Effect of Con fi ning Reinforcement

Chart 1.22: Concrete r–e Models

2.1 Compression Elements

2.1.2 Effect of Con fi ning Reinforcement

reinforcement contribution is predominant. In common situations, the presence of reinforcement can increase the load capacity of columns approximately by 20% or 30%, this being on average the mechanical reinforcement percentage.

It is eventually to be noted that several design codes impose to take into account a minimum eccentricity of the axial force, for example withe 0.05 h, wherehis the depth of the section. The verification therefore refers to combined action of axial force and bending moment (see Chap.6). Moreover, for moderate reinforcement ratios (approximatelyxs 0.8), such requirement remains implicitly fulfilled if, in the formula of verification of centred axial force, the concrete contribution is penalized attributing with 0.8f_cd. In such case, fixing the value of mechanical reinforcement ratio, the formula deduced here becomes

NRd¼f_cdAcð0:8þxsÞ

In the elastic range, the effect of confinement on the stress distribution is very low, as shown hereafter.

Having definedA_n= pD²/4 as the area of the core,A_lthe area of the longitudinal reinforcement and A_w= a_wpD/s the equivalent one of the spiral bar (of cross sectiona_w), the following relations are obtained. The isolated segment of the core of height s, subject to a vertical flux of stresses rv (see Fig.2.8a) exhibits a shortening

d^vo ¼rv

E_cs and, at the same time, an horizontal expansion

d^ho ¼ mrv

E_cD

The hoops oppose to such expansion with a horizontal stressrh(see Fig.2.8b, c) which can be considered as the unknown of the problem. For the equilibrium of the semicircular piece of bar of Fig.2.8c also the stress rw in the spiral can be expressed in terms ofrh:

2rwa_w¼rhDs whence

rw¼ Ds

2awrh¼ An

pD²=4 Ds

2awrh¼2An

Awrh Fig. 2.7 Details of a confined column

The relative horizontal expansion between spiral and core due to the unknownrh

is therefore obtained adding up the two deformation contributions of steel and concrete:

dhhrh¼D rw

Es

þ1v Ec rh

¼D 2A_n EsAw

þ1v Ec

rh

Eventually the compatibility of deformations is set between spiral and core:

dhhrhþdho¼0 from which one obtains

rh¼ dho

dhh

¼ m E_c 2An

EsAwþ1m Ec

rv

that, withqw= A_w/A_n,ae=E_s/E_cand ww=aeqw, becomes:

rh¼ 1 2 ww

þ 1 1m þ1

m 1mrv

Without the spiral (ww= 0) one hasrh0: it is the case of ordinary columns with stirrups. The maximum confining contribution is obtained instead at the limit situation of a spiral of so high size that it can be considered rigid with respect to concrete (ww=∞).

Fig. 2.8 Equilibrium conditions of concrete core and confining steel

In this case one obtains rh¼ m

1mrv

ðrhffi0:25r^v for mffi0:20Þ The vertical contraction of the confined column is therefore:

ev¼ 1

Ecðrv2mrhÞ ¼rv

Ec

1 2m² 1m

lesser than the one of the ordinary column, as if concrete had an effective elastic modulus

E⁰_c¼ E_c

1₁²_m^m2 ¼ 1m 1þm

ð Þð12mÞEc

With this effective elastic modulus, increased by about 10% with respect to the ordinary one as it can be deduced settingm ≅0.20, theelastic designcan be carried evaluating the stresses on the plane section of the column for a given axial force:

N¼AnrvþA1r1¼rv Anþa⁰eA1

¼rvA⁰_i

being, witha′e=E_s/E′c≅0.9aeand with,w⁰1¼a⁰eq1; A⁰_i¼An 1þw⁰1

the equivalent area. For longitudinal bars of about 1% with respect to the cross section of the core, with 6 ae 10, values increased by about 1% are obtained for the stressrvin concrete, values decreased by about 9% are obtained for stressr1

in the longitudinal reinforcement.

Considering that the actual elastic deformability of the spiral further reduces this effect, which remains still limited to the concrete core excluding the external cover layer of thicknessc, it can be seen how, in the elastic design, it can be neglected.

Theultimate resistanceis instead significantly increased by the confinement as indicated in the following formulation which is based on the experimental results.

First of all, the tests on confined columns exhibit early spalling outside the con- fining hoops. This occurs at level of the stresses close to the uniaxial strengthf_cof concrete.

As the loadNfurther increases, more significant transversal expansions of the core are observed, greatly increasing close to the ultimate limit, inducing tensions in the spiral reinforcement. If abnormal quantities of this reinforcement are excluded, the column failure occurs after the spiral yields. Failure itself, by crushing of the

concrete core, is characterized by high values of the contractionevto which also corresponds the yielding of the longitudinal reinforcement. The stressesrvrof the concrete core measured at ultimate limit state are much higher than the uniaxial strength f_c. The increase in strength appears to depend linearly on the confining stressrhrgiven by the spiral:

r^vr¼f_cþj r^hr

Actually the different tests lead to significantly discordant values ofj: an esti- mate precise and reliable enough for such coefficient is still not available, with the consequence of the need to penalize the resisting effect of spirals with greater factors of safety.

Integrating therefore the assumptions with what results from thefindings men- tioned above, the equilibrium of the section at the ultimate limit state is:

N_r ¼A_nðf_cdþjrhrÞ þA₁r1r

where thefirst term represents the contribution of the concrete core, the second one represents the contribution of the longitudinal reinforcement.

Since

rh¼1 2

Aw

Anrw

setting rwr= rlr=fy and introducing the design values of the semi-probabilistic method, one eventually obtains

NRd ¼f_cd Anþf_yd f_cdA1þj

2 f_yd f_cdAw

¼f_cdA⁰_ir

where the homogenization coefficients of the two types of reinforcement (longi- tudinal and transverse) are distinguished by the factorj=2. Assuming for example j¼4 (see formula r1¼1þr2 of Sect. 1.1.3 for triaxial stress states with

r1 r2¼r3), one obtains

N_Rd¼f_cdA_nð1þx1þ2xwÞ

where it can be noted that, in terms of resistance contribution, the mechanical ratio xw=f_sdA_w/f_cdA_nof the confining reinforcement is weighed twice as much as the one of the longitudinal reinforcement. However, there is a limitA_w 2A_lfor the confining reinforcement with respect to the longitudinal one, beyond which a failure by transverse shearing of the column occurs at lower load levels than the one deducible from the equation set above. The usual limitation to the longitudinal reinforcement ratio is to be eventually added, related to bond problems. Such limitation for confined columns can be set asA′ir 2A_n.

The comparison with the capacity of ordinary columns with stirrups can be deduced equating the contribution of reinforcement in the two cases:

x1¼x1þ2xw

which leads, for the same materials, to the relationship A_s = (5/3)A_t, having set A_w= 2A_l, having indicated with A_t=A_l+A_w the total reinforcement of the con- fined column and withA_sthe longitudinal reinforcement of the ordinary columns.

Taking into account the additional presence of the stirrups, one can deduce that the circular arrangement allows to roughly halve the amount of reinforcement for the same capacity and the same size of concrete. This does not contemplate possible problems of shape, which is limited to the circle or the equilateral polygon for confined sections, nor economical problems which normally lead to prefer, where permitted, an increase in the area of concrete instead of the confining reinforcement.

In order to take into account a minimum load eccentricity, introducing for the confined columns the same reduction in concrete resistance as for the ordinary columns, one eventually obtains the (conservative) formula of the design resistance

NRd¼f_cdAnð0:8þx¹þ1:6x^wÞ

which reevaluates the contribution of longitudinal reinforcement with respect to concrete and its confining reinforcement.