Bending Moment

Abstract

This chapter presents the design methods of reinforced and prestressed concrete sections subjected to bending moment. The criteria for cracking calculation are here extended to beams in flexure, for which the analysis of deformation is shown including creep effects. In the final section after the specific analysis of loads, the design of floors is shown with the pertinent serviceability and resistance verifications.

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: Actions and Bending Moment

3.1.1 Table 3.1: Partial Safety Factors For Actions

In the resistance verifications (ultimate limit states) the design values

$$ {F}_{\text{d}} = {\gamma }_{\text{F}} \,{F}_{\text{k}} $$

are adopted for actions, obtained with the pertinent partial safety factors. The factors shown in the following table are to be applied to the nominal values of actions deducible from the competent design codes (representative of F _k). The values are taken from Eurocode EN 1990. They refer to the resistance limit state of the structure “STR” including the foundation elements. For the verifications of the equilibrium ultimate limit state as rigid body “EQU” and the limit state of the resistance of the ground “GEO”, one can refer to Chart 9.6.

Usually thermal variations (Q _ε ) are not taken into account in the resistance verifications. The snow load is included in the variable actions Q. In the absence of more accurate analyses, the wind load W can be treated similar to the variable actions. For prestressing P the nominal value (specified in the design) is assumed.

The partial factors in the table are given for the analysis of actions to be carried with a linear elastic design, for structural situations with negligible second order effects, within the semi-probabilistic limit states method, assuming the safety factors of materials of Charts 2.2 and 2.3 for the subsequent resistance verification. In such analysis, the single load units should be distinguished, each one to be multiplied with the minimum or maximum value of the relative partial factor, depending whether it is favourable or unfavourable to the resistance for the verification under consideration.

Action (nominal value)		Factor	Min.	Max.
Structural self-weight (permanent actions)	G1	γ G1	1.00	1.30
Superimposed dead load (permanent actions)	G2	γ G2	0.00	1.50
Live loads (variable actions)	Q	γ Q	0.00	1.50
Internal action (prestressing)	P	γ P	1.00	1.20a

1. ^aRefers to local actions (e.g. on anchorages) with prestressing represented by a force

3.1.2 Chart 3.2: Formulas of Action Combination

The single load units, assumed with the respective design value F _d or with the respective nominal value F _k (see Table 3.1), should be used in the model for the structural analysis according to the combinations specified hereafter. Formulas and factors are deduced from the Eurocode 0 EN 1990 (Q ₁ = most critical load for the verification under consideration). For the meaning of symbols see Table 3.1.

Resistance Verifications (ULS)

$$ {F}_{\text{d}} = {\gamma }_{{{\text{G}1}}} {G}_{1} + {\gamma }_{{{\text{G}2}}} {G}_{2} + {\gamma }_{\text{P}} {P} + {\gamma }_{\text{Q}} {Q}_{1} + {\gamma }_{\text{Q}} \{ {\psi }_{02} {Q}_{2} + {\psi }_{{0{3}}} {Q}_{3} + \cdots\}. $$

Serviceability Verifications (SLS)

• characteristic combination (rare)

  $$ {F}_{\text{k}} = {G}_{1} + {G}_{2} + {\psi }_{{0{\varepsilon }}} {Q}_{\varepsilon } + {P} + {Q}_{1} + {\psi }_{{0{2}}} {Q}_{2} + {\psi }_{{0{3}}} {Q}_{3} + \cdots $$

• frequent combination

  $$ {F}_{\text{k}} = {G}_{1} + {G}_{2} + {\psi }_{{{1}{\varepsilon }}} {Q}_{\varepsilon } + {P} + {\psi }_{11} {Q}_{1} + {\psi }_{22} {Q}_{2} + {\psi }_{23} {Q}_{3} + \cdots $$

• quasi-permanent combination

  $$ {F}_{k} = {G}_{1} + {G}_{2} + {\psi }_{{{2}{\varepsilon }}} {Q}_{\varepsilon } + {P} + {\psi }_{21} {Q}_{1} + {\psi }_{22} {Q}_{2} + {\psi }_{23} {Q}_{3} + \cdots $$

Combination Factors

In the combination formulas shown above, the following values of the factors $\psi_{0}$, $\psi_{1}$ and $\psi_{2}$ can be used.

Category/variable action	${\varPsi }_{{0{j}}}$	${\varPsi }_{1j}$	${\varPsi }_{2j}$
Category A: domestic, residential areas	0.7	0.5	0.3
Category B: office areas	0.7	0.5	0.3
Category C: congregation areas	0.7	0.7	0.6
Category D: shopping areas	0.7	0.7	0.6
Category E: storage areas	1.0	0.9	0.8
Category F: traffic area, vehicle weight ≤ 30 kN	0.7	0.7	0.6
Category G: traffic area, 30 kN < vehicle weight ≤ 160 kN	0.7	0.5	0.3
Category H: roofs	0.0	0.0	0.0
Wind	0.6	0.2	0.0
Snow (altitude ≤ 1000 m a.s.l)	0.5	0.2	0.0
Snow (altitude > 1000 m a.s.l.)	0.7	0.5	0.2
Temperature	0.6	0.5	0.0

In the combinations at the SLS it is implied that the loads Q _i that give a favourable contribution with respect to the verifications are omitted.

For the allowable stresses of materials see Charts 2.2, 2.3–2.15 and Table 2.16.

3.1.3 Chart 3.3: Section in Bending: Elastic Design—Formulas

RC sections subject to pure uniaxial bending.

Symbols

M _Ek :

characteristic value of the applied moment

A _s :

area of the reinforcement in tension

${A}_{\text{s}}^{\prime }$ :

area of the reinforcement in compression

${A}_{\text{t}} = {A}_{\text{s}} + {A}_{\text{s}}^{\prime }$ :

total reinforcement area

b :

width of the edge in compression

d :

effective depth (see figures)

${d}^{\prime }$ :

concrete cover of the reinforcement in compression

${\rho }_{\text{s}} = {A}_{\text{s}} /{bd}$ :

geometric reinforcement ratio in tension

${\rho }_{\text{s}}^{\prime } = {A}_{\text{s}}^{\prime } /{bd}$ :

geometric reinforcement ratio in compression

${\alpha }_{\text{e}} = {E}_{\text{s}} /{E}_{\text{c}}$ :

ratio of elastic moduli (see Chart 2.3)

${\psi }_{\text{s}} = {\alpha }_{\text{e}} {\rho }_{\text{s}}$ :

elastic reinforcement ratio in tension

${\psi }_{\text{s}}^{\prime } = {\alpha }_{\text{e}} {\rho }_{\text{s}}^{\prime }$ :

elastic reinforcement ratio in compression

${\psi }_{\text{t}} = {\psi }_{\text{s}} + {\psi }_{\text{s}}^{\prime }$ :

total elastic reinforcement ratio

${\sigma }_{\text{c}}$ :

maximum compressive stress in concrete

${\sigma }_{\text{c}}^{\prime }$ :

maximum tensile stress in concrete

${\sigma }_{\text{s}}$ :

stress in the reinforcement in tension

See also Charts 2.2 and 2.3.

Serviceability Verifications in Phase I

(Uncracked section—see figure)

$$ {\sigma }_{\text{c}}^{\prime } = \frac{{{M}_{\text{Ek}} }}{{{I}_{\text{c}} }}{y}_{\text{c}}^{\prime } \quad \left( {{\sigma }_{\text{S}} = {\alpha }_{\text{e}} \frac{{{M}_{\text{Ek}} }}{{{I}_{\text{c}} }}{y}_{\text{S}} } \right), $$

with

$$ \begin{aligned} & {Ab} = {bt}\quad {A}_{\text{w}} = {b}_{\text{w}} {h}_{\text{w}} \quad {h}_{\text{w}} = {h} - {t} \\ & {A}_{i} = {A}_{\text{b}} + {A}_{\text{w}} + {\alpha }_{\text{e}} {A}_{\text{s}} + {\alpha }_{\text{e}} {A}_{\text{s}}^{\prime} \\ & {S}_{i} = {A}_{\text{b}} {t}/{2} + {A}_{\text{w}} \left( {{t} + {h}/{2}} \right) + {\alpha }_{\text{e}} {A}_{\text{s}} {d} + {\alpha }_{\text{e}} {A}_{\text{s}}^{\prime } {d}^{\prime } \\ & {y}_{\text{c}} = {S}_{i} /{A}_{i} \quad {y}_{\text{c}}^{\prime } = {h} - {y}_{\text{c}} \\ & {y}_{\text{s}} = {d} - {y}_{\text{c}} \quad {y}_{\text{s}}^{\prime } = {y}_{\text{c}} - {d}^{\prime } \\ & {y}_{\text{b}} = {y}_{\text{c}} - {t}/{2}\quad {y}_{\text{w}} = {t} + {h}_{\text{w}} /{2} - {y}_{\text{c}} \\ & {I}_{i} = {A}_{\text{b}} ({t}^{2} /12 + {y}_{\text{b}}^{2} ) + {A}_{\text{W}} ({h}_{\text{W}}^{2} /12 + {y}_{\text{W}}^{2} ) + {\alpha }_{\text{e}} {A}_{\text{S}} {y}_{\text{W}}^{2} + {\alpha }_{\text{e}} {A}_{\text{S}}^{\prime } {y}_{\text{S}}^{{{\prime }\,2}}, \\ \end{aligned} $$

for the verifications

$$ \begin{array}{*{20}l} {{\text{at}}\,{\text{the}}\,{\text{decompression}}\,{\text{limit}}} \hfill & {{\sigma }_{\text{c}}^{\prime } \le 0} \hfill \\ {{\text{at}}\,{\text{the}}\,{\text{limit}}\,{\text{of}}\,{\text{cracks}}\,{\text{formation}}} \hfill & {{\sigma }_{\text{c}}^{\prime } \le {\bar{\sigma}}_{\text{ct}}^{\prime } }. \hfill \\ \end{array} $$

(for the rectangular section, set t = h).

Serviceability Verifications in Phase II

(Cracked section—see figures)

Rectangular section—single reinforcement

${\sigma }_{\text{c}} = \frac{{2{M}_{\text{Ek}} }}{zbx} \le {\bar{\sigma }}_{\text{c}} ;\quad {\sigma }_{\text{S}} = \frac{{{M}_{\text{Ek}} }}{{{zA}_{\text{S}} }} \le {\bar{\sigma }}_{\text{S}}$ (see also Table 2.16)

with

$$ \begin{aligned} {x} & = {\psi }_{\text{S}} \left\{ { - 1 + \sqrt {1 + 2/{\psi_{s} }} } \right\}{d} \\ {z} & = {d} - {x/3}. \\ \end{aligned} $$

Rectangular section—double reinforcement

${\sigma }_{\text{c}} = \frac{{{M}_{\text{Ek}} }}{{{I}_{i} }} {x}\le {\bar{\sigma}}_{\text{c}} ;\quad {\sigma }_{\text{S}} = {\alpha }_{\text{e}} \frac{{{M}_{\text{Ek}} }}{{{I}_{i} }}{y}_{\text{S}} \le {\bar{\sigma}}_{\text{S}}$ (see also Table 2.16)

with

$$ \begin{aligned} & {x} = {\psi }_{\text{t}} \{ - 1 + \sqrt {1 + 2{\delta}/{\psi}_{\text{t}} } \} {d} \\ & {\delta } = ({\text{d}A}_{S} + {\text{d}}^{\prime } {A}_{\text{S}}^{\prime } {)/(}{\text{d}A}_{\text{t}} {)} \\ & {I}_{i} = {bx}^{3} /3 + {\alpha }_{\text{e}} {A}_{\text{S}} {y}_{\text{S}}^{2} + {\alpha }_{\text{e}} {A}_{\text{S}}^{\prime } {y}_{2}^{{{\prime }\,2}} \\ & {y}_{\text{s}} = {d} - {x}\quad {y}_{\text{s}}^{\prime } = {x} - {d}^{\prime }. \\ \end{aligned} $$

T-shape section—single reinforcement

${\sigma }_{\text{c}} = \frac{{{M}_{\text{Ek}} }}{{{I}_{i} }}{x} \le {\bar{\sigma }}_{\text{c}} ;\quad {\sigma }_{\text{S}} = {\alpha}_{\text{e}} \frac{{{M}_{\text{Ek}} }}{{{I}_{i} }}{y}_{\text{S}} \le {\bar{\sigma }}_{\text{S}}$ (see also Table 2.16)

with (a = b −b _w)

$$ \begin{aligned} {x} & = \frac{{{at + }{\alpha }_{\text{e}} {A}_{\text{S}} }}{{{b}_{\text{w}} }}\left\{ { - 1 + \sqrt {1 + \frac{{{at}^{2} + 2{\alpha }_{\text{e}} {A}_{\text{s}} {d}}}{{({at + }{\alpha }_{\text{e}} {A}_{\text{S}} )^{2} }}{b}_{\text{w}} } } \right\}\quad ( > {t}) \\ {I}_{i} & = ({bx}^{3} - {ay}^{3} )/3 + {\alpha }_{\text{e}} {A}_{\text{S}} {y}_{\text{S}}^{2} \quad ({y} = {x} - {t}). \\ \end{aligned} $$

T-shaped section—double reinforcement

${\sigma }_{\text{c}} = \frac{{{M}_{\text{Ek}} }}{{{I}_{i} }}{x} \le {\bar{\sigma }}_{\text{c}} ;\quad {\sigma }_{\text{S}} = {\alpha }_{\text{e}} \frac{{{M}_{\text{Ek}} }}{{{I}_{i} }}{y}_{\text{S}} \le {\bar{\sigma }}_{\text{S}}$ (see also Table 2.16)

with (a = b − b _w)

$$ \begin{aligned} & {x} = \frac{{{at + }{\alpha }_{\text{e}} {A}_{\text{t}} }}{{{b}_{\text{w}} }}\left\{ { - 1 + \sqrt {1 + \frac{{{at}^{2} + 2{\alpha }_{\text{e}} \left( {{A}_{\text{s}} {d} + {A}_{\text{s}}^{\prime } {d}^{\prime } } \right)}}{{({at + }{\alpha }_{\text{e}} {A}_{\text{t}} )^{2} }}{b}_{\text{w}} } } \right\}\quad ( > {t}) \\ & {I}_{i} = ({bx}^{3} - {ay}^{3} )/3 + {\alpha }_{\text{e}} {A}_{\text{S}} {y}_{\text{S}}^{2} + {\alpha }_{\text{e}} {A}_{\text{S}}^{\prime } {y}_{\text{S}}^{\prime 2} \quad ({y} = {x} - {t}). \\ \end{aligned} $$

3.1.4 Chart 3.4: Section in Bending: Resistance Design—Formulas

Symbols

M _Ed :

design value of the applied moment

M _Rd :

design value of the resisting moment

r = f _yd/f _cd :

design strengths ratio

${\omega }_{\text{s}} = {r}{\rho }_{\text{s}}$ :

mechanical reinforcement ratio in tension

${\omega }_{\text{s}}^{\prime } = {r}{\rho }_{\text{s}}^{\prime }$ :

mechanical reinforcement ratio in compression

${\omega }_{\text{t}} = {\omega }_{\text{s}} + {\omega }_{\text{s}}^{\prime }$ :

total mechanical reinforcement ratio

${\varepsilon }_{\text{yd}} = {f}_{\text{yd}} /{E}_{\text{s}}$ :

yield strain of steel

See also Charts 2.2, 2.3 and 3.3.

Resistance Verifications in Phase III

(Cracked section—see figure Chart 3.3)

Rectangular section—single reinforcement

$$ {M}_{\text{Rd}} = {A}_{\text{s}} \,{f}_{\text{yd}} \,{z} \ge {M}_{\text{Ed}} $$

with

$$ \begin{aligned} & {{\bar{x}}} = {\omega }_{\text{S}} {d} \\ & {z} = {d} - 0.5\,{{\bar{x}}} \\ & {\xi } = {\omega }_{\text{s}} /0.{8} \\ & {\varepsilon }_{\text{s}} = {\varepsilon }_{\text{cu}} {{\left( {1 - {\xi }} \right)} \mathord{\left/ {\vphantom {{\left( {1 - {\xi }} \right)} {{\xi } \ge }}} \right. \kern-0pt} {{\xi } \ge }}{\varepsilon }_{\text{yd}}. \\ \end{aligned} $$

Rectangular section—double reinforcement (case ${\sigma }_{\text{s}}^{\prime} = {f}_{\text{yd}}$)

$$ {M}_{\text{Rd}} = {A}_{\text{s}} \,{f}_{\text{yd}} \,{z}_{\text{s}} + {A}_{\text{s}}^{\prime } \,{f}_{\text{yd}} \,{z}_{\text{s}}^{\prime } \ge {M}_{\text{Ed}} $$

with

$$ \begin{aligned} & {{\bar{x}}} = ({\omega }_{\text{s }} - {\omega }_{\text{s}}^{\prime } ){ d} \\ & {z}_{\text{s}} = {d} - 0.5\, {{\bar{x}}} \\ & {z}_{\text{s}}^{\prime } = 0.5\,{{\bar{x}}} - {d}^{\prime } \\ & {\xi } = ({\omega }_{\text{s}} - {\omega }_{\text{s}}^{\prime } )/0.{8} \\ & {\varepsilon }_{\text{s}} = {\varepsilon }_{\text{cu}} ({ 1} - {\xi }{ })/{\xi } \ge {\varepsilon }_{\text{yd}} \\ & {\varepsilon }_{\text{s}}^{\prime } = {\varepsilon }_{\text{cu}} ({\xi } - {\delta }^{\prime } )/{\xi } \ge {\varepsilon }_{\text{yd}}. \\ \end{aligned} $$

Rectangular section—double reinforcement (case ${\sigma }_{\text{s}}^{\prime} < {f}_{\text{sd}}$)

$$ {M}_{\text{Rd}} = {A}_{\text{s}} \,{f}_{\text{yd}} \,{z}_{\text{s}} - {A}_{\text{s}}^{\prime } \,{\sigma }_{\text{s}}^{\prime } \,{z}_{\text{s}}^{\prime } \ge {M}_{\text{Ed}} $$

with ε _cu = 0.0035, ${\alpha }_{\text{o}} = {\varepsilon }_{\text{yd}} /{\varepsilon }_{\text{cu}} ,{\delta }^{\prime } = {d}^{\prime } /{d}$ and with

$$ \begin{aligned} & {\bar{\xi }} = \frac{1}{2}\left\{ {\left( {{\omega }_{\text{s}} - {\omega }_{\text{s}}^{\prime } /{\alpha }_{\text{o}} } \right) + \sqrt {\left( {{\omega }_{\text{s}} - {\omega }_{\text{s}}^{\prime } /{\alpha }_{\text{o}} } \right)^{2} + 3.2{\delta }^{\prime } {\omega }_{\text{s}}^{\prime } /{\alpha }_{\text{o}} } } \right\} \\ & {\bar{x}} = {\bar{\xi }}\,{d} \\ & {z}_{\text{s}} = {d} - 0.5\,{\bar{x}} \\ & {z}_{\text{s}}^{\prime } = 0.5\,{\bar{x}} - {d}^{\prime } \\ & {\xi } = {\bar{\xi }}/0.8 \\ & {\varepsilon }_{\text{s}} = {\varepsilon }_{\text{cu}} (1 - {\xi })/{\xi } \ge {\varepsilon }_{\text{yd}} \\ & {\varepsilon }_{\text{s}}^{\prime } = {\varepsilon }_{\text{cu}} ({\xi } - {\delta }^{\prime } )/{\xi < \varepsilon }_{\text{yd}} \\ & {\sigma }_{\text{s}}^{\prime } = {\varepsilon }_{\text{s}}^{\prime } {E}_{\text{s}}. \\ \end{aligned} $$

T-shaped section—single reinforcement

$$ {M}_{\text{Rd}} = {f}_{\text{cd}} {bt}\left( {{d} - {t}/{2}} \right) + {f}_{\text{cd}} {b}_{\text{w}} {y}({y}_{\text{s}} - {y}/{2}) \ge {M}_{\text{Ed}} $$

with a = b − b _w and with

$$ \begin{aligned} {{\bar{x}}} & = ({ rA}_{\text{a}} - {at})/{b}_{\text{w}} \quad ( > {t}) \\ {y} & = {{\bar{x}}} - {t}\;{(} > 0{)}\quad {y}_{\text{s}} = {d} - {{\bar{x}}} \\ {x} & = {{\bar{x}}}/0.8 \\ {\varepsilon }_{\text{s}} & = {\varepsilon }_{\text{cu}} ({d} - {x})/{x} \ge {\varepsilon }_{\text{yd}}. \\ \end{aligned} $$

T-shaped section—double reinforcement (case ${\sigma }_{\text{s}}^{\prime } = {f}_{\text{yd}}$)

$$ {M}_{\text{Rd}} = {f}_{\text{cd}} {bt}\left( {{d} - {t}/{2}} \right) + {f}_{\text{cd}} {b}_{\text{w}} {y}({y}_{\text{s}} - {y}/{2}) + {f}_{\text{yd}} {A}_{\text{s}}^{\prime } ({d} - {d}^{\prime } ) \ge {M}_{\text{Ed}} $$

with a = b − b _w and with

$$ \begin{aligned} {{\bar{x}}} & { = (}{rA}_{\text{s}} - {rA}_{\text{s}}^{\prime } - {at}{)/}{b}_{\text{w}} \quad ( > {t}) \\ {y} & = {{\bar{x}}} - {t}\;{(} > {0)} \\ {y}_{\text{s}} & = {d} - {{\bar{x}}} \\ {x} & = {{\bar{x}}}/0.8 \\ {\varepsilon }_{\text{s}} & = {\varepsilon }_{\text{cu}} ({d} - {x})/{x} \ge {\varepsilon }_{\text{yd}} \\ {\varepsilon }_{\text{s}}^{\prime } & = {\varepsilon }_{\text{cu}} ({x} - {d}^{\prime } )/{x}\; \ge {\varepsilon }_{\text{yd}}. \\ \end{aligned} $$

T-shaped section—double reinforcement (case ${\sigma }_{\text{s}}^{\prime } < {f}_{\text{sd}}$)

$$ {M}_{\text{Rd}} = {f}_{\text{cd}} {bt}\left( {{d} - {t}/{2}} \right) + {f}_{\text{cd}} {b}_{\text{w}} {y}({y}_{\text{s}} - {y}/{2}) + {\sigma }_{\text{s}}^{\prime } {A}_{\text{s}}^{\prime } ({d} - {d}^{\prime } ) \ge {M}_{\text{Ed}} $$

with ${\alpha }_{\text{o}} = {\varepsilon }_{\text{cu}} /{\varepsilon }_{\text{yd}} ,{a} = {b} - {b}_{\text{w}}$ and with

$$ \begin{aligned} & {{\bar{x}}} = \frac{1}{{2{b}_{w} }}\left\{ {({rA}_{s} - {at} - {rA}_{s}^{\prime } {\alpha }_{\text{o}} ) + \sqrt {({rA}_{\text{s}} - {at} - {rA}_{\text{s}}^{\prime } {\alpha }_{\text{o}} )^{2} + 3.2{d}^{\prime } {b}_{\text{w}} {rA}_{\text{s}}^{\prime } {\alpha }_{\text{o}} } } \right\} \\ & {y} = {{\bar{x}}} - {t}( > 0)\quad {y}_{\text{s}} = {d} - {{\bar{x}}} \\ & {x} = {{\bar{x}}}/0.8 \\ & {\varepsilon }_{\text{s}} = {\varepsilon }_{\text{cu}} ({d} - {x})/{x}\; \ge {\varepsilon }_{\text{yd}} \\ & {\varepsilon }_{\text{s}}^{\prime } = {\varepsilon }_{\text{cu}} ({x} - {d}^{\prime } {)}/{x} < {\varepsilon }_{\text{yd}} \\ & {\sigma }_{\text{s}}^{\prime } = {\varepsilon }_{\text{s}}^{\prime } {E}_{\text{s}}. \\ \end{aligned} $$

3.1.5 Table 3.5: Section in Bending: Viscous Redistribution of Stresses

For a section in reinforced concrete, symmetric for shape and reinforcement, subject to uniaxial bending in the uncracked phase I, the following table shows, for different ratios κ _s = E _s I _s/E _c I _c between the elastic stiffnesses of reinforcement and concrete and for three nominal coefficients ϕ _∞ of final viscosity, the following variation ratios with respect to the initial elastic values:

$$ \begin{aligned} {\alpha }_{\text{e}}^{*} & = {\alpha }_{{{\text{e}}\infty }} /{\alpha }_{\text{e}} \quad {\text{homogenization}}\,{\text{coefficient}}\,{\text{of}}\,{\text{reinforcement}} \\ {\sigma }_{\text{c}}^{*} & = {\sigma }_{{{\text{c}}\infty }} /{\sigma }_{\text{co}} \quad {\text{final}}\,{\text{stress}}\,{\text{in}}\,{\text{concrete}} \\ {\sigma }_{\text{s}}^{*} & = {\sigma }_{{{\text{s}}\infty }} /{\sigma }_{\text{so}} \quad {\text{final}}\,{\text{stress}}\,{\text{in}}\,{\text{steel}}\,({=}{\chi }_{\infty } /{\chi }_{\text{o}} ), \\ \end{aligned} $$

where stresses σ _c∞ and σ _so in the materials are assumed calculated with the competent formulas of serviceability verification of Chart 3.3 based on the actual value α _e = E _s/E _c of the elastic moduli. The variation ratio of stresses in steel coincides with the one χ _∞ /χ _o between final and initial curvatures of the section in bending.

The use of the table requires the calculation of the centroidal moments of inertia of the steel area I _s and of the one of the concretes I _c in order to deduce, from their ratio μ _s = I _s/I _c, the parameter

$$ {\kappa }_{\text{s}} = {\mu }_{\text{s}} {\alpha }_{\text{e}}. $$

The values of the table have been calculated with the following formulas:

$$ \begin{array}{*{20}l} {{\alpha }^{*} = \frac{{{\text{e}}^{\beta \phi }_\infty }}{\beta } - \frac{1}{{{\kappa }_{\text{s}} }}} \hfill & {{\text{with}}\,{\beta } = \frac{{{\kappa }_{\text{s}} }}{{1 + {\kappa }_{\text{s}} }}} \hfill \\ {{\sigma }_{\text{c}}^{*} = {\text{e}}^{{ - {\beta \phi }_{\infty } }} } \hfill & {{\sigma }_{\text{s}}^{*} = {\alpha }_{\text{e}}^{*} {\sigma }_{\text{c}}^{*} }, \hfill \\ \end{array} $$

valid for concretes loaded at early stages (extreme ageing theory).

χ s	ϕ ∞  = 2.4			ϕ ∞  = 2.9			ϕ ∞  = 3.4
	${\alpha }_{\text{e}}^{ * }$	${\sigma }_{\text{c}}^{ * }$	${\sigma }_{\text{s}}^{ * }$	${\alpha }_{\text{e}}^{ * }$	${\sigma }_{\text{c}}^{ * }$	${\sigma }_{\text{s}}^{ * }$	${\alpha }_{\text{e}}^{ * }$	${\sigma }_{\text{c}}^{ * }$	${\sigma }_{\text{s}}^{ * }$
0.00	3.40	1.00	3.40	3.90	1.00	3.90	4.40	1.00	4.40
0.05	3.54	0.89	3.16	4.11	0.87	3.58	4.69	0.85	3.99
0.10	3.68	0.80	2.96	4.32	0.77	3.32	4.98	0.73	3.66
0.15	3.82	0.73	2.79	4.52	0.69	3.10	5.28	0.64	3.39
0.20	3.95	0.67	2.65	4.73	0.62	2.92	5.57	0.57	3.16
0.25	4.08	0.62	2.52	4.93	0.56	2.76	5.87	0.51	2.97
0.30	4.21	0.57	2.42	5.13	0.51	2.63	6.16	0.46	2.81
0.35	4.33	0.54	2.32	5.32	0.47	2.51	6.46	0.41	2.67
0.40	4.45	0.50	2.24	5.52	0.44	2.41	6.75	0.38	2.55
0.45	4.56	0.47	2.17	5.70	0.41	2.32	7.03	0.35	2.45
0.50	4.68	0.45	2.10	5.89	0.38	2.24	7.32	0.32	2.36
0.55	4.79	0.43	2.04	6.07	0.36	2.17	7.60	0.30	2.27
0.60	4.89	0.41	1.99	6.24	0.34	2.10	7.88	0.28	2.20
0.65	5.00	0.39	1.94	6.42	0.32	2.05	8.15	0.26	2.14
0.70	5.10	0.37	1.90	6.59	0.30	2.00	8.42	0.25	2.08
0.75	5.19	0.36	1.86	6.75	0.29	1.95	8.69	0.23	2.02
0.80	5.29	0.34	1.82	6.91	0.28	1.91	8.95	0.22	1.97
0.85	5.38	0.33	1.79	7.07	0.26	1.87	9.20	0.21	1.93
0.90	5.47	0.32	1.75	7.23	0.25	1.83	9.46	0.20	1.89
0.95	5.56	0.31	1.73	7.38	0.24	1.80	9.70	0.19	1.85
1.00	5.64	0.30	1.70	7.53	0.23	1.77	9.95	0.18	1.82

3.1.6 Chart 3.6: Section in Bending: Additional Formulas

Reinforced concrete sections subject to uniaxial bending.

Symbols

M _ok :

characteristic value of the cracking moment

M _od :

design value of the cracking moment

See also Charts 2.2, 3.3 and 3.4.

Cracking Moment

Serviceability verifications

$$ {M}_{\text{ok}} = {\bar{\sigma}}_{\text{ct}}^{\prime} {I}_{i} /{y}_{\text{ct}}^{\prime} \le {M}_{\text{Ek}} \quad ({\bar{\sigma}}_{\text{ct}}^{\prime} = {\beta}{f}_{\text{ctk}}) $$

(for f_ctk see Table 1.2a, b)

Resistance verifications

$$ {M}_{\text{od}} = {\beta }{f}_{\text{ctd}} {I}_{i} /{y}_{\text{c}}^{\prime } \le {M}_{\text{Ed}} \quad ({\beta } = {1.3}) $$

(for $I_{i} ,y_{\text{c}}^{\prime }$ see Chart 3.3 with figure).

Minimum Reinforcement

For the longitudinal reinforcement at the edge of the beam in tension, a minimum area is to be set so that the force released by the concrete in tension when cracking occurs can be resisted by that reinforcement at the characteristic yield stress f _yk. This force should be conventionally calculated based on the triangular distribution of stresses with a maximum at the edge in tension equal to the mean value f _ctm of the concrete tensile strength.

For T-shape sections or similar, it can be set for example:

$$ {A}_{\text{s}} \ge \frac{1}{2}{y}_{\text{c}}^{\prime } {b}_{\text{w}} {f}_{\text{ctm}} /{f}_{\text{yk}} $$

(see figure Chart 3.3).