Beams in Bending

Abstract

This chapter presents the application of the basic tooth, truss and arch models to overall beam systems, showing the practical design procedure. The strut and ties balanced schemes are then applied to the resistance calculations of bearings, corbels, deep beams and slabs in punching shear. The criteria of nonlinear and collapse analysis are also presented. In the final section, the same beam examined in Chap. 4 is designed again with the different choice of flat shallow section.

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: Elements in Bending

5.1.1 Chart 5.1: Arch Behaviour: Formulas

RC elements in bending without transverse shear reinforcement.

Symbols

(see figure)

p _Ed :

design value of the applied distributed load

L :

distance of the section of maximum moment

R _Ed = p _Ed L :

design action on the support

R _Rd :

design value of the resistance at the support

d :

effective depth (flexural) of the element

z (≅ 0.9d):

lever arm of the internal couple

l:

length of the arch behaviour (z ≤ l ≤ L)

λ = l/z :

inclination of the lower strut (= ctg θ)

λ _o = λ/2:

inclination of the upper strut (= ctg θ _o)

b _w :

minimum web thickness of the element

A′_s :

longitudinal reinforcement of the arch behaviour

see also Charts 2.2 and 2.3.

Resistance Verification

$$ \begin{aligned} {R}_{\text{Rs}} & = \frac{{{A}_{\text{s}}^{\prime } {f}_{\text{yd}} }}{\lambda }\frac{2L}{2L - 1} \\ {R}_{\text{Rc}} & = 0.4{d \, b}_{\text{w}} {f}_{\text{cd}} \frac{{4{L}}}{{8{L} - 31}} \\ {R}_{\text{Rd}} & = {\min}({R}_{\text{Rs}} ,{R}_{\text{Rc}} ) \ge {R}_{\text{Ed}} \\ \end{aligned} $$

5.1.2 Chart 5.2: Bearing Details: Formulas

RC elements in bending

Symbols

R _Ed :

design action at the support

R _Rd :

design value of the resistance at the support

A _sl :

area of the longitudinal reinforcement at the support

A _st :

area of the bent bars at the support

α :

bending angle of the bars on the horizontal

l:

bending distance from the support

d :

effective depth (flexural) of the element

z(≅ 0.9d):

lever arm of the internal couple

b _w :

minimum web thickness of the element

θ :

angle of the compressions in the web at the limit of shear resistance

λ _c = ctg θ :

inclination of the transverse compressions in the web

λ _o = λ_c/2:

mean inclination of compressions at the support

γ _R = 1.25:

reliability coefficient of the model

see also Charts 2.2, 2.3, 4.1 and 4.2.

Elements Without Stirrups

$$ \begin{aligned} R_{\text{Rs}}^{\prime } & = A_{\text{s1}} f_{\text{yd}} \quad \left( { \ge R_{\text{rs}}^{\prime \prime } } \right) \\ R_{\text{Rs}}^{\prime \prime } & = A_{\text{st}} f_{\text{yd}} \sin \alpha \quad \left( {1 \le z/2} \right) \\ R_{\text{Rs}} & = R_{\text{Rs}}^{\prime } + R_{\text{Rs}}^{\prime \prime } \ge R_{\text{Ed}} \\ R_{\text{Rc}} & = 0.2\,d\,b_{\text{w}} f_{\text{cd}} R_{\text{Ed}} /R_{\text{Rs}}^{\prime } \ge \gamma_{\text{R}} R_{\text{Rs}}^{\prime } \\ \end{aligned} $$

Elements with Stirrups

$$ \begin{aligned} & R_{\text{Rs}}^{\prime } = A_{\text{s1}} {{f_{\text{yd}} } \mathord{\left/ {\vphantom {{f_{\text{yd}} } {\lambda_{\text{o}} }}} \right. \kern-0pt} {\lambda_{\text{o}} }}\quad \quad \left( { \ge R_{\text{Rs}}^{\prime \prime } } \right) \\ & R_{\text{Rs}}^{\prime \prime } = A_{\text{st}} f_{\text{yd}} \sin {\mkern 1mu} \alpha \quad \quad \left( {1 \le z/2} \right) \\ & R_{\text{Rs}} = R_{\text{Rs}}^{\prime } + R_{\text{Rs}}^{\prime \prime } \ge R_{\text{Ed}} \\ & R_{\text{Rc}} = 0.6\,d\,b_{w} f_{\text{cd}} R_{\text{Rs}} /\left[ {R_{\text{Rs}}^{\prime } \left( {1 + \lambda_{0}^{2} } \right)} \right] \ge \gamma_{R} R_{\text{Rs}}^{\prime } \\ \end{aligned} $$

For minimum stirrups see Chart 4.5.

5.1.3 Chart 5.3: Corbels: Formulas

RC elements in bending

Symbols

P _Ed :

design value of the applied load on the corbel

P _Rd :

design value of the resistance

l (≤ 2z):

distance of the load from the fixed-end (or of the resultant)

A _sl :

area of the longitudinal reinforcement of the corbel

A _st :

area of bent bars

α :

bending angle of bars on the horizontal

d :

effective depth (flexural) of the cantilever

z (≅ 0.9d):

lever arm of the internal couple

b _w :

minimum web thickness of the cantilever

λ = l/z:

inclination of transverse compressions in the web

b :

width of the compression flange

γ _R = 1.25:

reliability coefficient of the model

see also Charts 2.2, 2., 4.1 and 4.2.

Resistance Verification

$$ \begin{aligned} &{P}_{\text{Rs}}^{\prime} = {{{A}_{\text{s1}} {f}_{\text{yd}}} \mathord{\left/ {\vphantom {{{A}_{\text{s1}} {f}_{\text{yd}} } {\lambda }}} \right. \kern-0pt} {\lambda }}\quad \left( { \ge {P}_{\text{Rs}}^{\prime\prime } } \right)\\ & {P}_{\text{Rs}}^{\prime\prime } = {A}_{\text{st}} {f}_{\text{yd}} {\sin}\,{\alpha }\\ &{P}_{\text{Rs}} = {P}_{\text{Rs}}^{\prime } + {P}_{\text{Rs}}^{\prime\prime } \ge {P}_{\text{Ed}}\\ & {P}_{\text{Rc}}^{\prime\prime } = 0.2{d \, b \, f}_{\text{cd}} {P}_{\text{Rs}} /\left[ {{\lambda }{P}_{\text{Rs}}^{\prime\prime } } \right] \ge {\gamma}_{R} {P}_{\text{Rs}}^{\prime\prime } \end{aligned} $$

• Corbels without stirrups

  $$ {P}_{\text{Rc}}^{\prime } = 0.4{\text{db}}_{\text{w}} {f}_{\text{cd}} {P}_{\text{Ed}} /\left[ {{P}_{\text{Rs}}^{\prime } \left( {1 + {\lambda }^{2} } \right)} \right] \ge {\gamma }_{\text{R}} {P}_{\text{Rs}}^{\prime } $$

• Corbels with stirrups

  $$ {P}_{\text{Rc}}^{\prime } = 0.6{\text{db}}_{\text{w}} {f}_{\text{cd}} {P}_{\text{Ed}} /\left[ {{P}_{\text{Rs}}^{\prime } \left( {1 + {\lambda }^{2} } \right)} \right] \ge {\gamma }_{\text{R}} {P}_{\text{Rs}}^{\prime } $$

For the minimum stirrups see Chart 4.5.

5.1.4 Chart 5.4: Punching Shear in Slabs: Formulas

RC bidimesional elements, without transverse reinforcement, in bending in both directions x and y.

Symbols

R _Ed :

design action on the column

R _Rd :

design value of the punching shear resistance

a, b :

sides of the column along x and y

d _x, d _y :

effective depths (flexural) of the plate along x and y

A _x, A _y :

concerned flexural reinforcements along x and y

u _o :

perimeter of the column affected by stresses

u :

critical perimeter of the diffusion zone of the plate

A, B :

dimensions of the diffusion zone along x and y

see also Charts 2.2 and 2.3.

Resistance Verification

$$ \begin{aligned} {R}_{\text{Rc}} & = 0.123{d \, u}_{\text{o}} {f}_{\text{cd}} \\ {R}_{ct} & = 0.25{d \, u \, f}_{\text{ctd}} {\kappa }\,{r} \\ {R}_{\text{Rd}} & = {\min}({R}_{\text{Rc}} , {\text{R}}_{\text{ct}} ) \ge {R}_{\text{Ed}} \\ \end{aligned} $$

with

$$ \begin{aligned} &{d} = {{\left( {{d}_{x} + {d}_{y} } \right)} \mathord{\left/ {\vphantom {{\left( {{d}_{x} + {d}_{y} } \right)} 2}} \right. \kern-0pt} 2} \\ & {\kappa } = 1.6 - {d} \ge 1\quad \left( {d \,{\text{expressed in}}\, m} \right) \\ & {R} = 1.0 + 50{\rho }_{\text{s}} \le 2 \\ & {\rho }_{\text{s}} = \sqrt {{\rho }_{\text{sx}} {\rho }_{\text{sy}} } \\ & \rho_{\text{sx}} = {A}_{\text{sx}} /{\text{Bd}}\quad A_{\text{sx}} \, {\text{area of bars included in}}\, B \\ & \rho_{\text{sy}} = {A}_{\text{sy}} /{\text{Ad}}\quad A_{\text{sy}} \,{\text{area of bars included in}}\,A \end{aligned} $$

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

$$ \begin{aligned} {R}_{\text{Rd}} & = {\min}({R}_{\text{Rc}} , {\text{R}}_{\text{ct}} ) \ge {R}_{\text{Ed}} \\ {R}_{\text{ct}} & = {{0.18{\text{du}}{\kappa }(100{\rho }_{\text{s}} {f}_{\text{ck}} )^{1/3} } \mathord{\left/ {\vphantom {{0.18{\text{du}}{\kappa }(100{\rho }_{\text{s}} {f}_{\text{ck}} )^{1/3} } {{\gamma }_{\text{C}} \ge {\text{duv}}_{\min} }}} \right. \kern-0pt} {{\gamma }_{\text{C}} \ge {\text{duv}}_{\min} }} \\ \end{aligned} $$

with

$$ \begin{aligned}& {\kappa } = 1 { + }\sqrt {200/{d}} \le {2.0} \quad \left( {d\,{\text{expressed in mm}}} \right) \\ & {v}_{\min} = 0.035{\kappa }^{3/2} \sqrt {{f}_{\text{ck}} }\end{aligned} $$

(For the definition of d, ρ _s and R _Rs see above).

Shear Perimeters

For substantially symmetric arrangements of forces around the column, with constant uniform distribution of stresses along the shear perimeters, the following cases are possible.

• Internal column

  $$ \begin{aligned} {u}_{\text{o}} & = 2{a} + 2{b} \\ {A} & = {a} + 3{d}\quad {B}\,=\,{b}\,{ + }\, 3 {\text{d}} \\ {u} & = 2{a} + 2{b} + 3{\pi }{d} \\ \end{aligned} $$

• Edge column

  $$ \begin{aligned} {u}_{\text{o}} & = 2{a} + {b}\quad \left( {{\text{one side}}\,b\,{\text{on the edge}}} \right) \\ {A} & = {a} + 1.5{d}\quad {B}\,=\,{b}\,{ + }\, 3 {\text{d}} \\ {u} & = 2{a} + {b} + 1.5{\pi }{d} \\ \end{aligned} $$

• Corner column

  $$ \begin{aligned} {u}_{\text{o}} & = {a} + {b} \\ {A} & = {a} + 1.5{d}\quad {B}\,=\,{b}\,{ + }\,1.5{d} \\ {u} & = {a} + {b} + 0.75{\pi }{d} \\ \end{aligned} $$

5.1.5 Chart 5.5: Coefficients for Moment Redistribution

Within the Linear analysis with redistribution of hyperstatic moments of RC beams, the following values of the reduction coefficients δ of the elastic moments of critical sections can be adopted. It is implied that the bending moments of the other sections along the spans are to be modified accordingly, complying with the equilibrium with the applied load.

Symbols

M _ed :

design value of the hyperstatic moment of the linear elastic analysis

M _Ed = δM _ed :

reduced value of the moment for the design of reinforcement

ξ = x/d :

adimensionalized position of the neutral axis of the section

see also Chart 3.10.

Reduction Coefficients

High-ductility steel (as defined in Table 1.17)

${\delta } \ge 0.44 + 1.25{\xi }$ for concrete classes from C16/20 to C50/60

${\delta } \ge {0.56 + }{k}{\xi }$ for concrete classes from C50/60 to C70/75

where

$$ \begin{aligned} {k} & = 1. 2 5 { }(0. 6+ 0.00 1 4 { }/{\varepsilon }_{\text{cu}} ) \\ {\varepsilon }_{\text{cu}} & = 0.00 3 1 { };0.00 2 9 { };0.00 2 7 \,{\text{for C55}}/ 60 \, ;{\text{C6}}0/ 6 5 { };{\text{C7}}0/ 7 5\\ \end{aligned} $$

in the limits 0.70 ≤ δ ≤ 1.00.

NOTE: Concrete classes are the ones shown in Table 1.2a, b.

5.1.6 Chart 5.6: Allowable Deformations of elements in Bending

In the calculations of deformations with the appropriate serviceability load combinations, for RC and prestressed elements in bending of the decks of common buildings, one can make reference to the following allowable values indicating the ratio v/l between the maximum deflection and the span length.

• $\frac{v}{1} \le \frac{1}{200}\quad$ for any type of structural element

• $\frac{{{v}^{*}}}{1} \le \frac{1}{400}\quad$ for decks with non-structural walls

• $\frac{v}{1} \le \frac{1}{800}\quad$ for particular requirements of high stiffness

The viscous effects developed at the time of the verification should be included in the deflection v. The value v ^* refers to the maximum range of the viscoelastic deflection that non-structural walls have to accommodate.