Shear

Abstract

This chapter presents the design methods of reinforced concrete elements subjected to shear action. The basic resistance mechanisms are described and the related models are deduced, that is the tooth model for beams without shear reinforcement and the truss model for beams with shear reinforcement, with their more recent improvements. In the final section, after the completion of the floor design with the pertinent shear verifications, a complete design of a beam is developed, starting from the stress analysis and following with the serviceability and resistance verifications, both for bending moment and for shear.

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: Shear

4.1.1 Chart 4.1: Beams Without Shear Reinforcement: Formulas

RC elements in bending without transverse shear reinforcement.

Symbols

V _Ek :

characteristic value of the shear force

V _Ed :

design value of the shear force

V _od :

design resistance to shear cracking

V _ctd :

design resistance without shear reinforcement

b _w :

minimum web width

$\overline{z}$ :

lever arm of the internal couple (uncracked section)

z :

lever arm of the internal couple (cracked section)

d :

effective depth (flexural) of the section

ρ _l = A _s/db _w :

longitudinal geometric reinforcement ratio

a _l :

shifting of the longitudinal reinforcement on the beam axis

See also Charts 2.2, 2.3, 2.9, 3.3, 3.10, 3.18.

Serviceability Verifications

Uncracked section

(zones with M _Ek < M _ok—see Chart 3.18)

$$ \sigma_{\text{I}} = \frac{1}{2}\left( { - \sigma + \sqrt {\sigma^{2} + 4\tau^{2} } } \right) $$

with

$$ \tau = \frac{{V_{\text{Ek}} }}{{\overline{z} b_{\text{w}} }} $$

$$ \sigma = \frac{{N_{\text{Ek}} }}{{A_{i} }} \quad > 0{\text{ for compression}}, $$

that is

$$ \sigma_{\text{I}} = \frac{{V_{\text{Ek}} }}{{\overline{z} b_{\text{w}} }} \quad {\text{for uniaxial bending }}(N_{\text{Ek}} = 0), $$

where (see also Chart 3.3):

$$ \bar{z} = {{Ii} \mathord{\left/ {\vphantom {{Ii} {\bar{S}_{i} }}} \right. \kern-0pt} {\bar{S}_{i} }} $$

$$ \bar{S}_{i} = b_{\text{w}} y_{\text{c}}^{\prime 2} /2 + \alpha_{\text{e}} A_{\text{s}} y_{\text{s}} , $$

that is

$$ \overline{z} \cong 0.7d\quad {\text{for rectangular section}} $$

for the verification

at the shear cracking limit $\sigma_{\text{I}} \le f_{\text{ctk}}$

Cracked section

(zones with M _Ek > M _ok—see Chart 3.18)

For plate elements: no verification

(beams within floor depth, plates, slabs, …—with protected lateral edges).

For beam elements: minimum stirrups (see Chart 4.5)

(with exposed webs).

Resistance Verifications

Uncracked section

(zones with M _Ed < M _od—see Chart 3.18)

$$ V_{\text{od}} = \overline{z} b_{\text{w}} f_{\text{ctd}} \,\delta \ge V_{\text{Ed}} , $$

with

$$ \overline{z} \cong 0.7d $$

$$ \delta = \sqrt {1 - \sigma /f_{\text{ctd}} } $$

$\sigma = N_{\text{Ed}} /A_{i} > 0{\text{ for compression}}$

(δ = 1 for uniaxial bending).

Cracked section

(zones with M _Ed > M _ok—see Chart 3.18)

$$ V_{\text{ctd}} = 0.25db_{\text{w}} f_{\text{ctd}} \kappa r\delta \ge V_{\text{Ed}} $$

(V _ctd ≤ V _od) with

κ = 1.6 – d ≥ 1:

(d expressed in m)

$r = 1.0 + 50\rho_{ 1} \le 2$ :

.

$\delta = 1 + M_{\text{Ro}} /M_{\text{Ed}} \le 2$ :

for combined compression and bending

$M_{\text{Ro}} = \sigma I_{i} /y_{\text{c}}^{\prime }$ :

$\sigma = N_{\text{Ed}} /A_{i} > 0\quad {\text{for compression}}$

δ = 1:

for uniaxial bending

δ = 0:

for combined tension and bending

Zero value V _ctd = 0 should be set also for relevant alternated shear forces with inverted signs.

Alternatively, according to more recent codes, it can be set:

$$ \begin{aligned} V_{\text{ctd}} & = 0.18db_{\text{w}} \kappa \left( {100\rho_{ 1} f_{\text{ck}} } \right)^{1/3} /\gamma_{\text{C}} + 0.15b_{\text{w}} d\sigma_{\text{c}} \\ & \ge b_{\text{w}} dv_{\hbox{min} } + 0.15b_{\text{w}} d\sigma_{\text{c}} \\ V_{\text{ctd}} & \ge V_{\text{Ed}} , \\ \end{aligned} $$

where

κ = 1 + $\sqrt {200/d} \le$ 2.0:

(d in mm)

$\sigma_{\text{c}} = N_{\text{Ed}} /A_{\text{c}} \ge 0$ :

in compression

A _c :

area of the section

v _min = 0.035 $\kappa^{3/2} \sqrt {f_{\text{ck}} }$ :

(f _ck in N/mm²).

Construction Requirements

The longitudinal reinforcement at the face of the beam in tension, calculated based on the bending moment, should be shifted in increase by

$$ a_{ 1} = z \cong 0.9d $$

4.1.2 Chart 4.2: Resistance of Beams with Shear Reinforcement: Formulas

RC beams with transverse shear reinforcement.

Symbols

$V_{\text{cd}}$ :

design resistance for compression–shear

$V_{\text{sd}}$ :

design resistance per tension–shear

$V_{\text{Rd}}$ :

design value of the resistance with shear reinforcement

$A_{\text{w}}$ :

area of web transverse reinforcement

s :

spacing of transverse shear reinforcement

$a_{\text{w}} = A_{\text{w}} /s$ :

unit area of transverse reinforcement

α :

angle of transverse bar on the beam axis

${\rho }_{\text{w}} = {a}_{\text{w}} { \sin }{\alpha }/{b}_{\text{w}}$ :

geometrical web reinforcement ratio

${\omega }_{\text{w}} = {\rho }_{\text{w}} {f}_{\text{yd}} /{f}_{\text{c2}}$ :

mechanical web reinforcement ratio

${\theta }_{\text{I}}$ :

angle of initial shear cracking

${\theta }$ :

angle of web compressions on the beam axis

${\lambda }_{\text{s}} = {ctg}\,{\alpha }$ :

inclination of transverse shear reinforcement

${\lambda }_{\text{I}} = {ctg}\,{\theta }_{\text{I}}$ :

inclination of initial shear cracking

${\lambda }_{\text{c}} = {ctg}\,{\theta}$ :

inclination of web transverse compressions

(see also Charts 2.2, 2.3, 2.9, 3.3, 3.10, 3.18, 4.1).

Resistance with Assigned Truss

Assumed $\lambda_{c}$ in the interval $\lambda_{\text{I}} \le \lambda_{\text{c}} \le \lambda_{\hbox{max} }$, it is set

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

with $V_{\text{Ed}}$ evaluated in the middle of the considered segment and

$\lambda_{\text{I}} = \tau /\sigma_{\text{I}}$ :

(=1 for uniaxial bending)

$\lambda_{\hbox{max} } = \lambda_{\text{I}} + 1.5$ :

(=2.5 for uniaxial bending)

(for τ and σ _I see Chart 4.1)

Tension–shear (z ≅ 0.9d)

$V_{\text{sd}}^{\prime } = a_{\text{w}} zf_{\text{yd}} \sin \alpha \left( {\lambda_{\text{s}} + \lambda_{\text{c}} } \right)$ :

stirrups with α < 90°

$V_{\text{sd}}^{\prime } = a_{\text{w}} zf_{\text{yd}} \lambda_{\text{c}}$ :

stirrups with α = 90°

$V_{\text{sd}}^{\prime \prime } = 0.8A_{\text{w}} zf_{\text{yd}} \sin \alpha \left( {\lambda_{\text{s}} + \lambda_{\text{c}} } \right)/s$ :

bent bar

$V_{\text{sd}} = V_{\text{sd}}^{\prime } + V_{\text{sd}}^{\prime \prime }$ :

for a given λ _c

Compression–shear (z ≅ 0.9d)

$V_{\text{cd}} = zb_{\text{w}} f_{\text{c2}} \left( {\lambda_{\text{s}} + \lambda_{\text{c}} } \right)/\left( {1 + \lambda_{\text{c}}^{2} } \right)$ :

stirrups with α < 90°

$V_{\text{cd}} = zb_{\text{w}} f_{\text{c2}} \lambda_{\text{c}} /\left( {1 + \lambda_{\text{c}}^{2} } \right)$ :

stirrups with α = 90°

with stirrups orthogonal to the axis and bent bars:

$$ V_{\text{cd}} = zb_{\text{w}} f_{{{\text{c}}2}} \left( {\alpha^{\prime \prime } \,\lambda_{\text{s}} + \lambda_{\text{c}} } \right)/\left( {1 + \lambda_{\text{c}}^{2} } \right)\quad \alpha^{\prime \prime } = V_{\text{sd}}^{\prime \prime } /V_{\text{sd}} . $$

Resistance with Calculated Truss

$\omega_{\text{wa}} = 1/\left( {1 + \lambda_{\hbox{max} }^{2} } \right)$ :

(=0.138 for uniaxial bending)

$\omega_{\text{wc}} = 1/\left( {1 + \lambda_{\text{I}}^{2} } \right)$ :

(=0.5 for uniaxial bending)

Low reinforcement (ω _w ≤ ω _wa)

$\lambda_{\text{c}} = \lambda_{\hbox{max} }$ :

(z ≅ 0.9d)

$V_{\text{Rd}} = a_{\text{w}} zf_{\text{yd}} \lambda_{\text{c}}$ :

stirrups with α = 90°

$V_{\text{Rd}} = 2.5a_{\text{w}} zf_{\text{yd}}$ :

for uniaxial bending

Medium reinforcements (ω _wa < ω _w < ω _wc)

$\lambda_{\text{c}} = \sqrt {\left( {1 - \omega_{\text{w}} } \right)/\omega_{\text{w}} }$ :

(z ≅ 0.9d)

$V_{\text{Rd}} = a_{\text{w}} zf_{\text{yd}} \lambda_{\text{c}}$ :

stirrups with α = 90°

High reinforcements (ω _w  ≥ ω _wc)

$\lambda_{\text{c}} = \lambda_{\text{I}}$ :

(z ≅ 0.9d)

$V_{\text{Rd}} = zb_{\text{w}} f_{\text{c2}} \lambda_{\text{c}} /\left( {1 - \lambda_{\text{c}}^{2} } \right)$ :

stirrups with α = 90°

$V_{\text{Rd}} = zb_{\text{w}} f_{\text{c2}} /2$ :

for uniaxial bending.

Complementary Indications

Verification sections

The first verification section is usually located at

$$ z\left( {\lambda_{\text{s}} + \lambda_{\text{c}} } \right)/2\quad \quad (z \cong 0.9d) $$

from the contiguous support. The resistance related to the shear reinforcement can be referred to beam segments of finite length, not greater than 1/4 of the span, to be compared to the mean value of shear applied on the same segment.

Hung loads

In the case of loads applied on the lower part of the beam, an area of stirrups (orthogonal to the axis) should be added equal to

$$ p_{\text{d}} /f_{\text{yd}} \quad \quad \left( {{\text{per}}\,{\text{unit}}\,{\text{length}}} \right), $$

where p _d is the design value of hung distributed load.

Variable depth

In the case of beams with variable depth, for the verification of the web members a reduced value of the applied shear should be assumed with

$$ V_{\text{Ed}} - V_{\text{Cd}} - V_{\text{Zd}} , $$

where

V _Cd :

transverse component of the force in the compression chord

V _Zd :

transverse component of the force in the tension chord.

Such components can be positive or negative based on their direction (and therefore favourable or unfavourable for the resistance of the web mechanism) and should be calculated with the design value of the bending moment applied on the considered section.

4.1.3 Chart 4.3: Beams with Shear Reinforcement: Service Conditions and Construction Rules

RC beams with transverse shear reinforcement.

Symbols

$V_{{{\text{ctk}}\,}}^{\prime }$ :

tension contribution of the concrete of the web

$\sigma_{\text{w}}$ :

tensile stress in stirrups

$\sigma_{\text{c}}$ :

compression stress in the concrete of the web

see also Charts 2.2, 2.3, 4.1, 4.2.

Serviceability Verifications

Uncracked zones

The zones of the beam that, with uncracked sections, satisfy the cracking verification of Chart 4.1, do not require further verifications.

Cracked zones—stirrups

$$ \sigma_{\text{w}} = \frac{{V_{\text{Ek}} - V_{\text{ctk}}^{\prime } }}{{za_{\text{w}} \sin \alpha }}\frac{1}{{\lambda_{\text{s}} + \lambda_{\text{I}} }} \le \overline{\sigma }_{\text{w}} \quad {\text{stirrups}}\,{\text{with}}\,\alpha < 90^{ \circ } $$

$$ \upsigma_{\text{w}} = \frac{{{\text{V}}_{\text{Ek}} - {\text{V}}_{\text{ctk}}^{{\prime }} }}{{{\text{za}}_{\text{w}}\uplambda_{\text{I}} }} \le \overline{\upsigma}_{\text{w}} \quad {\text{stirrups}}\,{\text{with}}\,\upalpha < 90^{ \circ } $$

With stirrups orthogonal to the axis of unit area a _w and bent bars with equivalent area $a_{\text{w}}^{*} = A_{\text{w}} \sin \alpha \left( {\lambda_{\text{s}} + \lambda_{\text{I}} } \right)/s$

$$ \sigma_{\text{w}} = \frac{{\alpha^{\prime } \left( {V_{\text{Ek}} - V_{\text{ctk}}^{\prime } } \right)}}{{za_{\text{w}} \lambda_{\text{I}} }} \le \overline{\sigma }_{\text{w}} \quad \alpha^{\prime } = a_{\text{w}} /\left( {a_{\text{w}} + a_{\text{w}}^{*} } \right) \le 0.5, $$

with

$$ V_{\text{ctk}}^{\prime } = 0.60b_{\text{w}} zf_{\text{ctk}} \delta \quad \quad (z \cong 0.9\,d) $$

and where

$\delta = 1 + M_{\text{ro}} /M_{\text{Ek}} \le 2$ :

combined compression and bending

$M_{\text{ro}} = \sigma I_{i} /y_{\text{c}}^{\prime }$ :

$\sigma = N_{\text{Ek}} /A_{i} \ge 0$ in compression

$\delta = 1$ :

in uniaxial bending

$\delta = 0$ :

in combined tension and bending

A zero value of $\delta$ (with $V_{\text{ctk}}^{\prime }$ = 0) should be also set in the case of relevant alternate shear forces with inverted sign.

It should be assumed (see Chart 4.1):

$$ \lambda_{\text{I}} = \tau /\sigma_{\text{I}} \quad \quad \left( { = 1.0\,{\text{for}}\,{\text{uniaxial}}\,{\text{bending}}} \right), $$

whereas for the allowable stress $\bar{\sigma }_{\text{w}}$ one should refer to Chart 4.6.

Cracked zones—concrete

$$ \sigma_{\text{c}} = \frac{{V_{\text{Ek}} }}{{zb_{\text{w}} }}\frac{{1 + \lambda_{\text{I}}^{2} }}{{\lambda_{\text{s}} + \lambda_{\text{I}} }} \le 0.6\overline{\sigma }_{\text{c}} \quad \quad {\text{stirrups}}\,{\text{with}}\,\alpha < 90^{ \circ } $$

$$ \sigma_{\text{c}} = \frac{{V_{\text{Ek}} }}{{zb_{\text{w}} }}\frac{{1 + \lambda_{\text{I}}^{2} }}{{\lambda_{\text{I}} }} \le 0.6\overline{\sigma }_{\text{c}} \quad \quad {\text{stirrups}}\,{\text{with}}\,\alpha < 90^{ \circ } . $$

With stirrups orthogonal to the axis and bent bars (as before):

$$ \sigma_{\text{c}} = \frac{{V_{\text{Ek}} }}{{zb_{\text{w}} }}\frac{{\left( {\alpha^{\prime } \lambda_{\text{s}} + \lambda_{\text{I}} } \right)\left( {1 + \lambda_{\text{I}}^{2} } \right)}}{{\lambda_{\text{I}} \left( {\lambda_{\text{s}} + \lambda_{\text{I}} } \right)}} \le 0.6\overline{\sigma }_{\text{c}} \quad \quad \alpha^{\prime } \ge 0.5, $$

again with z ≅ 0.9d e λ _I = τ/σ _I.

‘Standard’ cracked zones

(uniaxial bending and orthogonal stirrups)

$$ \sigma_{\text{w}} = \frac{{V_{\text{Ek}} - V_{\text{ctk}}^{\prime } }}{{za_{\text{w}} }} \le \overline{\sigma }_{\text{w}} \quad \quad {\text{with}}\,V_{\text{ctk}}^{\prime } = 0.60\,b_{\text{w}} z\,f_{{_{\text{ctk}} }} $$

$$ \sigma_{\text{c}} = \frac{{2V_{\text{Ek}} }}{{zb_{\text{w}} }} \le 0.6\overline{\sigma }_{\text{c}} \quad \quad {\text{with}}\,z \cong 0.9d. $$

Construction Data

Shifting of moments

The longitudinal reinforcement at the edge of the beam in tension, calculated based on the bending moment, should be shifted in increase by

$$ a_{ 1} = z\left( {\lambda_{\text{c}} - \lambda_{\text{s}} } \right)/2 \ge 0\quad \quad {\text{with}}\,z \cong 0.9d $$

Spacing of stirrups

The spacing of stirrups orthogonal to the beam axis (α = 90°) should be limited with

$$ s \le 0.8d\quad \quad ( \le 300\,{\text{mm}}) $$

Minimum Stirrups

The minimum amount of peripheral stirrups close to the lateral faces of the web and encasing the longitudinal reinforcement, should be limited with

$$ a_{\text{w}} \ge 0.2b_{\text{w}} f_{\text{ctm}} /f_{\text{yk}} $$

Bent bars

In any case a quota of the shear force not less than 0.5 should be assigned to stirrups:

$$ V_{\text{sd}}^{\prime } \ge 0.50V_{\text{sd}} $$

The remaining shear force is to be resisted by the bent bars.

Flat beams

For large beams with b _w ≫ d the stirrups of each of the free lateral faces should be limited with

$$ a_{\text{w}}^{\prime } \ge 0.1df_{\text{ctm}} /f_{\text{yk}} , $$

whereas the total amount of transverse reinforcement should be uniformly distributed on the width resisting to shear with a spacing of links s ≤ 1.2d.

Flat beams with protected lateral faces can be reinforced with bent bars only; in this case in the resistance verifications it is assumed $\lambda_{\text{c}} = \lambda_{\text{I}}$ (=1 for uniaxial bending).

4.1.4 Table 4.4: Shear Cracking: Allowable Stresses in Stirrups

The following table shows, for different values of the longitudinal spacing s of stirrups, the allowable stresses $\bar{\sigma }_{\text{w}}$ to be used in the serviceability verifications of the cracked zones as per Chart 4.3.

The values are expressed in MPa and refer to the peripheral stirrups close to the lateral faces of the web and bent so that they include the longitudinal reinforcement.

Allowable stresses of this table are given in an experimental way.

s (mm)	50	100	150	200	250	300
$\overline{\sigma }_{\text{w}}$	200	150	100	75	60	50