# Design shear resistance (concrete beam: EC2)

The design value of the shear resistance of a concrete section with
vertical shear reinforcement, V_{Rd,max} is given by;

V_{Rd,max} =
0.9*α_{cw}*b_{w}*d*ν_{1}*f_{cwd}/(cotθ + tanθ)

where

θ = MIN {θ_{max}, MAX[0.5*sin^{-1}[2*V_{Ed,max}/(
α_{cw}*b_{w}*0.9*d*ν_{1}*f_{cwd})],
θ_{min}]}

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For design in accordance with **UK NA, Irish
NA, Malaysian NA** and **Singapore NA,**

f_{cwd}^{1} = α_{ccw}* min(f_{ck},50)/γ_{C}

` `

For design in accordance with **EC2 Recommendations, ****Finnish NA, Norwegian
NA** and **Swedish NA**;

f_{cwd} = α_{ccw}*f_{ck}/γ_{C}

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For design in accordance with **EC2 Recommendations, UK NA, Irish NA, Malaysian NA,
Singapore NA, ****Finnish NA, Norwegian NA** and **Swedish NA****;**

α_{cw} = 1.0 (assuming no axial load in the beam)

α_{cw} = 1.0

γ_{C} = 1.5

ν_{1} = 0.6*(1 - (f_{ck}/250))
*f*_{ck in N/mm2}

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The limits of θ are given by 1 ≤ cotθ ≤ 2.5 which gives;

θ_{max} = tan^{-1}1

θ_{min} = tan^{-1}(0.4)

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For design in accordance with **EC2 Recommendations, UK NA, Irish NA,
Malaysian NA, Singapore NA** and **Swedish NA**;

α_{ccw} = 1.0

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For design in accordance with **Finnish NA and Norwegian NA**;

α_{ccw} = 0.85

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IF V_{Ed,max} > V_{Rd,max}

where

V_{Ed,max} = the maximum design shear force acting anywhere on the beam

THEN the shear design process FAILS since the section size is inadequate for shear (the compression strut has failed at the maximum allowable angle).

The design shear capacity of the minimum area of shear links actually provided,
V_{nom} is given by^{2};

V_{nom} = (A_{sw,min,prov} /s_{l} ) * 0.9 * d *
f_{ywd} * cotθ

where

A_{sw,min, prov} is the area of shear reinforcement provided to meet the
minimum requirements.

f_{ywd} = f_{ywk}/γ_{S}

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For design in accordance with **EC2 Recommendations, UK NA, Irish NA, Malaysian NA,
Singapore NA, ****Finnish NA,****Norwegian NA** and **Swedish NA** the
limiting values of θ are given by;

1 ≤ cotθ ≤ 2.5

θ_{max} = tan^{-1}1

θ_{min} = tan^{-1}(0.4)

γ_{S} = 1.15

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The maximum possible value for the shear resistance provided by this area of shear
reinforcement will be when the angle of the compression strut is the minimum value
i.e. cotθ = 2.5 and therefore V_{nom} can be simplified to;

V_{nom} = (A_{sw,min,prov}/sl)*2.25*d*f_{ywd}

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In any region, *i*;

IF

V_{Ed,i} > V_{nom}

where

V_{Ed,i} = the maximum shear in region
*i* (see the next section for details of how this is determined in the support
region)

THEN shear links are required in the region.

For designed shear links in shear region S_{i}, first calculate the angle of
the compression strut from;

θ_{Si} = MIN{θ_{max}, MAX[0.5*sin^{-1}[2*V_{Ed,Si}
/( α_{cw}*b_{w}*0.9*d*ν_{1}*f_{cd})],
θ_{min}]}

The area of links required in shear region S_{i} is then given by;

(A_{sw,reqd}/s)_{Si} =
V_{Ed,Si}/(0.9*d*f_{ywd}*cotθ_{Si})

where

V_{Ed,Si} = the maximum shear force in shear region S_{i}

## Does Tekla Structural Designer design for the shear force at the support, or at a distance d from the support?

The provisions of Section 6.2.1(8) of BS EN 1992-1-1:2004 allow that for members subject to ‘predominantly uniformly distributed loading’ the design shear force need not be checked at a distance less than d from the face of the support.

Tekla Structural Designer automatically takes advantage of the provision of Section 6.2.1(8).

In this context, a ‘predominantly uniformly distributed loading’ is defined as one in
which the change in the shear force occurring at the face of the support,
V_{Ed,max} to that occurring at a distance d from the face of the support,
V_{Ed,d} is ≤ 25%*V_{Ed,max} i.e.

V_{Ed,max} – V_{Ed,d} ≤ 0.25*V_{Ed,max}

The limiting value of 25%*V_{Ed,max} is chosen to allow for variable and
relatively small point loads occurring in the length d from the face of the support
and yet not to exclude genuinely uniformly distributed loads in deeper beams.

The use of this limiting value effectively means that the rule is only taken advantage of for beams with uniformly distributed loads that have a span/depth ratio ≥ 8.

Since beams in most structures fall in the range 20 ≥ span/depth ≥ 14 this limit excludes only a very small percentage of beams from taking advantage of this enhancement.

## Does Tekla Structural Designer enhance the shear strength of the section near the support?

Section 6.2.3(8) of BS EN 1992-1-1:2004 states;

“For members with loads applied on the ** upper side** within a distance
0.5*d ≤ a

_{v}≤ 2.0*d

**of this load to the shear force V**

__the contribution___{Ed}may be reduced by β = a

_{v}/(2*d).”

Using this rule, the shear force resulting from any design point load of value P that is applied to the upper surface of the concrete beam within the valid distance can be taken as;

V_{Ed,PL,R} = β*V_{Ed,PL}

However, the background paper to the UK NAs for BS EN 1992-1 and BS EN 1992-3 published as PD 6687-1:2010 draws attention to the problem of making use of this enhancement when there are multiple point loads or when the beam is designed for an envelope of shear obtained from a number of load cases.

This document makes reference to a paper by Jackson et al “Enhancement at short shear spans in EN 1992” published in The Structural Engineer Vol. 85 No.23/24 in December 2007 which demonstrates that there are serious misgivings surrounding the use of this method which the authors claim to be fundamentally flawed.

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*Footnotes*

^{1}Eqn (3.15) EN 1992-1-1:2004 Section 3.1.6(1)P

^{2}EN 1992-1-1:2004 Section 6.2.3(3) Eqn (6.8)