This document summarizes a methodology for calculating steel reinforcement in reinforced concrete members subjected to both mechanical loads and thermal gradients. The methodology accounts for tension stiffening effects between cracks. Results from case studies show that using this direct computational method provides similar reinforcement ratios as the standard method of applying a factored thermal load with a factor of 0.22 to 0.3, lower than the commonly used factor of 0.5. Accounting for tension stiffening more realistically reduces required reinforcement ratios.

1. 3
rd
Conference on Technological Innovations in Nuclear Civil Engineering
Full paper Submission, TINCE-2016
Paris (France), September 5th
– 9th
, 2016
Steel reinforcement calculations in RC members with account of tension stiffening
Jacques Chataigner1
, Loïc Cloitre2
,
1
Civil Engineering Expert, Tractebel Engineering, Lyon, France
2
Senior Engineer, Tractebel Engineering, Lyon, France
Introduction
In the context of nuclear structures design, determination of reinforcement ratios in rein-
forced concrete members submitted to mechanical loads together with high through the wall
thermal gradients, either stationary or transient, is a recurring issue. Solution of this problem
needs to take into account, with sufficiently realistic methodology, the effect on thermo elastic
forces and moments of the reduction of RC section inertia due to concrete cracking. Though this
issue has been analyzed since long, typical methods proposed at steel reinforcement design
stage, to account for section cracking, were mainly limited to the use of reduction factors applied
to thermo elastic forces and moments, which was not ascertained to lead to optimized nor con-
servative results.
Improved calculation methodologies, which simultaneously account for concrete section
cracking together with tension stiffening effect associated with uncracked zones of the concrete
members have been tested by TRACTEBEL’s engineers prior to be implemented in specific soft-
ware dedicated to automatic steel reinforcement calculations.
Recalling existing methodologies
Optimizing automatic reinforcement calculations for sections simultaneously sustaining
mechanical loads and through the wall thermal linear or nonlinear gradients requires computa-
tional software in which the following input data shall be introduced:
- moments and forces from external loads (combined flexion) the studied concrete sections are
submitted to,
- thermal fields (linear or nonlinear gradients) that develop through these sections,
- realistic assumptions to modelize the stiffness brought by concrete in tension between the
cracks ("tension stiffening") in zone of the sections which are in tension (i.e. around tension re-
bars (zones usually named as tension ties ),

2. 3rd
Conference on Technological Innovations in Nuclear Civil Engineering
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to 9th
September
- geometry of the RC sections in which steel reinforcement has to be determined (dimensions)
and mechanical characteristics (and thermal) of concrete or rebars to be installed in section.
The most sensitive issues to be addressed in these calculations methodology may be listed
as:
• realistic modelization of the effect of tensile stresses in concrete tensile ties existing be-
tween cracked sections on effective inertia of the member sections (with correlated conse-
quence on actual thermal moments the section will finally undergo),
• combination of nonlinear strain-stress behavior of section concrete and steel rebars with
thermally induced stresses.
Methodologies to solve both these issues have been rather widely documented, though in most
cases mainly for stress checks rather than for actual steel sections computation purposes.
Implementation of tension stiffening in RC calculations
General
As a general commentary, it may be reminded that a significant difficulty arising when performing
direct steel reinforcement calculations in RC members submitted to mechanical loads together
with concomitant thermal gradients lies in the following observation:
• section stiffness increase depends of amount (ratio) of rebars met in tensile tie,
• in its turn stiffness of the tie brings more thermal stresses and therefore more equivalent
thermal moment
• the latter (thermal moment) combined to mechanical effects (moment and forces) governs
the steel reinforcement design (i.e. ratio of rebars in the tie)
Above listed interactions between needed steel reinforcement and section stiffness increase lead
necessarily to an iterative process that makes more complex, though still manageable, the steel
reinforcement calculations ; description more in detail of this process will be the object of follow-
ing paragraphs
Reinforcement
ratio R in tensile
tie
Section stiffness increase due to tensile tie
Thermal moment
Mth increases
Mth + {N,M}mechan. Reinf. ratio R

3. 3rd
Conference on Technological Innovations in Nuclear Civil Engineering
TINCE 2016, Paris 5th
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September
Taking into account tension stiffening and temperature gradients in RC calculations
The methodology that will be followed to carry out steel reinforcement calculations with simulta-
neous account of :
- temperature gradients through the section
- tensile ties representing the effect of uncracked concrete in tension between flexural cracks is
based on the following general assumptions:
→ Geometrical features:
Section in supposed to be fitted with one bottom reinforcement layer (section) with concrete cover
ci (see fig 1-1 below) ; for specific case when RC section could be in a full tensile state, an upper
reinforcement layer (section) with concrete cover cs is added
→ Deformations ε of section
Deformations, whether they are linked to thermal gradient or induced by mechanical loads
{M,N} acting on section are assumed to be linear through the section (see fig 1-2 below)
→ Stresses σ through RC section
Thermal stresses σc,th in concrete
Thermal stresses in concrete depend linearly from temperature gradient existing through
the section; they are obtained by means of following typical formula :
(1) σc,th= Ec,th αc T(x)

4. 3rd
Conference on Technological Innovations in Nuclear Civil Engineering
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to 9th
September
with
T(x) thermal gradient through the section translated from initial temperature field so that
T(x0) = 0,0 with x0 :distance of neutral axis from upper edge of the section (see fig 1-4)
Ec,th thermal Young modulus of concrete
αc thermal expansion coefficient of concrete
Thermal stresses σs,th in reinforcement steel
Thermal stresses in reinforcement steel also depend linearly from temperature gradient T(x)
and are obtained by means of following typical formula:
(2) σs,th= Esm αs T(h-c)
with
Esm Elastic Young modulus of steel with account of tension stiffening which develops
in a tensile tie around bottom reinforcement (or bottom and top reinforcement if section is in a full
tensile state) ; computation of Esm will be described in paragraph below,
αs thermal expansion coefficient of reinforcement steel
Mechanical stresses σc,m in concrete
Stresses in concrete in relation with mechanical loads {M,N} acting on section concomitant-
ly with thermal gradient are assumed to develop in agreement with a σc=f(ε) law complying with
[EC2], §3.1.7 ; simplified bi linear function σc=f(ε) is selected. It shall be recalled that overall com-
pressive stresses in concrete, σc,m + σc,th remain limited to fcd (see [EC2], §3.1.6 and fig 1-3).
Mechanical stresses σs,m in steel
Stresses in reinforcement steel induced by mechanical loads {M,N} are assumed to develop
in accordance with a σs=f(ε) law complying with simplified bi linear function as proposed [EC2],
§3.2.7 ; as for concrete stresses, at each step of the calculations it checked that overall tensile
stresses in reinforcement steel, σs,m + σs,th remain limited to fyd as specified in [EC2], §3.2.7,(see
also fig 1-5).
Tension stiffening.
As mentioned before, the methodology to determine steel reinforcement in section submit-
ted to thermal gradient that is proposed in this paper shall include a sufficiently realistic modeliza-
tion of the effect of tensile ties developing around tensile rebars between cracks in the tensile
zone of RC section. Numerous methods have been proposed in available documentation ; for
simplicity’s sake a simplified approach has been implemented in the software developed by
TRACTEBEL’s engineers ; it derives from a formulation in agreement with CEB Model Code :

5. 3rd
Conference on Technological Innovations in Nuclear Civil Engineering
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September
(3) Esm= Ea/(1-k(σscr/fs)2
)
Where
Ea elastic Young modulus of reinforcement steel
σscr stress in steel at crack occurrence in tensile tie
fs actual stress in steel
k coefficient to represent first or repeated loading of the section (1,0 or 0,5)
It is reminded that σscr in formula (3) depends of design tensile strength of concrete fctd to-
gether with assumed reinforcement section As ,which remains also an unknown in this computa-
tion methodology; this last comment to remind the rather complex iterative process that has to be
implemented (see § - General above).
Additionally, alternative formulation to reach an equivalent Young’s modulus that could rep-
resent tension stiffening may be obtained from [fibMC], §7.6.4.4, which would then provide
(4) E’sm= Ea/(1-β(σscr/fs))
with other limit values for β coefficient (0,4 to 0,6)
Examples of results and benefits gained
Method to compute directly steel reinforcement in RC sector submitted to thermal gradients
that has been described in previous § has been implemented in a dedicated program for auto-
matic steel reinforcement calculations.
We will than present, through a limited number of case studies, some interesting aspects of
the results that were obtained using this software.
Geometrical data
H =1,20 m
ci=cs =0,10m
Mechanical data
fck =45
Ec =35000 MPa
Ec,th =25000 MPa
γc =1.2
fy =500 MPa
Ea =200000 MPa
γs =1.15
Tension stiffening data
k = 0,50 (sustained load)

6. 3rd
Conference on Technological Innovations in Nuclear Civil Engineering
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Mth,elast= Ec,th*αc*(Ti-Te)*h2
)/12.
Thermal conditions the section is submitted are:
Thermal data
Thermal gradient δT= Ti-Te = 70°C, linear
αc= 1 10-5
A first step of the analysis consisted in a set of calculations in the frame of which a rather
wide range of applied {M,N} bending moment M, normal force N, was investigated with a single
thermal gradient, i.e. a 70°C linear gradient through the section. We focused the examination of
the results on the value of ratio:
βo = Mth/ Mth,elast
or ratio of the thermal moment given by the sum of elementary moments in the section, re-
sulting from the thermal stress in both concrete and rebars sections, to the thermal elastic mo-
ment that would arise in the section, assuming that it behaves elastically , i.e.
Mth,elast = Ec,th x αc x ∆T x
fig 2. presents the variation βo depending of the applied {M,N}; it shows that βo ,when δT=
Ti-Te is kept constant, varies inside a range of values extending approximately from 0.12 to 0.25,
i.e that βo has a mean value of 0.18.
This result indicates that, thought tension stiffening effect is taken into account, section
cracking leads to a significant reduction of thermal moment acting of the section.
Fig 2-β0 = Mth/Mth,elast. as a function of applied moment and normal force {M,N} - ∆T=70°C (linear)
In a second step of parametric study, we carried out calculations with same set of {M,N};
but for a set of two thermal gradients :
∆T = 70°C and ∆T1 = 0.5 ∆T0 = 35 °C
0,00
0,05
0,10
0,15
0,20
0,25
0,30
-2,0 -1,0 0,0 1,0 2,0
N (MN)
β0=Mth/Mth,elast.
M=2,0 MN m
M=1,5 MN m
M=1,0 MN m
M=0,75 MN m
M=0,5 MN m
M=0,25 MN m
M=0,35 MN m

7. 3rd
Conference on Technological Innovations in Nuclear Civil Engineering
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September
Figure 3 below summarizes the results that were obtained.
fig 3-β0 = Mth/Mth,elast. as a function of - ∆T0 =70°C (linear)or -∆T1= 0,5 ∆T0 =35°C
It comes from fig.3 that value of the thermal gradient ∆T has in itself little influence on coef-
ficient β ; as a matter of fact, reducing the thermal gradient by a factor 2 leads to mean diminution
coefficient β of 7%.
Other comparative analysis has been carried out, which is summarized in fig. 4. Steel rein-
forcement that were computed, for case studies, by means of direct computational methodology
(presented in this paper) were also computed using a typical EC2 methodology but with moment
and normal force {M, N} combined to a thermal moment computed as β'o Mth,elast , β'o being iter-
atively adjusted so as to obtain, same reinforcement ratio demand. Figure .4 shows that mean β'o
values shall then be higher by approximately 20% - 25% than equivalent β0 computed by direct
method.
This leads to conclude that proposed direct computational method provides steel rein-
forcement amounts which are similar to those that would be given by current methodology based
of the use of factored thermal load cases, with a factor F close to :
F = 0,18 x 1,225 = 0,22
from results from results
of Step 1 of Step 3
0,00
0,05
0,10
0,15
0,20
0,25
-1,5 -1,0 -0,5 0,0 0,5 1,0 1,5
N (MN)
β0=Mth/Mth,elast.
dt=35°C,M=2,0 MN m
dt=35°C,M=1,5 MN m
dt=70°C,M=2,0 MN m
dt=70°C,M=1,5 MN m

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Conference on Technological Innovations in Nuclear Civil Engineering
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September
fig 4- β0. as obtained with proposed direct computation method or with EC2 type calculation with ap-
plication of an equivalent thermal moment β’0 Mth,elast ( both calculations giving same reinforcement ratio) -
∆T0 =70°C (linear)
Additional step of results analysis has also been carried out to evaluate the influence, on β
= Mth/ Mth,elast , of the k coefficient introduced in formula (3) to modelize the nature of loads applied
to the section, i.e. first or repeated loading; comparative analysis of β values resulting from calcu-
lations assuming k=0,5 or 1,0 as per CEB Model Code is summarized in fig 5 below
0,00
0,05
0,10
0,15
0,20
0,25
-1,50 -1,00 -0,50 0,00 0,50 1,00 1,50
βMthe
k=0,5-M=1,25 MN m (direct computation)
k=0,5- M=1,25 MN m (EC2)
k=1,0-M=1,25 MN m (direct computation)
k=1,0- M=1,25 MN m (EC2)

9. 3rd
Conference on Technological Innovations in Nuclear Civil Engineering
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September
fig 5- β0. depending of the k coefficient value in formula (3)
It appears from a quick examination of fig 5 that β increases by a mean factor of 1,60 when
k takes a 1,0 value instead of 0,5 ; considering this remark, leads us to conclude that computing
steel reinforcement ratios with the proposed method, directly including thermal gradients, will pro-
vide, on average, results similar to those given by typical method making use of thermal loads
factored with a factor F = 1,3 x 0,22 y 0,3
Conclusions
Method to determine reinforcement calculations of RC sections sustaining concomitantly
normal forces and moments together with thermal gradients and taking into account tension stiff-
ening in cracked zone of the section has been implemented in TRACTEBEL’s steel reinforcement
softwares. Computations were based on assumed typical nonlinear stress-strain laws for both
concrete and reinforcement steel that were in agreement with [EC2] requirements. Results from a
rather wide set of case studies demonstrated that this computational methodology could signifi-
cantly reduce rebar sections demand.
As a matter of fact reinforcement ratios that were obtained with this methodology corre-
spond to those that result from standard computation method with thermal moments factored by
0,30, when common practice usually recommend or require an F factor equal at least to 0,50.
0,00
0,10
0,20
0,30
0,40
0,50
0,60
0,70
0,80
-1,5 -1,0 -0,5 0,0 0,5 1,0 1,5
N (MN)
β0=Mth/Mth,elast.
k=0,5, M=1,5 MN m
k=0,5, M=1,0 MN m
k=0,5, M=0,5 MN m
k=1,0, M=1,5 MN m
k=1,0, M=1,0 MN m
k=1,0, M=0,5 MN m

10. 3rd
Conference on Technological Innovations in Nuclear Civil Engineering
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to 9th
September
We might conclude that introducing this steel reinforcement calculation method in nuclear
structures design would be beneficial when thermal loads, in particular accident ones, are those
governing the concrete steel reinforcement design and subsequently lead to large reinforcement
ratios.
Despite the iterative process that has to introduced in calculations , as tension stiffening ef-
fect depends itself of the unknown reinforcement ratio, implementation of the method proved to
be feasible without being excessively time consuming at computation stage.
References
[KOGU75] -Optimal design of reinforced concrete for nuclear containments, including thermal
effects – T.D KOHLI, O.GURBUZ - 2nd
ASCE Specialty Conf. on Structural Design of Nuclear
Plant Facilities, Vol. 1-B, New Orleans, Dec. 1975.
[GURB06] -Thermal effects in concrete walls & slabs –O. Gurbuz- ACI 349 Committee Meeting-
(2006)
[BAE13] -Alternative design approach for thermal effects - S.BAE –SMIRT 22 - 2013
[fibMC] -fib Model Code for Concrete Structures – 2010
[GILB88] -Gilbert RI. - Time Effects in Concrete Structures, Elsevier, 1988.
[fib10] -fib Bulletin No. 10 - Bond of reinforcement in concrete (2000)
[EC2] -Eurocode 2: Design of concrete structures - Part 1-1: General rules and rules for buildings
[RCCW15] -RCC-CW– Appendix DL - Tensile prestressed and reinforced concrete tie stiffness
for thermal load advanced calculation (2015 )

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Preference: Poster Oral
Topic: 1 - Advanced Materials 2 - Design and Hazard Assessment
3 - Civil Works Construction 4 - Long Term Operation & Maintenance
5 - Dismantling of civil works & Civil Works in Hostile Environment
6 – Geotechnical Design & Construction & Fluid Structure Interaction
Corresponding author: jacques.chataigner@gdfsuez.com