cement and concrete research. vol. 4, pp. 567-579, …montrent que les courbes de fluage donnees...

CEMENT and CONCRETE RESEARCH. Vol. 4, pp. 567-579, 1974. Pergamon Press, Inc. Printed in the United States.

COMPUTATION OF AGE-DEPENDENT RELAXATION SPECTRA

Zden~k P. Ba~ant and Ali Asghari Department of Civil Engineering

Northwestern University Evanston, Illinois 60201, U.S.A.

(Communicated by F. H. Wittmann) (Received March 15, 1974; in f inal form Apri l I , 1974)

ABSTRACT A computer algorithm for the computation of discrete age-dependent relaxation spectra of concrete from creep or relaxation test data is presented and a full FORTRAN IV listing is given. The algorithm is based on a previously outlined method, consisting in the expansion of relaxation curves in series of real exponentials on the basis of a least square criterion. This method is refined herein by imposing suitable smoothing conditions upon the spectra in order to reduce spurious sensitivity of results to small changes in given creep data. Two variants are given; in one the spectrum is approximated as a cubic polynomial in the logarithm of age and the logarithm of the relaxation time, while in the other one a cubic polynomial in the logarithm of age alone is used. Numerical examples show that given smooth creep data can be recovered from the spectra with a negligible error, which demonstrates that the spectra fully characterize creep properties. Apart from being a fundamental characteristic of creep, the relaxation spectra convert a hereditary creep law into a history independent rate-type form, which is requisite for creep analysis of large struc-

tural systems.

Un algorithme pour le calcul ~ l'ordinateur des spectres de relaxation du b~ton a des ages diff~rents est presente, avec un programme

• I I ,

en FORTRAN IV. Les courbes de relaxations sont exprlmees en serle I I

d'exponentielles, utilisant la methode des plus petits carres et im- posant une condition de courbure minimum des spectres, ce qui r~duit la sensibilit~ aux petites variations des donn~es sur le fluage. Les

• l

modules de relaxation sont exprlmes en forme de polynomes cubiques de I .

l'age. Les exemples numerlques montrent que les courbes de fluage I donnees peuvent ~tre obtenues ~ partir des spectres. Les spectres

I . /

llnealre fluage equatzon~ differ- transforment un loi h~r~ditaire " " " de en entielles, ce qui est utile pour le calcul des constructions a grand

nombre d'ineonnues.

567

CEMENT and CONCRETE RESEARCH. Vol. 4, pp. 567-579, 1974. Pergamon Press, Inc. Printed in the United States.

ABSTRACT

COMPUTATION OF AGE-DEPENDENT RELAXATION SPECTRA

Zden~k P. Ba~ant and Ali Asghari Department of Civil Engineering

Northwestern University Evanston, Illinois 60201, U.S.A.

(Communicated by F. H. Wittmann) (Received March 15, 1974; in final form April 1, 1974)

A computer algorithm for the computation of discrete age-dependent relaxation spectra of concrete from creep or relaxation test data is presented and a full FORTRAN IV listing is given. The algorithm is based on a previously outlined method, consisting in the expansion of relaxation curves in series of real exponentials on the basis of a least square criterion. This method is refined herein by imposing suitable smoothing conditions upon the spectra in order to reduce spurious sensitivity of results to small changes in given creep data. Two variants are given; in one the spectrum is approximated as a cubic polynomial in the logarithm of age and the logarithm of the relaxation time, while in the other one a cubic polynomial in the logarithm of age alone is used. Numerical examples show that given smooth creep data can be recovered from the spectra with a negligible error, which demonstrates that the spectra fully characterize creep properties. Apart from being a fundamental characteristic of creep, the relaxation spectra convert a hereditary creep law into a history independent rate-type form, which is requisite for creep analysis of large struc-tural systems.

Un algorithme pour Ie calcul a l'ordinateur des spectres de relaxa-tion du beton a des ages differents est presente, avec un programme en FORTRAN IV. Les courbes de relaxations sont exprimees en serie d'exponentielles, utilisant la methode des plus petits carres et im-posant une condition de courbure minimum des spectres, ce qui reduit la sensibilite aux petites variations des donnees sur Ie fluage. Les modules de relaxation sont exprimes en forme de polynomes cubiques de l'age. Les exemples numeriques montrent que les courbes de fluage donnees peuvent etre obtenues a partir des spectres. Les spectres transforment un loi hereditaire lineaire de fluage en equations diff~rentielles, ce qui est utile pour Ie calcul des constructions a grand nombre d'inconnues.

567

568 Vol. 4, No. 4 COMPUTATION, RELAXATION SPECTRA, CONCRETE

Introduction

In the theory of time-decaying phenomena, the relaxation spectra play an

equally fundamental role as the frequency response spectra do for the periodic

phenomena. This has been recognized long ago in the field of viscoelasticity,

and the relaxation spectra of linearly viscoelastic materials have been exten-

sively studied (1-3). However, all of the classical works have been confined

to time-independent viscoelasticity, which is applicable to polymers but not

to concrete, except when the hydration process ceases as in the case of low

temperatures, completely dried samples, or completely hydrated samples. Re-

laxation spectra of materials with age-dependent properties have been recently

formulated (4,5) and a method of their determination has been outlined.

The aim of this paper is to present a detailed FORTRAN IV listing of the

computational algorithm, in order to facilitate the practical use. In addi-

tion, the method from (5) will be refined by introducing more appropriate

smoothing conditions which reduce the spurious sensitivity of the computed

spectrum to small changes in given creep data.

Review of the Basic Formulation

With the well known limitations and certain simplifying assumptions (4,5)

the creep law of aging concrete may be assumed to be linear, satisfying the

principle of superposition. Then it can be described with any desired accura-

cy by a sufficiently large system of first-order linear differential equations

whose structure can be conveniently visualized by the Maxwell chain model

(Fig. i).

I 2

Since generalization to multiaxial stress is straightforward (6),

3 m n

~~o_ ~ ~ ~ E o E~x I~ T,me Ronge ~ • A

i , : ; ; i

"E I "E 2 T 3 T m

FIG. 1

Maxwell Chain Model and the Associated

Relaxation Spectrum

attention will be restricted to uniaxial stress. The stress-strain law based

on Maxwell chain is then expressed in the form (4,5) n

d =~I op, ~ = ($~ + d /Tp)/Ep(t) (~ = i,... ,n) (I)

in which o = stress, e = s t r a i n , ~p = s t r e s s i n ~ - t h s p r i n g i n F i g . 1 , c a l l e d

568 Vol. 4, No.4 COMPUTATION, RELAXATION SPECTRA, CONCRETE

Introduction

In the theory of time-decaying phenomena, the relaxation spectra play an

equally fundamental role as the frequency response spectra do for the periodic

phenomena. This has been recognized long ago in the field of viscoelasticity,

and the relaxation spectra of linearly viscoelastic materials have been ext en-

sively studied (1-3). However, all of the classical works have been confined

to time-independent viscoelasticity, which is applicable to polymers but not

to concrete, except when the hydration process ceases as in the case of low

temperatures, completely dried samples, or completely hydrated samples. Re-

laxation spectra of materials with age-dependent properties have been recently

formulated (4,5) and a method of their determination has been outlined.

The aim of this paper is to present a detailed FORTRAN IV listing of the

computational algorithm, in order to facilitate the practical use. In addi-

tion, the method from (5) will be refined by introducing more appropriate

smoothing conditions which reduce the spurious sensitivity of the computed

spectrum to small changes in given creep data.

Review of the Basic Formulation

With the well known limitations and certain simplifying assumptions (4,5)

the creep law of aging concrete may be assumed to be linear, satisfying the

principle of superposition. Then it can be described with any desired accura-

cy by a sufficiently large system of first-order linear differential equations

whose structure can be conveniently visualized by the Maxwell chain model

(Fig. 1). Since generalization to multiaxial stress is straightforward (6),

:

c:r I 23m n

®@~'~&H Ep. c:r En r- --- Time Range

of l:lterest

FIG. 1

Maxwell Chain Model and the Associated

Relaxation Spectrum

attention will be restricted to uniaxial stress. The stress-strain law based

on Maxwell chain is then expressed in the form (4,5 ) n

CY = I CY)1 , E (a)1 + CY)1h)1)/E)1 (t) ()1 = 1, ... ,n) (1) )1=1

in which CY stress, E strain, CY)1 stress in )1-th spring in Fig. 1, called

Vol. 4, No. 4 569 COMPUTATION, RELAXATION SPECTRA, CONCRETE

hidden stress (or internal variable), ~ = relaxation time associated with

the ~-th spring, E~ = elastic modulus of the ~-th spring, depending on time t

which represents the age of concrete. Superimposed dots represent time deriv-

atives, e.g. ~ = d~/dt. Shrinkage strain is not included.

Eq. (i) belongs to the class of the so-called rate-type creep laws,

which have recently appeared to be preferable to integral-type creep laws

involving hereditary integrals, for several reasons. Firs~ thermodynamics of

time-dependent phenomena is much simpler in terms of current values of state

variables, including the hidden variables (~), than it is in terms of the

time-histories of stress and strain. Second, in numerical analysis of large

structural systems, the storage space in computer memory which is required

for the rate-type creep laws is only a fraction of that required for integral-

type laws. Third, a rate-type creep law allows establishing a correlation

with the physical theory of the processes in the microstructure, which can

provide valuable additional information (e.g. on the form in which tempera-

ture and water content should appear in the creep law (7,8)).

The plot of Ep versus log rp, which is discrete and is called relaxation

spectrum, depends on age t. The actual spectrum of the material is, of

course, essentially continuous, and the Ep- values are its approximate dis-

crete representation. Taking this point of view, it becomes clear that the

continuous spectrum can be approximated equally well for any set of ~U-values

which are sufficiently densely distributed in log-time over the whole range of

interest (e.g., the spectrum given by dashed lines in Fig. i is equivalent).

Therefore, one should not try to determine rp from the creep data (and indeed,

if attempted, a problem with a non-unique solution would result (ii~. Rather,

the T -values must be chosen. In particular they may be chosen time-constant

and a suitable choice is (5)

= T I i0 p-I (~ = l,...,n- i), • = ~ (~ = n) (2) n

where the last, infinite relaxation time is a convenient form of expressing the

fact that the last spring in the chain has no dashpot attached to it.

Eqs. (i) may be easily integrated when s(t) is a step function, i.e.,

= i for t ~ t', ~ = 0 for t < t' The corresponding o represents the relax-

ation function ER(t,t' ) and has the form

n -(t-t' ) /~ ER(t,t' ) = ~ E (t')e (3)

~=i P

Vol. 4, No.4 569 COMPUTATION, RELAXATION SPECTRA, CONCRETE

hidden stress (or internal variable), T~ = relaxation time associated with

the ~-th spring, E~ = elastic modulus of the ~-th spring, depending on time t

which represents the age of concrete. Superimposed dots represent time deriv-

atives, e.g. E = dE/dt. Shrinkage strain is not included.

Eq. (1) belongs to the class of the so-called rate-type creep laws,

which have recently appeared to be preferable to integral-type creep laws

involving hereditary integrals, for several reasons. Firs~ thermodynamics of

time-dependent phenomena is much simpler in terms of current values of state

variables, including the hidden variables (0~), than it is in terms of the

time-histories of stress and strain. Second, in numerical analysis of large

structural systems, the storage space in computer memory which is required

for the rate-type creep laws is only a fraction of that required for integral-

type laws. Third, a rate-type creep law allows establishing a correlation

with the physical theory of the processes in the microstructure, which can

provide valuable additional information (e.g. on the form in which tempera-

ture and water content should appear in the creep law (7,8)).

The plot of E~ versus log T~, which is discrete and is called relaxation

spectrum, depends on age t. The actual spectrum of the material is, of

course, essentially continuous, and the E~- values are its approximate dis-

crete representation. Taking this point of view, it becomes clear that the

continuous spectrum can be approximated equally well for any set of T~-values

which are sufficiently densely distributed in log-time over the whole range of

interest (e.g., the spectrum given by dashed lines in Fig. 1 is equivalent).

Therefore, one should not try to determine T~ from the creep data (and indeed,

if attempted, a problem with a non-unique solution would result (11)>. Rather,

the T~-values must be chosen. In particular they may be chosen time-constant

and a suitable choice is (5 )

T = T1 1O~-1 (~ = l, ... ,n-l), T (~ = n) (2 )

~ n

where the last, infinite relaxation time is a convenient form of expressing the

fact that the last spring in the chain has no dashpot attached to it.

Eqs. (1) may be easily integrated when £(t) is a step function, i.e. ,

E = 1 for t ;:: t' E = 0 , for t < t' . The corresponding 0 represents the relax-ation function ER(t,t') and has the form

n

L ~=l

-(t-t') /'T E (t')e u.

fl (3)


This is a series of real exponentials, called Dirichlet series. It is a

counterpart of the series of imaginary exponentials, or Fourier series,

arising in the study of periodic phenomena.

The time-dependent properties of concrete are usually characterized in

terms of the creep function, J(t,t'), which represents strain s at time t

caused by a constant unit stress acting since time t'. The relaxation func-

tion, ER(t,t'), may be computed from given J(t,t') by an algorithm which was

presented in (9). Later a more efficient version of this algorithm was pro-

posed in (i0). A FORTRAN IV subroutine RELAX, based on (i0), is presented in

the Appendix, because a conversion of J(t,t') into ER(t,t') is a necessary

step toward determining the relaxation spectra, unless sufficient data from

stress relaxation tests are available.

Function ER(t,t') which describes given test data is numerically charac-

= ER(t ~ + ~r,t~)~ where t'~ = chosen discrete terized by discrete values ERr s

at the start of relaxation, tr = chosen discrete times elapsed from the ages

start of relaxation. These discrete values must be sufficiently densely dis-

tributed in both log t'-~ and log tr-scales. A uniform distribution with

three ~r-values per log i0 and two t'-values per log i0 is normally suffi-

' and tr must completely cover the whole time range of interest. cient. Both t~

Using the method of least squares, the values E~ = E (t~) (for ~ =

l,...,n; ~ = 1,2,...,n ) may be found by minimizing the expression

n -tr/T ~ mR ) 2 = ~ ~( ~ E e - = Min. (4) r ~-i ~ r~

However, fitting of given data by Dirichlet series appears to be much more

difficult than it is in the case of Fourier series (ii). The difficulties

are due to the fact that an ill-conditioned system of linear equations arises

and that very different E -values can represent the same data nearly b

equally well, i.e., coefficients E~ are unstable functions of the data.

Identification of Relaxation Moduli from Given Relaxation Data

Method i. Improved Version of the Method from Ref. (5)

The afore-mentioned numerical difficulties have been circumvented in (5)

by imposing the condition that the E~-values as functions of ~ must be smooth,

which seems to be a physically plausible requirement. Using the least square

method, this is mathematically expressed by adding to ~ in Eq. (4) a sum of

squares of the first and second differences of E as functions of ~; see

570 Vo 1. 4, No. 4 COMPUTATION, RELAXATION SPECTRA, CONCRETE

This is a series of real exponentials, called Dirichlet series. It is a

counterpart of the series of imaginary exponentials, or Fourier series,

arising in the study of periodic phenomena.

The time-dependent properties of concrete are usually characterized in

terms of the creep function, J(t,t'), which represents strain E at time t

caused by a constant unit stress acting since time t'. The relaxation func-

tion, ER(t,t'), may be computed from given J(t,t') by an algorithm which was

presented in (9). Later a more efficient version of this algorithm was pro-

posed in (10). A FORTRAN IV subroutine RELAX, based on (10), is presented in

the Appendix, because a conversion of J(t,t') into ER(t,t') is a necessary

step toward determining the relaxation spectra, unless sufficient data from

stress relaxation tests are available.

Function ER(t,t') which describes given test data is numerically charac-

terized by discrete values ER = ER(t' + t ,t'), where t' = chosen discrete ra a r a a

ages at the start of relaxation, tr = chosen discrete times elapsed from the

start of relaxation. These discrete values must be sufficiently densely dis-

tributed in both log t~- and log tr-scales. A uniform distribution with

three tr-values per log 10 and two t~-values per log 10 is normally suffi-

cient. Both t~ and tr must completely cover the whole time range of interest.

Using the method of least squares, the values E~ = E (t~) (for ~ = a ~

1, •.. ,n; a 1,2, ... ,n) may be found by minimizing the expression a

-t /T' E e r ~ ~a

Min. (4)

However, fitting of given data by Dirichlet series appears to be much more

difficult than it is in the case o( Fourier series (11). The difficulties

are due to the fact that an ill-conditioned system of linear equations arises

and that very different E -values can represent the same data nearly I-L

equally well, i.e., coefficients E~ are unstable functions of the data.

Identification of Relaxation Moduli from Given Relaxation Data

Method 1. Improved Version of the Method from Ref. (5)

The afore-mentioned numerical difficulties have been circumvented in (5)

by imposing the condition that the E~-values as functions of ~ must be smooth,

which seems to be a physically plausible requirement. Using the least square

method, this is mathematically expressed by adding to ¢ in Eq. (4) a sum of

squares of the first and second differences of E as functions of ~; see ~


Eq. 8 from Ref. (5). In computing the fits of the extensive typical creep

data in (5) it was tacitly assumed that E~ had to be a smooth function of

up to the last spring ~ = n which is attached to no dashpot (~n = =)" Al-

though close fits of data were obtained in (5), it has later been found that

this assumption was unjustified and was responsible for the fact that similar

creep data yielded very different E~-vaiues (see Table 2 of (5)). Correctly,

the last spring modulus, Ep = En, can have no relationship to the preceding

spring moduli En_l, En_2, etc.,for the following reason. First one should

note that for the elapsed times t - t' that are less than T~/10, the displace-

ment that occurs in the p-th dashpot, as well as in the dashpots number p + i,

p + 2, etc., is insignificant and the dashpots are equivalent to rigid cou-

pling. The actual continuous relaxation spectrum normally extends far beyond

the range of interest (Fig. i), but all the dashpots to the right of this

range may be viewed as rigid couplings. Thus, the set of springs to the right

of the range is equivalent to a single spring (Fig. i), and the last spring

modulus in the chain, En, should be regarded as a sum of the moduli of all

springs to the right of the (n - l)st spring in the actual model for an un-

limited time range (Fig. i). Hence, no smooth change from En_ 1 to E n may be

enforced. (Note that on the left of Maxwell chain the situation is differ-

ent; the stress in all dashpots and springs to the left of the time range of

interest is virtually zero and therefore all springs to the left of the range

may be neglected.)

For other details the discussion in Ref. (5) is valid. A FORTRAN IV

subroutine for this method, called MAXWLI, is provided in the Appendix, and

ample comments within the program are inserted for clarity. After computing

all E~ -values, the dependence of E (t') upon log t' is fitted by a cubic

polynomial:

Ep(t') = Clp + C2px + C3 x2 + C4px 3, x = log t ' (5)

Method 2. Direct Identification of a Polynomial for Relaxation Moduli

The condition of smooth dependence of E~ upon p may be also enforced if

E is assumed in the form of a low-degree polynomial. A cubic polynomial P

appears to be suitable for concrete in the time range of t- t' from 0.001 day

to i0,000 days. A cubic polynomial is also suitable for the dependence on

log t', so that E~ may be assumed as full two-dimensional cubic polynomial,


Eq. 8 from Ref. (5). In computing the fits of the extensive typical creep

data in (5) it was tacitly assumed that E~ had to be a smooth function of ~

up to the last spring V = n which is attached to no dashpot (Tn = 00). Al-though close fits of data were obtained in (5), it has later been found that

this assumption was unjustified and was responsible for the fact that similar

creep data yielded very different E~-values (see Table 2 of (5». Correctly,

the last spring modulus, E~ = En, can have no relationship to the preceding spring moduli ~-l' En- 2 , etc.,for the following reason. First one should

note that for the elapsed times t-t' that are less than T~/lO, the displace-

ment that occurs in the ~-th dashpot, as well as in the dashpots number ~+l,

~+2, etc., is insignificant and the dashpots are equivalent to rigid cou-

pling. The actual continuous relaxation spectrum normally extends far beyond

the range of interest (Fig. 1), but all the dashpots to the right of this

range may be viewed as rigid couplings. Thus, the set of springs to the right

of the range is equivalent to a single spring (Fig. 1), and the last spring

modulus in the chain, ~, should be regarded as a sum of the moduli of all

springs to the right of the (n-l)st spring in the actual model for an un-

limited time range (Fig. 1). Hence, no smooth change from En- l to En may be

enforced. (Note that on the left of Maxwell chain the situation is differ-

ent; the stress in all dashpots and springs to the left of the time range of

interest is virtually zero and therefore all springs to the left of the range

may be neglected.)

For other details the discussion in Ref. (5) is valid. A FORTRAN IV

subroutine for this method, called MAXWLl, is provided in the Appendix, and

ample comments within the program are inserted for clarity. After computing

all E~a-values, the dependence of E~(t') upon log t' is fitted by a cubic

polynomial:

E (t') ~

Method 2. Direct Identification of a Polynomial for Relaxation Moduli

(5)

The condition of smooth dependence of E~ upon ~ may be also enforced if

E is assumed in the form of a low-degree polynomial. A cubic polynomial ~

appears to be suitable for concrete in the time range of t- t' from 0.001 day

to 10,000 days. A cubic polynomial is also suitable for the dependence on

log t', so that E~ may be assumed as full two-dimensional cubic polynomial,


E ( t ' ) = a 1 +a2~ +a3x+a4p2 +a5]Jx+a6x2 +a7p3 +a8P2 x+a9.~x2 +alOX3 10

= ~ ajfj(p,x), x = log t' (for ~ < n). j=l

(6)

According to the preceding discussion, this polynomial may not be extended to

the last spring of the chain, p = n.

From experience it has been found that the first discrete time tl from

loading should be about 0.2 TI, and the last one, ~N, about Zn-l" Setting

~N = Tn-i 'exactly' Eq. (3) for t = t' + tN yields

ER(t~+ tN't~)' = ERN~ = Eme e-I + En' (m = n- i) (7)

where all terms of the sum but the last two have been neglected since the -i0

exponentials have values less than e = 0.000045. Eq. (7) yields

-I = - E m e (m = n-l). E n ( t ' ) ERN e (8)

This provides a relation between E n and En_ 1 that should be satisfied a

priori. Thus the end values ERN ~ of stress relaxation are related to the

last two spring moduli E n and En_ 1 and, in conformity with Eq. (6), ERN ~

should be approximated as a cubic polynomial

ERN ~ = b I + b2x e + b3x ~ + b4x i x = log t'. (9) e

Now, expressing E n by means of E m and ER~ N from Eqs. (8) and (7), and substi-

tuting expression (6) into Eq. (i), one obtains

m 4

= ~ E e$~r + ~ bkxk-i (I0) ERrs D=I k=l e

-tr/Tu for p < m, -tr/Tp -i for p = m. (ii) where SPr = e SPr = e - e

Insertion of (6) into (I0) further provides

m i0 4 k-i

ER = I ~ a.H. + ~ bkX e r e ~=i j=l J Jar k=l

(12)

m

where Hj = ~ fj(p,x ) er p=l $~r"

(13)

Substituting Eq. (9) into the least-square expression (4) furnishes

I0 4 I0

@ = ~ f[ ~ H. a. + - f b k + f w.a2 = Min r j=l Jar j k=l j=l J j

(14)

572 Vol. 4, No.4

E (t') lJ

COMPUTATION, RELAXATION SPECTRA, CONCRETE

a +a2lJ+a x+a4lJ2+aslJx+a6x2+a7lJ3+aslJ2x+a9ilx2+a10xJ

10

L j=l

a.f.(lJ,x), J J

x = log t' (for lJ < n). (6)

According to the preceding discussion, this polynomial may not be extended to

the last spring of the chain, lJ = n.

From experience it has been found that the first discrete time tl from

loading should be about 0.2 '1' and the last one, tN, about Tn-I' Setting

tN = 'n_l,exactly, Eq. (3) for t = t' + tN yields

• -1 ER(t'+ tN,t') = ER = ~ e + E, (m = n-l)

a a Na a n

where all terms of the sum but the last two have been neglected since the

exponentials have values less than e-10

0.000045. Eq. (7) yields

-1 E (t') = ER - Em e

n Na a (m = n-l).

This provides a relation between En and En-l that should be satisfied a

priori. Thus the end values ER of stress relaxation are related to the Na

last two spring moduli ~ and En - l and, in conformity with Eq. (6), ERNa

should be approximated as a cubic polynomial

ER = b 1 + b 2x N + b x2 + b x 3

Na u 3 a 4 a' x = log t'.

a a

(7)

(8)

(9 )

Now, expressing En by means of Em and ER from Eqs. (8) and (7), and substi-aN

tuting expression (6) into Eq. (1), one obtains

where ¢lJ r

Insertion of

where Hj ar

e

m 4

L lJ ¢lJ L k-l

E E + bkxa Rra lJ=l a r k=l

-trhlJ for lJ

-trhlJ < m, ¢)1 e

r

(6) into (10) further provides

m 10 4

L L L k-l ER a.H. + bkxa r lJ=l j=l J Jar k=l a m

L f.()1,x)¢ . )1=1 J a)1r

-1 e for lJ m.

Substituting Eq. (9) into the least-square expression (4) furnishes

10 4

L L( L a r j=l

-H. a. + E - L Jar J ~a k=l

Hin

(10)

(11)

(12 )

(13)

(14)


where the sum with chosen weights w. has been added to achieve a reduction of J

slopes and curvatures of the relaxation spectrum, as a possible further

smoothing device. Applying the minimization conditions ~¢/~a. = 0 i

(i = I,...,i0), one obtains a system of ten linear equations,

i0 A. a = B (i = i, ,i0) (15)

j=l zj j i "'"

in which 4

Aij = ~ ~ H i H. + B i = ~ R - ~ bk )Hi ' r ~r J~r ~ r r~ k=l ~r

~.. = Kronecker delta. z]

The computation may now proceed as follows. First one fits the ER~ N- !

values for all t~ by the polynomial (8). According to the method of least

squares, coefficients bl,.., b 4 of this polynomial are solved from the

equations 4

k+~-2 , ! ER NX~-i ~gb~ = B~ (k= i,...,4), with ~% = ~ x , Bg = (17) ~=i

Subsequently coefficients Aij and B i are evaluated and parameters a.l are

solved from (15). To check how close is the approximation of given data

ERr, expression (12) may be evaluated.

A FORTRAN IV subroutine which implements the above algorithm, named

MAXWL2, is listed in the Appendix. Comments within the program should be

sufficient for easy use. After the polynomials for E~(t') are obtained, one

should integrate differential equations (i) to obtain the creep curves which

correspond to E~(t'), and compare them with the original creep curves,

J(t,t'). This integration cannot be done by standard step-by-step algorithms

because an increase of the time step with time would cause numerical insta-

bility. However, a general algorithm for efficient integration of structural

creep problems with a creep law in the form of Eq. (i) has been presented in

Ref. (5). Subroutine CRCURV applying this algorithm to the present problem

is listed in the Appendix. For the underlying mathematics see Eqs. 15 -19

from (5).

Choice of Method

Method i does not impose any specific analytical form on the spectrum

and is thus suitable for obtaining little distorted shapes of relaxation

spectra. Method 2 smoothes the spectra by a polynomial which gives more

distorted shapes of the spectra but leads to fewer material parameters and


where the sum with chosen weights w. has been added to achieve a reduction of J

slopes and curvatures of the relaxation spectrum, as a possible further

smoothing device. Applying the minimization conditions a~/aa. = 0 1

(i = 1, ... ,10), one obtains ~ system of ten linear equations,

in which

A .. 1J a r

10

I j=l

A .. a 1J j

H. H. + o .. w., 1ar Jar 1J 1

0 .. = Kronecker delta. 1J

B. 1

B. 1

(i = 1, ... ,10) (15)

(16)

The computation may now proceed as follows. First one fits the E -RaN

values for all t~ by the polynomial (8). According to the method of least

squares, coefficients b 1 , ••• b 4 of this polynomial are solved from the

equations 4

I Ak,Q,b,Q, ,Q,=l

B' k (k = 1, ..• ,4), wi th Ak,Q, a a

Subsequently coefficients Aij and Bi are evaluated and parameters a i are

solved from (15). To check how close is the approximation of given data

ER ,expression (12) may be evaluated. ra

(17)

A FORTRAN IV subroutine which implements the above algorithm, named

MAXWL2, is listed in the Appendix. Comments within the program should be

sufficient for easy use. After the polynomials for E~(t') are obtained, one

should integrate differential equations (1) to obtain the creep curves which

correspond to E~(t'), and compare them with the original creep curves,

J(t,t'). This integration cannot be done by standard step-by-step algorithms

because an increase of the time step with time would cause numerical insta-

bility. However, a general algorithm for efficient integration of structural

creep problems with a creep law in the form of Eq. (1) has been presented in

Ref. (5). Subroutine CRCURV applying this algorithm to the present problem

is listed in the Appendix. For the underlying mathematics see Eqs. 15 - 19

from (5).

Choice of Method

Method 1 does not impose any specific analytical form on the spectrum

and is thus suitable for obtaining little distorted shapes of relaxation

spectra. Method 2 smoothes the spectra by a polynomial which gives more

distorted shapes of the spectra but leads to fewer material parameters and


? 05 o c

%_ 0.4

0.3

/ / 0.7 0.7 - Given data

- From spectrum (method I) / j /~ 0.6 cubic in log t' 0nly ~ / 0.6

,//'// 0_00. 5 ,/

/~" xx

o

O A r L I ~ ~ i ~

I0 -s I0 -a I0 -I I I 0 10 2 I0 s 10 4

t - t ' in days

Fig. 2

Given creep curves and those recovered from spectrum in Fig. 4

-- Given data //

....... From spectrum (method 2) / ' cubic in log t' and log r/~

......... Same, but quadratic in ." log t' and log r/~ .//

/ / "~--0.4

~0 ~ .

0.2 .... ~ ~°Is "" ~ ~

I0 -3 I0 -a I0 -I I I 0 I 0 z I 0 s 104

t - t ' in days

Fig. 3


lessens the dependency of the spectra on small random components of given

creep data. Either case may be preferable, depending on the nature of

application. The cost of computing the spectra is minimal (about $i on

CDC 6400, at current rates).

Numerical Examples

Figs. 2 and 3 show by solid lines smoothed creep data of L'Hermite and

Mamillan (12),which are compactly presented in Fi~8, Ref. (5). The relaxa-

tion spectra computed from these data by Method 1 (MAXWLI, with weights w I =

.Ol, w 2 = .08) appear in Fig. 4 and those computed by Method 2 (MAXWL2, with

weights w i given in the comments in the program) appear in Fig. 5. The creep

curves integrated back from the relaxation spectra in Figs. 4 and 5 are in-

dicated in Figs. 2 and 3 by dashed lines. The fits are deemed to be satis-

factory. Closer fits can be obtained, of course, by choosing a denser dis-

tribution of T~-values and/or higher order polynomials. On the other hand,

even a fit obtained by Method 2 with a quadratic polynomial (which may be

computed by MAXWL2 if one assigns to w7,w8,w9,wl0 a very large value, 10 30 )

is acceptable (see dotted lines in Fig~ 3).

574 Vol.4,No.4

0.7

0.6

COMPUTATION, RELAXATION SPECTRA, CONCRETE

-- -- Given data

- - - -- From spectrum (method I) cubic in log t' only

7 1

I / 1

07

0.6

-- --- - Given data

- - From spectrum (method 2) cubic in log t' and log r J-L

.. Same, but quadratic in log t' and log r J-L

? j

I /1

':.... 0.4 /

/ 0.3

0.2

0.1 '--_--'-_--'-__ >--_--'-_--'-__ .L..._--'-- 0.1 '--_..L-_-'--__ '--_..L-_---L __ L-_--L-10-~ 10-2 10-1 10 102 10~ 104 10-~ 10-2 10-1

t - t' in days

Fig. 2


10

t-t' in days

Fig. 3


lessens the dependency of the spectra on small random components of given

creep data. Either case may be preferable, depending on the nature of

application. The cost of computing the spectra is minimal (about $1 on

CDC 6400, at current rates).

Numerical Examples

Figs. 2 and 3 show by solid lines smoothed creep data of L'Hermite and

Mamillan (12), which are compactly presented in Fig.8, Ref. (5). The relaxa-

tion spectra computed from these data by Method 1 (MAXWL1, with weights WI

.01, w2 = .08) appear in Fig. 4 and those computed by Method 2 (MAXWL2, with

weights Wi given in the comments in the program) appear in Fig. 5. The creep

curves integrated back from the relaxation spectra in Figs. 4 and 5 are in-

dicated in Figs. 2 and 3 by dashed lines. The fits are deemed to be satis-

factory. Closer fits can be obtained, of course, by choosing a denser dis-

tribution of T~-values and/or higher order polynomials. On the other hand,

even a fit obtained by Method 2 with a quadratic polynomial (which may be

computed by MAXWL2 if one assigns to w7,w S'w 9 ,w 10 a very large value, 1030)

is acceptable (see dotted lines in Fig .• 3).

Vo] . 4 , No. 4 575 COMPUTATION, RELAXATION SPECTRA, CONCRETE

.~ 1.0

% .E

0.5

Method i r~=r, x I0 P'- I ~ ,'==5000~oy, I 0 r~ -0004642 day, n-8 ~ / ~ ~ Er~ [ I End of spectrum / ] / , 1 0

t . s ~ / . # " A I ~°Y' ' I / I ,..~oo .... = .50 = ~ - - Z X ~ y s ~ ~ ,,~ ~o " ~

, o 5 ~ . . ~ r - ~ o o ~ ~ ! ' !,.~,~o.~,~ o.

I ' "'i [ ! 5 (En~ E7÷E6 ) I ~ ', [ I , I i I , ~ _

I 0 i 0 z i 0 s 10 4 i0 ~ i0 e m

r~/rl

Fig. 4

Relaxation Spectrum obtained by Method 1

Method 2 r = r x lO P'-I / , ' :5oooaa~

,,:'ooo . . . .

o 1/5(En~E 7 )

~o'J = , I o

/ - , I [ I E I ~'l E2 E3 E4 E5 ~6 E7

I 0 I 0 z 10 3 10 4 I 0 ~ I 0 e ' Qo

Fig. 5

Relaxation Spectrum obtained by Method 2

Time Range Limitation of Spectra at Various Ages

(t') for times ~ which are greater than about 30t' Moduli E

are irrelevant for the relaxation spectrum at age t', because at that

age the loading duration cannot exceed t', so that no appreciable de-

formation rate can occur in the corresponding dashpots at age t'

Therefore, these moduli E may be all lumped into one spring together

with E (t'), which has been done in Figs. 4 and 5= (This lumping, how- n

ever, need not be actually implemented in Eq. 6, in order to simplify

programming.) The physically meaningful relaxation spectrum at age t'

ends at log T ~ log (30t') and the E -values for higher log • are

meaningless; these portions of spectra are not shown in Figures 4

and 5.

It is also noteworthy that all E -values are positive at all times,

which is a condition that must be satisfied because of thermodynamic re-

strictions.

Acknowledgment

Support by the U.S. National Science Foundation under Grant No. GK-26030

is gratefully acknowledged.

References

i. M. L. Williams, The Structural Analysis of Viscoelastic Materials, AIAA Journal 2, 785 (1964).

Vo 1. 4, No. 4 575 COMPUTATION, RELAXATION SPECTRA, CONCRETE

Method i Method 2 '" 1.0 Til-' T, x '011-- 1 "=g5000dOYS._ 1.0 Til- = T, x 1011-- 1 " =~OOOda)'S 'iii 0-

"'Q

,£

::-w

05

0

! En ~ 51 .. ~

,'::.- 500 days :::.

1/5 ~n -:t.. ,'=50days W

! 1/5(~n.E7) 0.5

T, • 0,004642 day. n' 8 [ • End of spectrum

t = 5dayS 50 days

I i ,

E, ~< E4 Es E6 !

104 lOS

, 1 I

E, E2 E, E4 ~s ~6 ~7 i

! 10 2 10' 104 lOS 106

I ,'" 5 days

L...--_-k._--L-::-_7_--"-::"_---'--=-_E"-;' 7,-?l t,· 0 106


2. T. M. Cost, Approximate Laplace Transform Inversion in Viscoelastic Stress Analysis, AIAA Journal 2, 2157 (1964).

3. J. D. Ferry, Viscoelastic Properties of Polymers, 2nd ed., J. Wiley, New York (1970).

4. Z. P. Ba~ant, S. T. Wu, Dirichlet Series Creep Function for Aging Concrete, J. Eng. Mech. Div., Proc. Am. Soc. Civil Engrs. 99, 367 (1973).

5. Z. P. Ba~ant, S. T. Wu, Rate-Type Creep Law of Aging Concrete Based on Maxwell Chain, Materials and Structures 7 (Jan.-Feb. 1974), No. 37.

6. Z. P. Ba~ant, Theory of Creep and Shrinkage of Concrete: A PrEcis of Recent Developments, Mechanics Today, Vol. 2, Pergamon Press (in press).

7. Z. P. Ba~ant, S. T. Wu, Thermoviscoelasticity of Aging Concrete, J. Eng. Mech. Div., Proc. Am. Soc. Civil Engr. i00 (1974) (in press); also as ASCE Meeting Preprint 2110, Oct. 1973.

8. Z. P. Ba~ant, S. T. Wu, Creep and Shrinkage Law of Concrete at Variable Humidity, J. Eng. Mech. Div., Proc. ASCE (under review)°

9. Z. P. Ba~ant, S. T. Wu, Numerical Determination of Long-Range Stress History from Strain History in Concrete, Materials and Structures 5, 135 (1972).

i0. Z. P. Ba~ant, L. J. Najjar, Comparison of Approximate Linear Methods for Concrete Creep, J. Struct. Div., Proc. Am. Soc. Civil Engrs. 99, 1851 (1973).

ii. C. Lanczos, Applied Analysis, Prentice-Hall, Englewood Cliffs, N.J. (1964) (see pp. 272-280).

12o R. L'Hermite, M. Mamillan, C. Lef~vre, Nouveaux resultats de recherches sur la deformatlon et la rupture du beton, Annales de l'Institut Tech- nique du B~timent et des Travaux Publics, Vol. 18, No. 207-208, 325- 360 (1965).

576 Vol. 4, No.4 COMPUTATION, RELAXATION SPECTRA, CONCRETE

2. T. M. Cost, Approximate Laplace Transform Inversion in Viscoelastic Stress Analysis, AIAA Journal ~, 2157 (1964).

3. J. D. Ferry, Viscoelastic Properties of Polymers, 2nd ed., J. Wiley, New York (1970).

4. Z. P. Bazant, S. T. Wu, Dirichlet Series Creep Function for Aging Concrete, J. Eng. Mech. Div., Proc. Am. Soc. Civil Engrs. 22-,367 (1973).

5. Z. P. Bazant, S. T. Wu, Rate-Type Creep Law of Aging Concrete Based on Maxwell Chain, Materials and Structures I (Jan.-Feb. 1974), No. 37.

6. Z. P. Bazant, Theory of Creep and Shrinkage of Concrete: A Precis of Recent Developments, Mechanics Today, Vol. 2, Pergamon Press (in press).

7. Z. P. Bazant, S. T. Wu, Thermoviscoelasticity of Aging Concrete, J. Eng. Mech. Div., Proc. Am. Soc. Civil Engr. 100 (1974) (in press); also as ASCE Meeting Preprint 2110, Oct. 1973.

8. Z. P. Bazant, S. T. Wu, Creep and Shrinkage Law of Concrete at Variable Humidity, J. Eng. Mech. Div., Proc. ASCE (under review),

9. Z. P. Bazant, S. T. Wu, Numerical Determination of Long-Range Stress History from Strain History in Concrete, Materials and Structures 2, 135 (1972).

10. Z. P. Bazant, L. J. Najjar, Comparison of Approximate Linear Methods for Concrete Creep, J. Struct. Div., Proc. Am. Soc. Civil Engrs. ~, 1851 (1973) .

11. C. Lanczos, Applied Analysis, Prentice-Hall, Englewood Cliffs, N.J. (1964) (see pp. 272-280).

12. R. L'Hermite, M. Mamillan, C. Lefevre, Nouveaux r~sultats de recherches sur la deformation et la rupture du beton, Annales de l'Institut Tech-nique du Batiment et des Travaux Publics, Vol. 18, No. 207-208, 325-360 (1965).

APPENDIX-FORTRAN IV SUBROUTINES

SUBROUTINE RELAX (NDEC,JOEC,NA,JA,OOEF,T,TTtTP,TPL,NT,ER) C COMPUTES STRESS RESPONSE TO A GIVEN STRAIN HISTORY, AND RELAXATION C FUNCTION IN PARTICULAR. C INPUT- NDEC=NO. OF DECADES OF TIME I~ LOG, SCALE* JDEC=NO, OF STEPS C PER LOG(IO), NA=NO, OF AGES AT LOADING, JA=NO. OF AGES PER LOG(IO), C TII)=FIRST DISCRETE TIME, TPIII:FIRST AGE AT LOADING, DDEFIIT)=PRES- C CRIBEQ STRAIN INCKEMENT PER STEP FROM IT '1 TO IT,MINUS PRESCRIBED C SHRINKAGE INCREMENT PER STEP,(FOR RELAXATION ODEF(1)=I.,ALL OTHER C DOEF(IT)=O,) ( IF SHRINKAGE IS CONSIDERED,INSERT DEPP=DEPP'OSHR(IT) C AFTER STATEMENT NO.B0, OSHRIITI=SUPPLIEO SHRINKAGE INCREMENTS,) C SUBROUTINE CREEP FOR COMPUTING VALUES OF CREEP FUNCTION FROM T AND TP C MUST BE SUPPLIED(SEE TABLE Z OF REF,9), C OUTPUT- ER|IT, IA)=OISCRETE VALUES OF STRESS RELAXATION FUNCTION* C T(IT) AND TP(IA|:DISCRETE TIMES FROP LOADING AND DISCRETE AGES AT C LOADING, NT:NO, OF DISCRETE TINES. C DIMENSION MUST BE AT LEAST OSTR(NTItDOEF(NT)BTINTI,TT(NT),TP(NAI, C TPL(NA),ER(NT,NA)

OIMENSION D S T R I 3 0 ) , D O E F ( 3 0 ) , T ( 3 0 ) , T T ( 3 O I , T P ( B ) , T P L ( 8 ) , E R ( 3 0 , 8 ) XX:tO.~(1,1FLOAT(JA)I DO 11 IA:?,NA TP(IA)=TP(IA-I)~XX

11 TPL(IA)=ALOGtO(TP(IA)) T P L I 1 ) : A L O G I O ( T P ( I I ) DTR:IO,~(I,IFLOAT(JDEC)) NT:NDEC~JDEC DO 9 IT:?,NT TIIT)=T(IT-I I~DTR

9 T T I I T ) = S Q R T I T I I T I ' T ( I T - 1 ) ) PRINT 16, (TP(IA)~ IA=I,NA)

16 FORMATI7XIHTP( IA) : lOE1Z,~ / I tWXIOE%2,4) t DO 50 IA:i,NA

C COMPUTE THE FIRST STRESS VALUE USING EFFECTIVE MODULUS. DSTR(t)=ODEF(1)ICREEP(T(I),TP(IA)) ERII,IAI=OSTR(1) DO 40 I T : ~ , N T I I = I T - i DEPP=O, DO 30 J T = t , I i

30 DEPP=OEPP+DST~(JT)*ICREEP(T(ITI,TP(I A)÷TT(JT))'CREEP(T(I1)' I T P ( I A ) * T T ( J T ) ) )

C DSTR(JT)=ST~ESS INCREMENT (FROM JT-I TO JT ) , DEPP=INELASTIC C STRAIN INCREMENT,

DSTR(IT)={ODEFIITI-DEPP)ICREEP(T(IT),TP(IA)÷TT(IT)) 40 ER(IT,IA)=ER(II,IA)*OSTR(IT) SO CONTINUE

PRINT 52 52 FORNAT(4OHOIT I ( I T ) ER(IT,1) ~ R ( I T , 2 ) , , . )

DO 53 IT=I,NT 53 PRINT 54, IT, T(IT} .ER(IT,IA), IA=I,NA) 54 F O R M A T ( 1 X I 2 , F l Z , 3 , 1 J E I Z . 4 / ( 1 3 X l O E I Z , 4 ) )

RETURN END

SUBROUTINE MAXWLI(ER,TtTT,TPtTPL~NOEC,JDEC,NT,NA,W1,W2,TAU,ENU, IC,NHU)

C COMPUTES RELAXATION SPECTRA (VALUES OF ELASTIC MOOULI EMU IN MAXWELL C CHAIN) FOR VARIOUS AGES AT LOADINGtTP(IA). NO FORMULA FOR EMU IS C ASSUMED AT FIRST BUT INTERPOLATING CUEIC POLYNOMIALS IN AGE TP AT C LOADING ARE DETERMINED AT THE END FOR ALL MU, C INPUT- ERIIT,IA),T(IT)tTT(ITI,TP(IA),TPL(IAt,NDEC,JDEC,NT,NA FROM C RELAX(SEE SUBROUTINE RELAX FOR DEFINITION), C WItW2=CHOSEN WEIGHTS(SUITABLE VALUES WI=O.31,W2=Q,OB,E.G.) C OUTPUT- EMU(MUvIA)=RELAXATION MODULI, TAU(MU)=RELAXATION TIMES, C C=COEFFICIENTS OF CUBIC POLYNOMIALS Ik LOG(TP} FOR EMU,FOR MU=tt..,NMU

OIHENSION A(B,9},B(B),NJ(B),AW(4,5),BWIW),NK(W),D(8), 1 T ( 3 0 ) , T T ( 3 0 ) , T P ( 8 ) t C ( 4 , 1 O I , T A U ( I O | , E R ( 3 0 , B ) , T P L ( B | , E M U ( I O t B )

C CHOOSE RELAXATION TIMES TAU(HU) MMU=NDEC NMU=MMU~I T A U ( l I = T ( 1 ) ~ l O , J ' ( t . - % , / F L O A T ( J D E C ) ) DO 90 HU=2,MMU

90 TAU(MU)=TAU(MU "I)~10" TAU(NMU)=I.E30

C INFINITE IAU(NMU) MEANS THAT NO DASHPOT IS ATTACHED TO THE C LAST SPRING, C COMPUTE COEFFICIENTS A AND B OF LEAST SQUARE METHOO

DO 170 IA=I,NA DO 120 NU=I,NMU O0 110 MU=IBNMU SUM=O, DO 105 IT=I,NT

I05 SUM=SUM÷EXP(-T(IT)ITAU(MU)-T(IT)ITAU(NU)) 110 A(NU,MU)=SUM

SUM=O. DO 115 IT=I,NT

115 SUM=SUM+ER(IT,IA) ~ExP('T(IT)/TAU(NUD) 120 B(NU)=SUM

C ENFORCE A SMOOTH SPECTRUM BY ADDING WEIGHTS WL,W2 ON I-ST AND C 2-NO DIFFERENCES OF EMU AS FUNCTION OF MU.

W22:2,~W2 W24=4,~N2 A ( 1 , 1 I = A ( l t I ) + W 1 A ( 1 , 2 ) : A ( 1 , 2 ) - W 1 A(2,l)=A(2,1)'W1 A(2,2)=A(?,2)÷NI NMUI=MMU NMUZ=NMU-2 NMU3=NMU-3 DO 200 NU=I,NMU3 A(NU,NU)=A(NU,NU)÷W2 MU=NU+I A(NUtMU):A(NU,MU)-W22 NU=NU~Z

200 AINU,MU)=A(NU, MU)÷W2 O0 201 NU:2~NHUB MU=NU-t A(NU,MU)=A(NU~MU)-W22 A(NU,NU)=A(NU,NU)~Wl~W24 MU=NU~Z

201 A(NU,MU)=A(NU,HU)-WI-W22

C-) 0

"13

Z

i-rl

x

--I U:,

m

----I

E ' )

m .--t r ~

0 ...a

Z 0

(.~ "-4 -..J

APPENDIX-FORTRAN IV SUBROUTINES

SUBROUTINE RELAX INOEC,JOEC.NA.JA,OOEF,T,TT,TP,TPL,NT,ERI C COMPUTES STRESS RESPONSE TO A GIVEN STRAIN HISTORY, AND RELAXATION C FUNCTION IN PARTICULAR. C INPUT- NOEC=NO. OF OECAOES OF TIME I~ LOG. SCALE, JOEC=NO. OF STEPS C PER LOGll0), NA=NO. OF AGES AT LOAOING, JA=NO. OF AGES PER LOGll0l, C Tlll=FIRST OISCRETE TIME. TPll'=FIRST AGE AT LOAOING, ODEFIITI=PRES-C CRIBEO STRAIN INC~EMENT PER STEP FROM IT-l TO IT,MINUS PRESCRIBED C SHRINKAGE INCREMENT PER STEP.IFOR RELAXATION OOEFll'=l.,ALL OTHER C DOEFIITI=O.I IlF SHRINKAGE IS CONSIOEREO,INSERT DEPP=OEPP-OSHRIITI C AFTER STATEMENT NO.30. OSHRIITI=SUPPLIEO SHRINKAGE INCREMENTS. I C SUBROUTINE CREEP FOR CO~PUTING VALUES OF CREEP FUNCTION FROM T AND TP C HUST BE SUPPLIEDISEE TABLE 2 OF REF.91. C OUTPUT- ERIIT,IAI=OISC~ETE VALUES OF STRESS REL~XATION FUNCTION, C TIITI ANO TPIIAI=DISCRETE TIMES FRO~ LOADING AND OISCRETE AGES AT C LOADING, NT=NO. OF DISCRETE TIMES. C DIHENSION MUST BE AT LEAST DSTRINTI,DDEFINTI,TlNT),TTINTI,TPINAI, C TPLINAI ,ERINT,NAI

DIHENSION DSTRI 30 I ,OOEFIJO I. T 130 I. TT 130 I, TP IBI • TPL (8) .ER I 30. BI XX=10.··ll./FLOATIJAII 00 11 IA=2,NA TPIIAI=TPIIA-ll·XX

11 TPLIIAI=ALOG10ITPIIAII TPLll'=ALOG10ITPlll' OTR=10.··ll./FLOATIJOECII NT=NDEC·JOEC 00 9 IT=2.NT TIITI=TIIT-ll·DTR

9 TTIITI=SQRT (TIITI·TlIT-l11 PRINT 16. ITPIIAI, IA=l.NAI

16 FORHATI7X7HTPIIAI=10E1Z.~/11~x10E1Z.~11 00 50 IA=l,NA

C COHPUTE THE FIRST STRESS VALUE USING EFFECTIVE HODULUS. DSTRll'=OOEFllI/CREEPITlll,TPIIAII ERll.IAI=OSTRlll DO 40 Ih2.NT I1=IT-l DEPP=O. DO 30 JT=l,I1

30 DEPP=OEPP+DSTO(JTI·ICREEPITIIT),TP(IAI'TT(JTI'-CREEPIT(Ill, 1 TP IlAI.TT (JT III

C OSTRIJTI=STRESS INCREMENT IFROM JT-l TO JTI, DEPP=INELASTIC C STRAIN INCREHENT.

DSTR(IT)=IDDEFIITI-DEPP)/CREcPITIITI.TPIIAI+TTIITI I 40 ER(IT.IAI=ERlll,IAI+DSTRIIT) 50 CONTINUE

PRINT 52 52 FORHAT(40HOIT TlIT) ERIIT.lI ~RIIT,ZI ... 1

00 53 IT=l,NT 53 PRINT 54, IT, HITI .£RlndAl, IA=l.NA) 5 .. FORHATllXIZ,Fl1.3,lJ£12.~/113Xl0E12 ... ))

RETURN END

SUBROUTINE HAXWL1(ER.T.TT.TP.TPL.NOEC,JOEC.NT.NA,Wl,W2.TAU,EMU. lC.NMU)

C COMPUTES RELAXATION SPECTRA IVALUES OF ELASTIC MODULI EMU IN HAXWELL C CHAIN) FOR VARIOUS AGES AT LOAOING,TPIIAI. NO FORHULA FOR EMU IS C ASSUHEO AT FIRST BUT INTERPOLATING cueIC POLYNOHIALS IN AGE TP AT C LOADING ARE OETERHINEO AT THE ENO FOR ALL HU. C INPUT- ER(IT,IA),TIIT),TTIIT).TPIIA).TPLIIAI,NDEC.JDEC,NT.NA FROM C RELAXISEE SUBROUTINE RELAX FOR DEFINITION'. C Wl,WZ=CHOSEN wEIGHTS(SUITABLE VALUES Wl=0.Jl,WZ=~.06,E.G.) C OUTPUT- EMUIHU.IAI=RELAXATION MODULI. TAUIMU)=RELAXATIDN TIMES. C C=COlFFICIENTS OF CUBIC POLYNOMIALS I~ LOGITP) FOR EHU.FOR HU=l •••• NMU

DIMENSION A(B.9' .8IB) .NJIB) ,A'+14.51 .e41"'.NK(4) ,OIBI.

c

C C C

1 T(30) ,TT (30) ,TP(B) .CI4.10) ,TAUll0) .ERI3e,8) ,TPLlBI,EHUI10.6) CHOOSE RELAXATION TIMES TAUI~U)

HMU=NDEC NMU=MMU+l TAUll)=Tll)·10.··(1.-1./FLOATIJOEC)) DO 90 MU=Z,MHU

90 TAUIMU)=TAUIMU-11·1J. TAUINMUI=1.E30

INFINITE TAUINMU) MEANS THAT NO OASHPOT IS ATTACHEO TO THE LAST SPRING. COMPUTE COEFFICIENTS A AND B OF LEAST SQUARE HETHOD

DO 170 IA=l,NA DO 120 NU=l,NMU 00 110 MU=l.NMU SUM=O. DO 105 IT=l,NT

105 SUM=SUM +E XP (-T I IT) I TAU I ~U) -T I IT) I TAU INU) ) 110 AINU.MU)=SUM

SUH=O.

115 lZ0

C

DO 115 IT=l,NT SUH=SUH+ERIIT,IAI·EXPI-TIIT)/TAUINU)) BINU)=SUH

ENFORCE A SMOOTH SPECTRUH BY ADDING WEIGHTS Wl,WZ ON l-ST AND 2-ND DIFFERENClS JF EMU AS FUNCTION OF HU. C

WZZ=Z.·WZ WZ'+= ... ·WZ All,l'=All,ll+Wl All,Z)=A(l,Z)-Wl AIZ.l)=A(Z.ll-Wl A 12. Z) = A I Z, Z) + Wl NHU1=MMU NMUZ=NMU-Z NMU3=NMU-3 00 ZOO NU=l,NHU3 AINU,NU)=AINU,NU)'WZ HU=NU+l A(NU,MU)=A(NU,HU)-W22 "U=NU+Z

ZOO AINU.MU)=AINU,MU)+WZ DO ZOl NU=Z,NMUZ HU=NU-l AINU.HU)=A(NU,HU)-W22 A(NU,NU)=A(NU,NU)+Wl+WZ .. HU=NU+l

201 AINU.HUI=AINU,HU)-Wl-WZ2

n o 3: ""0 C --I

~ ..... o ::z

;:0 fT1 r :x:-X

~ ..... o ::z I/) ""0 fT1 n --I ;:0 :x:-

n o ::z n ;:0 fT1 --I fT1

(Jl '-I '-I

CO 202 NU=3,NMUI MU=NU-2 A(NU,MU)=A(NU,MU)mN2 MU=NU-I A I N U , H U ) = ~ ( N U , M ! ) ) - W I - ~ 2 ?

ZO~ A ( N U ~ N U ) = A ( N L J , N U ) ÷ N I ÷ R ~ C SOLVE THE FOUATIONS BY A LIBRARY SUBROUTINE (NJ:AUXILIARY ARRAY)

CALL SOISP(~,B,NMU,I, I .~-6,1ERR. NMU,NJ) ~0 165 ~U=I,NMU

165 EMU(MU,IA)=B(MU) 170 CONTINUE

PRINT I l l , T(1)~T(WT) l T l FOP~AT(~CHG RELA×~TION ~OCIULI FOP TIME RANGE F~OM E I I . 3 , 3 H TO .

I [ I I . 3 / ~ O H C I A TP(IA) EMUII , IA) EMU(2,IA) . . . ) DO 172 IA=I,NA

172 PRINT IT3, I A , T P ( I A ) , (EMN(HU,IA), HU=I,NMUI 173 FOP~AT(IXI2,FIG.3, |~E11.~))

C FINO COEFFICIENTS C(I,Md} OF AN INTERPOLATING CUBIC POLYNOMIAL C IN LOG(AGE TP AT LOAOIN5) FOR EMU(IT, IA)

IF(NA °LT. ~) STOP PRINT 80~

~06 FO~MATI53HUCOOFFICIENTS OF CUBIC POLYNOMIAL IN LOG(~GE) FOR EMU/) O0 860 MU=I,NMU DO ~] I = I , 4 SUM=G. 00 810 IA=I,NA

BI0 SUM=SUM+EMU(MU,IA)~IPL(IA)~(I-I) ~ ( I )=SUM OO 8~0 J= l ,~ SUM=O, DO 815 IA=I,NA

815 SUM=SUM*TPL(IA)~'(J*I-2) BWO A~(J,I)=SUM

C SOLVE THE EOUATIONS BY A LIBRARY SUBROUTINE (NK=AUXILIARY ARRAY) CALL SOISP(Aq,B~,W,I, I .E-6,1ERR,A,NK) O0 85O i = i , ~

8BO C(I,MU)=BW(I) 860 PRINT 859,MU, IC( I ,MU) , I=I ,W) 859 FORMAT(~H MU=I2,wXBMC(I.MU)=WEt2.3)

PRINT 861 861 FOPMAT(B3HOCHECK EMU(HU,IA)-VALUES COMPUTED FROM THE POLYNOMIAL/}

DO 865 IA=I,NA O0 863 MU=I,NMU

~63 O(MU):C(I~MU)÷(C(2~MU}÷(C(3,MU}wC (~ ,MU)~TPL( IA) )~TPL( IA I )~TPL( IA) SUM=30.*TP(IA) DO 86~ MU=I,MMU IF(TAU(MU) °LT. SUM) GO TO BB~ D(NMU)=O(NMU)÷O(MU) D(MUI=O.

86~ CONTINUE B65 PRINT 173. I A , T P { I A ) , (O(MU), MU=I,NMU)

C AS A CHECK, COMPUT~ CREEP FUNCTION VALUES FROM EHU. CALL CRCU~V(T,TT,TP,IAU~NT,NA,MMU,NMU~I,C) kETURN END

SUBROUTINE M A X N L 2 ( E R , T , T T , T P . T P L , N O E C , J D E C , N T , N A , M U E × , I X E X t W , T A U , 1 M M U , N M U , C E , A A A )

C COMPUTES R E L A X A T I O N MOOULI EMU ASSUMED (FOR M U = I , . . . , M M U ) AS A TWO- C OIMENSIONAL CU~IC POLYNOMIAL IN L O G ( T I M E T F~OM LOADING) AND LOG(AGE C TP AT LOAOING) WITH COEFF. AAA, AN9 EMU FOR FU=NMU AS ONE-D IMENSIONAL C CUBIC POLYNOMIAL IN LOG(TP l WITH COEFF. CE. C I N P U T - E R ( I T , I A ) , T ( I T ) , T T { I T ) , T ~ ( I A ) , T P L ( I A i , N O E C , J O E C , N T , N A FRQM C RELAX(SEE SUBROUTINE RELAX FOR 3 E F I N I T I O N ) . C EXPONENTS MUEX ANO I × E X OF POLYNOMIAL FOR EMU MUST BE SUPPLIED AS C MUEX=O,I,O,~,I,G~3t2,1,O, IXEX=O,O,l,0,1,2,0,1,2,3, WEIGHTS W MUST 3E C SUPPLIED(SUITABLE IS W=O.,I.,G.,I.,,5,0,,I.,,67,,33,O,)° C OUTPUT- COEFFICIENTS AAA ANO CE OF CUeIC POLYNOMIALS, TAUIMU}= C RELAXATION TI~ES.

CIMPNST'N A2I~,5),A3IIO,II),WIIO)~ERMAIB)oHIIO,8,21i,PHI(IO), 1 N L ( 1 . ) , A B ( 8 ) , A B ( 2 1 ) , T I 3 0 ) , T T ( 3 O I , T P ( ~ ) , T A U ( I O ) , E P ( 3 O t B ) , N I ( ~ ) , 2 T P L ( ~ l , E M I ] ( l O , B I , C E I ~ ) , A A A I l O ) , M U E X I l O ) , I X E X ( l O } , C ( ~ , 1 0 )

C CHOOSE RELAXATION TIMES TAUIHU) T A U I I I = T I 1 ) ' I O , ' ' ( 1 , - I , / F L O A T ( J D E C } ) DO 90 MU=2,MMU

9G T A U ( M U I = T A U ( M U - 1 1 ~ I O , T A U ( N M U ) = I , E 3 0

C INFINITE IAU(NMU) MEANS THAT NO OASHPOT IS ATTACHED TO THE LAST C SPRING. ALSO NOTE IHAT TAU(MMU) MUST EQUAL T(NT) IN THIS METHOD. C FIRST FIT EG(NT,IA) BY A CUBIC POLYNOMIAL IN LOG(AGE TPIIA) AT C LOADING)

DO 51J L= I ,~ CEIL)=O. O0 505 IA=I ,NA

505 CE(L)=CE(L)+ER(NT, IA)~TPL( IA)mm(L ' I } DO 5~0 K=/,W A2(K.L)=O. O0 510 IA=I,NA

BIO A 2 ( K , L ) = A 2 I K , L ) m T P L I I A } ~ ( K e L - 2 } C SOLVE THE EQUATIONS BY A LIBRARY SUBROUTINE (HI=AUXILIARY ARRAY}

CALL S01SP(A2wCE,W,I , I°E-6, IERRv~,NI) PRINT B l l , CE

B l l FORMAT(WIMOCOEFF.OF CUBIC POLYNOMIAL FOR ER(NT,IA)=WEI2.3) C NOW COMPUTE COEFFICIENTS AAA(J} OF CUBIC POLYNOMIAL IN MU AND C LOG(AGE TP(IA} AT LOADING)

P : E X P ( - 1 , ) O0 5BO IT=I ,NT DO 560 MU:I~MMU PHI(MU)=EXP(-T( IT) / IAU(MU)) IF(MU .EQ. MMU) PMI(HU)=RHI(MU}-P

560 CONTINUE 00 580 I=l,lO 00 BBO IA=I,NA SUM=O. DO 570 MU=%vMMU

570 SUM=SUM*PHI(MU)~(FLOAT(MU'~MUEX(1))~TPL(IA}*~IXEX(1)) 580 M( I , IA , IT )=SUM

C NOW COMPUTE EQUATION COEFFICIENT MATRICES A,B DO 650 l = I , l O SUM=O. OO 610 IA=I ,NA

C ERMA(IA)=SMOOTHEO END VALUE OF ER(NT~IA)=RELAXATION FUNCTION ERMA( IA ) :CE(1 ) * (CE(2 ) * ICE(3 I *CE(Wt~TPL( IA ) )~TPL( IA ) )~TPL( IA )

O0

C~

0 Z

FTI

x

0

(/)

(~

;7o

C-) 0

- - t

0

Z 0

202 C

165 170

171

172 173

C C

3J d

810

815 840

C

85,J 8&0 859

861

%3

864 865

C

[0 202 NU~3.NHUl HU~NU-2

A{NU.MUI =td NU,HUI +W2 HU~NU-l

A (NU,"'1U):::l! (NU,t-'IJI-I'H-1'42? f,1N-U,NU):::A( NLJ,NUJ +r11+W2

SCLVE THE E~UATIO~S BY A LI~~ARY SUBROUTINE INJ=IUXILIARY ARRAY) CALL SOlSP IA,8.!\;"1Ij.l,1,t:-6.IE~R, NHU,NJ) CO Ib5 ~U=ltNML rMU (HI). IA'=R(!'1UI CONTli~U::

j:J ~ I NT 17 1, T ( 11 t T ( I~ T J F00~ATI.CHC QELAXATIO'

1fll.3/.uHCIA TPII.) [0 172 IA=l.NI

·OOULI FOR TIME RI~GE FROM E~(J(t,IA) tMU(2,IAI ' •• )

PRINT 173, IA,TD(IAI, (["1U('iU,IAI, HU=l,NP1UI FOD~AT{l'.(I2 ,FIC.3, (RElt.3) I

El1.3,3H TO ,

FINO C~EFFICIENTS CII.M01 OF AN INTE~POLITING CUQIC POLYNOMIAL Jr' LOGIAGE TP AT LOADIN~) FO~ EMUIIT.!.)

II=(NA .LT. 41 STOiJ P, -l ....... o z

;AJ rn r ):> >< ):> -l

o z Vl -0 rn n -l ;AJ ):>

n o z n ;AJ rn -l rn

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'-l 00

00 610 IT: I ,NT 610 SUM=SUM+(ER(IT,IA)-ERMA(IA))~HII, IA , I T|

AAA(I|=SUM DO 650 J : 1 , 1 0 SUM:C. IF(J .EO, I ; SUM:WIT) DO 6~G IA:I,NA DO 6wO IT=I.NT

640 SUM=SUH+H(I,IA,IT)°HKJ,IA.IT) 650 A3(I,J)=SUM

C SOLVE THE EQUATIONS BY A LIBRARY SUBROUTINE (NL=AUXILIARY ARRAY) CALL SOXSP(A3,AAA,XO,X,I.E-6,IERR.iO,~L) PRINT 751, AAA

751 FORMAT(?3HOCOEFFICIENTS AAAII| OF CUBIC POLYNOMIAL FOR EMU(MU,IA) I IN MU AND LOGITP)/XXluE12.3/) PRINT 761

761 FORMAT(?OHGVALUES OF E M U ( M U , I A ) COMPUTED FROM CUBIC POLYNOMIAL IN 1MU AND L O G ( T P ) ) DO 7?0 IA:I,NA O0 766 MU:I,MMU SUM=O. DO 765 I=1,10

?65 SUM:SUM+AAA(I)~(FLOAT(MU ~MUEX(I)|~TpLKIAI~*IXEX(I)| ?66 AG(MU)=SUM

AG(NMU)=ERMA(IA)-P~AGKMMU) SUM=30.°TP(IA) DO 767 MU=I,MMU IF(TAU(MU) .LT* SUM) GO TO ?67 AG(NMU)=AG(NMU)÷aG(MU) AG(MU):O.

767 CONTINUE PRINT 768, IA~TP(IA)~ (AG(MUI, MU:I,NMU)

768 FORMAT(WH IA=I2,WH TP=FXC.3,GH EMU= BE11.3) 770 CONTINUE

PRINT 78~ T8W FORMAT(T5HOCOMPARE RELAX. FUNCTIO~ VALUES ER(IT,IA) COMPUTEO FROM

ICOEFF. AAA(I),CEKJ)) DO 790 IA=I,NA D6 788 IT=I,NT SUM=ERMa(IA) DO 785 J = l , i O

?85 SUM:SUM÷AAA(J)~H(J,IA.IT ! ?88 AG(IT)=SUM ?90 PRINT 7BEIIA,(ABIIT|, IT:X.Nr) TB6 FORMAT(WM IA=I2,lGH ERIIT.IA)-FIT=lOEll .3/(21XlOE11.3|)

C AS A CHECK, COMPUTE CREEP FUNCTION VALUES FROM COEFFICIENTS G AAAII} AND CE(J|

CALL CRCURV(T,TT,TP,TAU,NT,NA.MMU,NMU,2,C,CE,AAA,MUEX,IXEX| FETURN END

SUBROUTINE C~CURVIT.TT,TP~IAU.NT.NA,MMU~NMU,IALT.C,CE,AAA.NUEX. XIXEX)

C COMPUTES DISCRETE VALUES AJ(IT, IA| OF CREE ~ FUNCTION J FROM GIVEN C RELAXATION MOOULI OF MAXWELL CHAIN. C INPUT- T,TT,TP,NT,NA,TAU,MMU,NMU FROM SUBROUTINES RELAX C AND MaXWLI OR MAXWL2(SEE THERE FOR OEFINITIONS). USER MUST SUPPLY C SUBROUTINE FUNCTION EMUF THAT COMPUTES EMU-VALUES FOR ANY AGE TP AND C ANY MU. HERE EMUF IS COMPUTED FROM EITHER G (MAXWLIt IALT=I; OR C CE,AAA,MUEX,IXEX(HAXWL2,IALT:2).KIALT ON INPUT SELECTS THE OPTION.) C ARGUMENTS OF EMUF APPEAR BELOW.

DIMENSION T(3O),TT(30),TP(B)eTAU(Z0),EMU(IO,B),C(&~IOI.ELAN(XO), 1 CE(W),AAA(IC),MUEX(t3),IXEX(IOI.SMU(%]),EX(XO),AJ(30,8),DJ(30,8)

C LOOP ON CREEP CURVES FOR VARIOUS AGES TP(IA) AT LCAOING O0 30 IA=I,NA,2 DEF=C. DO 16 MU:I,NMU

16 SMU(MU):O. CT=T(%}

C VARIABLES HAVE NOW BEEN INITIALIZEO C LOOP ON DISCRETE TIMES T ( I I ) FROM LOADING(OT=TIME STEP)

DO 30 IT : I ,NT IF( IT .GT. 1) OT=T(I I ) -T{ IT-X) ERR:O° SUM=O. DO 20 MU=I,NNU X:DT/TAU(MU) EX(MU)=EXP(-X) BD=I.°ZXKMU) 60A:BO/X

C WHEN X TENDS TO 0., LIMIT(BOA)=O*/O. COMPUTE BOA FROM TAYLOR C SERIZS aBOUT X:O*

IF(X .LT. IoE-6) 30A=1.'(.5-.166666667 ~x|~x ELAM(MU)=eOA~EMUF(~U,TTKIT|+TP(IA),MMU,NMU,IALT,CtCE,AAA,MUE X •

1 IXEX) EPP=EPP+ELAM(MU|

20 SUM=SUMfBO~SMU(MUJ IF{ IT .ED. I | SUtI:SUM+$.

C %.=VALUE OF STRESS INCREMENT AT TIME OF LOAOINGILATER INCREMENTS C :Oo),DE=STRAIN INCREMENT, OEF:STRAIN. SMUiMU):STFESS IN M-TH C SPRING CF CHAIN, ~DA=LA~BA, EPP:E-DOUBLE PRIME, DT:TIME STEP

DE=SUH/EPP OEF=OEF+DE AJ(IT.IA)=DEE DO 30 MU=I~NMU

30 SMUIMUi=SMU(MU)~EX(4U)÷ELA M(MU)~OE PRINT 3g

39 FORMAT(61MCCREEP FUNCTION VALUES COMPUTEO FROM MAXWELL CHaI~ ¥OOUL 1I EMU/W2 H IT T(IT) AJ(IT~I) AJ(IT,2) . . . |

DO WO IT : I ,NT WO PRINT WL,IT,T( IT) , (AJKIT.IA), IA:I .NA,2} WI FORMATKlXI2,F10.3~I3EX2.3/KI3XIOEX2.3|)

PRINT ~9 ~9 FORMAT(SOHCCOMPARE CRZEP FUNCTION VALUES AS ORIGINALLY GIVEN/)

60 52 IT : I ,NT DO 50 IA=%,Na,2

50 DJ( IT , I~) :CR~EP(T( I I ) .TP( Ia | | 52 PRINT ~1, IT ,TKIT | . (OJKIT,IA), IA=I,NA,2)

~ETURN END

0

Z 0

0 ..~ -13 ,...-

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.--I

0

I - "

X

0

m

--4

0

m .--I m

(.n

to

610

6~0

650 C

C C

751

761

765 16&

167

768 170

78~

785 788 790 786

00 610 IT=l,NT SUM=SUM+I[RIIT,IAI-ERMAllAll'HII,IA,ITI AAA III =SUH 00 650 J=l,10 SUH=C. IFlJ .EO. II SUH=,j1I1 00 64" IA=l,NA 00 6~a H=l.NT SUH=SUH+HII,IA.ITI·HIJ.IA.ITI A3II.JI=SUH

SOLVE THE EDUATIO~S BY A LIB~AQY SUB~OUTINE INL=AUXILIARY A~RAYI CALL SOlSPIA3.AAA.l0.l.1.E-6.IER~.10.NLI PRINT 751. AAA FORMATI73HOCOEFFICIENTS AAAIII OF CUBIC POLYNOHIAL FOR EHUIHU.IAI

lIN MU AND LOGITPlflXluE12.3/1 PRINT 761 FORHATI70H~VALUES OF EHUI~u.IAI COMPUTED FROH CUBIC POLYNOMIAL IN

lHU AND LOG I TPI I 00 770 14=1. NA 00 766 MU=l.HHU SUM=O. 00 765 1=1.10 SUH=SUM+AAAIII'IFLOATIMU"MUEXIIII'TPLIIA1"IXEXII11 A5IHUI=SUH A5INHUI=ERMAIIAI-P'A5IHMUI SUH=30.·TPIIAI DO 767 HU=l.MHU IFITAUIHUI .LT. SUi'll GO TO 767 A5INHUI=A5INHUI+A5IMUI A5IHUI=0. CONTINUE PRINT 768. IA.TPIIAI, IA5IHUI. ~U=l.NHUI FORHATI~H IA=I2.~H TP=Fl0.3.5H EHU= 8El1.31 CONTINUE PRINT 78~ FORHATI75HOCOMPARE QELAX. FUNCTIO~ VALUES ERIIT.IAI COMPUTEO FRuM

1COEFF. AAAIII,CEIJII 00 790 14=l,NA DO 788 IT=1,NT SUM=ERMA 1141 00 785 J=l,10 SUM=SUH+AAA IJI'H IJ.IA. lTl A6IITI=SUM PRINT 786,IA,IA6IITI, IT=l.NTI FORMATI~H IA=I2,15H ERIIT.IAI-FIT=10E11.3fI21X1DE11.311

AS A CHECK. COMPUTE CREEP FUNCTION VALUES FROH COEFFICIENTS AAAIlI AND CEIJI

CALL CRCURVIT,TT,TP,TAU.NT,NA,HHU.NHU,2,C,CE,AAA,HUEX,IXEXI FETURN ENO

SUBROUTINE C'CURVIT.TT,TP,TAU,NT,NA,HMU,NMU,IALT.C,CE,AAA,MUEx, lIXEXI

C COMPUTES OISCRETE VALUES AJIIT.IAI OF CREEP FUNCTION J FROM GIVEN C RELAXATION HOOULI OF MAXWELL CHAIN. C INPUT- T,TT,TP,NT.NA,TAU,MMU,NMU FROH SUBROUTINES RELAX C ANO HAXWLl O~ MAXWL21SEE THERE FOR DEFINITIONSI. USER MUST SUPPLY C SUBROUTINE FUNCTION EMUF THAT COMPUTES EMU-VALUES FOR ANY AGE TP AND C ANY ~U. HC:"E EMUF IS COMPUTED FROH EITHER C IMAXWL1, IALT=lI OR C CE.AAA,MUEX,IX~XIHAXWL2,IALT=ZI.IIALT ON INPUT S~LECTS THE OPTION.I C ARGU~ENTS OF EHUF APPEAQ BELOW.

DIMENSION T 1301 ,TT130 I ,TPI81.TAU (101 ,EHUll0 ,81 ,CI4,101 ,ELAHI101, 1 C~I~I.AAAI1CI.HUEXI1l1.IXEXll01.SMUI1J1 ,EXll0I,AJI30.81 ,DJ130,81

C lOOP ON CREEP CURVES FO~ VA~IOUS AGES TPIIAI AT LOADING DO 30 IA=l,NA,2 DEF=C. DO 16 HU=l.NHU

16 SMUIHU1=0. CT=1Ill

C VARIABLES HAVE NOW 9EEN INITIALIZED C LOOP ON DISCQETE TI~fS TIITI FRCM LOADINGIOT=TIME STEP I

DO 30 H=l,NT IFliT .GT. 11 DT=TllTl-TlIT-ll EPP=O. SUH=O. DO 20 HU=l,NHU X=DTfTAUI~Ul

EXIMU'=EXPI-XI BD=l.-~XI~UI

BDA=BOfX C WHEN X TENDS TO ~ •• LIHITlqDAI=O./O. COMPUTE BOA FROH TAYLOR C SERI"S ABOUT X=O.

IFIX .LT. 1.E-&1 3DA=1.-1.5-.166666&67·XI·X ELAHIMUI=eDA·EMUFI~U,TTIITI+TPIIAI.MHU,N~U.IALT,C,CE,AAA.HUEX.

1 IXEXI EPP=EPP+ELAM IMUI

20 SUM=SUM.BD·S~UIMUI IFIIT .EO. 11 SUH=5U~.1.

C 1.=~AlU~ OF STQESS INCREMENT AT TIHE OF LOADINGILATER INCRE~ENTS C =O.I,OE=STRAIN I~CREM~NT, DEF=STRAIN. SHUIHUI=ST"ESS IN ~-TH C ,PRING CF CHAIN. ~OA=LA~9A. EPP=E-DDURLE PRIHE. DT=TIME STEP

OE=SUM/EPP DEF=DEF+OE AJ I IT .IAI =DEF DO 30 MU=l,NHU

3C SMUIMUI=SMU(t1UI'"XI1UI+ElAMIHUI'DE PRINT 39

39 FORMATI61"CCREEP FUNCTION VALUES COHPUTEO FROH MAXwELL CHAI~ ~OJUl 11 EMU/42H IT TIITI A)IIT,ll AJIIT,2' ••• 1

DO ~O IT=l.NT ~O PRINT 41,IT,TIITI, IAJIIT.IAI, IA=1.NA,21 ~1 FDRHATIIXIZ,FI0.3,lJE12.3/113X10E12.311

PRlNT ~9 ~9 FO~MATI5uHCCOMPARE C""EP FUNCTION VALU~S AS ORIGINALLY GIVENfl

DO 52 IT=l,NT DO 50 IA=l.NA,Z

50 DJIITolAI=CR"EPITIITI.TPIIAII 52 PQINT ~1, IT.TIITI. IDJltT,IAI. IA=1,NA,21

Rr::TURN END

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cement and concrete research. vol. 4, pp. 567-579, …montrent que les courbes de fluage donnees...

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