1.4 Behaviour of Reinforced Concrete Sections
1.4.2 Basic Assumptions for Resistance Calculation
steel type Fe 1860, Fe 1960 and Fe 2060 with nominal diameters between 5.2 and 7.5 mm
• Strands‘7W’(7-wires)
steel type Fe 1770, Fe 1860 and Fe 2060
with nominal diameters between 7.0 and 18.0 mm
• Strands‘7WC’(7-wires compacted) steel type Fe 1770, Fe 1820 and Fe 1860
with nominal diameters between 12.7 and 18.0 mm.
The data relative to the type of products for R.C. and P.C. mentioned above are reported in Tables1.7,1.8,1.9,1.10,1.11,1.12,1.13,1.14,1.15,1.16,1.17,1.18, 1.19,1.20,1.21.
It is to be noted that the quality of steel of the current industrial production is with good approximation constant and reliable, with a relative dispersion of rep- resentative values much smaller than what can be expected for concrete.
whereeois the strain at the beam axis taken as reference (‘design axis’) andvis a constant that represents the curvature of the beam at the considered section.
The second assumption concerns theperfect bondbetween concrete and rein- forcement steel, basically verified as long as appropriate design rules are followed in reinforcement detailing. For the longitudinal rebars crossing the section of Fig.1.27, this assumption leads to the equality
es¼ec;
between expansions (or contractions) of the two materials at their contact points.
The third assumption refers to both serviceability and strength calculations and leads to neglect completely the small tensile strength fct of concrete. This is equivalent to assume in ther–ediagram of this material:
Ect 0:
A consequence is the so-called partialization of the cross section (see again Fig.1.27), which is assumed to be cracked in the tension part of concrete. In opposition to the acting force, the effective section remains to resist, reduced in general with respect to the entiregeometrical section, and consisting of the entire area of compression and tension rebars, and the only compression part of concrete.
In the calculations of deformation and cracking, this assumption will have to be opportunely integrated.
The fourth assumption eventually refers to the behaviour of materials and it is expressed by the appropriate models that represent the constitutive relationships r–e. Two areas of application have to be distinguished: elastic analysis under moderate loads and nonlinear analysis as for ultimate limit states.
In the elastic analysis of cross sections, the elastic relationships represented by Hooke’s laware assumed:
rc¼Ecec rs¼Eces;
thefirst one, relative to concrete, is only valid in compression. For the same strain ec= es=eone has
rs¼Ese¼ ðEs=EcÞrc¼aerc;
which means that, in the elastic range, steel is stressed ae times more than the concrete around it, whereaeis equal to the ratio between the elastic moduli of the two materials.
In the nonlinear analysis of cross sections, appropriate analytical models suitable for numerical applications have to be defined to represent the real relationshipsr–e of the materials.
r–eModels for Concrete
The Saenz’s model has already been discussed for concrete, which reproduces quite well the behaviour in compression under loads of small duration (see Fig.1.4). For strength verifications such model could be used referring it to the design strength reduced by the partial safety factorcC = 1.5 of concrete
fcd¼fck=cC:
This strength value is to be further reduced to take into account the fraction of long-term loads.
Given that, in strength design of cross sections, magnitude and position of the resultant of compressions in concrete are to be calculated, it is possible to simplify the model with the assumption of simplified diagrams. It is sufficient to reproduce with good approximation the area and the centre of the surface covered by the diagram, without caring about the exact local slope of the curve. The three diagrams of Fig.1.28have been defined with these criteria: they represent the most widely used models.
Thefirst model is theparabola–rectangleshown in Fig.1.28a. The second one is thetriangle–rectangleshown in Fig.1.28b. The most simple is represented by thestress blockshown in Fig. 1.28c. The valuesec2= 0.20%,ac3= 0.15%,ec4= 0.07% and ecu= 0.35% are conventionally assumed for the three models, as a mean of the ones of the different strength classes up to C50/60. For the application of the semi-probabilistic limit state method, the long-term design strength is assumed equal to
fcd¼acc
fck
cC¼acc
0:83Rck
cC ;
where, starting from the characteristic value of the cubic strength experimentally determined, the value of the characteristic prismatic strength is obtained with the already mentioned correlation formula and from this value to the design value is obtained with the pertinent coefficientcC.
An appropriate cut of the short-term strengths is eventually applied, based on the duration features of the loading combination examined. In the case of permanent loads only, it can be assumedacc= 0.80 (see Fig.1.2). If the combination includes short-term loads, one can assume acc= 1.00. Strictly speaking, the verifications
Fig. 1.28 Simplifiedr–emodels for concrete in compression
under the two load combinations mentioned above should be repeated. Some regulations allow to take the average value acc= 0.85 for a unique verification under a global loading combinations.
The extension of the models presented above the higher strength classes requires the adoption of modified values for the parametersec2,ec3,ec4andecufor which one can refer to Chart1.22.
r–eModels for Steel
For steel, thebilinear modelof Fig. 1.29reproduces with good accuracy the beha- viour of the material, straightening the plastic-hardening part after the yield point.
The analytical expression of the model is set with r¼Eoe foreey
r¼fyþE1 eey
fore[ey
with
Eo ¼fy
ey
E1¼ ftfy
euey
:
For strength calculations of cross sections thisfinite bilinear model with hard- eningis used setting in the previous expressions (see Fig.1.30a—model A):
fy¼fyd ¼fyk
cS withcS¼1:15 ft¼ftd¼kfyd withk¼1:2 ey¼eyd¼fyd
Es eu¼euk
and cutting it off at the limit
eud¼0:9euk; where one has
ftd0 ¼fyþE1 eudey
:
ε εt
εu
εy
ft
fy
σ Fig. 1.29 Bilinear model for reinforcing steel
With a conservative approximation that leads to a simplification of the calcu- lations, theindefinite elastic–perfectly plasticmodel can be adopted, settingE1= 0 and removing any limit to strainse(see Fig.1.30a—model B).
The alternative models of Fig.1.30b are proposed by certain authors who see the elastic modulusEsas a resisting characteristic of the material to be reduced with the pertinent coefficientcS(Esd= Es/cS). This model therefore sets the discontinuity of the bilinear curve in
ey¼eyk¼fyk
Es
:
For pre-stressing steel a bilinear model can still be assumed as shown in Fig.1.31, where the yield strength fpy (or fp0.2) of the bars is substituted by the stressesfp0.1orfp1, respectively, for wires and strands.
For strength calculations of cross sections, in the bilinear model relationships it is therefore set (v. Fig.1.32a—model A):
kfyd
ftk
fyd
fyk
fyd
fyk
εyd εyk εud εuk
kfyd
ftk
f′td
B A
εyd=εyk εud εuk
f′td
B A σ
σ
ε ε
(a) (b)
Fig. 1.30 Reinforcing steelr–emodels for strength calculations
fpy
εpu ε εpy
fpt
σ Fig. 1.31 Bilinear model for
pre-stressing steel
ft¼fptd¼fptk=cS concS¼1:15 fy¼fpyd¼jfptk conj¼ fpy=fpt
ey¼epyd¼fpyd=Ep k
eu¼epuk
and the model itself is cut off at the limit epud¼0:9epuk; where
fptd0 ¼fpydþE1 epudepyd
:
If more accurate values are not available, it can be conservatively assumed j= 0.9 andepdu= 0.02.
With a conservative approximation the model can be simplified settingE1= 0 and removing any limit to strains e (see Fig.1.32a—model B). Even for pre-stressing reinforcement the alternative model exists that penalizes the elastic modulus with Epd= Ep/cS (see Fig.1.32) and that sets the discontinuity of the bilinear curve in
eyk ¼epyk¼fpyk
Ep : fpyk
fpyd
εpud ε εpyd
σ
εpyk εpuk
B A
fptk
fptd
f′ptd fpyk
fpyd
εpud ε εpyd=εpyk σ
εpuk B A
fptk
fptd
f′ptd
(a) (b)
Fig. 1.32 Pre-stressing steelr–emodels for strength calculations