👋 Introduction

Shear design is one of the essential check in reinforced concrete beam design, as shear force existed in most situation. This article aims to demonstrate the steps involved in calculating the shear links to be provided of a section with given shear force through worked numerical examples.

📐 Design Data

Below are the design information of the shear analysis of the section.

Geometry
Section
	Width	400mm
	Height	600mm
Reinforcement
	Tension Area	1470 mm²

Design Action
	Shear Fy	500kN

Material
Concrete
	Strength	30 MPa
Rebar
	Strength	500 MPa

Assumption
$cover, cc$	=	$50mm$
$tension \ reinforcement \ dia., \phi_{tension}$	=	$25mm$
$shear \ links \ dia., \phi_{shear}$	=	$8mm$

🔢 Calculation Steps

*Unless otherwise specified, all clauses and equations quoted herein are based on BS EN 1992-1-1:2004.

1. Determine concrete shear resistance without shear reinforcement

1.1Formula to be used

From Cl 6.2.2(1),
$V_{Rd,c}$	=	$[C_{Rd,c}k(100\rho_{1}f_{ck})^{0.33}]b_{w}d > V_{Rd,cmin}$

1.2Compute required parameters

Effective Depth
$d$	=	$h - cc - \phi_{shear} - \phi_{tension}/2$
	=	$600 mm - 50 mm - 8 mm - 25 mm /2$
	=	$529 mm$

Depth factor(Cl.6.2.2(1))
$k$	=	$1 + \sqrt{\frac{200}{d}}$
	=	$1 + \sqrt{\frac{200}{529}}$
	=	$1.61$

Reinforcement Ratio(Eq. 6.6N)
$\rho_{1}$	=	$min(\frac{A_{sl}}{d * b}, 0.02)$
	=	$min(\frac{1470 mm²}{529 mm * 400 mm}, 0.02)$
	=	$0.007$

Concrete Factor(N.A. Cl.6.2.2(1))
$C_{Rd,c}$	=	$\frac{0.18}{\gamma_{c}}$
	=	$\frac{0.18}{1.5}$
	=	$0.12$

Minimum Shear Resistance(Eq. 6.3N)
$V_{Rd,min}$	=	$0.035k^{1.5} \cdot \sqrt{f_{ck}} \cdot b_{w} \cdot d$
	=	$0.035k^{1.5} \cdot \sqrt{30 MPa} \cdot 400 mm \cdot 529 mm$
	=	$82 kN$

1.3Compute Concrete Shear Resistance

Concrete Shear Resistance(Eq. 6.2a)
$V_{Rd,c}$	=	$[C_{Rd,c} \cdot k \cdot (100 \cdot \rho_{1} \cdot f_{ck})^{1/3}]b_{w} \cdot d$
	=	$[0.12 \cdot 1.61 \cdot (100 \cdot 0.007 \cdot 30 MPa)^{1/3}]400 mm \cdot 529 mm$
	=	$112 kN > V_{Rd,min}$

1.4Check against provided shear force

Check Concrete Shear Resistance
$V_{Rd,c}$	<	$V_{Ed}$
$112kN$	<	$500kN$

∴ Solely concrete shear resistance is not sufficient, shear links are required.

2. Determine compression strut angle

2.1Formula for compression strut angle

To find V_{Rd,max},consider
$V_{Rd,max}$	=	$\frac{\alpha b_{w}zv_{1}f_{cd}}{cot\theta + tan\theta}$
$\frac{sin2\theta}{2}$	=	$\frac{V_{Ed}}{(\alpha_{cw} v f_{cd})b_{w}z}$
$\theta$	=	$\frac{1}{2} sin^{-1} \frac{2V_{Ed}}{(\alpha_{cw} v f_{cd})b_{w}z}$

2.2Compute required parameters

Strength Reduction Factor of Concrete(Eq. 6.6N)
$v$	=	$0.6 \cdot (1 - \frac{f_{ck}}{250})$
	=	$0.6 \cdot (1 - \frac{30 MPa}{250})$
	=	$0.53$

2.3Compute compression strut angle

Inclined angle of compression strut
$\theta$	=	$\frac{1}{2}sin^{-1}\frac{2 \cdot V_{Ed}}{\alpha_{cw} \cdot v \cdot f_{cd} \cdot b_{w} \cdot z}$
	=	$\frac{1}{2}sin^{-1}\frac{2 \cdot 500 kN}{1 \cdot 0.53 \cdot 17MPa \cdot 0.9 \cdot 400mm \cdot 529mm}$
	=	$17.8° < 21.8°, take 21.8° where 21.8° < \theta < 45°(Eq. 6.7N)$

3. Calculate shear links

3.1Calculate minimum shear links

Min. Shear Rebar req.(Eq. 9.4 & 9.5N)
$\frac{A_{sw,min}}{s}$	=	$\frac{0.08 \cdot \sqrt{f_{ck}}}{f_{yk}} \cdot b \cdot sin\alpha$
	=	$\frac{0.08 \cdot \sqrt{30 MPa}}{500 MPa} \cdot 400 mm \cdot sin 90°$
	=	$0.35 mm^{2}/m$

3.2Determine required shear links

Shear Reinforcement required(Eq. 6.8)
$A_{sw}/s$	=	$\frac{V_{Ed}}{z \cdot f_{ywd} \cdot cot\theta}$
	=	$\frac{500 kN}{0.9 \cdot 529mm \cdot 500 MPa \cdot cot21.8°}$
	=	$0.84 mm^2/2 > \frac{A_{sw,min}}{s} = 0.35 mm^{2}/m, OK!$

3.3Calculate maximum shear links spacing

Max. spacing(Eq. 9.7)
$s_{max}$	=	$0.75 \cdot d \cdot (1 + cot\alpha)$
	=	$0.75 \cdot 529 mm \cdot (1 + cot90°)$
	=	$397mm$

Suggested Shear Rebars Arrangement

Bar Dia.	nos. of legs	spacing	As/s(mm2/m)	notation
8	2	100	1.00	R8@100 2 legs
12	2	200	1.13	R12@100 2 legs
16	2	200	2.01	R16@100 2 legs

∴ The Minimum Shear Rebar Area is 0.84 mm2/m and suggested shaer rebar arrangement is R8@100 2legs.

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#rc design #shear design #work example #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.