CONCRETE LIBRARY OF JSCE NO. 25, JUNE 1995
THE FINITE ELEMENT ANALYSIS OF HEAT CONDUCTION FOR CONCRETE
STRUCTURES WITH UNCERTAIN MATERIAL PROPERTIES
(Translation from Proc. of JSCE, No.496/V-24, August 1994)
*“R
:= '>,.f€<4:'r-»>.-:/ ~'
'555R>‘353:(’Zi§Z7l°5;l
¢>;.»>‘;<'
.~:<».=;=;-:~
-='
<4,4-:A><<~-4<,>;.-/
.-;-;,.,¢;_~ _;;-.
F ¢>'>=:'.>~¥"/.-‘4:
». .I.’:'
11-¢-,--.- .-.
.- -'1'
~>.=- =4+'»=; -' ,':=¢-,‘¢-=='.*-:=¢5-.<§:=s§.‘~@‘-:r:'11
: i.-:-:
.-. I-.-.-:-;r:;";::;;:;=§;' -‘Bi;-:1;;:;-;;;;::=:=;=:-.-1-.§,.,.-.
<":===‘-~-:-<.
=-=$’:-'- =i-@=:€<¢§=4>
:1Eszatfirzr:=§:z=:=:=:2=-a=:=:
L-!'?A¢-2-I-21 .-0 .-, -._ .ll-._<;.,.;.
-IE1315I.=:El-'-;.If-§:=;%@g1-=+"‘~ ' :-:¢.¢£¢;= .5:i:§:§:§:§:;:§2:§:;:=:§:§:;:5§:;;1;:;:,._.,,.
.
-.:-;-,>- I:-,.-;-.-.~<.,.-;,, ,.-_.;.;.;.<.;.;.;.;.m-_.;.,;.;.;.;.;.;.;.;.;.,,.i
-1:-:-1-:-:-‘i:-“<5--' .-‘-;v:-.».~.¢
Y.
;.>g.;.,.;.;.;.;.;.;i;.; ;.;.;.;4_-;-;¢-:-:\;-:-;<'—_“5'_ ‘ I" ‘IIZ‘iIII" .1. . _.::_.;:_1';
_:":i:::"::'
':\;,;I-‘:!;I;Z;I-'1;g.;.;.:
1 :;:;1;;:;:j;17,g,;@%4;i§§;%;§-1 -:$;:;:;i;:;:;:;:;:;;g;_-¢g:;:;:;z:;:;:;:;:;'-'-' ‘
3 3:5:3J:5I3.5:576:Z-"'§};.;,;I;;-I ';:;:;:;:;‘;:;:;1;.-£;Z;£;!;:-25!;I;2;I;
EIE151:IE1515I51:11Q-E§$‘
__
1/ ff
¥:§'?.-:?i:§°§§1'1'_I5.,<;:¢:;:;:1:;:;;;$5:;5:§;:z:;:5:5:
-.= ¢==¢>:: =
=’éili==E<‘?;=1
5:1; 5.1»5.1;‘5.3;‘;:§;‘.;.;:j;;;;?§$;;g3 ‘===L5;:5:1;E=E:i=E;E;5;E;5;€32555132555555‘
,9-.-:-.-.;(.~ * —.;.. . ._. .
:14.-;l:=.1;'.1:1:‘:=;€‘:-:1:‘:I:1: ;I:I:~I:I-~:j».i:~:2:;: - ~‘§$:_.-"‘§1:~i11=E1E1==:1E1‘-11.1.--I>.l:I
<v:1 -:¢:1 -:?;i.~:3£5:3;'.'i‘:3}:3I'.5i‘.5I5f‘I"5“I" - ‘-r-';f-=3:--e,--2;=:!=:¢;1:=:=;1.=:=;I.1:I;I:!;1:=:!:1:=:=.
-'-I-2-:1-1:-:1-I:‘:=:=:‘:=-I:1:=;=.1:=;=.15;;‘:=;1-‘ ,‘.-.3‘.=----:-:-.-;-:;1-.»;-;;Z;;;;;»:;:;;L;;:;:;:;:;:;:;:;;-q:;.,
€lE3‘5I=E?'5£=§15;15155353552‘E‘E5€‘E‘E5?‘E"’"
"?l?%EE1?%iE'-2%:E'-
i=i;=rE;5,;;';5;;§3is :;.j5ji;I¢Ej§’.,'
"2<jE5.';E5E5':5E§5‘==‘
' 5§==E
1'£§I;5f£512552555:=5;5; =,..
..-===:=-==BE;=;E3:g:;53E1:5:§;:z;:¢5é;;5='_‘.;.j;‘:;;§:‘.§;'
_,_.;g.'->;j:j:;;_-_=_ =:3,'¢;:<r<;<s -,=;=:=;‘-====:1=15;>=;:; <; .; r:;:; 5:
.:=;z€=;i=E=2=E£=ié=E=;=:y:=¢:€=:I====;=;====1;§;;>1:-:1:¢:=;=':I:%=i;§‘-é=F=¥1:=:=:=-=:= -Ei=E=E-‘&1E=E=E=E=E=E=;r:=:r;1:1:===:=:=
I
M
12
6
59‘
» Y5;5
1
-1~!;1\-I~;.1:;._‘_'-1. -M;
5:1§: §: §:?§ :Q;3§'l: $'-.j§ =I-E$;= E1=53$EI= E = €E=§§i=¥$E§==§ 1E=i1§%:=¢=§1:=E:=5§=-E:I;=1P{I§~.§Z;=I1:-Z;I:Z=;5I i§'I=;-12:$=s : = :1=$2= a =i5'é=E5i-=IE5=k 3i=E i£:§=!E"'-~‘I1-'E~=3;%:§- :§ :§ :§ :§;E :§:§:§I-:I2--Ztj5:§1;!I:§-;I}C:"jc;I5§-£':1§$=:§ :===31=
!1;.?:< =-: r:=:fr =: := :1:":-§.a;1.".<_~;.i‘.€i_‘l\I‘IQ’
§{:§ :i§I:i=:i :lY1$E:§=-5:':=-5s::mnic--=:>:R-4s:-§=:.>- :“‘-~:'15-t:i<:I§lg1:: ¥§Ii:;=i:I-2-: >:q-' 1:I2$I:=7:§5E:5" =:" 5: i:§;$§ 3:i§. :i I §: §: 15 :-§ I -:§'=; -:$zrzi
1?_:£‘Er =E~<>.-:\=1;¢E:'j;€I; .2;5:I %§:3.9l5‘E;-Ig.=:?2:5"‘1:=5.’=3‘,:;-si;.-z~,1.-':=:=".-ri-=*1-=-':‘?.='15E= ?§E=_§.-=:T.~:'-.~? '7£551E
<°*>:- ~.1:-1=;1.¢=E:;=E513=:é;1FE=i;:sE$=; 5§:E$=
EEi';E§E§éi E§éiE§i §i
- -.-.-"'.<.¢-""='4;-»2..>-;:1~_.‘E-‘\:,-;>5=_:.'s;\5>-<._.r=<-§><'
‘$-.=\s.:<>-->.:--s.<-:»=.<->.4g;: =: >- -55E1=:-=-:-:-~.-:-:-‘
:?:I>?:15:i
:ZK:1i::4-:»:-:-:-:-1-:-I4;-'1;-:-=-=-:-=-=1-=4-I-1-1-=-1-1-1'1¢<': _i>==-'—l;='.\-If-I-=—
- -’:-:-:':-:-:I:-:-:V:-:-:':-:-:':-:-:-:-:-:-:-:-:-:-'-:-:-‘-:-:-;-:-‘-:—:-L-;-;-r-;-:-L»2-2-I4-I-9I~71»I'I'I'I'IC"'Il~f‘I‘f‘f‘I-I
-'-.4:~.-:1 .<:c-:-'\>:-'-:-:-:-:-:-'-:-#1->:-:-:-:-:-.<:-7-:-:-'-:-:»:':-'1-:-:1:-:-:6 115;:
- =:§‘.».v-,;-.<:<:-<~'. <:71" ‘»"j1I'i-51-1-'.
\ "41:=:'53;
331225T5>i5J$Z3Z=i.4"€5i5'5}:I1I::l754:-‘~:;L-;5;-§!;1-!-I;J!4I>T;I5I§;IZég!;fij{21§S51"l!3‘T;-15:19:11?‘ _ ~ =:=:I:1:§:§:§:§:§:§:§:§:;:§:3:-:5:;:;é£;:;:¢:;:;:<:;;;:1:;‘;:;:;:§i£:;;-;;;;:-:;:
Hideaki NAKAMURA
Sumio HAMADA
An analytical procedure which deals influencing factors affecting to temperatures in the
structure is herein proposed.
Factor analyses were conducted against the thermal
characteristics affecting to thermal analytical results by means of the method of sensitive
analysis.
The variances of temperature in the structure having variable thermal
characteristics of concrete and environment were evaluated from the sensitivity obtained by
the sensitive analysis and the approximation theory using Taylor’s expansion.
Keyword:
sens1't1'v1'ty analysis, first-order appr0X1'1nat1'0n, FEM, probabil/istjc transient heat
conduction analysis
Hideaki Nakamura is a Research Associate in the Department of Civil Engineering at
Yamaguchi University, Yamaguchi, Japan. He obtained his M.Eng. from Yamaguchi
University in 1986. His research interests include thermal stress in mass concrete structure
and earthquake engineering.
Sumio Hamada is a Professor in the Department of Civil Engineering at Yamaguchi University,
Yamaguchi, Japan. He received his Ph.D. from the University of Alberta in Canada in 1973.
His research interests include thermal stress in mass concrete structure and behavior and
design of reinforced and prestressed concrete structure.
-"-Z67-
1.
INTRODUCTION
Thermal analysis is essential in the design of massive concrete structures these days. As the
scale of concrete structures becomes larger, cracks are induced by thermal stresses during
curing. Experimental and analytical studies of thermal stresSeS in mass concrete structures
have been camied out, and several procedures have been proposed for the analysis of heat
conduction and thermal stress. A Study of cracking probability has also been made by the
committee on mass concrete thermal stresses at the Japan Concrete Institute[1], and this
method is practiced in the estimation of cracking during design. The JSCE's Standard
Specifications[2]for concrete were substantially reviewed as regards mass concrete structures
in 1986, and an analytical method for the estimation of cracking was included. Using these
methods[3,4], the cracking probability can be obtained from the relationship between thermal
cracking index and cracking probability, and this relationship has been determined fhm the
latest field surveys of mass concrete Structures. A study by Morimoto[5]has shown how to
carry out a more detailed evaluation of cracking, and the subcommittee for the revision of the
Standard Specificationsis now accumulating new data on cracking in mass concrete structures
with the aim of improving the relationship between thermal cracking index and cracking
probability.
Predictions ofthermal crackingbegin with the determination ofthe temperature distribution in
a structure. Designs of pipe cooling systems to prevent initial cracking and pipe heating
systems for road de-icingrequire temperature distributions of the same type. Finite element
approaches to the analysis of Suchtemperature distributions have becomemore widespread, as
the power of workStations and personal computer develops. The analytical results, however,
are considerably affected by the input data on material and environmental properties, and
precise results cannot be obtained without precise data. The main thermal properties of
concrete required in thermal analysis are the heat generation, thermal conductivity, heat
transfer coefficient,and speciflCheat. These specificproperties all exhibit uncertain resulting
from variations in concrete materials, mix proportion, age, drying condition, atmospheric
temperature, and so on. Consequently, an appropriate analytical method is now required for
the analysis of temperature distribution that takes into account these uncertain physical
properties.
Pipe cooling[6] and pipe heating systems[7] are now relatively common in maSS concrete
structures and roads in coldregions. The temperature of the heating or coolingmedium in the
pipe is not always stable, and engineers workingin the field need to analytically determine the
allowablescatter of temperature in the pipe.
Analytical results obtained in previous studies do not always agree well with experimental
results, and inverse analysis[8]is often employedto obtain the appropriate specificProperties of
concrete. Ono[9]investigated the parameters influencingtemperature rises in mass concrete.
Based on his study, the parameters affecting temperature rise can be evaluated by numerical
experiment. That is, the values affecting temperature rise are obtained from an analysis in
which the parameters are varied by small amounts. Matsui et al[10]. presented analytical
results, including the scatter of specificproperties and environmental conditions, obtained in a
Monte Carlo simulation. Sensitivity analysis[11] is often required in structural analysis,
where the specificstructural properties are vary substantially.
This study proposes a sensitivity analysis method for the thermal stresses in structures and
obtains information on the innuence of scatter in material and environment properties on the
temperature rise in concrete structures. The analysis is based on a transient heat conduction
analysis of the structure with probability parameters, and is a method which yields the mean
values and scatter of temperatures in a structure. It is based on sensitivity analysis and
Taylor's expansion theory. A Monte Carlo Simulationcan also be used to determine mean and
scatter values, but enormous computational time is required for moderately precise results as
compared with the this method. In order to confirm the results obtained by this new
analytical method, its results are compared with those of a Monte Carlo simulation. The
-168-
analysis is also applied to thick wall and slab
concrete structures well as t,o st,ructures
requiring a pipe coolingsystem.
2. ANALYTICAL THEORY
2 .1
FINITE ELEMENT PROCEDURE FOR
TRANSIENT HEAT CONDUCTION
s ensitivity[11,12]
is expressed as A in Fig.1,
aX,
where 0(i) is a nodal temperature, and Xi
represents parameters such as the specific
thermal properties of the material.
This
sensitivity gives the innuence of the parameter
on the nodal temperature.
^
O
0V
O
L<
3
･J-CO
7O<
D<
E
E<O
/
-cd
7T3
O
Z
/
/
G ra dient(S ensitivity)
/
/
/
S p e c ific th e rm a l p ro p e rtie s
Fig. I Sensitivity
The transient heat conduction equation discretized by the finite element method can be
expressed as below[13].
[K](OlI [c]i6l- (F)
(I)
where [K],(0),[C],i6l, and (Fi are the heat conductivity matrix, temperature vector, heat
capacity matrix, vector for temperature slope with respect to time, and heat source vector,
respectively.
The first derivative of the temperature vector with respect to parameter X, is obtained by
differentiating Eq.(1), yielding the followingequation:
%{o}.[K]%.#(6).[c]#
(2)
9ifi
aXI
Rewriting Eq.(2),
[K'%.[C]#
(3)
aX,
9ffL%{o}-#(6)
The end derivative can be determined by differentiating both sides of Eq.(3) with respect to the
parameter X,.
¥K]#.[C]S%-&2T#{0}-A(6)
a[K]a(Ola[K]a(0) a[C]a(6) a[c]a(6)
aXi aXj
Eqs.(1),
(3), and (4) are
aXj aXi
aX, aXj
(4)
aXj aXi
discretized with respect to Space, and not with respect to time.
,:1at6l
Derivativeswith respect to ti-e, t6i%
.
a2(6)
U
LVJ
and if3Ure
-169-
. m ,Jt
n.
present in Eq.'4'. Discretizing
these variables by the Crank-Nicolson method, the temperature at time
i+-
At
2
derivative
oftef.p(etra:fe))ca-n;b(e{
:Etp;eAStS,e}d.r{e:;te,c,t,ively
as
(o(i+At)i-(o(i))
6(i.%))
At
(
Substituting
and the
(5)
(6)
Eqs.(5) and (6) into Eq.(1), the following equation is obtained:
(![K].ii[C])(0(i.At))- (-i[K]. i;[c])(o(i)).(FI
(7)
The first and secondsensitivities at time i. 4t
are expressed as
2
a(o(i.%))1 a(o(i' At))
aXi
a(6(i.
%))
2
(
¥#)
aXi
(8)
a(o(i' At)i a(o(i))
aXi
aX,.
(9)
At
aXi
S
ubstituting EqS.(8) and (9) into Eq.(3), yields the following:
(
+
a(o(i))
a(o(i' At)i
;[K] L[c])
- (-;[K].ii[C])
aXi
aX,.
¥%-%(o(i.%))-#(6(i.=)) (10,
Similarly Eq.(4) can be written aS
a
2(o(i+ At)i
(;[K] L[c])
+
aXlaXj
a
2(o(i))
- (-;[K].=[C]) aX,.aX]
¥&S-%(o(i.%))
-g%(6(i.=))
a[K]a(0(i.%))a[K]
a(0(i.%))
aXi
-17V-
aX]
aX)
aX,.
a[c]a(6(i.%)) a[c]a(6(i.%))
aXl
o(i.%)and6(i.%)on the right-hand
from Eq.(7).
aXj
(ll)
aX,.
side of Eq.(10) can be determined by taking values
a(o(i
iW
A and
aXi
aX]
aX,.
on the right-hand side of Eq.(ll)
are also
d etermined in the same manner from Eqs.(ll) and (10) by substituting the values obtained
from Eq.(10) into these equations.
aBm
The nodal temperatures in a structure with scattered specificproperties can be determined
using the sensitivities evaluated in the previous section. Since each nodal temperature in the
structure is a function ofprobability
variables
X.,X2,......,Xn,
it is given by
o - g(x1,X2,...,Xn)
(12)
The following equation is obtained by Taylor's expansion near to the expectation X- -(X-1,X-2,
¥..,x-ni,
as follows,
o-
g(x-1,X-2,...,X7. a(i)x_(xi
¥;ag(A)x_(x,
-X-i).......
- x1(xj - X-i).......
(13)
where (. )x-implies derivatives with respect to (X-"X-2,...,X-n).
Neglecting higher orders than the second order in Eq.(13), the expectation E[4,] and variance
Var[0] are represented respectively by
(14)
E[0] - g[X-1,X-2,...,X-n]
and
,xj]
ar[0]- ZZ]i(%)x_(A)x_cov[xz
V
c
a
(15)
ov[x"xj] impliesthe secondmomentwith respecttothe expectationand is nominatedas the
covariance. This procedure is called the first-Orderapproximation.
Neglecting higher orders than the third order on the right-hand side of Eq.(13), the secondorder approximation can be obtained as
-171-
Table 1 The Properties for Analysis
Form(t=12mm)
(U=6.6kcal/m2hroC)
diabatic Boundary
tA
i
4
A
h
cd
1
N e w ly C a s t
Tj
F]
j
[ / ( I
]
O
Fq
Ei
•`
Le
U)
F]
cd
•`
•`
I
JJJ
cd
O
73 ;?;_";.
TJAu1 (.Jb
/JJ/.r
{,--{t>/.I/_i:_
(<v1,
l1(EI.
._A.1 :-':t,
y" /
:i,-ii=tglrt3J
OLr)
I[
tST,∼
cd
qG
N
@
9
0
P3
u
o .30 2 ( 12 6 4 )
T h e rm a l C on d u ctiv ity ( A a)
k ca l/m .h r .oC (W /m .oC )
2 .42 4 (2 ,8 19 )
H eat-tran sfer C oefficien t( h c)
k ca 1/m 2.h r.oC (W /m 2.oc )
l2 .0 ( 14 .0 )
D en sity ( p c)
kdm 8
2 8 00
In itial
Am
T em p er atu re
oC
2 9 .8
2 4 .8
b ien t T e m p e ra tu r eCr out)
oC
A diab a t,i c H ea t G e n era tion
Tad - K (1 - e -d )
･:r^) :1./
T.i;-,k} ∼
, ,= tb_=/ 5::A?l
it:;..J' H
.I_'TJJ,.
;.i;-/i"6I.':
./-.*lJ. 7S･:i+/>3>l;il
vJi:y l^ -JplJy,.I,:1J J
･∼3)t},A; rL-_T... t,
l
x is t in g ;;(1gSi1;!;i a
S p ecific H ea t(C c)
k cal n (g .oC (J n(g .oC )
K - 4 8 .5 oC
a - 1A 2 6
lf
5
EE
Table 2 Values Based on the SI System
u
ThermalFixed
Boundary(24-8oC)
1000
(mm)
S p e cific H ea t
1k ca 1/k g .oC
T h erm a l
C on d u c t,iv it,y
H ea t-tra n sfer
C o efficien t.
lk ca 1/m .h r .oC
-
1k cal /m 2.h r .oC
4 .186 X lO8J n tg .oC
-
i.163 W /m .oC
1 .16 3W /m 2.oc
Fig.
2Finite
EleEm[t.T:s.h(
:-;;x-B2?:7,dxlnr;.cZngi;o(nk)x_
cov[
x"xj]
(16,
var[o]
- i A(%)xl%)x_cov[x,,
xj].aAa(%)x_(A)x_
xE[(X,
-X-i)(Xj
- X-j)(Xk
-X-k)].iaiaa((A)x_(A)x_
x(E[(Xz.-X-i)(Xj -X-j)(Xk -X-k)(Xl -X-I)])-Cov[Xi,Xj]Cov[Xk,Xl] )
(17,
The mean and expectation values can be determined by substituting the sensitivities obtained
from Eqs.(1) and (ll) into EqS.(14) and (15) or Eqs.(16) and (17).
3. NUMERICAL EXAMPLES OF SENSITIVITY ANALYSIS
Analytical temperature results cannot be determined to great accuracy without the exact
values ofspecificproperties, yet these specificProperties are presumably Scatteredin the case of
concrete. The specificproperties of concrete needed in thermal analysis are heat generation,
thermal conduction,heat transfer coefficient,and specificheat, and these have uncertainties
relating to age, mix proportion, drying conditions, and temperature. The other causes of
scatter are the environment surrounding the concrete,and differencesin testing methods. In
this chapter, we indicate how to evaluate the factorsinnuencingconcrete temperature fromthe
sensitivity analysis.
-172-
3.I
TEMPERATURE
ANALYSIS
a
iW
The wall structure described in reference [1] is used as an example in this analysis. The
specific heat properties which influence temperature in concrete may be regarded as the
coef6lCientsof adiabatic heat generation (K,ct), the specificheat (Cc),the concrete density (pc),
the thermal conductivity (Lc),the heat transfer coefncient (hc),the coefficientof overall heat
transmission of form (U), and the ambient temperature (Tout). Their innuence on nodal
temperature canbe determined by evaluating the sensitivities
,#
%
and 9i9i.
a(o) a(o) a(o) a(o) a(o)
aK'
aa'
aCc'
apc'
ale
Fig-e 2 illustrates the 6lniteele-ent -esh andboundary con&tions.
aTout
Table 1gives data for the analysis ofthe wall structure.
analysis is assumed to be
The adiabatic heat generation in this
F
ad- K(1 -e-at) - K(1 -e-%t')
(18)
where TodiS rise in temperature at age t CC) , K and a are coefficients, and i and i/ are ages
of concrete in day and hour, respectively. Differentiating Eq.(18) with respect to t'gives the
adiabatic rise in temperature, as follows.
4W
dt I
= E9e-Zt,
(19)
24
Consequently, the heat generated per unit time and unit volume is represented by
B@
(20)
a - pccc%eT&'
which is equivalent to the internal heat generated in an element of the concrete shown in Fig.2.
We evaluate sensitivity at six nodes at intervals of one hour.
The temperature histories for the six nodes in the structure marked in Fig.2 are illustrated in
Fig.3. The sensitivities ofcoef61CientsK and a, thermal conductivity, heat tranSmiSsionrate,
and atmospheric temperatures are also illustrated in Figs.4 to 9,respectively. The ordinate in
these flgureS represents Sensitivity, which reflects nodal temperature change due to a unit
change in each specificProperty from the standard value given in Table 1. The temperature
rises with increasing sensitivity, and falls as it decreases. The unit of the ordinate is
temperature divided by the unit of the specificcoefficient. Values based on the SI system are
also given in the table. Figure 4 points out that sensitivity varies with location and time.
The values of sensitivity are larger for newly cast concrete, and the peak value is reached after
approximately one day. On the other hand, the sensitivity values are fairly small for existing
concrete, which implies that the heat generated in newly cast concrete influences sensitivity.
Figure 5 shows that the coefficient a innuences the temperature more at an early stage
compared with the other coefficients. According to Fig.6, the temperature is sensitive to
thermal conductivityin both the positive and negative directions, depending on whether the
concrete is newly cast or old. The sensitivity to heat transfer coefficient,as shown in Fig.7,
decreases with passing time. Since the differencebetween atmospheric temperature and the
temperature in a concrete structure iS Small,the coefficientdoes not affectnodal temperatures
-173-
,1
O
O
U
d)
h
=
•`
cd
h
C)
EI
E=
a)
•`
r-•`
CCI
TC)
70
lVode I
60
llTode 2
JI
50
Node3
40
hTode4
30
Node5
S)
1LJ
2
Node I
0
Node 2
E
i
a
O
J<
i
P
Node 3
-2
Node 4
4
V
Node6
20
>
4J
2
a0
3
4
5
6
7
-6
>
8
*r1
Node 5
0
1
O
m
0.8
4
5
6
7
8;';;:-;I
Curing time (day)
Fig.6 The Sensitivity Histories of
Thermal Conductivity
hrode
0
3
4J
C
Q)
CD
Curing time (day)
Fig.3 The Temperature Histories
^
2
i
tLhhh
O
O
0.6
Node2
0.4
lVode3
Eh
•`
¥r1
>
0.2
Node 4
0.05
A
U
U
0.0
E
tLLLl
eO
U
¥rl
•`
Node 5
0
¥rl
U)
C
Q)
Node 6
m
.i
hhht
3
4
5
6
7
8
Node
10
A
JP
5
-TI
0
83
-5
E
0
1
2
3
4
5
6
7
4
5
6
7
8;A:;+;I
Fig.7 The Sensitivity Histories of
Heat Transfer Coefficient
i
Node2
.a
1.2
Node3
s)tt
o.8
Node4
v>
0.6
¥=
0.4
Node 5
:i
0.2
Node 6
a -o.2
u'
0
-•`
'E;
3
Curing time (day)
a)
C
a)
U?
15
2
•`
FigA The Sensitivity Histories of K
'>rl
hrode4
1
>
Curing time (day)
,1
O
o-
Node3
-0.
•`
2
I
lVode2
Node 5
i
1
-0.
0.1
.U
V
Node
0
5
1
5
L<
.=
1
Node
I
Node 2
hrode3
Node 4
iVode5
0
8
1
2
3
4
5
6
7
8;':;{;u
Curing time (day)
Curing time (day)
Fig.5 The Sensitivity Histories of a
Fig.8 The Sensitivity Histories of
Ambient Temperature
at an early stage. As time advances, sensitivity
to the coefEICient
becomesgreater due to the large
difference in temperature.
Negative values
imply a heat transfer from the concrete to the
atmosphere. The sensitivity to atmospheric
temperature becomes greater as time advances,
asymptotically approaching a constant value of
unity, as indicated in Fig.8.
!g
•`E; 2
K
a
a 1
Cc
BE o
^c
¥E: -1
hc
GgL-2
U
8Q
a-3
In order to obtain information on the degree of
2
3
4
5
6
7
8Tout
influence of specific heat properties and
Curing time (day)
environmentalproperties on the temperature, the
change in temperature at node 6 are shown in
Fig.9for the case where each parameter varies by Fig.9 The Change in Temperature at node 6
-174-
Table 3The Properties for the Analysis
135000
2000
6000
H
3000
eat-transfer Boundary
O
Concrete
0
4
O
H
2
rq
eat-transfer Boundary
A
LJ
a
1
a
TED
Ej
j
O
C
3
O
Fq
O
O
O
O
O
tJI
T<
Rock
O
A
ed
q
a
TT3
i
i
O
C
O
cq
a
C=
O
Cq
R o ck
0 .2 l
(8 7 1 )
i.9
(2 .2 1 )
0 .38
(1 59 l)
lO .0
( ll .6 3 )
10 .0
(1 l.6 3 )
18 0 0
1.1
(1 .2 8 )
D en sity k g /m 8
2 30 0
2 0 .0
A d ia b atic H ea t,
G en eratio n
K - 4 0 .OoC
a - 0.7 5 5
T1
C
O
ln
cq
C o n crete
In itial T em p era tu re oC
Am b ien t T e m p era tu re oC
0
Fq
Ed
J3
O
O
Ln
S p ecific H e at
k cau k g . oC (Jn {g . oC )
T h erm a l C on du ctiv ity
k ca l/m .h r . OC O V /m . oC )
H ea t-tr an sfer C o eff icien t
k ca 1/m 2.h r .OC O V /m 2. oc )
20 .0
20 .0
Jn
Node i
oO 50
Node 2
U
O
O
Vl
Node 3
S40
rq
•`
# Thermal FixedBoundary
Fig. 10 Finite Element Mesh
+
value.
Node5
B 20
lVode6
TI
cd
and Boundary Conditions
10% fTrom the standard
lNode4
i;co
aE 30
rC
O
a
0
1
2
3
4
5
6
7
8
Curing time (day)
Since 10% of
Fig. ll The Temperature histories
atmospheric temperature is meaningless (It would
be 2oC at 20oC and OoCat OoC),the parameter for
e 2
the environment is varied by loC.
Jl
V
L
=
•`
K
a
Ccone
Figure 9 illustrates that the speci61C
adiabatic rise in S 1
Acone
temperature, K, and the specific heat affect the •`
h cone
¥E:
0
temperature in the concrete. Parameter a has an
Tout
innuence at an early stage. The influence due to a-1
4?A
heat transfer coefficientis small within the flrSt five
8 fAOA?tn
0
2
3
4
5
6
Arock
days because ofthe forms, which are removed at the
Curing time (day)
5th day. A sudden change in sensitivity to heat
transfer coefficient occurs at the 5th day fTorthe Fig.12 The change in temperature at node 4
same reason. The computation results indicate that the influence of atmospheric temperature
is not insignificant and a diftTerencein the atmospheric temperature between that assumed in
the analySiS and that of the actual curing period may aftTectthe concrete temperature
considerably. Assuming that these various SenSitivitiesdo not correlate, the temperature in
structures with various different properties can be easily determined by superimposition of the
senSitivitiesin Fig.9 multiplied by the diftbrencesin the parameters.
A
E=
Q)
C)
tLO
E=
3.2THERMALWALYSIS OF CIRCULARSLABS
8iaiW
The second analytical example is a circular slab 6m in diameter and 2m in thickness cast on
rock. An axis-symmetric finite element model is employed in the analysis, as shown in Fig.10.
Table 3 provides the properties for the analysis.
aL2i2W
Figure ll shows the temperature histories of the six nodes marked in Fig.10. Figure l2 shows
the temperature differences at node 4 for a lO% change in each parameter.
-175-
&+7
T
On
T<
m
0@Ou0T1lZ 3W
u
1
Cooling Pipe
W
(m m )
il
0.2 1
(87 9 )
T h erm a l C on du ctiv ity
k ca 1/m .h r .oC (W /m .oC )
2.0 0
(2 .3 5 )
H ea t-tra n sfe r C o effic ien t
k ca l/m 2.h r .oC (W /m 2.oc )
lO .0
( l l.6 3 )
D en sity
1
2
,r
,
able 4 Analytical Conditions
S p ecific ll ea t
k c au k g .oC (Jn (g .oC )
k g/m 3
In itial T en t)era tu r e
A diab a tic H ea t,
G en era tion
2300
3 0 .0
oC
K - 4 0 .O oC
a - 0 .88 9
H ea t-tran sfer C oeff icie n t of P ip e
k cal /m 2.h r .OC O V /m 2. oc )
2 33
(27 1)
C oo lin g W a te r T em p era tu r e
10 .0
0 .02 5
P ip e D ia m et.er
cC
m
NodeI
P60
U
ll'ode 2
A)
B 50
lVode3
q3
LI
Fig.13 Finite Element Mesh
a)
a
Node 4
BE 40
and Boundary Conditions
eQ
2 30
A similar tendency to that in the wall structure is
seen, where are parameter, K, and the specific
heat innuence the change in nodal temperature for
equal variations in specific heat properties. The
coefficient ct influences the initial nodal
temperature during adiabatic temperature rise.
The Specificheat properties of the rock base do not
influence nodal temperatures.
This study
demonstrates that fairly precise data for K and ct ._>
are required in order to obtain accurate analytical
results.
0
2
4
6
8
Curing time (day)
Fig. 14 The Temperature Histories
S8ii
Node
lVode3
0.3
:i a:i
a _o.?0
Bm
I
Node 2
.Node 4
1
2
3
4
5
6
7
8
Curing time (day)
Pipe cooling systems are often adopted in mass
concrete structures. In such eased, the pipe size,
pipe spacing, now rate, water temperature, now
time, and soon must be known to suitable accuracy.
Accordingly, the factors influencing concrete
temperature are investigated by employing the
sensitivity analysis procedure on an existing
structure.
Fig. 15 The Sensitivity Histories
of Cooling Water Temperature
^
O
V
q}
LJ
5
•`
a
L
C)
a
E
•`V
G
BW
C)
tLO
=
O3
E
U
4
3
2
1
0
1
2
0
K
a
C conc
Acone
h pipe
water teTtlP.
Analysisis carried out on a concrete structure with
2
4
6
8pipe diameter
coolingpipes, part of which is shown in Fig.13,
under the conditions given in Table 4. A pipe
Curing time (day)
element is assumed to be a heat transfer boundary
Fig. 16 The Change in Temperature
with a temperature equal to that of the cooling
water. The heat transfer coefficientbetween the
cooling pipe and cooling water, at a velocity of 40cm/s, is determined using the equation
proposed by Tanabe[6]. The analysis cares eight days with an interval ofO.1 hour for first 10
hours and an interval of one hour thereafter.
3.3.2 Analvtical Results
-176-
Temperature histories fTorthe four nodes shown in Fig.13 are obtained as exhibited in Fig.14.
Figure 15 indicates that the sensitivity to cooling water is higher as the temperature rises.
The sensitivity increases linearly after the end day. This implies that the influence of cooling
by water exceeds that of the heat of hydration on the end day. Figure l6 shows the mean
variance of the four node temperatures at for 10%change in all parameters, except cooling
water temperature, which changes by loC. The coefBICients
of adiabatic temperature rise, K
and cL,also innuence the nodal temperatures. The speci61C
heat and heat conductivity have a
considerable innuence on the temperature. The innuence of heat transfer coefficientis almost
the same as that of pipe diameter, which implies that increasing the pipe diameter wouldhave
an effect equal to that of reducing the heat transfer coefrlCient.
4i&
HEAT PROPERTIES
A thermal analysis was conducted to determine the scatter in temperature based on the
sensitivities and approximate Taylor expansion, when each specific heat property and
environmental property varies. The Monte Carlo simulation is well known in sensitivity
analysis, but the method requires numerous transient thermal analyses in order to obtain
accurate and reliable results. Here, we carryout a sensitivity analysis on a wall structure with
seven scattered parameters, and the results are compared with the results obtained by a Monte
Carlo simul.ation. Taking the eight stochastic variables used in a Monte Carlo simulation as
normal stochaSticvariables, transient thermal analysis was conducted for 5,000 cases of each
stochastic variable. The sampling method proposed in reference [15] is employed in
evaluating correlative StochaSticvariables. Table I provides the specific thermal properties
adopted as fundamental data. The analysis is conductedbased on the data given in Table i,
assuming that each specific property follows a normal distribution function. The expected
value and variance of nodal temperatures with the eight stochastic variables are determined
from the fTollowing
equations.
E[0] - g(K-,5,
C-"P"*c,
h-a,U-,T-ou,)
(2l)
var[o]- (#var[K]. (%)(a)cov[K,a].......
¥......(A)(%)cov[Tou"U]. (A)2var[TouE]
(22,
where g(.), Var[X,.], and Cov[X,] are functions for the evaluation of nodal temperature,
variance of a specific thermal property, and covariance of the mutual specific thermal
properties, respectively.
The variance and covariance of mutual specificthermal properties can be determined from the
followingequations.
var[xi]
- (Vx,.E[Xi])2
c
u
(24)
ov[xi, Xj]
(25)
xL'Xj
- JWJW
where, vx,.and pxiXjare the coefncients of variation of the speciflCthermal properties, Xz., and
coefficients of correlation, respectively. The expected value and variance to a second-order
approximation can be evaluated from Eqs. (16) and (17), and the third-order moment is 0 and
-177-
^
P•`
S)
V
Q}
L
3
•`
aL
q)
A
=
C}
25
Node I
20
Node 2
15
Node 3
10
Node4
+
Node 5
5
cd
0
G
lVode6
tH
O
2
q)
U
G
cd
..-Ej.-
1.2
4
6
--.."A...""
-+-
-"-"I..-...
First-order
Approximation
O
¥rl
i.J
JU
Node 1 Node 2 hTode3 Node4 Node5 Node 6
Second -order
Approximation
-P
1.i
cd
h
'C
Q,)
8
N
¥rl
T<
aE=
Curing time (day)
L
cO
>
JI
.U
V
C}
LJ
=
•`
ed
i<
d)
25
cd
U
C
E
0
10 20 30 40 o 10 20 30 40
Coefficient of variation(%)
)!onte Carlo
15
Ncde6
a
=
Q)
JJ
a
hnte Carlo
Ncde5
20
. E5'
I
0
L
O
Fig. 17 The Variance of Nodal Temperature
P
•`
-.JL.-.
Fig. 19 The Accuracy of Analysis
First-order
10
Ncde5
First-order
5
Node 1 Node 2 Node 3 Node4 Node5 Node 6
Ncde 6
+
..-E}-.
---LA.......
--+-
-..-..L"-...
-.-.JL"..
Lid
O
q)
U
=
a
L
ed
>
Second-order
2
4
6
Curing time (day)
8
1.2
Ncde 5
Second-order
Ncde 6
Fig.18 The Scattering History at Node 5 and 6
L}=10%
u
0
=20%
¥TI
i
h
1.
rTj
Q)
•`
1
1
¥TI
the fourth can be determined as a function of the
'l''''.E]
T<
cd
ilE= o.
second-order moment[14].
aO
4iiyAAiAW
9
0.
0.5
1
0
0.5
1
Correlation
Correlation
Figure 17 shows the variance of nodal
temperature obtained from the first-order
Fig.20 The Accuracy of Analysis
approximation method, when the eight specific
thermal properties have a coefncient of variation
of 10%. Figure 18 shows the scattering history at nodes 5 and 6 evaluated using the Monte
Carlo simulation, the first-order approximation method, and second-order approximation
method. The period of the maximum scatter is the 2nd to 4th days, and is longer than the
time at the maximum temperature. This is due to the influence of properties with the later
maximum sensitivity shown in FigsA to 8. Figure 18 shows that the scatter in nodal
temperature evaluated by the first-Order and second-order approximation methods are close to
those evaluated by the Monte Carlo simulation.
4i2m
MONTE CARLOMETHOD
I
n order to investigate the applicabilityof the new methodto sensitivity analysis, its results are
comparedwith those of a Monte Carlo simulation. The normalized ratio of these results t,othe
Monte Carlo results is employedfTorthis purpose. This normalization is given aS
i (Tn-mz.)
z'=1
e
(25)
-
where n, Tn.,and Tmiare the number of computations, the scatter in the results, and the scatter
in Monte Carlo results.
-178-
Figure 19 depicts the normalized values for first-Orderand second-order approximations, when
the correlation is zero. In the case of the flrSt-Order approximation, it gradually decreases
with increasing coefficientof variation, whereas fTor
the second-order approximation it increases
with increasing coef61Cientof variation. Since these normalized values are both fairly close to
unity, the first-Order approximation method is sufficient for sensitivity analysis. Figure 20
shows the normalized value when each parameter has a correlation, which provides better
accuracy.
5. CONCLUSION
This study describes a procedure fTorobtaining the quantitative influence of specific thermal
properties and environmental properties on concrete temperature due to hydration heat, as
well as a procedure for determining the scatter of temperature in a concrete structure when
there is scatter in the specific thermal properties and environmental properties. The
analytical procedure was applied to wall and fTootingStructure and a pipe cooling structure.
The fTollowing
conclusions have been reached:
(I) This method of sensitivity analysis provides a quantitative measure of the influence of
random specificproperties on concrete temperature.
(2)Results obtained with this analysis agree well with the results of a Monte Carlo simulation
within the range of a 35% coefficientof variation.
(3) Both first- and second-order approximation methods give similar results for the scatter,
implying that a first-order approximation method is adequate for determining the scatter.
(4)This new analysis method oftTerS
fairly accurate results for specificproperties that correlate
with each other.
Acknowledgment
The authors express their appreciation to the members of the Subcommittee on Mass Concrete
(Chairman: Kokubu; Secretary general: Ono ) in the concrete Standard Specifications
Committee, who gave valuable suggestions.
References
[1]JCI: "Recommendation for the Control of Cracking of Mass Concrete,"JCI, l986
[2]JSCE: "Concrete Standard Specifications,"JSCE, 1991
[3]H. Morimoto and W. Koyanagi: "Study on the EStimation of Thermal Cracking in Concrete
Structures ," Proc. of the 39th Annual Conference ofthe JSCE, 5, pp.291-292, 1984
[4]H. Morimoto and W. Koyanagi: "Study on the Estimation of Thermal Cracking in Concrete
Structures,"
Proc. ofJCI,
Vol.8, pp.53-56,
1986
[5] H. Morimoto and W. Koyanagi: "Study on the Estimation of Thermal Cracking in Concrete
Structures,"
Proc. of JSCE,
No.390N-8,
pp.67-75,
l988
[6] T. Tanabe, H. Yamakawa and A. Watanabe: "Determination of Convection Coef61Cientat
Cooling Pipe Surface and Analysis of Cooling Effect," Proc. ofJSCE, No.343, pp.171-179, l984
[7] S. Tanaka, H. Nakamura and S. Hamada: "Snow-melting and Antifreeze System for
Prevention of Auto Accidents on Slippery Bridge Decks," Proc. of JCI, Vol.15, No.1, pp.12011206,1993
[8] H. Chikahisa, J. Tsuzaki, Y. Arai and S. Sakurai: "Estimation of Heat Transfer CoefrlCients
ofMasS Concrete Structures by Means of Back Analysis," Proc. ofJSCE, NoA51N-17, pp.39-47,
1992
[9] S. Ono: "Studies on Various Factors Affecting the Temperature Rise of Mass Concrete and
Proposal of a Calculation Method for the Temperature Rise," Proc. of JSCE, No.348N-1,
pp.123-132,
1984
[10] K. Matsui, N. Nishida, Y. Dobashi and K. UShioda: "Thermal Stress in Massive Concrete
-179-
under Uncertain
Parameters,"
Proc. ofJCI,
Vo1.15, No.1, pp.1143-1148,
[11] H. Otsubo, el al.: "Sensitive Analysis Po.i),"
No.714,
pp.23-31,
1988
[12] H. Otsubo, el al.: "Sensitive Analysis
No.716,
pp.16-22,
1993
Journal of the Japan Marine Engineers,
OJo.2)," Journal of the Japan Marine Engineers,
1989
[13] G. Yagawa: "Finite Element Analysis in Fluid Dynamics and Heat Transfer," Baifukan,
1983
[14] S. Nakagiri and T. Hisada: "Stochastic Finite Element Method," Baifukan, 1985
[15] M. Hoshiya and K. Ishii: "Reliability Design for Structures," Kajima Syuppan-kai, pp.88-90,
1986
-180-