Centred Axial Force

Abstract

This chapter presents the design methods of reinforced concrete elements subjected to axial action, starting from the columns under compression and proceeding with the tension members, for which in particular the criteria for cracking calculation are given. In the final section the structure of a multi-storey building is described, assumed as applicative example for the design calculations. The analysis of loads is developed and the complete design of a column is shown.

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: General Aspects and Axial Force

2.1.1 Table 2.1: Environmental Conditions: Exposure Classes

The following table, reproduced in a summarized form, is extracted from the European Norm EN 206-1 “Concrete specification, performance, production and conformity”. Classes XC, XD and XS refer to corrosion of reinforcement, classes XF and XA refer to the surface attack of concrete.

Minimum concrete covers for the protection of reinforcement against corrosion for different degrees of aggressiveness are given in Table 2.17.

Class	Description	Examples
1. No risk of corrosion or attack
X0	Plain concrete without attacks Reinf. concrete in very dry environ	Concrete inside buildings with very low air humidity
2. Corrosion induced by carbonation
XC1	Dry or permanently wet	Concrete inside buildings
XC2	Wet, rarely dry	Many foundations
XC3	Moderate humidity	External concrete sheltered
XC4	Cyclic wet and dry	Structures in water line
2. Corrosion induced by chlorides
XD1	Moderate humidity	Airborne salt
XD2	Wet, rarely dry	Swimming pools
XD3	Cyclic wet and dry	Bridges, outdoor pavements
3. Corrosion induced by chlorides from sea water
XS1	Exposed to airborne salt	Structures near to the coast
XS2	Permanently submerged	Parts of marine structures
XS3	Tidal, splash and spray zones	Parts of marine structures
4. Freeze/thaw attack
XF1	Wet surfaces, without de-icing agents	Vertical surf. exposed to rain
XF2	Wet surfaces, with de-icing agents	Vertical surfaces of bridges
XF3	Soaked surf. without de-icing agents	Horiz. surfaces open to rain
XF4	Soaked surf. with de-icing agents	Horiz. surfaces of bridges
5. Chemical attack
XA1	Slightly aggressive chemical environ.	Natural soils-groundwater
XA2	Moderate aggressive chem. environ.	Natural soils-groundwater
XA3	Highly aggressive chemical environ.	Natural soils-groundwater

For design applications, with reference to corrosion of reinforcement, the exposure classes can be grouped as follows:

Aggressiveness	Exposure classes
Low	X0, XC1, XC2, XC3
Medium	XC4, XD1, XS1
High	XD2, XD3, XS2, XS3

With reference to freeze/thaw attack or chemical attack (classes XF and XA) an adequate concrete mix design should be adopted.

2.1.2 Chart 2.2: Concrete: Design Strength

In the resistance verifications (ultimate limit states) the following values (in MPa) are adopted:

$$ {f}_{\text{cd}} = {\alpha }_{\text{cc}} \frac{{{f}_{\text{ck}} }}{{{\gamma }_{\text{C}} }}\quad {\text{design}}\;{\text{compressive}}\;{\text{strength}} $$

with α _cc = 1.00 for short term loads and α _cc = 0.85 for ordinary loads.

$$ {f}_{{{\text{c}}2}} = 0.6\left( {1.0 - \frac{{{f}_{\text{ck}} }}{250}} \right){f}_{\text{cd}} \cong 0.50{f}_{\text{cd}} \quad {\text{reduced}}\;{\text{design}}\;{\text{strength}} $$

$$ {f}_{\text{ctd}} = \frac{{{f}_{\text{ctk}} }}{{{\gamma }_{\text{C}} }}\quad {\text{tensile}}\;{\text{design}}\;{\text{strength}} $$

where the partial safety factor should be assumed equal to:

γ _C = 1.5:

for concrete of ordinary production

γ _C = 1.4:

for concrete of controlled production with δ ≤ 0.10

(δ coefficient of variation = ratio of the standard deviation to the mean value).

In the verifications of stresses (serviceability limit state) the following values are adopted for concrete:

${\bar{\sigma }}_{\text{c}} = \text{0.45}\,{f}_{\text{ck}}$ :

allowable tensile stress for quasi-permanent combination

${\bar{\sigma }}_{\text{cj}} = \text{0.60}{f}_{\text{ckj}}$ :

allowable tensile stress for characteristic combination

${\bar{\sigma }}_{cj}^{\prime } = \text{0.70}{f}_{\text{ckj}}$ :

allowable tensile stress at prestressing.

The value of the ultimate tensile stress ${\bar{\sigma }}_{\text{ct}}^{\prime } = {\beta }{f}_{\text{ctk}}$ refers to the limit of crack formation, with:

$\beta = 1.0$ :

for constant distribution

$\beta = 1.3$ :

for triangular distribution.

For parts with a thickness t < 5d _a, where d _a is the maximum aggregate size, all values of resistance and allowable stresses should be reduced by the factor (0.5 + 0.1t/d _a).

2.1.3 Chart 2.3: Steel: Design Strength

In the resistance verifications (ultimate limit states) the values indicated below are adopted for steel of reinforced and prestressed concrete.

For passive reinforcement

$$ {f}_{\text{yd}} = \frac{{{f}_{\text{yk}} }}{{{\gamma }_{\text{S}} }}\quad {\text{design}}\;{\text{value}}\;{\text{of}}\;{\text{yield}}\;{\text{stress}} $$

$$ {f}_{\text{td}} = \frac{{{f}_{\text{tk}} }}{{{\gamma }_{\text{S}} }}( {=} {kf}_{\text{yd}} )\quad {\text{design}}\;{\text{strength}}\;\left( {{k} = 1. 20} \right) $$

For prestressing reinforcement

$$ {f}_{\text{ptd}} = \frac{{{f}_{\text{ptk}} }}{{{\gamma }_{\text{S}} }}\quad {\text{design}}\;{\text{strength}}\;{\text{for}}\;{\text{prestressing}}\;{\text{strands}} $$

$$ {f}_{\text{ypd}} = 0.9{f}_{\text{ptd}} \quad {\text{design}}\;{\text{value}}\;{\text{of}}\;{\text{the}}\;{\text{yield}}\;{\text{stress}} $$

The partial safety factor for all reinforcements should be assumed equal to $\gamma_{\text{S}} = 1.15$.

In the verification of stresses (serviceability limit state) the values indicated as follows are adopted for steel.

For passive reinforcement

$$ {\bar{\sigma }}_{\text{s}} = 0.80{f}_{\text{yk}} \quad {\text{allowable}}\;{\text{stress}}\;{\text{of}}\;{\text{passive}}\;{\text{reinforcement}} $$

For prestressing reinforcement

• ${\bar{\sigma }}_{\text{p}} = 0.80{f}_{\text{pyk}}$ allowable stress after losses

• ${\bar{\sigma }}_{\text{pi}} = 0.85{f}_{\text{pyk}} \le 0.75{f}_{\text{ptk}}$ initial admis. stress in post-tensioned tendons

• ${\bar{\sigma }}_{\text{pi}} = 0.90{f}_{\text{pyk}} \le 0.80{f}_{\text{ptk}}$ initial admis. stress in pre-tensioned tendons

where f _py is to be substituted by f _p0.1 or f _p1, respectively for wires and strands.

In the calculations of stresses in service, the viscoelastic effects can be approximated with the assumption in the elastic formulas of

$$ {\alpha }_{\text{e}} = \frac{{{E}_{\text{s}} }}{{{E}_{\text{c}} }} = 15 $$

as homogeneizaton coefficient of the steel areas in the resisting section.

For tensile stresses in the reinforcement, the allowable limits given by Table 2.16 are also to be considered, for the cracking verifications.

2.1.4 Chart 2.4: Reinforcement: Shaping and Detailing

The schemes of the present chart refer to the bending of steel bars for reinforced concrete with a diameter ϕ ≤ 24 mm, unless noted otherwise.

• Bending radius (at the axis) for end hooks and stirrups:

  $$ {r} \ge 3 . 0\,{\phi } $$

• Mandrel diameter

  $$ {d}_{\text{o}} = {2r - }{\phi }\quad ( \ge 5\phi ) $$

• Bending radius (at the axis) for bent bars and continuous reinforcement

  $$ {r}^{\prime } = 2{r}\quad ({d}^{\prime }_{\text{o}} \ge 11{\phi }) $$

• Developed length of the end hook for r = 3ϕ with α = 90°÷135°÷180° for bent bars and continuous reinforcement

  $$ {u} \ge 8{\phi },\, 10 {\phi} {\text{ and }}\, 1 2{\phi } $$

• Detailing referred to the rectified axial polygon line (with a* nominal sides dimensioned in the reinforcement drawings)

  $$ {a} = {a}^{*} + 1.0{\phi }\quad {\text{actual}}\;{\text{dimensions}}\;{\text{for}}\;{r} = 5{\phi } $$

2.1.5 Chart 2.5: Reinforcement: Positioning Tolerances

The following deviations δ from the nominal dimensions shown in the design refer to the position of the longitudinal bars (passive bars or pretensioned strands) in the cross section. The relevant dimension (height or width) of the section is indicating with l.

• $\pm {\bar{\delta }} = 0.04\;l$ (≥5 mm) for l < 500 mm

• $\pm {\bar{\delta }} = 15 + 0.01\;l$ (≤30 mm) for l > 500 mm

The partial safety factors already take into account such tolerances in the resistance verifications.

For the values of cover, given that spacers adequately distributed on the formwork surfaces are used, it can be assumed:

$$ \pm {\delta } = 10\;{\text{mm}} $$

The above-mentioned positioning tolerances can be halved in the case of industrial production in which the verification of bars positioning is part of the quality control system. In this case the tolerance of the cover becomes

$$ \begin{array}{*{20}l} {\delta = - 0} \hfill & {{\text{and}}\quad {\delta = + 5}\;{\text{mm}}} \hfill \\ \end{array} $$

2.1.6 Table 2.6: Bond: Design Strength

The following table shows, for different codified classes of concrete and for the design of end anchorages of the bars, the following values:

f _bk :

characteristic bond strength

l _b :

anchorage length

Ordinary production $\varDelta {f} = 8\,{\text{MPa}}$			Controlled production $\varDelta {f} = 5 {\text{ MPa}}$
Concrete class	f bk	l b/ϕ	Concrete class	f bk	l b/ϕ
			C30/37	5.2	28
C16/20	3.6	41	C35/43	5.6	26
C20/25	3.8	38	C40/50	6.1	24
C25/30	4.3	34	C45/55	6.5	22
C30/37	4.7	31	C50/60	7.0	21
C35/43	5.2	28	C55/67	7.4	20
C40/50	5.6	26	C60/75	7.6	19
C45/55	6.1	24	C70/85	8.1	18

The values are expressed in MPa and are deduced from the formulas:

• ${f}_{\text{bk}} = {\beta }_{\text{b}} \,{f}_{\text{ctk}}$ (see Table 1.2)

• ${\beta }_{\text{b}} = 2.25$ for ribbed bars

• ${\text{l}}_{\text{b}} = \frac{\phi }{4}\frac{{{f}_{\text{yd}} }}{{{f}_{\text{bd}} }}\quad ({f}_{\text{yd}} = {f}_{\text{yk}} /{\gamma }_{\text{S}} ;{f}_{\text{bd}} = {f}_{\text{bk}} /{\gamma }_{\text{C}} )$

In particular, the anchorage length refers to the ribbed bars in steel of the type B450C, with γ _S = 1.15 and γ _C = 1.5 and it is expressed as a ratio to the diameter ϕ of the bar (l _b/ϕ). For anchorages in surface zones in tension, the bond strength should be halved.

2.1.7 Chart 2.7: Reinforcement: Anchorages and Overlaps

It can be assumed that bond stresses at the end of a bar in tension are distributed along the anchorage length l _b with a constant value and that the effectiveness of the bar in tension increases linearly starting from its end up to the full value (=1.0) of its capacity, reached at the distance l _b. The first segment equal to 10ϕ is to be considered ineffective. For the anchorage, hooks are to be calculated with reference to their developed length and they are to be considered ineffective up to the tangent point. The following scheme refers to an end anchorage in uncracked zone.

The overlapping on the tension side corresponds to a double end anchorage of the consecutive bars and it should be done with a segment ${l}_{\text{b}} < (20{\phi + }{i})$ of straight overlapping, where i is the distance between bars to be joint, plus an end segment of length u ≥ 10ϕ bent inwards, towards the compression zone. For bond, the effectiveness of the surface straight segment l _ob should be halved; the full capacity of the bar is therefore reached at:

$$ {l}_{\text{ob}} = 2({l}_{\text{b}} - {u}) $$

with l _b defined in Table 2.6. The following scheme gives the complementary growth of the effectiveness of the two joint bars. The capacity of the joint, indicated by the dotted line, can be enhanced increasing the overlapping.

Reinforcement joints in tie elements should be done with a full confinement, introducing transverse stirrups in the segment of bars overlapping, commensurate to the axial force to be transferred.

2.1.8 Chart 2.8: Concrete Structures: Minimum Dimensions

Structural elements in plain, reinforced and prestressed concrete should be designed with the minimum dimensions given by the most restrictive minimum values of the following cases:

Absolute Minimum Thicknesses

Technological limits to ensure a sufficient compact mass:

components for extruded or vibrocompacted floors	t ≥ 30 mm
components for cast in situ floors	t ≥ 40 mm
parts of main structural elements	t ≥ 50 mm
wall panels and plain concrete	t ≥ 80 mm

Relative Minimum Thicknesses

Requirement of good homogeneity of concrete for a uniform strength (d _a = maximum aggregate size):

• walls in plain concrete (unreinforced)	t ≥ 5.0 d a
• structural elements reinforced on both sides	t ≥ 4.0 d a
• slabs and ribs reinforced on one layer	t ≥ 2.4 d a
• reinforced toppings sitting on permanent blocks	t ≥ 1.6 d a

Minimum Bar Spacing (Concrete)

Guarantee of the passage of aggregates for good compaction of concrete (d _a = maximum aggregate size):

	Spacing		Cover
	Horizontal i oh	Vertical i ov	c o
• stirrups and links	≥ 1.6 d a	≥ 1.6 d a	≥ 0.8 d a
• passive reinforcement	≥ 1.0 d a	≥ 0.8 d a	≥ 1.0 d a
• pretensioned reinforc.	≥ 1.2 d a	≥ 1.0 d a	≥ 1.0 d a

Minimum Bar Spacing (Steel)

Requirement of good encasing of bars for effective bond (ϕ reinforcement diameter):

	Spacing i o	Cover c o
• passive reinforcement	≥ 1.0 ϕ	≥ 1.0 ϕ
• pretensioned reinforc.	≥ 2.0 ϕ	≥ 1.5 ϕ

For cover see also Table 2.17.

2.1.9 Chart 2.9: Ordinary Columns: Formulas and Construction Rules

Reinforced concrete sections subject to compression axial force.

Symbols

N _Ek :

characteristic axial force

N _Ed :

design axial force

b :

smaller side dimesion of section

ϕ :

diameter of longitudinal bars

ϕ′:

stirrups diameter

i :

centre-to-centre distance of longitudinal bars

s :

stirrups spacing (current part)

s′:

stirrups spacing (column ends)

A _c :

concrete area

A _s :

area of longitudinal reinforcement

ρ _s = A _s/A _c :

geometrical reinforcement ratio

α _e = E _s/E _c :

elastic moduli ratio (see Chart 2.3)

ψ _s = α _e ρ _s :

elastic reinforcement ratio

f _cd :

concrete design strength

f _yd :

reinforcement design strength

r _s = f _yd/f _cd :

design strength ratio

ω _s = r _s ρ _s :

mechanical reinforcement ratio

σ _c :

concrete stress

σ _s :

steel stress

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

concrete allowable stress (see Chart 2.2)

N _Rd :

design resisting axial force

Verifications

Service

$$ \begin{array}{*{20}l} {\sigma_{\text{c}} = \frac{{{N}_{\text{Ed}} }}{{{A}_{\text{c}} (1 + {\psi }_{\text{s}} )}} \le 0.7{\bar{\sigma }}_{\text{c}} } \hfill & {(\sigma_{\text{s}} = {\alpha }_{\text{e}} \sigma_{\text{c}} )} \hfill \\ \end{array} $$

Resistance

$$ {N}_{\text{Rd}} = {f}_{\text{cd}} {A}_{\text{c}} (0.8 + {\bar{\omega }}_{\text{s}} ) $$

Construction requirements

b ≥ 200 mm :

(≥150 mm in prefabrication)

A _s ≥ 0.10 N _Ed/f _yd :

i ≤ 300 mm

ρ _s ≥ 0.003 :

s ≤ b

ρ _s ≤ 0.04 :

s ≤ 300 mm

ϕ ≥ 12 mm :

s ≤ 12 ϕ

ϕ′ ≥ ϕ / 4:

s′ ≤ 0.6 s

2.1.10 Chart 2.10: Confined Columns: Formulas and Construction Requirements

Symbols

D :

diameter of spiral

n :

number longitudinal bars

s :

pitch of spiral or spacing of hoops

a _w :

area of spiral or hoops

A _n = πD ²/4:

area of confined core

A _w = a _w πD/s :

equivalent area of spiral or hoops

ω ₁ = r _s A ₁/A _n :

longitudinal mechanical reinforcement ratio

ω _w = r _s A _w/A _n :

spiral or hoops mechanical reinforcement ratio

See also Chart 2.9.

Verifications

Service

$$ \begin{array}{*{20}l} {\sigma_{\text{c}} = \frac{{{N}_{\text{Ed}} }}{{{A}_{\text{c}} (1 + {\psi }_{\text{s}} )}} \le {\bar{\sigma }}_{\text{c}} } \hfill & {(\sigma_{\text{s}} = {\alpha }_{\text{e}} \sigma_{\text{c}} )} \hfill \\ \end{array} $$

Resistance

$$ {N}_{\text{Rd}} = {f}_{\text{cd}} {A}_{\text{c}} (0.8 + {\omega }_{1} + 1.6{\omega } ) \ge {N}_{\text{Ed}} $$

Construction Requirements

n ≥ 6

s ≤ D/5

0.8 + ω ₁ +1.6 ω _s ≤ 2

A ₁ ≥ A _w/2

Data of Chart 2.9 are also valid except s′.

2.1.11 Chart 2.11: RC Walls: Construction Requirements

Walls reinforced on both sides with internal vertical bars and external horizontal bars.

Symbols

t :

wall thickness

ϕ :

diameter of vertical bars

ϕ′:

diameter of horizontal bars

i :

centre-to-centre distance between vertical bars

s :

spacing of horizontal bars

c :

edge axis distance

a _v :

area of vertical reinforcement per unit length

a _h :

area of horizontal reinforcement per unit height

Construction requirements

a _v ≥ 0.0030 t :

(total on both sides)

a _v ≤ 0.04 t :

(total on both sides)

a _h ≥ 0.0015 t :

(total on both sides) i ≤ 300 mm

ϕ ≥ 8 mm:

i ≤ 2 t

ϕ′ ≥ ϕ/3:

s ≤ 300 mm

c ≥ 2 ϕ :

s ≤ 25 ϕ

The end parts of the walls are to be reinforced with longitudinal (vertical) and transverse bars according to the requirements for ${\phi }$, ${\phi }^{\prime }$ and s of Chart 2.9.

The requirements above are to be applied if the vertical reinforcement is taken into account in the calculation of the capacity of the wall according to the verification formulas of Chart 2.9.

2.1.12 Table 2.12: Creep in Columns: Stress Redistribution

The following table shows, for different elastic reinforcement rations and for the three nominal coefficients of final creep given for RC in Table 1.16, the stress variation ratios with respect to the initial elastic values:

${\psi }_{\text{s}}$	${\varphi }_{{\infty }} = 1.9$			${\varphi }_{{\infty }} = 2.5$			${\varphi }_{{\infty }} = 3.1$
	${\alpha }_{\text{e}}^{*}$	$\sigma_{\text{c}}^{*}$	$\sigma_{\text{s}}^{*}$	${\alpha }_{\text{e}}^{*}$	$\sigma_{\text{c}}^{*}$	$\sigma_{\text{s}}^{*}$	${\alpha }_{\text{e}}^{*}$	$\sigma_{\text{c}}^{*}$	$\sigma_{\text{s}}^{*}$
0.00	2.90	1.00	2.90	3.50	1.00	3,50	4.10	1.00	4.10
0.02	2.94	0.96	2.83	3.56	0.95	3.39	4.20	0.94	3.95
0.04	2.97	0.93	2.76	3.62	0.91	3.29	4.29	0.89	3.81
0.06	3.01	0.90	2.70	3.69	0.87	3.20	4.39	0.84	3.68
0.08	3.04	0.87	2.64	3.75	0.83	3.11	4.48	0.79	3.56
0.10	3.07	0.84	2.59	3.81	0.80	3.03	4.58	0.75	3.46
0.12	3.11	0.82	2.53	3.87	0.77	2.96	4.68	0.72	3.36
0.14	3.14	0.79	2.49	3.93	0.74	2.89	4.77	0.68	3.26
0.16	3.17	0.77	2.44	3.99	0.71	2.82	4.87	0.65	3.17
0.18	3.20	0.75	2.40	4.04	0.68	2.76	4.96	0.62	3.09
0.20	3.24	0.73	2.36	4.10	0.66	2.70	5.06	0.60	3.02
0.22	3.27	0.71	2.32	4.16	0.64	2.65	5.15	0.57	2.95
0.24	3.30	0.69	2.28	4.22	0.62	2.60	5.25	0.55	2.88
0.26	3.33	0.68	2.25	4.27	0.60	2.55	5.34	0.53	2.82
0.28	3.36	0.66	2.21	4.33	0.58	2.50	5.44	0.51	2.76
0.30	3.38	0.65	2.18	4.38	0.56	2.46	5.53	0.49	2.70
0.32	3.41	0.63	2.15	4.44	0.55	2.42	5.62	0.47	2.65
0.34	3.44	0.62	2.13	4.49	0.53	2.38	5.71	0.46	2.60
0.36	3.47	0.60	2.10	4.54	0.52	2.34	5.80	0.44	2.56
0.38	3.50	0.59	2.07	4.60	0.50	2.31	5.90	0.43	2.51
0.40	3.52	0.58	2.05	4.65	0.49	2.28	5.99	0.41	2.47

• ${\alpha }_{\text{e}}^{*} = {\alpha }_{{{\text{e}}\infty }} /{\alpha }_{\text{e}}$ homogeneization coefficient of reinforcement

• $\sigma_{\text{c}}^{*} = \sigma_{{{\text{c}}\infty }} /\sigma_{\text{co}}$ final stress in concrete

• $\sigma_{\text{s}}^{*} = \sigma_{{{\text{s}}\infty }} /\sigma_{\text{so}}$ final stress in steel (${=} {\varepsilon }_{\infty } /{\varepsilon }_{\text{o}}$)

where the stresses σ _co, σ _so in the materials are intended to be calculated with the service verification formula of Chart 2.9 based on the actual ratio ${\alpha }_{\text{e}} = {E}_{\text{s}} /{E}_{\text{c}}$ of elastic moduli.

The values of the table are calculated with the formulas:

$$ {\alpha }_{\text{e}}^{*} = \frac{{{e}^{{\beta \phi_{{\infty }} }} }}{\beta } - \frac{1}{{{\psi }_{\text{s}} }}\quad {\text{with}}\;\beta = \frac{{{\psi }_{\text{s}} }}{{1 + {\psi }_{\text{s}} }} $$

$$ \sigma_{\text{c}}^{*} = {e}^{-\beta \phi \infty} $$

$$ \sigma_{\text{s}}^{*} = {\alpha }_{\text{e}}^{*} \sigma_{\text{c}}^{*} $$

valid for concrete loaded at an early stage (extreme ageing theory).

2.1.13 Table 2.13: Shrinkage in RC: Stress Effects

The following table shows, for the different elastic reinforcement ratios ${\psi }_{\text{s}} = {\alpha }_{\text{e}} {\rho }_{\text{s}}$, the coefficients

$$ {\beta } = \frac{{{\psi }_{\text{s}} }}{{1 + {\psi }_{\text{s}} }}\quad {\beta }^{\prime } = \frac{1}{{1 + {\psi }_{\text{s}} }} $$

for the calculation of shrinkage self-induced stresses in concrete and steel

$$ \sigma_{\text{cs}} =\upbeta \upsigma _{\text{ce}} \quad \sigma_{{\text{ss}}} = - {\beta }^{\prime } \sigma_{\text{se}} $$

with

$$ \sigma_{\text{ce}} = {E}_{\text{c}} {\varepsilon }_{\text{cs}} \quad \sigma_{\text{se}} = {E}_{\text{s}} {\varepsilon }_{\text{cs}} $$

in the doubly symmetric RC sections (ε _cs = concrete shrinkage).

ψ s	β	β′	ψ s	β	β′
0.00	0.00	1.00
0.02	0.02	0.98	0.22	0.18	0.82
0.04	0.04	0.96	0.24	0.19	0.81
0.06	0.06	0.94	0.26	0.21	0.79
0.08	0.07	0.93	0.28	0.22	0.78
0.10	0.09	0.91	0.30	0.23	0.77
0.12	0.11	0.89	0.32	0.25	0.76
0.14	0.12	0.88	0.34	0.25	0.75
0.16	0.14	0.86	0.36	0.26	0.74
0.18	0.15	0.85	0.38	0.28	0.72
0.20	0.17	0.83	0.40	0.29	0.71
			∞	1.00	0.00

2.1.14 Chart 2.14: Ties in Reinforced and Prestressed Concrete

Reinforced concrete sections subject to axial tension force with possible centred precompression.

Symbols

A _p :

area of prestressing reinforcement

ρ _p = A _p/A _c :

geometric prestressing reinforcement ratio

ψ _p = α _e ρ _p :

prestressing elastic reinforcement ratio

A _t = A _s + A _p :

total area of passive plus active reinforcement

ψ _t = ψ _s + ψ _p :

total elastic reinforcement ratio

α = A _s/A _p :

passive to active reinforcement ratio

σ _po :

prestressing stress in the tendon

N _po = σ _po A _p :

prestressing force in the tendon

See also Charts 2.2, 2.3 and 2.9.

Verifications

Service

• Uncracked section

$$ \begin{aligned} \sigma_{\text{c}} & = \frac{{{N}_{\text{ak}} - {N}_{\text{po}} }}{{{A}_{\text{c}} (1 + {\psi }_{\text{t}} )}} \\ \sigma_{\text{s}} & = {\alpha }_{\text{e}} \sigma_{\text{c}} \quad \sigma_{\text{p}} = {\alpha }_{\text{e}} \sigma_{\text{c}} + \sigma_{\text{po}} \\ \end{aligned} $$

Verification of decompression of concrete $\sigma_{\text{c}}$ ≤ 0

Verification of cracks formation $\sigma_{\text{c}}$ ≤ ${\bar{\sigma }}_{\text{ct}}^{\prime }$

• Cracked section

  • $\sigma_{\text{s}} = \frac{{{N}_{\text{ak}} - {N}_{\text{po}} }}{{{A}_{\text{t}} }} \le {\bar{\sigma }}_{\text{s}}$ (see also Table 2.16)

  • $\sigma_{\text{p}} = \frac{{{N}_{\text{ak}} - \alpha {N}_{\text{po}} }}{{{A}_{\text{t}} }} \le {\bar{\sigma }}_{\text{s}}$ (see also Table 2.16)

Resistance

$$ {N}_{\text{Rd}} = {f}_{\text{sd}} {A}_{\text{s}} + {f}_{\text{pd}} {A}_{\text{p}} \ge {N}_{\text{ad}} $$

Minimum reinforcement

$$ {A}_{\text{s}} \ge {A}_{\text{c}} {f}_{\text{ctm}} /{f}_{\text{yk}} $$

For technological data see Chart 2.15.

2.1.15 Chart 2.15: Cracking in RC and PC: Verification Scheme

The following scheme shows the verifications required in the different service conditions of the elements in reinforced and prestressed concrete. The symbols are defined here under:

σ′_s :

stress in passive reinforcement calculated in the cracked section;

σ _c :

stress in concrete in tension calculated in the uncracked section;

$\sigma_{\text{P}}^{*}$ = σ′_p − σ _po :

stress increment in the pretensioned reinforcement calculated in the cracked section with respect to the decompression of concrete;

Type of reinforce	Load combinations	Environment aggressiveness
		Low	Medium	High
Passive	Rare	–	–	–
	Frequent	${\sigma }_{\text{s}}^{\prime } \le {\bar{\sigma }}_{\text{s3}}^{\prime }$	${\sigma }_{\text{s}}^{\prime } \le {\bar{\sigma }}_{\text{s2}}^{\prime }$	${\sigma }_{\text{s}}^{\prime } \le {\bar{\sigma }}_{\text{s1}}^{\prime }$
	Quasi perman.	${\sigma }_{\text{s}}^{\prime } \le {\bar{\sigma }}_{\text{s2}}^{\prime }$	${\sigma }_{\text{s}}^{\prime } \le {\bar{\sigma }}_{\text{s1}}^{\prime }$	${\sigma }_{\text{s}}^{\prime } \le {\bar{\sigma }}_{\text{s}}^{\prime }$
Pretensioned	Rare	–	$\sigma_{\text{C}} < {\beta }{\text{f}}_{\text{ctk}}$	$\sigma_{\text{C}} < {\beta }{\text{f}}_{\text{ctk}}$
	Frequent	$\sigma_{\text{p}}^{*} \le \bar{\sigma }_{{{\text{s}}2}}^{\prime }$	$\sigma_{\text{p}}^{*} \le \bar{\sigma }_{\text{s1}}^{\prime }$	$\sigma_{\text{p}}^{*} \le \bar{\sigma }_{\text{s}}^{\prime }$
	Quasi perman.	$\sigma_{\text{p}}^{*} \le \bar{\sigma }_{\text{s1}}^{\prime }$	$\sigma_{\text{p}}^{*} \le \bar{\sigma }_{\text{s}}^{\prime }$	$\sigma_{\text{C}} \le 0$

For the allowable stresses ${\bar{\sigma }}_{\text{s1}}^{\prime }$, ${\bar{\sigma }}_{\text{s2}}^{\prime }$, ${\bar{\sigma }}_{\text{s3}}^{\prime }$, see Table 2.16. The passive reinforcement is made of ribbed bars; the pretensioned reinforcement is made of adherent smooth or indented wires or strands. For the classification of environments see Table 2.1.

${\bar{\sigma }}_{\text{s}}^{{^{\prime } }}$ = 0.5σ′sr:

with $\sigma_{\text{sr}}^{\prime }$ stress corresponding to cracking of the section (${\bar{\sigma }}_{\text{sr}}$ = ${(A}_{\text{c}} + {\alpha }_{\text{e}} {A}_{\text{s}} ){f}_{\text{ctk}} /{A}_{\text{s}} \,{\text{for}}\,{\text{ties)}};$

β ${f}_{\text{ctk}}^{\prime }$ :

characteristic tensile strength of concrete (with β = 1.0 for constant distribution and β = 1.3 for triangular distribution of stresses);

${\sigma }_{\text{D}}^{*} = \sigma^{\prime }_{\text{D}} - \sigma_{\text{DO}}$ :

increment of tension in pretensioned reinforcement calculated in the cracked section with respect to decompression in concrete.

2.1.16 Table 2.16: Cracking in RC and PC: Allowable Stresses

The following table shows, for different values of the diameter ϕ, the allowable stresses in passive and pretensioned reinforcement to be used in cracking verifications of Chart 2.15. Stresses ${\bar{\sigma}}_{\text{s1}}^{\prime}$, ${\bar{\sigma}}_{\text{s2}}^{\prime}$, ${\bar{\sigma}}_{\text{s3}}^{\prime}$ correspond respectively to crack widths ${\bar{w}}_{1}$ = 0.2 mm, ${\bar{w}}_{2}$ = 0.3 mm, ${\bar{w}}_{3}$ = 0.4 mm.

The values are expressed in MPa and refer to longitudinal reinforcement with ribbed bars distributed along the edges in tension of the section, with a centre-to-centre distance

${i} \le 5{\phi}$ :

for pure tension (ties)

${i} \le 8{\phi}$ :

for pure bending (beams)

and to alternate or long duration loads.

Pure tension (Ties)			ϕ (mm)	Pure bending (beams)
${\bar{\sigma}}_{\text{s1}}^{\prime}$	${\bar{\sigma}}_{\text{s2}}^{\prime}$	${\bar{\sigma}}_{\text{s3}}^{\prime}$		${\bar{\sigma}}_{\text{s1}}^{\prime}$	${\bar{\sigma}}_{\text{s2}}^{\prime}$	${\bar{\sigma}}_{\text{s3}}^{\prime}$
240	320	360	8	280	360	400
225	280	320	10	260	320	360
210	250	290	12	240	280	320
195	235	295	14	220	260	300
180	220	260	16	200	240	280
165	205	235	18	190	230	260
150	190	210	20	180	220	240
140	175	190	22	170	210	230
130	165	180	24	160	200	220

This table is deduced from the analogous table of the standard EN 1992-1-1:2004 with adequate adaptations.

2.1.17 Table 2.17: Durability: Minimum Cover of Reinforcement

The following tables give, for the different combination of environmental aggressiveness (see Table 2.1), the values of minimum reinforcement cover for the protection against corrosion. The values of the table are expressed in mm and refers to the actual concrete cover required for constructions of a nominal life of 50 years. The nominal values of cover to be shown in the drawings would have to be increased by the positioning tolerances of reinforcement assumed equal to ±10 mm for ordinary production, equal to ±5 mm for controlled production.

Concrete classes		Environment	Bars for plate elements		Bars for other elements
C min	C o	Aggressiv.	C ≥ C o	C min ≤ C < C o	C ≥ C o	C min ≤ C < C o
C25/30	C35/45	Low	15	20	20	25
C30/37	C40/50	Medium	25	30	30	35
C35/45	C45/55	High	35	40	40	45

Concrete classes		Environment	Strands for plate elements		Strands for other elements
C min	C o	Aggressiv.	C ≥ C o	C min ≤ C < C o	C ≥ C o	C min ≤ C < C o
C25/30	C35/45	Low	25	30	30	35
C30/37	C40/50	Medium	35	40	40	45
C35/45	C45/55	High	45	50	50	50

For constructions with a nominal life of 100 years, the values of the table should be increased by 10 mm. For strength classes lower than C _min, such values are to be increased by 5 mm. For elements of controlled production they can be reduced by 5 mm.