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Creep damage analysis and its application to nonlinear creep of reinforced concrete beam
Engineerhg Fracrure Mechanics Vol. 34, No. 4, pp. 851-860, 1989 Printed in Great Britain.
0013-7944/89 $3.00 + 0.00 0 1989 Pergamon Press pk.
CREEP DAMAGE ANALYSIS AND ITS APPLICATION TO NONLINEAR CREEP OF REINFORCED CONCRETE BEAM LI ZHAOXIAt
and QIAN JICHENG$
Dept of Mechanics, Hohai University, Nanjing, P.R.C. Ahstraet-Gn the basis of submacro-analysis on the creep process in concrete, the evolution equation of creep damage is obtained. At the same time, the constitutive equation of elastic creep coupled with damage can be given. Using the equations, the authors calculate the long term nonlinear creep and creep failure when stresses are over the long-term sustained strength. In addition, the creep damage equations are applied to calculating nonlinear creep and damage for reinforced concrete beam. The results fit with the experimental data.
1. INTRODUCTION creep of plain and structural concrete have been studied over a period of several years. There are many investigations on this problem. Roll[l] suggested a rheologic model for concrete sustained high compressive stresses which is composed by a Maxwell and three Voigt elements, his model summarized nonlinear creep under high stresses and linear creep under low stresses. Aputyunyan[21 studied nonlinear creep by simply constructing a nonlinear function of stress which replaced the stress in linear creep theory. In addition, Lin Nanxun[3], Bazant[4,5] et al. also studied nonlinear creep theory of concrete. The test data has cotirmed that creep under high sustained loading is mainly due to development of microcracks inside concrete or concrete specimen. So, it is a new way of studying nonlinear creep of concrete that investigating development of microcracks associated with creep. The continuum damage mechanics present another viable framework for the investigation. In this paper, we studied creep damage of concrete and its influence on nonlinear creep and creep failure under higher sustained compressive stresses with the aid of the continuum damage mechanics. Creep damage of concrete mainly formed by microcracks is described by a scalar variable, and evolution equation of creep damage is obtained. At last, the equations are applicable to calculating nonlinear creep and creep damage for a reinforced concrete beam. NONLINEAR
2. SUBMACRO-ANALYSIS OF CREEP DAMAGE Concrete is a composite material being composed of the aggregate and hydrated cement paste. Its behavior depends on the particular mechanical behavior of aggregate and mortar and the interaction between these two phases. From creep test in uniaxial compression, it appears that the creep increases finally at a constant rate when stress level is over the limit of proportionality and below the long-term strength, and that the creep will cause failure within a very short time when stress level is over the long-term strength. In analogy to the metal, the stress-creep curve for concrete recorded before failure also have stage III which is associated with an increase of creep rate. It has been estimated that internal microcracking in concrete is responsible for creep deformation in the majority of cases. So we should directly observe and examine the progressive microcracking process in concrete as induenced by sustained loading and time-dependent deformation. Only a few studies were carried out in the past along this direction. Smadi et a1.[6] examined directly the internal microcrack propagation of high-, medium- and low-strength concrete tGraduate student. SAssociate Professor. EFM 34/4-E
851
852
LI ZHAOXIA and QIAN JICHENG
Fig. 1. Crack types on the creep process of concrete.
specimens subjected to sustained compressive stresses of 40-95% of ultimate stress. From their work, the following observations are made: (1) Sustained load increases the amount of microcracking in concrete, and the formation of these additional cracks infIuences the concrete’s behavior with time. (2) In analogy to the short-term behavior, microcracks at sustained stress can be classified into two types, namely bond or interfacial cracks occurring at the interface between aggregate and mortar, and mortar cracks occurring through mortar as shown in Fig. 1. (3) The long-term stress-strain curve may be divided into three portions which are associated with three stages of microcracking. Stage I and stage II take place at sustained stresses below the long-term sustained strength, while stage III occurs at higher stresses, where concrete eventually fails under load. (4) In the initial stage (stage I), the increase rate of interfacial bond cracks is relatively small up to a stress level of 40%. Stage II of crack growth is associated with sustained stress up to about 80 or 75%. During this stage, creep strain caused a substantial increase in the number and length of interfacial bond cracks. Within the above stress levels, no significant amount of mortar cracks is initiated by creep strain. On the other hand, few combined cracks connecting mainly bond cracks around adjacent aggregate particles are initiated during the Srst stage and their rate of growth increases with increasing stress during the second stage as shown in Fig. 2. (5) In the third stage (stage III) and up to maximum stress, the test results have shown a sharp increase in bond cracks, mortar cracks and those combined cracks connecting bond and mortar cracks with increasing stress at short time after the beginning of sustained loading.
f& 20.9 - 24.3 MI% Age at loading : 28 days
Comb&l crack length ( mm ) (a) Fig. 2. Relations between stress-strength
creep strain ( lo+ ) fb)
ratio and (a) combined crack length, (b) creep strain under tong-term loading.
Creep damage analysis
853
From the above observations, we get the following idea on the creep damage for concrete. Firstly, in the creep process, the interior damage of concrete increases with increasing time, even though sustained loading is not changed with time. Secondly, the creep damage is composed of two parts, one is the damage corresponding to the interfacial bond cracks which can be denoted by DB, another is the damage corresponding to the mortar cracks which can be denoted by DM, When sustained stresses are below the long-term sustained strength, there are few mortar cracks, so we assume that D M = 0 at the sustained stresses below the long-term sustained strength. Generally, we assume that there is no interaction between DB and D,, and the total damage can be written as follows: D(o, t) = D(c, 0) + &(a, &(a, where D(cr, 0) is instantaneous
0
(2.1)
0 = &,(c, 2) + &,(a, 0
(2.2)
or initial damage.
3. EVOLUTION
EQUATIONS
FOR CREEP DAMAGE
According to eqs (2.1) and (2.2), the following expressions can be obtained: .&(a, t) = d(a, 0) + &(a, &*(~, 0 = &@, t) + &(6 Generally, the constitutive
t) 0.
(3.1) (3.2)
equation for creep damage can be given by &(c,
t) = F(%
c, t)+
(3.3)
3.1, Evolution equation at the sustained stresses below the long-term strength Based on the analysis presented above, when the sustained stresses are below the long-term strength, the damage corresponding to mortar cracks can be neglected, and the microcracking process is clearly stable since it does not cause failure and because of the strengthening effect due to creep and aging. Thus the damage rate at stresses below the long-term strength decrease with time and damage as well. We can assume that the damage rate is a function of the damage D and constant stress (T, so eq. (3.3) can be expressed in the form: & = - YD,, +f(a).
(3.4)
Considering the boundary conditions of eq. (3.4), we have D&, 0) = 0 1 Dn(c, t)[,-, = D,(a) - D(o, 0)
(3.5)
where D,(a) is the maximum value of damage for the certain constant stress b. Solving eq. (3.4), we obtained: &(o,
t) = [D,(a) - D(a, 0)] (1 -e+“)
3.2. Evolut~an equat~an at the staled
(3.6)
stresses over the Lang-term strength
In stage III where concrete eventually fails under load, the amount of bond cracks, mortar cracks and combined cracks increase sharply a short time after loading. In contrast to the stableness in stage I and stage II, stage III of microcracking may be classified as unstable. It is mainly due to the development of mortar cracks. Therefore, eq. (3.3) at the sustained stresses over the long-term strength may be expressed in the form: (3.7) where
fiw=I:+g(t)
: ;:i:
(3.8)
LI ZHAOXIA and QIAN JICHENG
854
k is a constant dependent on the materials and tR is the time at which failure occurs. Considering the boundary conditions of eq. (3.7), we have D&r, 0) = 0, &((a, 0) = 0 &(a, ZR)= Q&r) - No, 0) %(a, la) = DR - D (0, 0) i
(3.9)
where &(o) is the maximum value of bond damage at the failure time, and DR is the damage threshold at failure. With eqs (3.7)-(3.9), it is now possible to derive the expression for the evolution equation at the sustained stresses over the long-term strength as follows:
D&, t) = “-‘p’--,~~
‘)(l -e-y?
+ DRt9Q(‘)
[t + G(t)int (;>I
R
(3.10)
R
where Q = G(IR). In the above equation, the function G(t) describes the damage increasing sharply before failure. Based on the test data, G(t) can be expressed in the form:
c(t)=[l-~-~)~(l -k)r where n is a large constant dete~ined
(3.11)
by the materials.
3.3. General form of evolution equation From eqs (3.6) and (3.10), we can write a general expression of damage evolution for constant stress as follows:
Dcrh t) = Dm(f!-e:$
')(l
- e+‘) + DmyV+Dl(‘)
[t f G(t)int (&)I
R
(3.12)
R
where &(a) is a maximum value of bond damage Dg, that is damage corresponding to the interfacial bond cracks, and D, is a maximum value of total damage. If a/f; < 0.8, we have = D&C), tR-03 and eq. (3.12) becomes eq. (3.6). If b/f: 2 0.8, thus DW = DR and DM=O,Deq. (3.12) is the same as (3.10). Based on the hypothesis of damage summation proposed by Piechnik et a1.[7], the damage law for concrete under the stresses which vary with time can be given as follows: D(a, t)= D[a(t),O]+
‘[D,(a)s0
D(o,O)];
(1 -e-Y’)dz,
o/f; < 0.8
(3.13)
and D(a, t) = D[a(t), O] + Q,“~ z 0.8. It must be noted that D(a, t) = Ho@, ), 51I.
when
a(t) cc@,),
(t > r,),
the
damage
is irreversible,
(3.14) i.e.
3.4. The determination of the co&icients and fwtctions in evolution equations There are some coefficients and functions which must be determined by special creep test. They are coefficients related with cr: y, t, and functions D(a, 0) and D,(a), expressing respectively the instantaneous or initial damage and the maximum value of bond damage DB. The coefficient y can be expressed by the stress u and the partial derivative of damage about time at the beginning of sustained loading which is denoted by (X)/at),. From equation (3.12) we can obtain
Creep damage analysis
855
Since
where e1 is the long-term sustained strength, thus
K’,t~)-Dtp,011, o/!f;
Y
DR - D&a)
Y =r
tR + a
(3.15)
1
(1 -e -““M&,(e) - He, 011. a/Y;2 CT~ lff
(3.16)
The partial derivative (&l/c%), expresses the damage rate at the be&ming of sustained loading, it is a function of the stress-strength ratio a/f 6.(i?D/c%),,and D,(o) express respectively the damage characteristics on the process of stable creep, so they may be called the characteristical parameters for creep damage. Through measu~ment and calculation of ins~taneous elastic modulus on the creep process, the characteristical parameters D, and @D/i%), under different stresses can be measured indirectly. Their functional relations with the stress-strength ratio have been obtained experientially by the authotjll], namely L>,(a)=k,(offr--0.4)
(3.17)
(>
(3.18)
aD
= A (aff; - 0.4)
dr,
where k,, A and c are the material constants. Based on the data of creep failure test, the relation between failure time t, and stress-strength ratio o/f 6 can be given as follows: tR =
al
(a/f6)“’
(3.19)
where a, and 6, are the material constants. Similar to the analysis for damage by Loland[9], the ins~n~~eo~ assumed as:
damage law can be
08
D,,=D(o,O)=D,
(3.20)
0fr
in above equation, Do is the instantaneous damage under the ultimate stress a,, and it can be measured indirectly using the following equation:
where E. = (u&~) is the secent modulus for the ultimate stress. 4. THE CONSTIT~~
EQUATION AND C~C~AT~G PLAIN CONCRETE
RESULTS FOR
For the time-dependent problem, the effective stress can be defined as follows:
W) =
o(t) 1
-D(t)
where o(t) is the nominal stress. Based on the hypothesis of strain equivalence proposed by Lemaitre{lO], the constitutive equation of elastic creep coupled with damage can be given as follows: c(t)
=
*a
a(t) E(t)[l
-D(t)]
-
i
I E -
tl i% E(z)
‘t’) + C(t, 7) 1 - tt(r) 1
&
where C(t, r) is the specific creep at time t under a constant stress applied at time r.
(4.1)
LI ZHAOXIA and QIAN JICHENG
856
-calculation I 0
I
10
I
,
/
/
f
20
30
40
50
60
0
IO
/
/
I
I
8
,
20
30
40
50
60
70
Time (mm)
Time ( days ) Fig. 3. Comparison of calculation with experimental data for creep under the stress below long-term strength (a) u/f{ = 0.70, (b) o/f: = 0.75.
Fig. 4. Comparison of calculation with experimental data for creep under the stress over long-term strength (a) a/f; = 0.90 (b) a/f; = 0.85.
Using the evolution equation (3.12) and constitutive equation (4.1), we can calculate the long-term nonlinear creep and creep failure for plain concrete when stresses are below or over the
long term sustained strength. The calculating results and the comparison of calculation with experimental data[6] are shown in Figs 3 and 4, we find that calculating results fit well with the creep data.
5. NONLINEAR
CREEP AND DAMAGE FOR REINFORCED
CONCRETE
BEAM
5.1. Problem
The idealized beam model shown in Fig. 5(a) is a singly-reinforced rectangular beam subjected to a pure bending moment M. The concrete in the tension zone of the beam is assumed to be cracked and these cracks are assumed to be uniformly distributed along the length of the beam, i.e. every section is cracked. The steel reinforcing bar is considered to be in perfect bond with the concrete (see Fig. 5b). In addition to the idealizations described above, it is also assumed that the strain distribution across the beam depth is linear, i.e. plane sections remain plane after loading (see Fig. 5~). Creep damage equations and the nonlinear constitutive equation for creep proposed above can be used in predicting creep and damage of concrete for load magnitudes at which concrete is nonlinear.
M
(b)
(cl
Fig. 5. (a) Beam (b) stress and (c) strain distribution across the beam depth.
Creep damage analysis
857
5.2. The governing equations According to the assumptions above, we can write following equations: Equilibrium equations:
‘crdy
A,a,+b
=0
s0
(5.1)
c
I
A,o,d+b
yady=M s 0
in which A, = steel area, b = beam width, a, and L, = stress and strain in the steel, respectively. Geometry equation: d-c p=T=- 8
y-c
(5.2)
fz
or
fz=eY
--cl
(5.3)
cs= t$(d - c)
in which C$= curvature, c = depth of the neutral axis. Stress-strain-time relations: (5.4) for steel in the case of an elastic idealization, and c(t, ?) =
I*,t
N>
J(t, T)
Jw-m-
a(z)dr + S,
1 -D(z)
1
J(t, 2) = + C(t, r) E(r)
(5.5)
for concrete, in which, damage D(a,t’)=D(a,O)+
I’[D,(a)-D((a,O)]ye-Y’ds s0
(54
and shrinkage S = So(e-@l - e-6’) eliminating L, and CT,from eqs (5.1X5.4),
the following equations are obtained:
A&&d-c)-b
1
(5.7)
A,E&(d-c)d-b
‘trdy=O s0
(5.8)
‘yady=M. s 0
With the differentiation of eqs (5.8) and (5.5) with respect to time and rearrangement of them[8], a set of nonlinear differential equations for the rate of curvature and the rate of change of the location of the neutral axis is obtained: a,,d +a,,k
I a2,f$ +a,t
= A, =4?,
(5.9)
8.58
LI ZHAOXIA and QIAN JICHENG
in which, a,, =A,EJd-c)+bE
‘(1 -r>)@ -c)dy s0
(5.10)
~~~=A,E;(d--c)d+bE
a22= -4A,E,dfbE
‘(l-D)o,-c)ydy s0 ‘(1
-r>)ydy
f0 (T(r) dr I -D(r) - s -+;J(r,
r&r 1 I;$)]
dv (5.11)
. B,=b
a(r)
o’ --aa l_,+E(l-D
1 -D(r)
S[
dz
where (5.12)
Equations (5.9) are the go~e~ing equations for the creep deformation of the beam. They may be solved by numerical method@]. 5.3, Example of applications and results
A simple example is the beam subjected to a pure bending moment MAwhich is slightly greater than MY,the moment at initial yielding of the steel. In this case, E, = const, and the stress in the steel is constant, The condition of static ~ui~b~urn requires a instant stress in concrete and a tied position of neutral axis on the creep process for concrete. The creep of concrete in the comp~ssive zone of the beam may be calculated by using eq. (5,5), on the basis of initial stress a(~~) at the ~ginni~g of loading, With the strain at the top of the beam, the curvature of the beam can be obtained as follows: # =(so/:
(5.14)
where co is the strain at the top of the beam, i.e. the strain at y = 0 point. When the moment MA= 1.004MY(MY= 4630.32Nm), the stress on the top of the beam at the beginning of loading is over 40% of the short-term strength of the concrete, i.e. ~(t,)l,_~ > 0.4fL, thus, the concrete on the top of the beam has suffered compressive damage. Based on measured
Creep damage analysis
859
data from high sustained load tests of plain concrete cylinders by Mozer [II] and the method described in Section 3.4 of this paper, the functions expressed in eqs (3.17), (3.18) and (3.20) can be determined as follows: & ((i) = 1*6(alff - 0.4) 80
(at,>
= 114.4(a/ff - 0.4)6
D(o, 0) = 0.27(a/~)z.7.
Based on the data from low sustained load tests of plain concrete by Mozer, the specific creep at time t under a constant stress applied at time r is c(t, z) = 1.22[1 -e-
0.4(-r)] x
lo-6
+
44(e-0.03t
_
e-O.O3l) x
lo-6
and the shrinkage strain is S = 1_38(e-“03’- e-0.03’)x 10-3. Some constants needed in calculation are J; = 35.8 MPa,
E = 21.2 x lo3 MPa,
Es = 1.89 x 10’ MPa,
d = 127 mm,
A, = 128.8 mm’,
b = 76.2 mm,
A comparison between experimental data by Mozer[l I] and calculated curvature histories are given in Table 1 and Fig. 6. In Fig. 6, a satisfactory accuracy of the calculation with experimental data is observed. In addition, at the same time of calculating creep and curvature, we observed that the damage values of concrete at the top of the beam are all less than the damage threshold at failure DR(DR = OS-O.8 for medium- and low-strength concrete). Thus, although the concrete in the tension zone of the beam has cracked and the steel initially yielded, the beam will not fail due to creep under a constant load. The reason is that the concrete in the compressive zone in the beam is not failed.
Table I. Curvature histories and a comparison experimental data[ 111 Time after loading (days)
between them and
Curvature (10e3 rad/mm)
Error (%)
Damage
0.02180 0.05070 0.06503 0.07335 0.07792 0.08039 0.08175
11.22 7.92 3.02 1.84 6.31 6.37 3.37
0.0776 0.2163 0.2789 0.3070 0.3196 0.3253 0.3279
2:: 40 60 80
XTest [ll] -Calculation I 0
20
&
40
,
I
I
t
60
80
100
120
Time ( days ) Fig. 6. Beam curvature histories.
860
LI ZHAOXIA and QIAN JICHENG
6. CONCLUSIONS
(1) In the stage of stable creep, the damage rate decrease with time, and the damage is tending toward a constant below the damage threshold at failure DR with the time increase, thus it will not cause failure; When the sustained stresses are over the long-term strength, the damage rate does not decrease with time, and the damage increase sharply in short time after loading up to the damage threshold DR. (2) It is successful that the evolution equations of creep damage and the constitutive equation of elastic creep coupled with damage are applied to calculating nonlinear creep and damage for plain concrete and reinforced concrete beam. REFERENCES [l] A. M. Frendenthal and F. Roll, Creep and creep recovery of concrete. under high compressive stress, J. ACI., 1114-I 142 (1958). [2] N. H. Arutyunyan, Some Prob1em.r in rhe Theory of Creep. National Press of Technico-Theoretical Literatures (1952) (in Russian). [3] Lin Nanxun, The theory for nonlinear creep of concrete. J. Civil Engng, 16, 14-20 (1983). 141Z. P. Bazant and A. A. Asghari, Constitutive law for nonlinear creep of concrete, J. Engng Me& Dht. H&133-124 (February 1977). [S] Z. P. Bazant and S. S. Kim, Nonlinear creep of concrete-adaptation and flow, J. Engng Mech. Div. 429-446 (June 1979). (61 M. M. Smadi, F. 0. Slate and A. H. Nilson, Time-dependent behavior of high-strength concrete under high sustained compressive stresses. ~~83-227041, NSFICEE-82 142 (November 1982). [7] S. Piechnik and H. Pa&la, The continuous field of damage and its influence on the creep process in concrete under tensile loading. IUTAM, 3rd Synrp. on Creep in Sirttctures, pp. 202-219 (8-12 September 1980). [S] Li Zhaoxia, Analyses of creep damage and nonlinear creep for concrete. Ph.D. thesis presented to Hohai University at Nanjing (1988). 191 _ _ K. E. Loland. Continuous damage model for load-response estimation of concrete. Cem. Concr. Res. 10, 395-402 (1980). :lO] J. Lemaitre, How to use damage mechanics. Nucl. Engng Des. 80, 233-245 (1984). ,111 J. D. Mozer et al., Time-dependent behavior of concrete beams. Proc. ASCE, ST3, No. 96, pp. 597-612 (1970). (Receioed 27 October 1988)
0013-7944/89 $3.00 + 0.00 0 1989 Pergamon Press pk.
CREEP DAMAGE ANALYSIS AND ITS APPLICATION TO NONLINEAR CREEP OF REINFORCED CONCRETE BEAM LI ZHAOXIAt
and QIAN JICHENG$
Dept of Mechanics, Hohai University, Nanjing, P.R.C. Ahstraet-Gn the basis of submacro-analysis on the creep process in concrete, the evolution equation of creep damage is obtained. At the same time, the constitutive equation of elastic creep coupled with damage can be given. Using the equations, the authors calculate the long term nonlinear creep and creep failure when stresses are over the long-term sustained strength. In addition, the creep damage equations are applied to calculating nonlinear creep and damage for reinforced concrete beam. The results fit with the experimental data.
1. INTRODUCTION creep of plain and structural concrete have been studied over a period of several years. There are many investigations on this problem. Roll[l] suggested a rheologic model for concrete sustained high compressive stresses which is composed by a Maxwell and three Voigt elements, his model summarized nonlinear creep under high stresses and linear creep under low stresses. Aputyunyan[21 studied nonlinear creep by simply constructing a nonlinear function of stress which replaced the stress in linear creep theory. In addition, Lin Nanxun[3], Bazant[4,5] et al. also studied nonlinear creep theory of concrete. The test data has cotirmed that creep under high sustained loading is mainly due to development of microcracks inside concrete or concrete specimen. So, it is a new way of studying nonlinear creep of concrete that investigating development of microcracks associated with creep. The continuum damage mechanics present another viable framework for the investigation. In this paper, we studied creep damage of concrete and its influence on nonlinear creep and creep failure under higher sustained compressive stresses with the aid of the continuum damage mechanics. Creep damage of concrete mainly formed by microcracks is described by a scalar variable, and evolution equation of creep damage is obtained. At last, the equations are applicable to calculating nonlinear creep and creep damage for a reinforced concrete beam. NONLINEAR
2. SUBMACRO-ANALYSIS OF CREEP DAMAGE Concrete is a composite material being composed of the aggregate and hydrated cement paste. Its behavior depends on the particular mechanical behavior of aggregate and mortar and the interaction between these two phases. From creep test in uniaxial compression, it appears that the creep increases finally at a constant rate when stress level is over the limit of proportionality and below the long-term strength, and that the creep will cause failure within a very short time when stress level is over the long-term strength. In analogy to the metal, the stress-creep curve for concrete recorded before failure also have stage III which is associated with an increase of creep rate. It has been estimated that internal microcracking in concrete is responsible for creep deformation in the majority of cases. So we should directly observe and examine the progressive microcracking process in concrete as induenced by sustained loading and time-dependent deformation. Only a few studies were carried out in the past along this direction. Smadi et a1.[6] examined directly the internal microcrack propagation of high-, medium- and low-strength concrete tGraduate student. SAssociate Professor. EFM 34/4-E
851
852
LI ZHAOXIA and QIAN JICHENG
Fig. 1. Crack types on the creep process of concrete.
specimens subjected to sustained compressive stresses of 40-95% of ultimate stress. From their work, the following observations are made: (1) Sustained load increases the amount of microcracking in concrete, and the formation of these additional cracks infIuences the concrete’s behavior with time. (2) In analogy to the short-term behavior, microcracks at sustained stress can be classified into two types, namely bond or interfacial cracks occurring at the interface between aggregate and mortar, and mortar cracks occurring through mortar as shown in Fig. 1. (3) The long-term stress-strain curve may be divided into three portions which are associated with three stages of microcracking. Stage I and stage II take place at sustained stresses below the long-term sustained strength, while stage III occurs at higher stresses, where concrete eventually fails under load. (4) In the initial stage (stage I), the increase rate of interfacial bond cracks is relatively small up to a stress level of 40%. Stage II of crack growth is associated with sustained stress up to about 80 or 75%. During this stage, creep strain caused a substantial increase in the number and length of interfacial bond cracks. Within the above stress levels, no significant amount of mortar cracks is initiated by creep strain. On the other hand, few combined cracks connecting mainly bond cracks around adjacent aggregate particles are initiated during the Srst stage and their rate of growth increases with increasing stress during the second stage as shown in Fig. 2. (5) In the third stage (stage III) and up to maximum stress, the test results have shown a sharp increase in bond cracks, mortar cracks and those combined cracks connecting bond and mortar cracks with increasing stress at short time after the beginning of sustained loading.
f& 20.9 - 24.3 MI% Age at loading : 28 days
Comb&l crack length ( mm ) (a) Fig. 2. Relations between stress-strength
creep strain ( lo+ ) fb)
ratio and (a) combined crack length, (b) creep strain under tong-term loading.
Creep damage analysis
853
From the above observations, we get the following idea on the creep damage for concrete. Firstly, in the creep process, the interior damage of concrete increases with increasing time, even though sustained loading is not changed with time. Secondly, the creep damage is composed of two parts, one is the damage corresponding to the interfacial bond cracks which can be denoted by DB, another is the damage corresponding to the mortar cracks which can be denoted by DM, When sustained stresses are below the long-term sustained strength, there are few mortar cracks, so we assume that D M = 0 at the sustained stresses below the long-term sustained strength. Generally, we assume that there is no interaction between DB and D,, and the total damage can be written as follows: D(o, t) = D(c, 0) + &(a, &(a, where D(cr, 0) is instantaneous
0
(2.1)
0 = &,(c, 2) + &,(a, 0
(2.2)
or initial damage.
3. EVOLUTION
EQUATIONS
FOR CREEP DAMAGE
According to eqs (2.1) and (2.2), the following expressions can be obtained: .&(a, t) = d(a, 0) + &(a, &*(~, 0 = &@, t) + &(6 Generally, the constitutive
t) 0.
(3.1) (3.2)
equation for creep damage can be given by &(c,
t) = F(%
c, t)+
(3.3)
3.1, Evolution equation at the sustained stresses below the long-term strength Based on the analysis presented above, when the sustained stresses are below the long-term strength, the damage corresponding to mortar cracks can be neglected, and the microcracking process is clearly stable since it does not cause failure and because of the strengthening effect due to creep and aging. Thus the damage rate at stresses below the long-term strength decrease with time and damage as well. We can assume that the damage rate is a function of the damage D and constant stress (T, so eq. (3.3) can be expressed in the form: & = - YD,, +f(a).
(3.4)
Considering the boundary conditions of eq. (3.4), we have D&, 0) = 0 1 Dn(c, t)[,-, = D,(a) - D(o, 0)
(3.5)
where D,(a) is the maximum value of damage for the certain constant stress b. Solving eq. (3.4), we obtained: &(o,
t) = [D,(a) - D(a, 0)] (1 -e+“)
3.2. Evolut~an equat~an at the staled
(3.6)
stresses over the Lang-term strength
In stage III where concrete eventually fails under load, the amount of bond cracks, mortar cracks and combined cracks increase sharply a short time after loading. In contrast to the stableness in stage I and stage II, stage III of microcracking may be classified as unstable. It is mainly due to the development of mortar cracks. Therefore, eq. (3.3) at the sustained stresses over the long-term strength may be expressed in the form: (3.7) where
fiw=I:+g(t)
: ;:i:
(3.8)
LI ZHAOXIA and QIAN JICHENG
854
k is a constant dependent on the materials and tR is the time at which failure occurs. Considering the boundary conditions of eq. (3.7), we have D&r, 0) = 0, &((a, 0) = 0 &(a, ZR)= Q&r) - No, 0) %(a, la) = DR - D (0, 0) i
(3.9)
where &(o) is the maximum value of bond damage at the failure time, and DR is the damage threshold at failure. With eqs (3.7)-(3.9), it is now possible to derive the expression for the evolution equation at the sustained stresses over the long-term strength as follows:
D&, t) = “-‘p’--,~~
‘)(l -e-y?
+ DRt9Q(‘)
[t + G(t)int (;>I
R
(3.10)
R
where Q = G(IR). In the above equation, the function G(t) describes the damage increasing sharply before failure. Based on the test data, G(t) can be expressed in the form:
c(t)=[l-~-~)~(l -k)r where n is a large constant dete~ined
(3.11)
by the materials.
3.3. General form of evolution equation From eqs (3.6) and (3.10), we can write a general expression of damage evolution for constant stress as follows:
Dcrh t) = Dm(f!-e:$
')(l
- e+‘) + DmyV+Dl(‘)
[t f G(t)int (&)I
R
(3.12)
R
where &(a) is a maximum value of bond damage Dg, that is damage corresponding to the interfacial bond cracks, and D, is a maximum value of total damage. If a/f; < 0.8, we have = D&C), tR-03 and eq. (3.12) becomes eq. (3.6). If b/f: 2 0.8, thus DW = DR and DM=O,Deq. (3.12) is the same as (3.10). Based on the hypothesis of damage summation proposed by Piechnik et a1.[7], the damage law for concrete under the stresses which vary with time can be given as follows: D(a, t)= D[a(t),O]+
‘[D,(a)s0
D(o,O)];
(1 -e-Y’)dz,
o/f; < 0.8
(3.13)
and D(a, t) = D[a(t), O] + Q,“~ z 0.8. It must be noted that D(a, t) = Ho@, ), 51I.
when
a(t) cc@,),
(t > r,),
the
damage
is irreversible,
(3.14) i.e.
3.4. The determination of the co&icients and fwtctions in evolution equations There are some coefficients and functions which must be determined by special creep test. They are coefficients related with cr: y, t, and functions D(a, 0) and D,(a), expressing respectively the instantaneous or initial damage and the maximum value of bond damage DB. The coefficient y can be expressed by the stress u and the partial derivative of damage about time at the beginning of sustained loading which is denoted by (X)/at),. From equation (3.12) we can obtain
Creep damage analysis
855
Since
where e1 is the long-term sustained strength, thus
K’,t~)-Dtp,011, o/!f;
Y
DR - D&a)
Y =r
tR + a
(3.15)
1
(1 -e -““M&,(e) - He, 011. a/Y;2 CT~ lff
(3.16)
The partial derivative (&l/c%), expresses the damage rate at the be&ming of sustained loading, it is a function of the stress-strength ratio a/f 6.(i?D/c%),,and D,(o) express respectively the damage characteristics on the process of stable creep, so they may be called the characteristical parameters for creep damage. Through measu~ment and calculation of ins~taneous elastic modulus on the creep process, the characteristical parameters D, and @D/i%), under different stresses can be measured indirectly. Their functional relations with the stress-strength ratio have been obtained experientially by the authotjll], namely L>,(a)=k,(offr--0.4)
(3.17)
(>
(3.18)
aD
= A (aff; - 0.4)
dr,
where k,, A and c are the material constants. Based on the data of creep failure test, the relation between failure time t, and stress-strength ratio o/f 6 can be given as follows: tR =
al
(a/f6)“’
(3.19)
where a, and 6, are the material constants. Similar to the analysis for damage by Loland[9], the ins~n~~eo~ assumed as:
damage law can be
08
D,,=D(o,O)=D,
(3.20)
0fr
in above equation, Do is the instantaneous damage under the ultimate stress a,, and it can be measured indirectly using the following equation:
where E. = (u&~) is the secent modulus for the ultimate stress. 4. THE CONSTIT~~
EQUATION AND C~C~AT~G PLAIN CONCRETE
RESULTS FOR
For the time-dependent problem, the effective stress can be defined as follows:
W) =
o(t) 1
-D(t)
where o(t) is the nominal stress. Based on the hypothesis of strain equivalence proposed by Lemaitre{lO], the constitutive equation of elastic creep coupled with damage can be given as follows: c(t)
=
*a
a(t) E(t)[l
-D(t)]
-
i
I E -
tl i% E(z)
‘t’) + C(t, 7) 1 - tt(r) 1
&
where C(t, r) is the specific creep at time t under a constant stress applied at time r.
(4.1)
LI ZHAOXIA and QIAN JICHENG
856
-calculation I 0
I
10
I
,
/
/
f
20
30
40
50
60
0
IO
/
/
I
I
8
,
20
30
40
50
60
70
Time (mm)
Time ( days ) Fig. 3. Comparison of calculation with experimental data for creep under the stress below long-term strength (a) u/f{ = 0.70, (b) o/f: = 0.75.
Fig. 4. Comparison of calculation with experimental data for creep under the stress over long-term strength (a) a/f; = 0.90 (b) a/f; = 0.85.
Using the evolution equation (3.12) and constitutive equation (4.1), we can calculate the long-term nonlinear creep and creep failure for plain concrete when stresses are below or over the
long term sustained strength. The calculating results and the comparison of calculation with experimental data[6] are shown in Figs 3 and 4, we find that calculating results fit well with the creep data.
5. NONLINEAR
CREEP AND DAMAGE FOR REINFORCED
CONCRETE
BEAM
5.1. Problem
The idealized beam model shown in Fig. 5(a) is a singly-reinforced rectangular beam subjected to a pure bending moment M. The concrete in the tension zone of the beam is assumed to be cracked and these cracks are assumed to be uniformly distributed along the length of the beam, i.e. every section is cracked. The steel reinforcing bar is considered to be in perfect bond with the concrete (see Fig. 5b). In addition to the idealizations described above, it is also assumed that the strain distribution across the beam depth is linear, i.e. plane sections remain plane after loading (see Fig. 5~). Creep damage equations and the nonlinear constitutive equation for creep proposed above can be used in predicting creep and damage of concrete for load magnitudes at which concrete is nonlinear.
M
(b)
(cl
Fig. 5. (a) Beam (b) stress and (c) strain distribution across the beam depth.
Creep damage analysis
857
5.2. The governing equations According to the assumptions above, we can write following equations: Equilibrium equations:
‘crdy
A,a,+b
=0
s0
(5.1)
c
I
A,o,d+b
yady=M s 0
in which A, = steel area, b = beam width, a, and L, = stress and strain in the steel, respectively. Geometry equation: d-c p=T=- 8
y-c
(5.2)
fz
or
fz=eY
--cl
(5.3)
cs= t$(d - c)
in which C$= curvature, c = depth of the neutral axis. Stress-strain-time relations: (5.4) for steel in the case of an elastic idealization, and c(t, ?) =
I*,t
N>
J(t, T)
Jw-m-
a(z)dr + S,
1 -D(z)
1
J(t, 2) = + C(t, r) E(r)
(5.5)
for concrete, in which, damage D(a,t’)=D(a,O)+
I’[D,(a)-D((a,O)]ye-Y’ds s0
(54
and shrinkage S = So(e-@l - e-6’) eliminating L, and CT,from eqs (5.1X5.4),
the following equations are obtained:
A&&d-c)-b
1
(5.7)
A,E&(d-c)d-b
‘trdy=O s0
(5.8)
‘yady=M. s 0
With the differentiation of eqs (5.8) and (5.5) with respect to time and rearrangement of them[8], a set of nonlinear differential equations for the rate of curvature and the rate of change of the location of the neutral axis is obtained: a,,d +a,,k
I a2,f$ +a,t
= A, =4?,
(5.9)
8.58
LI ZHAOXIA and QIAN JICHENG
in which, a,, =A,EJd-c)+bE
‘(1 -r>)@ -c)dy s0
(5.10)
~~~=A,E;(d--c)d+bE
a22= -4A,E,dfbE
‘(l-D)o,-c)ydy s0 ‘(1
-r>)ydy
f0 (T(r) dr I -D(r) - s -+;J(r,
r&r 1 I;$)]
dv (5.11)
. B,=b
a(r)
o’ --aa l_,+E(l-D
1 -D(r)
S[
dz
where (5.12)
Equations (5.9) are the go~e~ing equations for the creep deformation of the beam. They may be solved by numerical method@]. 5.3, Example of applications and results
A simple example is the beam subjected to a pure bending moment MAwhich is slightly greater than MY,the moment at initial yielding of the steel. In this case, E, = const, and the stress in the steel is constant, The condition of static ~ui~b~urn requires a instant stress in concrete and a tied position of neutral axis on the creep process for concrete. The creep of concrete in the comp~ssive zone of the beam may be calculated by using eq. (5,5), on the basis of initial stress a(~~) at the ~ginni~g of loading, With the strain at the top of the beam, the curvature of the beam can be obtained as follows: # =(so/:
(5.14)
where co is the strain at the top of the beam, i.e. the strain at y = 0 point. When the moment MA= 1.004MY(MY= 4630.32Nm), the stress on the top of the beam at the beginning of loading is over 40% of the short-term strength of the concrete, i.e. ~(t,)l,_~ > 0.4fL, thus, the concrete on the top of the beam has suffered compressive damage. Based on measured
Creep damage analysis
859
data from high sustained load tests of plain concrete cylinders by Mozer [II] and the method described in Section 3.4 of this paper, the functions expressed in eqs (3.17), (3.18) and (3.20) can be determined as follows: & ((i) = 1*6(alff - 0.4) 80
(at,>
= 114.4(a/ff - 0.4)6
D(o, 0) = 0.27(a/~)z.7.
Based on the data from low sustained load tests of plain concrete by Mozer, the specific creep at time t under a constant stress applied at time r is c(t, z) = 1.22[1 -e-
0.4(-r)] x
lo-6
+
44(e-0.03t
_
e-O.O3l) x
lo-6
and the shrinkage strain is S = 1_38(e-“03’- e-0.03’)x 10-3. Some constants needed in calculation are J; = 35.8 MPa,
E = 21.2 x lo3 MPa,
Es = 1.89 x 10’ MPa,
d = 127 mm,
A, = 128.8 mm’,
b = 76.2 mm,
A comparison between experimental data by Mozer[l I] and calculated curvature histories are given in Table 1 and Fig. 6. In Fig. 6, a satisfactory accuracy of the calculation with experimental data is observed. In addition, at the same time of calculating creep and curvature, we observed that the damage values of concrete at the top of the beam are all less than the damage threshold at failure DR(DR = OS-O.8 for medium- and low-strength concrete). Thus, although the concrete in the tension zone of the beam has cracked and the steel initially yielded, the beam will not fail due to creep under a constant load. The reason is that the concrete in the compressive zone in the beam is not failed.
Table I. Curvature histories and a comparison experimental data[ 111 Time after loading (days)
between them and
Curvature (10e3 rad/mm)
Error (%)
Damage
0.02180 0.05070 0.06503 0.07335 0.07792 0.08039 0.08175
11.22 7.92 3.02 1.84 6.31 6.37 3.37
0.0776 0.2163 0.2789 0.3070 0.3196 0.3253 0.3279
2:: 40 60 80
XTest [ll] -Calculation I 0
20
&
40
,
I
I
t
60
80
100
120
Time ( days ) Fig. 6. Beam curvature histories.
860
LI ZHAOXIA and QIAN JICHENG
6. CONCLUSIONS
(1) In the stage of stable creep, the damage rate decrease with time, and the damage is tending toward a constant below the damage threshold at failure DR with the time increase, thus it will not cause failure; When the sustained stresses are over the long-term strength, the damage rate does not decrease with time, and the damage increase sharply in short time after loading up to the damage threshold DR. (2) It is successful that the evolution equations of creep damage and the constitutive equation of elastic creep coupled with damage are applied to calculating nonlinear creep and damage for plain concrete and reinforced concrete beam. REFERENCES [l] A. M. Frendenthal and F. Roll, Creep and creep recovery of concrete. under high compressive stress, J. ACI., 1114-I 142 (1958). [2] N. H. Arutyunyan, Some Prob1em.r in rhe Theory of Creep. National Press of Technico-Theoretical Literatures (1952) (in Russian). [3] Lin Nanxun, The theory for nonlinear creep of concrete. J. Civil Engng, 16, 14-20 (1983). 141Z. P. Bazant and A. A. Asghari, Constitutive law for nonlinear creep of concrete, J. Engng Me& Dht. H&133-124 (February 1977). [S] Z. P. Bazant and S. S. Kim, Nonlinear creep of concrete-adaptation and flow, J. Engng Mech. Div. 429-446 (June 1979). (61 M. M. Smadi, F. 0. Slate and A. H. Nilson, Time-dependent behavior of high-strength concrete under high sustained compressive stresses. ~~83-227041, NSFICEE-82 142 (November 1982). [7] S. Piechnik and H. Pa&la, The continuous field of damage and its influence on the creep process in concrete under tensile loading. IUTAM, 3rd Synrp. on Creep in Sirttctures, pp. 202-219 (8-12 September 1980). [S] Li Zhaoxia, Analyses of creep damage and nonlinear creep for concrete. Ph.D. thesis presented to Hohai University at Nanjing (1988). 191 _ _ K. E. Loland. Continuous damage model for load-response estimation of concrete. Cem. Concr. Res. 10, 395-402 (1980). :lO] J. Lemaitre, How to use damage mechanics. Nucl. Engng Des. 80, 233-245 (1984). ,111 J. D. Mozer et al., Time-dependent behavior of concrete beams. Proc. ASCE, ST3, No. 96, pp. 597-612 (1970). (Receioed 27 October 1988)