3} } & {\chi = \frac{T}{GJ}} \\ {W = k_{1} hb^{2} } & {\tau = \frac{T}{W}} \\ {k_{1} = \frac{1}{3 + 1.8\beta }} & {k_{2} = \frac{1}{{3 + 4.1\sqrt {\beta^{3} } }}} \\ \end{array} $$

with $\beta = b/h \le 1.$

Section composed of rectangles (h _i , b _i  = sides of the i-th rectangle)

$$ \begin{aligned} J_{i} & = k_{2i} h_{i} b_{i}^{3} \\ J & = \sum {J_{i} } \quad \chi = \frac{T}{GJ} \\ T_{i} & = \frac{{J_{i} }}{J}T \\ W_{i} & = k_{{{\text{l}}i}} h_{i} b_{i}^{2} \quad \tau = \frac{T}{{W_{i} }} \\ \end{aligned} $$

Thin hollow section (t, t _o = current and minimum thicknesses)

$$ J = 4A^{2} /\oint {\frac{dl}{t}} \quad \chi = \frac{T}{GJ} $$

where A is the area enclosed by the mid-fiber and l is the abscissa along the mid-fiber

$$ W = 2At_{\text{o}} \quad \tau = \frac{T}{W} $$

In particular for a thickness t = const.:

$$ J = 4A^{2} t/L\quad L = {\text{perimeter on the mid-fiber}} $$

For sides with thickness t _i  = cost.:

$$ J = 4A^{2} /\sum {L_{i} /t_{i} } \quad L_{i} = {\text{length}}\,{\text{of}}\,{\text{the}}\,i{\text{-th}}\,{\text{side}}. $$

Rectangular Cracked Section

RC straight beam with constant cross section, subject to simple torsion in the cracked elastic stage, reinforced with longitudinal bars and transverse stirrups.

Stress in stirrups

$$ \sigma_{\text{s}} = \frac{T}{{2Aa_{\text{s}} }}\quad {\text{circulatory tension}} $$

Stress in longitudinal bars

$$ \sigma_{l} = \frac{T}{{2Aa_{l} }}\quad {\text{longitudinal}}\,{\text{tension}} $$

Peripheral concrete stress

$$ \sigma_{\text{c}} = \frac{T}{{A{\text{t}} }}\quad {\text{compression}}\,{\text{inclined}}\,{\text{at}}\, 4 5{^\circ } $$

with

$$ \begin{aligned} A_{c} & = bh\quad u_{c} = 2b + 2h \\ t & = A_{c} /u_{c} \ge 1.5c \\ b_{s} & = b - t\quad h_{s} = h - t \\ A & = b_{s} h_{s} \\ \end{aligned} $$

where b and h are the longer and shorter sides of the section and c is the concrete cover at the axis of the bar placed at the corners.

8.1.2 Chart 8.2: Torsion: Resistance Design—Formulas

Reinforced concrete straight elements with constant cross section, subject to circulatory torsion, at the resisting ultimate limit state of the cracked phase, reinforced with longitudinal bars and transverse stirrups.

Symbols

T _Ed :

Design value of torsion

T _Rd :

Design value of torsional resistance

T _sd :

Torsional resistance from stirrups

T _ld :

Torsional resistance from longitudinal bars

T _cd :

Torsional resistance from concrete

θ _I :

Angle of initial cracking due to torsion

θ :

Angle of peripheral compressions on the beam axis

$\lambda_{\text{I}} = ctg\theta_{\text{I}}$ :

Inclination of initial cracking due to torsion

$\lambda_{\text{c}} = ctg\theta$ :

Inclination of peripheral compressions in concrete

σ _I :

Tensile principal stress corresponding to τ

see also Charts 2.2, 2.3, 8.1.

Resistance with Isostatic Truss

With $\lambda_{\text{I}} = \lambda_{\text{c}} = 1$ it is set

$$ T_{\text{Rd}} = \hbox{min} (T_{\text{sd}} ,T_{l\text{d}} ,T_{\text{cd}} ) \ge T_{\text{ad}} $$

where

$$ \begin{aligned} T_{\text{sd}} & = 2Aa_{\text{s}} f_{\text{yd}} \quad {\text{tension-torsion}}\,{\text{from}}\,{\text{stirrups}} \\ T_{l\text{d}} & = 2Aa_{l} f_{\text{yd}} \quad {\text{tension-torsion}}\,{\text{from}}\,{\text{longitudinal}}\,{\text{bars}} \\ T_{\text{cd}} & = Atf_{{{\text{c}}2}} \quad {\text{compression-torsion}}\,{\text{from}}\,{\text{concrete}} \\ \end{aligned} $$

Resistance with Given Truss

Assumed $\lambda_{\text{c}}$ in the interval $\lambda_{\hbox{min} } \le \lambda_{\text{c}} \le \lambda_{\hbox{max} }$, one can set

$$ T_{\text{Rd}} = \hbox{min} (T_{\text{sd}} ,T_{l\text{d}} ,T_{\text{cd}} ) \ge T_{\text{Ed}} $$

where

$$ \begin{aligned} T_{\text{sd}} & = 2Aa_{\text{s}} f_{\text{yd}} \lambda_{\text{c}} \quad {\text{tension-torsion}}\,{\text{from}}\,{\text{stirrups}} \\ T_{l\text{d}} & = 2Aa_{l} f_{\text{yd}} /\lambda_{\text{c}} \quad {\text{tension-torsion}}\,{\text{from}}\,{\text{longitudinal}}\,{\text{bars}} \\ T_{\text{cd}} & = 2Atf_{\text{c2}} \lambda_{\text{c}} /\left( {1 + \lambda_{\text{c}}^{ 2} } \right)\quad {\text{compression-torsion}}\,{\text{from}}\,{\text{concrete}} \\ \end{aligned} $$

and where for simple torsion one has:

$$ \begin{aligned} {\lambda }_{\text{I}} & = {\tau /\sigma }_{\text{I}} = 1.0 \\ {\lambda }_{\hbox{min} } & = {\lambda }_{\text{I}} /2.5 = 0.4 \\ {\lambda }_{\hbox{max} } & = 2.5\,{\lambda }_{\text{I}} = 2.5 \\ \end{aligned} $$

Resistance with Calculated Truss

$$ \lambda_{\text{r}} = \sqrt {a_{{l}} /a_{\text{s}} } $$

• High shear reinforcement ratio $(\lambda_{\text{r}} < \lambda_{\hbox{min} })$

  $$ T_{\text{Rd}} = \hbox{min} (T_{l\text{d}} ,T_{\text{cd}} ) \ge T_{\text{Ed}} $$

  where

  $$ \begin{aligned} T_{l\text{d}} & = 2Aa_{{l}} f_{\text{yd}} /\lambda_{\hbox{min} } = 5Aa_{{l}} f_{\text{yd}} \quad ( < T_{\text{sd}} ) \\ T_{\text{cd}} & = 2Atf_{{{\text{c}}2}} \lambda_{\hbox{min} } /\left( {1 + \lambda_{\hbox{min} }^{2} } \right) = 0.69Atf_{{{\text{c}}2}} \\ \end{aligned} $$

• Balanced reinforcement $\text{(}\lambda_{\hbox{min} } \le \lambda_{\text{c}} \le \lambda_{\hbox{max} } )$

  $$ T_{\text{Rd}} = \hbox{min} (T_{\text{sd}} ,T_{\text{cd}} ) \ge T_{\text{Ed}} $$

  where

  $$ \begin{aligned} T_{\text{sd}} & = 2Aa_{\text{s}} f_{\text{yd}} \lambda_{\text{r}} \quad ( = T_{l\text{d}} ) \\ T_{\text{cd}} & = 2Atf_{\text{c2}} \lambda_{\text{r}} /\left( {1 + \lambda_{\text{r}}^{ 2} } \right) \\ \end{aligned} $$

• Low shear reinforcement ratio $(\lambda_{\text{r}} > \lambda_{\hbox{max} })$

  $$ T_{\text{Rd}} = \hbox{min} (T_{\text{sd}} ,T_{\text{cd}} ) $$

  where

  $$ \begin{aligned} T_{\text{sd}} & = 2Aa_{\text{s}} f_{\text{yd}} \lambda_{\hbox{max} } = 5Aa_{\text{s}} f_{\text{yd}} \quad ( < T_{l\text{d}} ) \\ T_{\text{cd}} & = 2Atf_{\text{c2}} \lambda_{\hbox{max} } /\left( {1 + \lambda_{\hbox{max} }^{2} } \right) = 0.69Atf_{{{\text{c}}2}} \\ \end{aligned} $$

8.1.3 Chart 8.3: Torsion: Interaction Formulas

Reinforced concrete elements subject to torsion, uniaxial bending, shear and axial force.

Symbols

N _Ed :

Design value of applied axial force

M _Ed :

Design value of applied bending moment

M _Rd :

Design value of resisting bending moment

V _Ed :

Design value of applied shear force

V _cd :

Design value of resistance by compression-shear

z :

Distance between tension and compression chords

y _s :

Distance between tension chord and axial force axis

y _c :

Distance between compression chord and axial force axis

$\lambda_{\text{c}}^{{\prime }}$ :

Inclination of higher compressions

$\lambda_{\text{c}}^{{\prime \prime }}$ :

Inclination of lower compressions

λ _c :

Mean inclination of web compressions

A _s :

Area of longitudinal reinforcement in tension under M _Ed

$A_{\text{s}}^{{\prime }}$ :

Area of longitudinal reinforcement in compression under M _Ed

$a_{\text{s}}^{'}$ :

Unit area of stirrups on the side under higher tension

$a_{\text{s}}^{{\prime \prime }}$ :

Unit area of stirrups on the side under lower tension

$\bar{x}$ :

Depth of compression chord

b :

Width of compression chord

b _w :

Width of web

see also Charts 2.2, 2.3, 3.11, 4.2, 6.12, 8.1 and 8.2.

Notes on Truss Model

In the following interaction formulas M _Ed, V _Ed, T _Ed are assumed with the absolute value, whereas N _Ed is intended positive in tension and of small magnitude $(M_{\text{Ed}} \gg |N_{\text{Ed}} |z )$. The inclination $\lambda_{c}^{{\prime \prime }}$ is assumed positive if rising like $\lambda_{c}^{{\prime }}$, negative if falling.

For the elements of the truss model the resistance verifications are therefore set as shown hereby.

• Tension chord (due to M _Ed)

  $$ Z_{\text{Ed}} = N_{\text{Ed}} \frac{{y_{\text{c}} }}{z} + M_{\text{Ed}} \frac{1}{z} + V_{\text{Ed}} \frac{{\lambda_{\text{c}} }}{2} + T_{\text{Ed}} \frac{{\lambda_{\text{c}} }}{4A} < A_{\text{s}} f_{\text{yd}} $$

• Compression chord (due to M _Ed)

  $$ C_{\text{Ed}} = + N_{\text{Ed}} \frac{{y_{\text{s}} }}{z} - M_{\text{Ed}} \frac{1}{z} + V_{\text{Ed}} \frac{{\lambda_{\text{c}} }}{2} + T_{\text{Ed}} \frac{{\lambda_{\text{c}} }}{4A} < A_{\text{s}}^{\prime } f_{\text{yd}} $$

  if from the formula above one obtains C _Ed < 0, it is set:

  $$ ( * )C_{\text{Ed}} = - N_{\text{Ed}} \frac{{y_{\text{s}} }}{z} + M_{\text{Ed}} \frac{1}{z} - V_{\text{Ed}} \frac{{\lambda_{\text{c}} }}{2} < b\overline{x} f_{\text{cd}} + A_{\text{s}}^{\prime } f_{\text{yd}} $$

• Transverse stirrup (side with higher tension)

  $$ q_{\text{sd}}^{\prime } = \left[ {V_{\text{Ed}} \frac{1}{2z} + T_{\text{Ed}} \frac{1}{2A}} \right]\frac{1}{{\lambda_{\text{c}}^{\prime } }} < a_{\text{s}}^{\prime } f_{\text{yd}} $$

• Transverse stirrup (side with lower tension)

  $$ q_{\text{sd}}^{\prime \prime } = \left[ {V_{\text{Ed}} \frac{1}{2z} - T_{\text{Ed}} \frac{1}{2A}} \right]\frac{1}{{\lambda_{\text{c}}^{\prime \prime } }} < a_{\text{s}}^{\prime \prime } f_{\text{yd}} $$

• Diagonal in compression (more stressed side)

  $$ (*)\bar{q}_{\text{cd}}^{\prime } = \left[ {V_{\text{Ed}} \frac{1}{2z} + T_{\text{Ed}} \frac{1}{2A}} \right]\frac{{1 + \lambda_{\text{c}}^{\prime 2} }}{{\lambda_{\text{c}}^{\prime } }} < tf_{{{\text{c}}2}} $$

• Diagonal in compression (least stressed side)

  $$ ( * )\bar{q}_{\text{cd}}^{\prime \prime } = \left[ {V_{\text{Ed}} \frac{1}{2z} - T_{\text{Ed}} \frac{1}{2A}} \right]\frac{{1 + \lambda_{\text{c}}^{\prime \prime 2} }}{{\lambda_{\text{c}}^{\prime \prime } }} < tf_{{{\text{c}}2}} $$

Based on what mentioned above, the suggested values are

$$ \begin{aligned} & 1.0 \le {\lambda }_{\text{c}}^{\prime } \le 2.5 \\ & \quad +\, {\lambda }_{\text{c}}^{\prime } /2 \le {\lambda }_{\text{c}}^{\prime \prime } \le +\, {\lambda }_{\text{c}}^{\prime } \quad {\text{if}}\quad {\text{V}}_{\text{Ed}} /2{\text{z}} > {\text{T}}_{\text{Ed}} /2{\text{A}} \\ & \quad -{\lambda }_{\text{c}}^{\prime } \le {\lambda }_{\text{c}}^{\prime \prime } \le - {\lambda }_{\text{c}}^{\prime } /2\quad {\text{if}}\quad {\text{V}}_{\text{Ed}} /2{\text{z}} < {\text{T}}_{\text{Ed}} /2{\text{A}} \\ {\lambda } &_{\text{c}} = ({\lambda }_{\text{c}}^{\prime } + {\lambda }_{\text{c}}^{\prime \prime } )/2 \\ \end{aligned} $$

The formulas indicated with (*) can be substituted by the more reliable empirical ones of the next section. In the verification of the chords the term $V_{\text{Ed}} \lambda_{\text{c}} /2$ introduces the rule of shifting of moments already shown in the construction requirements of Chart 4.3.

Application to the Project

For the interaction formulas reported above, the following practical interpretations are given.

Reinforcement in tension

The necessary longitudinal reinforcement can be designed separately for bending moment (see Chart 3.11) including the possible axial force (see Chart 6.12), and for torsion (see Chart 8.2) placing along the tension side of the beam all the flexural reinforcement plus half of the torsional reinforcement.

If on the edge in compression due to the bending moment, the effect of torsion is predominant, a longitudinal reinforcement designed for the residual tension is introduced, equal to half of the global torsional one minus the flexural compression.

On the contrary, if the effect of bending is predominant, no torsional reinforcement is to be added on the edge in compression, whereas the effect of the circulatory flux of shear stresses on its resistance limit can be evaluated with the empirical formula shown below.

The orthogonal stirrups necessary on each of the two sides of the section can be designed separately for half of the shear force (see Chart 4.2) and for torsion (see Chart 8.2), again for a given inclination $\lambda_{\text{c}}^{\prime } \,o\,\lambda_{\text{c}}^{\prime \prime }$ of the web compressions; the two sets of stirrups are therefore to be added or deduced, depending on whether it is the case of the side with higher or lower stresses.

Concrete in compression

For compressions in concrete, the bending-torsion and shear-torsion interaction of beams with no significant axial forces can be evaluated, respectively, with the following empirical formulas of resistance verification:

$$ \left( {\frac{{M_{\text{Ed}} }}{{M_{\text{Rd}} }}} \right)^{2} + \left( {\frac{{T_{\text{Ed}} }}{{T_{\text{cd}} }}} \right)^{2} \le 1\quad \left( {\frac{{V_{\text{Ed}} }}{{V_{\text{cd}} }}} \right)^{2} + \left( {\frac{{T_{\text{Ed}} }}{{T_{\text{cd}} }}} \right)^{2} \le 1. $$

8.1.4 Chart 8.4: Torsion: Construction Requirements

For the symbols see Chart 8.1.

Stirrups

Stirrups should be bent to follow, without outward pressures, the entire resisting peripheral perimeter of the section.

Stirrups should be closed, with adequate anchorages to ensure an effective circulatory continuity.

The stirrups spacing should be limited, other than with what shown for shear in Chart 4.5, also with s ≤ u/8.

For the minimum amount of stirrups, one can follow what mentioned for shear in the mentioned Chart 4.5.

Longitudinal bars

The longitudinal bars should be effectively continuous, well anchored at the ends and enclosed in the stirrups.

At each stirrup bending there should be a longitudinal bar, whose diameter should be sufficient, with respect to the stirrup spacing, to deviate the flux of compressions in the concrete.

Along the resisting peripheral perimeter there should be at least one longitudinal bar every 350 mm.

For the minimum longitudinal reinforcement one can follow what mentioned for beams in bending in Chart 3.19.