Transverse reinforcement design

Transverse reinforcement designSymbolsCalculation effortsElements without transverse reinforcementMaximum shear forceTensile force in the longitudinal reinforcementCalculation of transverse reinforcementChecking the compression of concrete connecting rodsCalculation of shear reinforcementMinimum percentage of transverse reinforcement

Symbols

Symbol	Unit	Description
$V_{Ed}$	kN	Shear force acting
$V_{Rd,c}$	kN	Design resistant shear force for an element without shear force reinforcement
$V_{Rd,s}$	kN	Calculation value of the shear force equilibrated by shear reinforcement in the ultimate limit state
$V_{Rd,max}$	kN	Calculation value of the maximum shear force an element can withstand before the concrete rods are collapsed.

Calculation efforts

The transverse reinforcement is design at ULS in accordance with Eurocode 2.

Elements without transverse reinforcement

Resistant shear force for an element without shear reinforcement:

V_{Rd,c}=max\left\{\begin{array}{l}V_{Rd,c1}=\left[C_{Rd,c}k\sqrt[3]{100\rho_1f_{ck}}+k_1\sigma_{cp}\right]b_wd\\V_{Rd,c2}=\left[v_{min}+k_1\sigma_{cp}\right]b_wd\end{array}\right.

With:

C_{Rd,c}=\frac{0.18}{\gamma_c}

Percentage of longitudinal reinforcement :
$\rho_1=\frac{A_{sl}}{b_wd}\not>2‰$

With:

k=min\left\{2;\;1+\sqrt{\frac{0.2\;m}d}\right\}

Compressive stress of the concrete :
$\sigma_{cp}=\frac{N_{Ed}}{A_c}\leq0.2f_{cd}$

With:

v_{min}=\frac{0.053}{\gamma_c}k^{3/2}\sqrt{f_{ck}}

k_1=0.15

$V_{Ed}\leq V_{Rd,c}$ .

Maximum shear force

Even if no transverse reinforcement is required, the following condition must be respected :

V_{Ed}\leq V_{Rd,max}

$V_{Rd,max}$ is the design value of the maximum shear force that an element can support before the concrete rods are crushed:

V_{Rd,max}=\alpha_{cw}v_1f_{cd}b_wz\frac1{\tan\theta+co\tan\theta}

With :

z=0.9d

\alpha_{cw}=\left\{\begin{array}{lc}0<\sigma_{cp}<0.25f_{cd}&\alpha_{cw}=1+\frac{\sigma_{cp}}{f_{cd}}\\\begin{array}{l}0.25f_{cd}<\sigma_{cp}<0.5f_{cd}\\0.5f_{cd}<\sigma_{cp}<f_{cd}\end{array}&\begin{array}{l}\alpha_{cw}=1.25\\\alpha_{cw}=2.5\left(1-\frac{\sigma_{cp}}{f_{cd}}\right)\end{array}\end{array}\right.

Tensile force in the longitudinal reinforcement

The tensile force in the longitudinal reinforcement is calculated from the following expression. The force to be taken up by the reinforcements cannot be exceeded.

F_{td}=\frac{M_{Ed}}z+\frac12V_{Ed}\left(co\tan\theta-co\tan\alpha\right)\leq\frac{M_{Ed,max}}z

Calculation of transverse reinforcement

The transverse reinforcement is designed at the ULS in accordance with Eurocode 2.

$V_{Ed}>V_{Rd,c}$ , the transverse reinforcements are required

$1.0\leq co\tan\theta\leq2.5$

Checking the compression of concrete connecting rods

It should be checked that the compression force of the concrete rods do not exceed :

V_{Ed}\leq V_{Rd,max}=b_wzv_1f_{cd}\frac{co\tan\theta+co\tan\alpha}{1+\left(co\tan\theta\right)^2}

With:

$z=0.9d$
$v_1=0.6\left(1-\frac{f_{ck}}{250}\right)$

Calculation of shear reinforcement

The force taken up by the transverse reinforcement is calculated as follows:

V_{Rd,s}=\rho_wf_{ywd}\sin^2\alpha\left(co\tan\theta+co\tan\alpha\right)b_wz=\frac{A_{sw}}{sb_w\sin\alpha}f_{ywd}\sin^2\alpha\left(co\tan\theta+co\tan\alpha\right)b_wz

$V_{Ed}$ .

\frac{A_{sw}}s=\frac{V_{Ed}}{zf_{ywd}\left(co\tan\theta+co\tan\alpha\right)\sin\alpha}

Minimum percentage of transverse reinforcement

The percentage of transverse reinforcement is calculated as follows:

\rho_w=\frac{A_{sw}}{sb_w\sin\alpha}

Where:

$A_{sw}$ is the section of transverse reinforcements at a given cross-section
$s$ is the spacing of the transverse reinforcement
$b_w$ is the width of the section
$\alpha$ is the inclination of the transverse reinforcements with respect to the longitudinal reinforcements.

Even when no shear reinforcement is necessary, a minimum transverse reinforcement is required.

\rho_{w,min}=0.08\frac{\sqrt{f_{ck}}}{f_{yk}}