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Steady creep bending in a beam with random material parameters
Steady Creep Bending in a Beam with Random Material Parameter? by W. N. HUANG Theoretical and APplied Mechanics Section, Engineering and Technology Division Argonne .National Laboratory, Argonne, Illinois and
F. A. COZZARELLI
Department of Engineering Science, Faculty of Engineering and Applied Sciences State University of New York at Bu$alo, Bu$alo, New York A statistical
ABSTRACT:
the material parameters $uctuationa
in temperature
are presented found
analysis
for
is presented for steady
creep bending
and imperfection
the statistical
properties
density.
Analytical
is highly random. The statistics
and numerical
of the norm&l stress and lateral velocity.
that whereas the normal stress shows very little ra&Juctuation
on the nominal
in a beam, where
are random processes alon,g the neutral axis arising from ran&an
and on the position
It is
th.elateral velocity
of the lateral velocity generally show a sign&ant
value of the creep power
results
dependence
along the neutral axis.
Notation
A” b B
BPV41 c 11 d
6
f( 1 ww,) Wd h H f J L M n 6 N P sgn( 1 u:;
ti
statistical constant, Eqs. (43) cross-sectional area of beam width of rectangular beam temperature and imperfection insensitive creep constant, Eq. (16b) function of n(X), Eq. (14) autocovariance of indicated quantity nondimensional correlation distance, Eq. (33) nondimensional creep viscosity parameter, Eq. (17) expected value operator probability density function function of z and n,,, Eq. (28b) function of n,,, Eq. (27b) depth of rectangular beam root of Eq. (48) property of rectangular beam, Eq. (8b) property of beam cross-section, Eq. (5) length of beam bending moment, Eq. (4) creep power, Eq. (1) negative of the nominal slope, Eq. (21b) imperfection density lateral point force signum function, Eq. (2) temperature unit step function lateral velocity, Eq. (12)
* This research was supported in part by the National Science Foundation under Grant No. GK-1834X, and in part by the Office of Naval Research under Contract No. N 00014-71-C-0108.
323
W. N. Huang and F. A. Cozzarelli right-hand side of Eq. (47) nondimensional space coordinate centroidal axis of beam function in stress solution, Eqs. (43) transverse coordinate creep viscosity parameter, Eq. (1) normal strain rate, Eq. (1) argument for autocovariance of homogeneous random process, Eqs. (19~) and (23~) rate of curvature of beam, Eq. (3) right-hand side of Eq. (35) standard deviation of indicated quantity arbitrary deterministic constant reference stress (chosen aa 2PL/bh2), Eq. (1) normal stress, Eq. (1) maximum normal stress, Eq. nondimensional temperature, indicates nominal value, i.e. density equal mean value (T, designates random variable designates random process in
(9) Eq. (16b) value when temperature or N,)
or imperfection
x
I. Introduction
Creep parameters are highly dependent on temperature, and some effects of randomness in temperature have been considered by Soong and Cozzarelli (l), Parkus (2), Parkus and Bargmann (3) and Cozzarelli and Huang (4). Creep parameters are also highly dependent on imperfection density, and the effect of randomness in imperfection density was also considered in Ref. (4). In that reference the authors have studied the effect of random material parameters on nonlinear steady creep in a 3-bar truss. The stresses and velocities analyzed were simply random variables, and it would be instructive to also consider a problem where they are random processes in some coordinate. Thus, in this investigation we shall study the problem of steady creep bending in a beam, where the temperature, imperfection density, creep parameters, normal stress and lateral velocity are all in general random processes along the neutral axis. The basic approach is similar to that used in Ref. (a), i.e. the problem is “random temperature problem” separated into two uncoupled parts -the and the “random imperfection problem”. In the random temperature problemonlythe creep viscosity parameter 4 is random, whereas in the random imperfection problem only the creep power n is random. The derivation of the governing equations for the normal stress and lateral velocity in the beam, with arbitrary statistics for .$ and n, is presented in Section 2. Then, the density functions and statistical moments of the stress and velocity are given in Section 3, with the temperature and imperfection density taken as independent homogeneous normal random processes along the neutral axis. The final section contains a summary of the results obtained.
324
Journal of The Franklin
Institute
Steady Creep Bending in a Beam with Random Material Parameters ZZ. Governing
Equations
Consider an initially straight beam subjected to prescribed deterministic loads. The cross-section of the beam is uniform along the neutral axis of the beam (X), and symmetric with respect to the plane of the applied loads (2X). Using an asterisk with a symbol to indicate a random quantity, we employ a steady creep power law with random material parameters in the form
(1) Here, the random material parameters i,*(X) random processes in X, the normal stress o,*(X) $(X) are random processes in X for a designated deterministic constant reference stress and sgn defined as
1,
sgn(u,*) =
and n*(X) are in general and the normal strain rate value of 2, a, is an arbitrary (a:) is the Signum function
a,*> 0,
(2)
u,*
i -I,
The Bernoulli assumption of plane sections yields the expression strain rate at the arbitrary position X as s;(x)
= G*(x)
for the (3)
Here, a*(X) is the curvature rate of the beam, which is a random process in X and independent of 2. Introducing Eqs. (1) and (3) into the definition of bending moment, we obtain M =
I’
u,*(X)ZdB
= a,J*(X)
where A is the cross-sectional
w (k*),
(4)
area of the beam. The quantity
J”(X)
=
]Z
~l+wn*(X))
&j
s
(5)
is a property of the beam cross-section and reduces to the ordinary second moment of area for n*(X) = 1. For example, for a beam of rectangular cross-section of width b and depth h we have J*(X)
=
2n*(X) 2n*(X)+l
bh2 h l/n*(X) -4 02 .
When n*(X) = 1 Eq. (6) reduces to bh3/12, which is the ordinary moment of area for a rectangular beam. Combining Eqs. (l), (3) and (4), we obtain for the stress u*(x)
z
Vol. 294, No. 5, November 1972 19
=
M 12I1’n*(X) sgnm* J*(X)
(6) second
(7)
325
W. N. Huang and F. A. Cozzarelli For a rectangular form
cross-section,
this may be rewritten as in Hult (5) in the M
22
f*(x)
The maximum
=
l/n*(x)
II
u,*(X) = I*(x)
-
h
n,*(x)
w Vlf (sb)
bh‘2.
2 + 4n*(X)
normal stress at 2 = h/2 is then given by = M/f*(X)
ok(X)
The rate of curvature of the beam is obtained from Eq. (4) as g*(X)
=
‘:tx) 1M in*(X)sgn (M)
[J*(x)
u$-)
*
If the slope of the deformed beam is such that d~*(-V I --ax--
<1 I
(11)
’
where C*(X) is the lateral velocity of the beam, Eq. (10) may then be written as d2 C*(X) dX2
= _ 9,*(X) ( M In*(X) [J*(X) ,c]n*(X) sgn (M)*
(12)
The maximum normal stress U&(X) and the lateral velocity G*(X), as specified by Eqs. (9) and (12), are the quantities of greatest practical interest. A statistical analysis of Eq. (9) does not require a specification of the particular form of the bending moment M. However, Eq. (12) is more complex and a choice of a particular form for M would facilitate the statistical Thus, let us consider a cantilever rectangular analysis of this equation. cross-section beam of length L loaded at its free end (X = L) with a prescribed deterministic constant point force P. We then have M = - P(L - X), and obtain from Eq. (12) together with the boundary conditions at the clamped end (X = 0)
Here, the reference stress a, in Eq. (12) has been chosen as 2PL/bh2 and B*[n*(X)] is a function of n*(X) defined as B*[n*(x)]
=
y$;
‘1n’(X),
Equations (9) and (13) will form the basis of the statistical the next section.
326
(14) analysis in
Journal of The Franklin
Institute
Steady Creep Bending in a Beam with Random Material Parameters III.
Statistical
Analysis
of
Stress
and
Velocity
Material Parameters We assume that the temperature T*(X) and imperfection density N*(X) are independent homogeneous normal random processes in X, with means !Z” and NO respectively. Also, assuming that &z(X) is a function of T*(X) only while n*(X) is a function of N*(X) only, we can show that i:(X) and n*(X) are independent random processes obeying the lognormal statistics given in Ref. (4). And, we may treat the “random temperature problem” and the “random imperfection problem” as two uncoupled problems. For convenience, we summarize some of the statistical results pertaining to i,*(X) and n*(X) obtained in Ref. (4). The first-order density function of 5”*(X) is given by
f(T) = &-
exp [-(‘2-p2], T
where u$, is the variance. The random parameter 9,*(X) is related to the nondimensional random temperature T*(X) by the expression S,*(X) = &exp
[T*(X)],
Wa)
where T*(X)
=
W’*(X) - To1 T;
In Eqs. (16), kc,,is the nominal value of $(X) deterministic constant. Letting 8*(X)
= 9,*(X)/&
Wb)
*
(value at T* = To) and B is a
= exp [7*(X)]
we have the first-order density function for a*(X)
(17)
as
(18) where U(&‘) is the unit step function. Equation (18) is the lognormal probability density function (6). The mean E(&*}, variance U$ and autocovariance C8(r]) of a*(X) are given by E{&*} = exp (e/2)
> 1,
0: = exp (a:) [exp (~9) - l] > C&d
=
exp
(4)
{exp
(1% CT:,
W%I)I- $3
Ugb)
P9c)
where C,(q) is the autocovariance of T* and 17= XI -X2. Similarly, the first-order density function of N*(X) is given by
Vol. 294, No. 5. November
1972
327
W. N. Huung where +
F. A. Cozzarelli
and
is the variance. n*(X)
Also
= l+(n,--
1)exp
1
-
n,EN*(X) -
41 1’
(%I-1)
(214
where n, = n* IN.=NO>nh = - dn*/dN* INesNo
(21b)
and thus
f(n) =
(no- 1) 4274 ~.&A=-
1)
exp
i
- (720-1)2 [In(~)]2)U(n-l). 2(UNnh)2
This is a lognormal density function with two parameters and n*(X) has the following statistical properties: E{n*j = l+(n,-1)exp
(2($14”l;l)
ui = (no- 1)2exp [Es]
C,,(q) = (no- l)2exp
(no and CYST;) (6),
(234
> no>
(exp [m2]
- 1) > (aAV$J2,
[H2] (w [&2C,0] - 1).
Finally, if uNnh/(no- 1) is small enough Eq. (22) by the normal density function f(n)=hmexp
(22)
(e.g.
[-VI
Wb) (23c)
< 0.2) we may approximate
, --$J+ Small,
(24)
where now E{n*) = no and u”, = (uAr$J2. Since it is more convenient to work with nondimensional equations, we introduce the nondimensional distance 5 = X/L and the nominal values u,,, and zip,(values at T * = Toand N* = No). These follow from Eqs. (9) and (13) as (25)
G,(x) =
(1 - x)m+2 2L2 B,, B, -(no+l)(no+2)+no~l h
1 (no+l)(no+2)
1’ (26)
where B, is the nominal value of B*[a*(x)]. We can then obtain from Eqs. (9) and (13) in conjunction with Eqs. (25) and (26) the expressions G@)
_ 2+
uX?ILO
G(n,)
328
=
2+
l?/~“(~H Gbo) ’ P/no)
VW
W’b)
Journal ofTheFranklin Institute
Steady Creep Bending in a Beam with Random Material Parameters and %5”(x)
-=
?&l(x)
ss f&
1:
0 0
B*[n*(al)]
d*(q)
dorrdor, &, F(x, no) I
(1 - c#‘(~~)
1
(1 - @of2
(29a)
(28b)
F(s,no)=(no+l)(no+2)+noY-1-(no+I)(no+2)*
Random Temperature Problem Stress. If only the temperature Eq. (27a) yields
T*(s)
u=
is random
we set n*(x) = no, and
1.
(29)
=3X&0
Thus, the normal stress is a deterministic constant. Velocity. Turning now to the lateral velocity, we obtain from Eq. (28a) [with n*(x) = no] the expression (1 --1)nodolld~2
I
F(x,n,).
(30)
The mean of the lateral velocity ratio follows directly from Eqs. (19) and (30) and the homogeneity of 8*(x) as E(E)
=exp($)>l.
We see that the mean lateral velocity velocity. Proceeding now to the determination from Eq. (30) the relation ti*(x,) E -i ho
ti*(z,)
(31)
always exceeds the nominal lateral of second-order
statistics, we obtain
=
Co@,) 1
In obtaining Eq. (32), the autocovariance of L?*(x) was used in accordance with Eq. (19c). Choosing for C,(x, - x2) the physically plausible exponential expression C,(X,-2,)
= a:exp(-
where d is the nondimensional correlation velocity ratio follows from Eq. (32) as c+;,
Vol.
294, No. 20
5, Novemkr
1972
= exp (04) (A-
Ix,-x,I/a). distance, l),
the variance
(33) of the (34)
329
W. N. Huang and F. A.
Cozxarelli
where
s
A = O~(x-y)(l-y)~o X
,2(x (s
2) (1 - .z)noexp [ofexp
( -v)]dz)d?l/F(z,n,J2.
(35)
Then, the variance ratio is obtained from Eqs. (19b) and (34) as
4Jhim
A-1 (~3) - 1’
(36)
20s = exp
The following limiting values are obtained from this result : lim a~iz;o = 0, d+OT
limd$i!?= d+x 08
1.
(37)
We note from the above that the mean of the lateral velocity ratio ti*/ti,, is independent of the parameter no, while its variance is a function of n,. This differs from the result obtained for a 3-bar truss in Ref. (4), where both the variance and the mean of the velocity ratio are independent of n,,. In the present problem the bending moment is a function of the space coordinate z; thus the quantity 1MI no is not factorable from the double integral for velocity [Eq. (12)] and the velocity ratio is a function of n,. However, for the special case of d* a random variable (d-+co) we see from Eqs. (30) and (37) that @*/zi+, = b*, with statistics independent of n, and x. In Fig. 1, the variance ratio at the free end X = L (x = 1) as given by Eq. (36) has been plotted vs. the nondimensional correlation distance d for several values of CT,and n,. Consider the typical values ur = 0.3 and us = O-321; Eqs. (31) and (34) then yield the values E{ti*/ti,) = 1.046, UJ~~ = 0.289 I.0
06
d
FIG. 1. Variance of velocity (x fixed)-random
330
temperature problem.
Journal of The Franklin
Institute
for n, = 1 and o,J~~ = O-306for n, = 7. This is clearly a significant effect and it becomes more significant as no increases. The dependence of the variance ratio on position along the beam is illustrated in Fig. 2, where the variance ratio for n, = 3 has been plotted vs. d for several values of or and x. Note that the variance ratio increases as x decreases, and in faot it approaches one as x approaches zero (clamped end).
0.8
d
FIG.2. Varianceof
velocity
()F,~fixed)-random
temperature
problem.
Random Imperfection Problem We set i:(x) = &,, in this problem, and Eq. (28a) then becomes ti*(z)
-=
‘Li&)
ss *
CQ
0 0
B*[n*(cz,)]
(1 - o#*@l) da1dat, B. H(x, n,) I
(33)
The statistical analysis of Eqs. (27) and (38) is considerably more difficult than the statistical analysis of the corresponding equations in the random temperature problem [Eqs. (29) and (30)]. Thus, we shall in the random imperfection problem consider the special case N*(x) = N* and n*(z) = n* (i.e. random variables). Equations (27) and (38) then simplify to a& 2+(1/n*) = @no) cZmO
(39)
and
~“(xf _ B”(n”) P”(x, n*) ’ Bo J’(x,no) @o(x)
Vol. 294, NO. 5, November 1972
(40)
331
W. N. Huang and P. A. Coxzarelli where F*(z,n*)
=
(1 -X)%*+2 (n*+l)(n*+2)+n*+l
--
1 (n*+l)(n*+2)
X
(41)
and where B*(n*) is given by Eq. (14). Stress. The density function of u?&,Juzmo follows from Eqs. (22) and (39) as f(EJ
= G(“$!~n~~n~
2 ‘12exp
[ - a2(ln Y)2]
(42) where y =
W(no)b,,/~xmo)- 21-Y- 1 no-l
’
(43)
a = J(;;v:;.
Equation (42) has been plotted in Fig. 3 (solid curves) for typical values of no and a,$,. The lower cut off on stress [~,/a,,, = 2/G(n,)] corresponds with n-too, while the upper cut off [o,/u,,, = 3/G(n,)] corresponds with n = 1. At both of these points ~(u,/u,~ ) equals zero. It is seen from Fig. 3 that the density function for the normal stress ratio is almost symmetric with respect to ~,,/a,, = 1; in fact it is exactly symmetric when no = 2. This behavior is analogous to the behavior of the axial stress in the truss of Ref. (4). The mean and the standard deviation of the stress ratio are easily obtained by numerical integration, and typical curves are shown in Fig. 4. Note that for values of the material parameters within the range of practical interest is very close to one and Use/,,, is very small. For example, ~~~*zWJ%??%o1 consider the rather extreme values no = 3, uNnh = 0.8 ; Fig. 4 then yields E{u$Ju,,~> = 1.002 and uWnn/_, = O-037. Thus, we may conclude that the stress ratio u?&Ju,, is almost insensitive to fluctuations in the imperfection density N*(x) and may be assumed to be deterministic with little loss of accuracy. For small enough values of a, nA/(n, - 1) (e.g. < O-2), we may approximate the lognormal density function for n * by a normal density function [Eq. (24)], and then obtain in place of Eq. (42) the approximate expression
‘(2)
,NJ(2n) u,La(n??Jumo)
- 21”
XexP[-&([a(n,)(~)-2]-1-no)P~,
nNorma1.
(44)
This equation has also been plotted in Fig. 3 (dashed curves). The interval outside of 2/G(n,) < u~~,Ju,,~ 6 3/G(n,) should be ignored.
332
Journal of The Franklin
Institute
Xteady Creep Bending in a Beam with Random Material Parameters
n log normal ----
FIG. 3. Density function of &Jo,,,--random
Finally, using Taylor series expansions may obtain the following approximate statistical moments :
n Normal
approx.
imperfection problem.
valid for uNnh/(n,- 1) small, we closed-form expressions for the
Table I gives several values of the statistical moments with uNni = O-5 as calculated by numerical integration (Fig. 4) and by the approximate Taylor series formulas [Eqs. (45)]. TABLE I
Mean
and standard deviation.
of u$,Jasmo
agz; = 0.5
%GJ~,,o>
no
Numerical integration
%ln I%nLO Taylor series
Numerical integation
Taylor series
3
1.00097
1.00397
0.02360
0.02381
5
1.00034
1.00091
0.00911
0.00909
7
1*00014
1.00034
0.00477
0.00476
10
1*00005
1.00012
0.00239
0.00238
Vol.294,No.5,November 1972
333
W. N. Huang
and F.
A. Cozzarelli
0.08
Results
:
b’ B 3 b
by
numerical
integration
O-04
0
I.5
2
3
5
4
6
7
6
9
10
no FIG. 4. Mean and standard deviation of a~,,,/a,,,--random
imperfection problem.
Velocity. We now consider the statistical analysis of the lateral velocity ratio [Eq. (40)] for arbitrary x. Note that ti*/ti, is a random process in x of first degree of randomness, since it is a function of the single random variable n* . Accordingly, only the density function of first order is of interest, and this follows from Eqs. (22) and (40) as
= (nO--l)expU-[(nO- l)2/2(w4J21 {InUn- l)/(n,- l)ll”I13 n= H .\1(277) UN nib - 1) WC4 *U$-
[’ wo
22(3-x) 24,3kno)
(tk/tio)
(46)
I ’
Here
W(x) =
~n(2+~)-&] F(s,n)-&
.Z?,~~~~n,)(
x
+(n+t~?n3+2)2+(n(tY)(i+2)
n+2
2n+3
ln(l-x)-
(n+l)(n+2)
[
(47)
and n = H (ti/tio) is the root of the expression B(n) F(x, n) - (zi~/zi)~) I?, F(x, no) = 0. The lower cut off [ti/ti, = x2(3 -z)/2B, F(x, no)] corresponds Equation (46) has been plotted in Fig. 5 (solid curves).
334
(48)
with n = 1.
Journal of The Franklin
Institute
Steady Creep Bending in a Beam with Random Material
-I
$7
5
----
n
3 log
-/I
”
Parameters
normal,
Normal
0.394
Q E
approx.
4 .& .
3
2
I
0
FIG.
5. Density
function
20
I.0
05
of ti*/zir,--random
imperfection
problem.
We find that the improper integrals defining the statistical moments of the velocity ratio ti*lti, as obtained from density function (46) are unbounded. This is due to the fact that the tail of the lognormal density function [Eq. (22)] d oes not decay toward zero rapidly enough to offset the growth of the velocity ratio toward infinity as n approaches infinity. However, the contribution to the statistical moments arising from the lognormal tail is physically meaningless, since the velocity ratio can in fact never be infinite in an actual beam. We can avoid this difficulty by using the normal approximation to n* [Eq. (24)], since this density function decays more rapidly toward zero. The normal density function is a good approximation to the lognormal density function for small enough value of crNn~/(nO- 1) (e.g. < O-2), except in the region of the tail. Adopting this procedure, Eq. (46) is replaced by the approximate implicit relation exp [ - (n - QP41 J(2n) an W(z)
n Normal. (49) n-iYki&~ This approximate equation has been plotted in Fig. 5 (dashed curves). We ignore the interval less than ti/zi+, = x3( 3 - z)/2B, F(x, n,). Figure 6 gives some typical curves for the mean and standard deviation of the lateral velocity ratio. Consider the values for the normal approximation arr = olvnh = 0.5 and E{n*} = n, = 5; then Fig. 6 gives E{ti*/2i(,,) = 1.0569 and UJZ;~= O-3620 for x = O-1 and E{ti*/zi~,} = l-0418 and ot;l~~ = O-2964 for x = 1. This is clearly a significant effect. We observe that the curves level off rapidly and in the range of practical interest there is little dependence on n, and x. Also, the mean velocity exceeds the nominal velocity.
Vol. 294, No. 5, November 1972
335
W. N. Huang and F. A. Coxxarelli Finally, we obtain approximate Taylor series formulas for the mean and variance which are valid for ~~n,$/(n,, - 1) small. The results show a very slight dependence on n,, and thus we may let n,-+co and obtain
E 2 (
1
z l+$(ln2)2+*(ln2)4,
uzjdOz o2,(ln 2)2 + $i(ln
(504 2)4
for arbitrary x. Table II gives a comparison of the statistical moments as obtained by numerical integration (Fig. 6) and by approximate formulas (50) for u,, = O-5 and x = 1. 0.7 0.6
-
04
-
n
.1 .> b
---
0.2 -
0
a,
I
2
Normal
approx.
0” - 03 ------------------
0.2
I
I
I
4
-
6
6
I
IO
I
I2
I
I
14
16
IE
1.15-
I.12
I
20 7
-
< ,P 2 f
-
x-1
---
x
-
0.1
4 I.04
-
I .o 1 2
0”
I
=
0.2
I 6
I 4
I IO
I 6
I I2
I I4
1 I6
I 16
20
no FIG. 6. Mean and standard deviation of ti*/ti,-random TABLE
Mean
II
and standard deviation (T, = 0.5; z = 1
of ti*/ti,
E{ti*/ti,}
336
imperfection problem.
ui&
%
Numerical integration
Taylor series
Numerical integration
Taylor series
3.5
1.03821
1.06186
0.27697
0.37650
5
1.04177
0.29638 0.31328
7
1.04523
10
1.04871
0.32878
15
1.05215
0.34300
20
1.05418
0.35098
Journal of The Franklin
Institute
Steady Creep Bending in a Beam with Random Material Parameters Summary
of Results
A statistical analysis has been presented for steady creep bending in a beam, where the material parameters are random processes along the neutral axis due to random fluctuations in temperature and imperfection density. The “random temperature problem” and the “random imperfection problem” are uncoupled problems, and the statistics of the normal stress and lateral velocity have been found for both problems. In the random temperature problem with the temperature T* taken as a homogeneous normal random process, the normal stress is deterministic whereas the lateral velocity is very sensitive to variations in temperature. We find that the mean velocity always exceeds the nominal velocity, and that the mean of the velocity ratio is independent of n, and x, while its variance is a function of n, and x. However, if the temperature is taken as a random variable the variance is independent of n, and x. In the random imperfection problem with the imperfection density N* taken as a normal random variable, we find that the normal stress is only very slightly random, whereas the velocity is again highly random. For values of the parameters of greatest practical interest, we find that the statistics of the lateral velocity ratio exhibits only a minor dependence on n, and x. The mean velocity in this case again always exceeds the nominal velocity. In the present problem the stress and velocity are in general random processes, in contra.st with the problem of Ref. (4) where they were random variables. We again find that the stress shows very little random fluctuation whereas velocity is highly random. However, we now find that in general the statistics of the velocity may show a significant dependence on n, and x, which was not the case in Ref. (4). If the temperature and imperfection density are restricted to random variables, this dependence on n, and x becomes insignificant. References
(1) T. T. Soong and F. A. Cozzarelli, “Effect of random temperature distributions on
(2)
(3)
(4)
(5) (6)
Vol.
creep in circular plates”, Int. J. Non-Linear Mech., Vol. 2, No. 1, pp. 27-38, March 1967. H. Parkus, “On the lifetime of viscoelastic structures in a random temperature field”, Recent Progress iw Applied Mechanics--PoZbe Odqvist Volume, pp. 391397, John Wiley, New York, 1967. H. Parkus and H. Bargmann, “Note on the behavior of thermorheologically simple materials in random temperature fields”, Acta Me&an&a, Vol. 9, pp. 152-157, 1970. F. A. Cozzarelli and W. N. Huang, “Effect of random material parameters on nonlinear steady creep solutions”, Int. J. solid8 Structures, Vol. 7, No. 11, pp. 1477-1494, Nov. 1971. Section 5.3, Blaisdell, Waltham, J. Hult, “Creep in Engineering Structures”, Massachusetts, 1966. J. Aitchison and J. A. C. Brown, “The Lognormal Distribution”, Cambridge University Press, London, 1957.
294, No.
5. November
1972
337
F. A. COZZARELLI
Department of Engineering Science, Faculty of Engineering and Applied Sciences State University of New York at Bu$alo, Bu$alo, New York A statistical
ABSTRACT:
the material parameters $uctuationa
in temperature
are presented found
analysis
for
is presented for steady
creep bending
and imperfection
the statistical
properties
density.
Analytical
is highly random. The statistics
and numerical
of the norm&l stress and lateral velocity.
that whereas the normal stress shows very little ra&Juctuation
on the nominal
in a beam, where
are random processes alon,g the neutral axis arising from ran&an
and on the position
It is
th.elateral velocity
of the lateral velocity generally show a sign&ant
value of the creep power
results
dependence
along the neutral axis.
Notation
A” b B
BPV41 c 11 d
6
f( 1 ww,) Wd h H f J L M n 6 N P sgn( 1 u:;
ti
statistical constant, Eqs. (43) cross-sectional area of beam width of rectangular beam temperature and imperfection insensitive creep constant, Eq. (16b) function of n(X), Eq. (14) autocovariance of indicated quantity nondimensional correlation distance, Eq. (33) nondimensional creep viscosity parameter, Eq. (17) expected value operator probability density function function of z and n,,, Eq. (28b) function of n,,, Eq. (27b) depth of rectangular beam root of Eq. (48) property of rectangular beam, Eq. (8b) property of beam cross-section, Eq. (5) length of beam bending moment, Eq. (4) creep power, Eq. (1) negative of the nominal slope, Eq. (21b) imperfection density lateral point force signum function, Eq. (2) temperature unit step function lateral velocity, Eq. (12)
* This research was supported in part by the National Science Foundation under Grant No. GK-1834X, and in part by the Office of Naval Research under Contract No. N 00014-71-C-0108.
323
W. N. Huang and F. A. Cozzarelli right-hand side of Eq. (47) nondimensional space coordinate centroidal axis of beam function in stress solution, Eqs. (43) transverse coordinate creep viscosity parameter, Eq. (1) normal strain rate, Eq. (1) argument for autocovariance of homogeneous random process, Eqs. (19~) and (23~) rate of curvature of beam, Eq. (3) right-hand side of Eq. (35) standard deviation of indicated quantity arbitrary deterministic constant reference stress (chosen aa 2PL/bh2), Eq. (1) normal stress, Eq. (1) maximum normal stress, Eq. nondimensional temperature, indicates nominal value, i.e. density equal mean value (T, designates random variable designates random process in
(9) Eq. (16b) value when temperature or N,)
or imperfection
x
I. Introduction
Creep parameters are highly dependent on temperature, and some effects of randomness in temperature have been considered by Soong and Cozzarelli (l), Parkus (2), Parkus and Bargmann (3) and Cozzarelli and Huang (4). Creep parameters are also highly dependent on imperfection density, and the effect of randomness in imperfection density was also considered in Ref. (4). In that reference the authors have studied the effect of random material parameters on nonlinear steady creep in a 3-bar truss. The stresses and velocities analyzed were simply random variables, and it would be instructive to also consider a problem where they are random processes in some coordinate. Thus, in this investigation we shall study the problem of steady creep bending in a beam, where the temperature, imperfection density, creep parameters, normal stress and lateral velocity are all in general random processes along the neutral axis. The basic approach is similar to that used in Ref. (a), i.e. the problem is “random temperature problem” separated into two uncoupled parts -the and the “random imperfection problem”. In the random temperature problemonlythe creep viscosity parameter 4 is random, whereas in the random imperfection problem only the creep power n is random. The derivation of the governing equations for the normal stress and lateral velocity in the beam, with arbitrary statistics for .$ and n, is presented in Section 2. Then, the density functions and statistical moments of the stress and velocity are given in Section 3, with the temperature and imperfection density taken as independent homogeneous normal random processes along the neutral axis. The final section contains a summary of the results obtained.
324
Journal of The Franklin
Institute
Steady Creep Bending in a Beam with Random Material Parameters ZZ. Governing
Equations
Consider an initially straight beam subjected to prescribed deterministic loads. The cross-section of the beam is uniform along the neutral axis of the beam (X), and symmetric with respect to the plane of the applied loads (2X). Using an asterisk with a symbol to indicate a random quantity, we employ a steady creep power law with random material parameters in the form
(1) Here, the random material parameters i,*(X) random processes in X, the normal stress o,*(X) $(X) are random processes in X for a designated deterministic constant reference stress and sgn defined as
1,
sgn(u,*) =
and n*(X) are in general and the normal strain rate value of 2, a, is an arbitrary (a:) is the Signum function
a,*> 0,
(2)
u,*
i -I,
The Bernoulli assumption of plane sections yields the expression strain rate at the arbitrary position X as s;(x)
= G*(x)
for the (3)
Here, a*(X) is the curvature rate of the beam, which is a random process in X and independent of 2. Introducing Eqs. (1) and (3) into the definition of bending moment, we obtain M =
I’
u,*(X)ZdB
= a,J*(X)
where A is the cross-sectional
w (k*),
(4)
area of the beam. The quantity
J”(X)
=
]Z
~l+wn*(X))
&j
s
(5)
is a property of the beam cross-section and reduces to the ordinary second moment of area for n*(X) = 1. For example, for a beam of rectangular cross-section of width b and depth h we have J*(X)
=
2n*(X) 2n*(X)+l
bh2 h l/n*(X) -4 02 .
When n*(X) = 1 Eq. (6) reduces to bh3/12, which is the ordinary moment of area for a rectangular beam. Combining Eqs. (l), (3) and (4), we obtain for the stress u*(x)
z
Vol. 294, No. 5, November 1972 19
=
M 12I1’n*(X) sgnm* J*(X)
(6) second
(7)
325
W. N. Huang and F. A. Cozzarelli For a rectangular form
cross-section,
this may be rewritten as in Hult (5) in the M
22
f*(x)
The maximum
=
l/n*(x)
II
u,*(X) = I*(x)
-
h
n,*(x)
w Vlf (sb)
bh‘2.
2 + 4n*(X)
normal stress at 2 = h/2 is then given by = M/f*(X)
ok(X)
The rate of curvature of the beam is obtained from Eq. (4) as g*(X)
=
‘:tx) 1M in*(X)sgn (M)
[J*(x)
u$-)
*
If the slope of the deformed beam is such that d~*(-V I --ax--
<1 I
(11)
’
where C*(X) is the lateral velocity of the beam, Eq. (10) may then be written as d2 C*(X) dX2
= _ 9,*(X) ( M In*(X) [J*(X) ,c]n*(X) sgn (M)*
(12)
The maximum normal stress U&(X) and the lateral velocity G*(X), as specified by Eqs. (9) and (12), are the quantities of greatest practical interest. A statistical analysis of Eq. (9) does not require a specification of the particular form of the bending moment M. However, Eq. (12) is more complex and a choice of a particular form for M would facilitate the statistical Thus, let us consider a cantilever rectangular analysis of this equation. cross-section beam of length L loaded at its free end (X = L) with a prescribed deterministic constant point force P. We then have M = - P(L - X), and obtain from Eq. (12) together with the boundary conditions at the clamped end (X = 0)
Here, the reference stress a, in Eq. (12) has been chosen as 2PL/bh2 and B*[n*(X)] is a function of n*(X) defined as B*[n*(x)]
=
y$;
‘1n’(X),
Equations (9) and (13) will form the basis of the statistical the next section.
326
(14) analysis in
Journal of The Franklin
Institute
Steady Creep Bending in a Beam with Random Material Parameters III.
Statistical
Analysis
of
Stress
and
Velocity
Material Parameters We assume that the temperature T*(X) and imperfection density N*(X) are independent homogeneous normal random processes in X, with means !Z” and NO respectively. Also, assuming that &z(X) is a function of T*(X) only while n*(X) is a function of N*(X) only, we can show that i:(X) and n*(X) are independent random processes obeying the lognormal statistics given in Ref. (4). And, we may treat the “random temperature problem” and the “random imperfection problem” as two uncoupled problems. For convenience, we summarize some of the statistical results pertaining to i,*(X) and n*(X) obtained in Ref. (4). The first-order density function of 5”*(X) is given by
f(T) = &-
exp [-(‘2-p2], T
where u$, is the variance. The random parameter 9,*(X) is related to the nondimensional random temperature T*(X) by the expression S,*(X) = &exp
[T*(X)],
Wa)
where T*(X)
=
W’*(X) - To1 T;
In Eqs. (16), kc,,is the nominal value of $(X) deterministic constant. Letting 8*(X)
= 9,*(X)/&
Wb)
*
(value at T* = To) and B is a
= exp [7*(X)]
we have the first-order density function for a*(X)
(17)
as
(18) where U(&‘) is the unit step function. Equation (18) is the lognormal probability density function (6). The mean E(&*}, variance U$ and autocovariance C8(r]) of a*(X) are given by E{&*} = exp (e/2)
> 1,
0: = exp (a:) [exp (~9) - l] > C&d
=
exp
(4)
{exp
(1% CT:,
W%I)I- $3
Ugb)
P9c)
where C,(q) is the autocovariance of T* and 17= XI -X2. Similarly, the first-order density function of N*(X) is given by
Vol. 294, No. 5. November
1972
327
W. N. Huung where +
F. A. Cozzarelli
and
is the variance. n*(X)
Also
= l+(n,--
1)exp
1
-
n,EN*(X) -
41 1’
(%I-1)
(214
where n, = n* IN.=NO>nh = - dn*/dN* INesNo
(21b)
and thus
f(n) =
(no- 1) 4274 ~.&A=-
1)
exp
i
- (720-1)2 [In(~)]2)U(n-l). 2(UNnh)2
This is a lognormal density function with two parameters and n*(X) has the following statistical properties: E{n*j = l+(n,-1)exp
(2($14”l;l)
ui = (no- 1)2exp [Es]
C,,(q) = (no- l)2exp
(no and CYST;) (6),
(234
> no>
(exp [m2]
- 1) > (aAV$J2,
[H2] (w [&2C,0] - 1).
Finally, if uNnh/(no- 1) is small enough Eq. (22) by the normal density function f(n)=hmexp
(22)
(e.g.
[-VI
Wb) (23c)
< 0.2) we may approximate
, --$J+ Small,
(24)
where now E{n*) = no and u”, = (uAr$J2. Since it is more convenient to work with nondimensional equations, we introduce the nondimensional distance 5 = X/L and the nominal values u,,, and zip,(values at T * = Toand N* = No). These follow from Eqs. (9) and (13) as (25)
G,(x) =
(1 - x)m+2 2L2 B,, B, -(no+l)(no+2)+no~l h
1 (no+l)(no+2)
1’ (26)
where B, is the nominal value of B*[a*(x)]. We can then obtain from Eqs. (9) and (13) in conjunction with Eqs. (25) and (26) the expressions G@)
_ 2+
uX?ILO
G(n,)
328
=
2+
l?/~“(~H Gbo) ’ P/no)
VW
W’b)
Journal ofTheFranklin Institute
Steady Creep Bending in a Beam with Random Material Parameters and %5”(x)
-=
?&l(x)
ss f&
1:
0 0
B*[n*(al)]
d*(q)
dorrdor, &, F(x, no) I
(1 - c#‘(~~)
1
(1 - @of2
(29a)
(28b)
F(s,no)=(no+l)(no+2)+noY-1-(no+I)(no+2)*
Random Temperature Problem Stress. If only the temperature Eq. (27a) yields
T*(s)
u=
is random
we set n*(x) = no, and
1.
(29)
=3X&0
Thus, the normal stress is a deterministic constant. Velocity. Turning now to the lateral velocity, we obtain from Eq. (28a) [with n*(x) = no] the expression (1 --1)nodolld~2
I
F(x,n,).
(30)
The mean of the lateral velocity ratio follows directly from Eqs. (19) and (30) and the homogeneity of 8*(x) as E(E)
=exp($)>l.
We see that the mean lateral velocity velocity. Proceeding now to the determination from Eq. (30) the relation ti*(x,) E -i ho
ti*(z,)
(31)
always exceeds the nominal lateral of second-order
statistics, we obtain
=
Co@,) 1
In obtaining Eq. (32), the autocovariance of L?*(x) was used in accordance with Eq. (19c). Choosing for C,(x, - x2) the physically plausible exponential expression C,(X,-2,)
= a:exp(-
where d is the nondimensional correlation velocity ratio follows from Eq. (32) as c+;,
Vol.
294, No. 20
5, Novemkr
1972
= exp (04) (A-
Ix,-x,I/a). distance, l),
the variance
(33) of the (34)
329
W. N. Huang and F. A.
Cozxarelli
where
s
A = O~(x-y)(l-y)~o X
,2(x (s
2) (1 - .z)noexp [ofexp
( -v)]dz)d?l/F(z,n,J2.
(35)
Then, the variance ratio is obtained from Eqs. (19b) and (34) as
4Jhim
A-1 (~3) - 1’
(36)
20s = exp
The following limiting values are obtained from this result : lim a~iz;o = 0, d+OT
limd$i!?= d+x 08
1.
(37)
We note from the above that the mean of the lateral velocity ratio ti*/ti,, is independent of the parameter no, while its variance is a function of n,. This differs from the result obtained for a 3-bar truss in Ref. (4), where both the variance and the mean of the velocity ratio are independent of n,,. In the present problem the bending moment is a function of the space coordinate z; thus the quantity 1MI no is not factorable from the double integral for velocity [Eq. (12)] and the velocity ratio is a function of n,. However, for the special case of d* a random variable (d-+co) we see from Eqs. (30) and (37) that @*/zi+, = b*, with statistics independent of n, and x. In Fig. 1, the variance ratio at the free end X = L (x = 1) as given by Eq. (36) has been plotted vs. the nondimensional correlation distance d for several values of CT,and n,. Consider the typical values ur = 0.3 and us = O-321; Eqs. (31) and (34) then yield the values E{ti*/ti,) = 1.046, UJ~~ = 0.289 I.0
06
d
FIG. 1. Variance of velocity (x fixed)-random
330
temperature problem.
Journal of The Franklin
Institute
for n, = 1 and o,J~~ = O-306for n, = 7. This is clearly a significant effect and it becomes more significant as no increases. The dependence of the variance ratio on position along the beam is illustrated in Fig. 2, where the variance ratio for n, = 3 has been plotted vs. d for several values of or and x. Note that the variance ratio increases as x decreases, and in faot it approaches one as x approaches zero (clamped end).
0.8
d
FIG.2. Varianceof
velocity
()F,~fixed)-random
temperature
problem.
Random Imperfection Problem We set i:(x) = &,, in this problem, and Eq. (28a) then becomes ti*(z)
-=
‘Li&)
ss *
CQ
0 0
B*[n*(cz,)]
(1 - o#*@l) da1dat, B. H(x, n,) I
(33)
The statistical analysis of Eqs. (27) and (38) is considerably more difficult than the statistical analysis of the corresponding equations in the random temperature problem [Eqs. (29) and (30)]. Thus, we shall in the random imperfection problem consider the special case N*(x) = N* and n*(z) = n* (i.e. random variables). Equations (27) and (38) then simplify to a& 2+(1/n*) = @no) cZmO
(39)
and
~“(xf _ B”(n”) P”(x, n*) ’ Bo J’(x,no) @o(x)
Vol. 294, NO. 5, November 1972
(40)
331
W. N. Huang and P. A. Coxzarelli where F*(z,n*)
=
(1 -X)%*+2 (n*+l)(n*+2)+n*+l
--
1 (n*+l)(n*+2)
X
(41)
and where B*(n*) is given by Eq. (14). Stress. The density function of u?&,Juzmo follows from Eqs. (22) and (39) as f(EJ
= G(“$!~n~~n~
2 ‘12exp
[ - a2(ln Y)2]
(42) where y =
W(no)b,,/~xmo)- 21-Y- 1 no-l
’
(43)
a = J(;;v:;.
Equation (42) has been plotted in Fig. 3 (solid curves) for typical values of no and a,$,. The lower cut off on stress [~,/a,,, = 2/G(n,)] corresponds with n-too, while the upper cut off [o,/u,,, = 3/G(n,)] corresponds with n = 1. At both of these points ~(u,/u,~ ) equals zero. It is seen from Fig. 3 that the density function for the normal stress ratio is almost symmetric with respect to ~,,/a,, = 1; in fact it is exactly symmetric when no = 2. This behavior is analogous to the behavior of the axial stress in the truss of Ref. (4). The mean and the standard deviation of the stress ratio are easily obtained by numerical integration, and typical curves are shown in Fig. 4. Note that for values of the material parameters within the range of practical interest is very close to one and Use/,,, is very small. For example, ~~~*zWJ%??%o1 consider the rather extreme values no = 3, uNnh = 0.8 ; Fig. 4 then yields E{u$Ju,,~> = 1.002 and uWnn/_, = O-037. Thus, we may conclude that the stress ratio u?&Ju,, is almost insensitive to fluctuations in the imperfection density N*(x) and may be assumed to be deterministic with little loss of accuracy. For small enough values of a, nA/(n, - 1) (e.g. < O-2), we may approximate the lognormal density function for n * by a normal density function [Eq. (24)], and then obtain in place of Eq. (42) the approximate expression
‘(2)
,NJ(2n) u,La(n??Jumo)
- 21”
XexP[-&([a(n,)(~)-2]-1-no)P~,
nNorma1.
(44)
This equation has also been plotted in Fig. 3 (dashed curves). The interval outside of 2/G(n,) < u~~,Ju,,~ 6 3/G(n,) should be ignored.
332
Journal of The Franklin
Institute
Xteady Creep Bending in a Beam with Random Material Parameters
n log normal ----
FIG. 3. Density function of &Jo,,,--random
Finally, using Taylor series expansions may obtain the following approximate statistical moments :
n Normal
approx.
imperfection problem.
valid for uNnh/(n,- 1) small, we closed-form expressions for the
Table I gives several values of the statistical moments with uNni = O-5 as calculated by numerical integration (Fig. 4) and by the approximate Taylor series formulas [Eqs. (45)]. TABLE I
Mean
and standard deviation.
of u$,Jasmo
agz; = 0.5
%GJ~,,o>
no
Numerical integration
%ln I%nLO Taylor series
Numerical integation
Taylor series
3
1.00097
1.00397
0.02360
0.02381
5
1.00034
1.00091
0.00911
0.00909
7
1*00014
1.00034
0.00477
0.00476
10
1*00005
1.00012
0.00239
0.00238
Vol.294,No.5,November 1972
333
W. N. Huang
and F.
A. Cozzarelli
0.08
Results
:
b’ B 3 b
by
numerical
integration
O-04
0
I.5
2
3
5
4
6
7
6
9
10
no FIG. 4. Mean and standard deviation of a~,,,/a,,,--random
imperfection problem.
Velocity. We now consider the statistical analysis of the lateral velocity ratio [Eq. (40)] for arbitrary x. Note that ti*/ti, is a random process in x of first degree of randomness, since it is a function of the single random variable n* . Accordingly, only the density function of first order is of interest, and this follows from Eqs. (22) and (40) as
= (nO--l)expU-[(nO- l)2/2(w4J21 {InUn- l)/(n,- l)ll”I13 n= H .\1(277) UN nib - 1) WC4 *U$-
[’ wo
22(3-x) 24,3kno)
(tk/tio)
(46)
I ’
Here
W(x) =
~n(2+~)-&] F(s,n)-&
.Z?,~~~~n,)(
x
+(n+t~?n3+2)2+(n(tY)(i+2)
n+2
2n+3
ln(l-x)-
(n+l)(n+2)
[
(47)
and n = H (ti/tio) is the root of the expression B(n) F(x, n) - (zi~/zi)~) I?, F(x, no) = 0. The lower cut off [ti/ti, = x2(3 -z)/2B, F(x, no)] corresponds Equation (46) has been plotted in Fig. 5 (solid curves).
334
(48)
with n = 1.
Journal of The Franklin
Institute
Steady Creep Bending in a Beam with Random Material
-I
$7
5
----
n
3 log
-/I
”
Parameters
normal,
Normal
0.394
Q E
approx.
4 .& .
3
2
I
0
FIG.
5. Density
function
20
I.0
05
of ti*/zir,--random
imperfection
problem.
We find that the improper integrals defining the statistical moments of the velocity ratio ti*lti, as obtained from density function (46) are unbounded. This is due to the fact that the tail of the lognormal density function [Eq. (22)] d oes not decay toward zero rapidly enough to offset the growth of the velocity ratio toward infinity as n approaches infinity. However, the contribution to the statistical moments arising from the lognormal tail is physically meaningless, since the velocity ratio can in fact never be infinite in an actual beam. We can avoid this difficulty by using the normal approximation to n* [Eq. (24)], since this density function decays more rapidly toward zero. The normal density function is a good approximation to the lognormal density function for small enough value of crNn~/(nO- 1) (e.g. < O-2), except in the region of the tail. Adopting this procedure, Eq. (46) is replaced by the approximate implicit relation exp [ - (n - QP41 J(2n) an W(z)
n Normal. (49) n-iYki&~ This approximate equation has been plotted in Fig. 5 (dashed curves). We ignore the interval less than ti/zi+, = x3( 3 - z)/2B, F(x, n,). Figure 6 gives some typical curves for the mean and standard deviation of the lateral velocity ratio. Consider the values for the normal approximation arr = olvnh = 0.5 and E{n*} = n, = 5; then Fig. 6 gives E{ti*/2i(,,) = 1.0569 and UJZ;~= O-3620 for x = O-1 and E{ti*/zi~,} = l-0418 and ot;l~~ = O-2964 for x = 1. This is clearly a significant effect. We observe that the curves level off rapidly and in the range of practical interest there is little dependence on n, and x. Also, the mean velocity exceeds the nominal velocity.
Vol. 294, No. 5, November 1972
335
W. N. Huang and F. A. Coxxarelli Finally, we obtain approximate Taylor series formulas for the mean and variance which are valid for ~~n,$/(n,, - 1) small. The results show a very slight dependence on n,, and thus we may let n,-+co and obtain
E 2 (
1
z l+$(ln2)2+*(ln2)4,
uzjdOz o2,(ln 2)2 + $i(ln
(504 2)4
for arbitrary x. Table II gives a comparison of the statistical moments as obtained by numerical integration (Fig. 6) and by approximate formulas (50) for u,, = O-5 and x = 1. 0.7 0.6
-
04
-
n
.1 .> b
---
0.2 -
0
a,
I
2
Normal
approx.
0” - 03 ------------------
0.2
I
I
I
4
-
6
6
I
IO
I
I2
I
I
14
16
IE
1.15-
I.12
I
20 7
-
< ,P 2 f
-
x-1
---
x
-
0.1
4 I.04
-
I .o 1 2
0”
I
=
0.2
I 6
I 4
I IO
I 6
I I2
I I4
1 I6
I 16
20
no FIG. 6. Mean and standard deviation of ti*/ti,-random TABLE
Mean
II
and standard deviation (T, = 0.5; z = 1
of ti*/ti,
E{ti*/ti,}
336
imperfection problem.
ui&
%
Numerical integration
Taylor series
Numerical integration
Taylor series
3.5
1.03821
1.06186
0.27697
0.37650
5
1.04177
0.29638 0.31328
7
1.04523
10
1.04871
0.32878
15
1.05215
0.34300
20
1.05418
0.35098
Journal of The Franklin
Institute
Steady Creep Bending in a Beam with Random Material Parameters Summary
of Results
A statistical analysis has been presented for steady creep bending in a beam, where the material parameters are random processes along the neutral axis due to random fluctuations in temperature and imperfection density. The “random temperature problem” and the “random imperfection problem” are uncoupled problems, and the statistics of the normal stress and lateral velocity have been found for both problems. In the random temperature problem with the temperature T* taken as a homogeneous normal random process, the normal stress is deterministic whereas the lateral velocity is very sensitive to variations in temperature. We find that the mean velocity always exceeds the nominal velocity, and that the mean of the velocity ratio is independent of n, and x, while its variance is a function of n, and x. However, if the temperature is taken as a random variable the variance is independent of n, and x. In the random imperfection problem with the imperfection density N* taken as a normal random variable, we find that the normal stress is only very slightly random, whereas the velocity is again highly random. For values of the parameters of greatest practical interest, we find that the statistics of the lateral velocity ratio exhibits only a minor dependence on n, and x. The mean velocity in this case again always exceeds the nominal velocity. In the present problem the stress and velocity are in general random processes, in contra.st with the problem of Ref. (4) where they were random variables. We again find that the stress shows very little random fluctuation whereas velocity is highly random. However, we now find that in general the statistics of the velocity may show a significant dependence on n, and x, which was not the case in Ref. (4). If the temperature and imperfection density are restricted to random variables, this dependence on n, and x becomes insignificant. References
(1) T. T. Soong and F. A. Cozzarelli, “Effect of random temperature distributions on
(2)
(3)
(4)
(5) (6)
Vol.
creep in circular plates”, Int. J. Non-Linear Mech., Vol. 2, No. 1, pp. 27-38, March 1967. H. Parkus, “On the lifetime of viscoelastic structures in a random temperature field”, Recent Progress iw Applied Mechanics--PoZbe Odqvist Volume, pp. 391397, John Wiley, New York, 1967. H. Parkus and H. Bargmann, “Note on the behavior of thermorheologically simple materials in random temperature fields”, Acta Me&an&a, Vol. 9, pp. 152-157, 1970. F. A. Cozzarelli and W. N. Huang, “Effect of random material parameters on nonlinear steady creep solutions”, Int. J. solid8 Structures, Vol. 7, No. 11, pp. 1477-1494, Nov. 1971. Section 5.3, Blaisdell, Waltham, J. Hult, “Creep in Engineering Structures”, Massachusetts, 1966. J. Aitchison and J. A. C. Brown, “The Lognormal Distribution”, Cambridge University Press, London, 1957.
294, No.
5. November
1972
337