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Beams on random elastic supports

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Beams on random elastic supports
A. A. Mahmoud
Department of Engineering Mathematics and Physics, Cairo University, Cairo, Egypt
Magdy A. El Tawil
Department of Engineering Mathematics, Cairo University, Giza, Egypt
The fourth-order differential equation of a beam resting on elastic supports is highly dependent upon the modulus of subgrade reaction,
k. In
general, the value of
k
is
random; consequently,
a beam resting on random supports is represented by a random differential equation of random coefficient. This dtfferential equation has no exact solution; accordingly, approximate solution is recommended. The statistical moments
of the
solution process are expanded
in terms of the stochastic scale parameter 4’ and evaluated by using successive approximations.
Keywords
stochastic differential equations, stochastic beams, stochastic subgrade reactions
Introduction
Some
mathematical models are influenced
by the
ran-
dom uncertainties included in the applied excitation or in the coefficient of the represented differential equa- tion. Stochastic analysis is of great interest for engi- neers, especially structural and geotechnical ones. Casciati et al.’ analyzed the three-dimensional frames under dynamic stochastic excitation, using the sto- chastic equivalent linearization. Li and Ibrahim2 in- vestigated the autoparametric interaction of two nor- mal modes in a 3 degrees of freedom (3-DOF) structural model under a wideband random excitation. Muscolino3 evaluated the mean square response of a linear system subjected to a multicorrelated stationary or nonsta- tionary process. Branstetter and Thomas4 estimated the first- and second-order statistical response mo- ments for linear multi-degree of freedom dynamical systems having random loads and random structural characteristics. Vanmarcke and Grigorit? discussed the analysis of shear beams of random rigidity. Spanos and Ghanem discussed the solution of problems involving material variability. The authors’ assumed cantilev- ered beams with continuous stochastic rigidity and de- veloped the stochastic second-order differential equa- tion with its random uncertainty in the excitation function.
Address reprint requests to Dr. El Tawil at Department of Engi- neering Mathematics, Faculty of Engineering, Cairo University, Giza, Egypt. Received 8 July 1991; revised 7 January 1992; accepted IS January 1992
330
Appl. Math. Modelling, 1992, Vol. 16, June
In this paper the fourth-order differential equation with a stochastic coefficient of a beam resting on ran- dom elastic supports (Winkler’s model) is analyzed. The stochastic uncertainty in the subgrade reaction
k
is scaled by a deterministic parameter 5. The estimation of this parameter depends on the amount of the un- certainty in the Winkler’s subgrade model. The suc- cessive approximation method proved that the random deflection can be obtained as a power series in the parameter 5. The first and the second statistical mo- ments of the random deflection are obtained by as- suming the first-order approximation.
Random deflection of beams resting on random elastic supports
Beams resting on the ground are often modelled as beams supported by elastic springs (Winkler sup- ports). In general, the properties of the ground differ from one point to another, which involves uncertainty in the characteristics of equivalent elastic springs. In other words, the elastic supports are of random nature. Consequently, the fourth-order differential equation of a beam resting on random elastic supports, shown in
Figure I,
is random and is given as
Ez
d4y(x
w)
dx4 + k x;
w)y(x; WI = q(x)
where
E
is the Young modulus of elasticity, I is the moment of inertia of the cross section, y(x; w) is the random deflection, w is the random outcome which belongs to (Cn, 93,
P)
in which IR is a sample space, 93 is a a-algebra associated with 0, and
P
is a probability measure, q(x) is the applied load, and
k x; w)
is the random modulus of subgrade reaction. In the case of
0 1992 Butterworth-Heinemann
Beams on random elastic supports: A. A. Mahmoud and M. A. El Tawil
.
b. x
III
d
random outcomes. This term is the solution of the de- terministic system (i.e., 5 = 0). By using ye(x) as the first approximation the second approximation is ob- tained as
Figure 1.
A beam on elastic supports
a simply supported beam the following boundary con- ditions for all random outcomes are applied:
y(0) = y”(0) =
0
(24
y L) =
y” L) =
0
(2b)
where
L
is the beam length. The modulus of subgrade reaction
k
can be represented as
k = kO + l&x; W)
(3) where
kO
s the mean value of
k
or it is the subgrade reaction for the deterministic system. Here, 5 is a de- terministic scale parameter which represents the de- gree of uncertainty in
k,
and 77(x; w) is any zero mean random process. Substituting equation (3) into (1) yields
ZYk
WI =
q(x) - [7/(x; w)y(x; w)
(4) where .JZ s a deterministic differential operator given as
2% +k,,
(5)
The random deflection of the beam can be expanded in power series expansion as Cc Y(K w) = 2 C’Y;(G w) (6) i=o It should be noted that the first term of the power series, ye(x), is deterministic and independent of the
~Y(‘)(X. w) = 4(x) - 5yoc477(x; WI
(7) The ith approximation is computed as LEy”‘(X; w) = q(x) - 57&V; w),(j- “(X; w) i>o (8) which leads to the existence of the assumed series solution. The first term in the series in equation (6) is obtained when solving the deterministic system and is given by
Yo(X) = 4(x)
(9) When equation (6) is substituted into equation (I), the ith term is computed as
~yj X;W = -yi_l X;W77 X;W
i>O 10)
Using the deterministic boundary conditions given in equation (2), we apply the following conditions for each term of equations (9) and (10): yJ0) = y;(o) = 0 (1 la) y;(L) = y::‘(L) = 0 (llb) The impulse response function associated with the de- terministic operator 5 ? s known to be* h(x) = cash (7x). sin (yx) - sinh (yx)
. cos (yx) (12)
Assuming uniformly applied load and solving equations (9) and (10) together with boundary conditions (lla) and (11 b), the first and second terms of the series are cash
(yx) . cos
y L - x) +
cash
y L -
x) .
cos (yx)
cash
yL) + cos yL)
(13) and yi(x; w) = Q(CZ +
P/m;)
cash
(yx) .
sin (yx) + Q((Y -
Plmo)
sinh (yx)
* cos (yx) - Q/W > i h(x - sh(s;
w)yob) ds
0
(14) where Q = ll(fimo) (15)
L a w) = aI
I
h L - s)v s;
w)y,,(s)ds
0 L + a2
J
4 L - sh s;
w)Yo(s)~~ (16)
0 L P W = PI Ih L - dv s;
w)yo(s)ds
0 L + P2
_f
+ L - S)T S;
w)yo(s)ds
(17)
0
in which (Y~
y2/(Elmo).
(sinh
(yL)
.
cos
yL) +
cash
(yL)
.
sin
yL))
(18) a2 = l.l(2EZmg
r)
*
(sinh
yL)
* cos
yL) -
cash (-&). sin
yL))
(19) PI =
y2/ EZmg r).
(cash
(yL)
.
sin
yL) -
sinh
(yL)
.
cos
-yL))
(20) p2 =
l.l 2EZmg)
*
(cash
(yL)
.
sin
yL) +
sinh
(yL)
.
cos
yL))
(21)
Appl. Math. Modelling, 1992, Vol. 16, June 331
Beams on random elastic supports: A. A. Mahmoud and M. A. El Tawil ma = (kolEZ)1’4 (22)
The variance of the random deflection is given as y = molti (23) f =
(2y2/mo)
*
(cash* (7L)
*
sin* (@) Vary@; w) = 5 [*j vary& w)
i=l +
sinh* ($_).
co9 (-yL))
(24) <b(x) = 2y2 (cash (yx)
.
sin (yx) + 22 2 ~(R+l)C~~{yn(~;~),yj(~;~)} (27) + sinh (yx)
* cos (yx))
(25)
j=l n=l
Variance of random deflection
Since 77(x; w) is a zero mean process, then the expec- tation of y;(x; w), i > 0, is also zero. Consequently, the mean function of the random deflection is where the covariance term, Cov (y,, yj), is the expec- tation of y,, * yi. The first approximation of the variance is Var y(x; w) = t*
.
Var y,(x; w) (28)
EYCG
WI =
yaw
(26)
where Vary,(x; w) = yll(x) Var (Y + y12(x) Varp
+ k; x
2E2Z2mg I h(x - sJ.h(x -
s*)*Y0 ~1)~Y0 ~2) 0
(2
in which X
Zn = I h(x -
s)yo(sMs; WI ds
(30) 0 yll(x) = (l/~mo)~(cosh(yx)~sin(yx) + sinh(yx).cos(yx)) (31) y,*(x) = (l/V%zo)~ (cash (7x). sin (yx) - sinh (7x).
cos (yx)) (32)
COV( P) =
2
4k2 -21--94...., ma + $*(iWR3 - WR3) - &WIv.R,>
1 (33) Cov ((.u, n) = ti Ezm8 k;
.P
ti mar (2y*M.
& - N. Rd ki Cov (P, InI = ti kzrn8. ti moT Cb*M.
R4 + M. Rd
where (34) (35)
L
M =
~movldU
(36)
N
=
Yhm2y12(L)
(37)
R, = I h2(L - s)y;(s)ds (38)
L
R2 = I +*(I, - s)y;(s)ds
(39)
L
R3 = c Q. s)ya(s)h(l - s)ds
(40) R4 = h(L - s)y;(s)h(x - s)
ds
(41)
Results and discussion
The above formulation is applied to the case of a simply supported beam of rectangular cross section resting on a random elastic foundation. The modulus of subgrade reaction is taken as a deterministic value,
ko,
plus a modulated white noise. The general mathematical expression for the modulated white noise9 is 77(x; w) = e(x)
.
n(x; w)
(42) 332
Appl. Math. Modelling, 1992, Vol. 16, June
Beams on random elastic
supports: A. A.
Mahmoud and
M.
A. El Tawil
where e(x) is a deterministic envelope and n(x; w) is white noise with the following statistical properties: En(x; w) = 0 (43) and ulus of elasticity 210 t/cm2, width of 25 cm, and depth, d, 60 cm, 80 cm, and 100 cm. The beam rests upon a soil of deterministic portion of the modulus of subgrade reaction, which is made to vary from 2 kg/cm2 to 18 kg/cm2. The load intensity is assumed to be 1 t/m.
Figures 2-4
show the expected deflection along the beam length for different beam depths and soil subgrade reactions. In general, the expected deflection is de- creased as the beam rigidity and the modulus of subgrade reaction increased.
Figures 5-7
show the change of
Eqbl; wM2; w) = &A. 4.~2) . I - x2)
(44) where S( -) is the Dirac delta function. The Romberg technique is used to calculate the resultant determin- istic integrals. The solution obtained above will be ap- plied for simply supported beams of length 10 m, mod-
X
-0
loo 200 300 400 500 5ao 700 800 900 moo
Figure 2. The expected deflection along the beam length for k0 = 2.0 E(x)
0.121 I
X
0
100
200 300 400 500 000 700 BOOso0 lG0
Figure 3. The expected deflection along the beam length for Figure 6. ko = 10.0 The change of the variance of the deflection along the beam length fork,, = 10.0 Figure 4. The expected deflection along the beam length for ko = 18.0 Figure 7. The change of the variance of the deflection along the beam length for k,, = 18.0
S(x) - * -
too.0
0
100 200 300 400 500 500 700 800 so0 ll
50
X
Figure 5. The change of the variance of the deflection along the beam length for k0 = 2.0
.-_--
Y
-0
100 200 300 400 500 500 700 800 900 moo -
Appl.
Math. Modelling, 1992, Vol. 16, June 333
Beams on random elastic
supports:
A. A. Mahmoud and M. A. El Tawil
S(x) (lE-04) or
X
Figure 8. The relationship between the variance of the deflec- tion and the deterministic portion of the modulus of subgrade reaction for depth = 60.0 S(x) (1E -04) 7 6 6 Figure 9. The relationship between the variance of the deflec- tion and the deterministic portion of the modulus of subgrade reaction for depth = 80.0
the variance of the deflection along the beam length for the above mentioned beam and soil properties. It can be noticed that the variance increases as the ri- gidity of the beam increases for soft soils (up to 12 kg/cm*) and decreases as the rigidity of the beam de- creases for stiff soils (k, 2 I2 kg/cm*). It is also noticed that the variance of the deflection increases along the beam up to 60-80% of its length and decreases again to zero at the end support.
Figures 8-10
show the relationship between the modulus of subgrade reaction and the variance of the deflection at different points along the beam length. It is clear that the variance becomes constant when the modulus of subgrade re- action exceeds the value of 10 kg/cm2. It is noticed that the minimum variance is obtained for soils of mod- ulus of subgrade reaction between 4 kg/cm* and 12 kg/cm2.
-I>
-IL
- x 2 lo 111
Figure IO. The relationship between the variance of the de- flection and the deterministic portion of the modulus of subgrade reaction for depth = 100.0
Conclusions
The following conclusions are derived:
I.
2. For soft soils the variance of the deflection in- creases as the stiffness of the beam increases while the expected value decreases. For stiff soils the expected value of the deflection together with its variance decreases as the stiffness of the beam increases.
References
Casciati,
Fabio, Faravelli, and Lucia. Hysteretic 3-dimensional frames under stochastic excitation. Int. J.
Struct. Mech. Mater. Sci.
1989, 26, 193-213 Li, W. and Ibrahim, R. A. Principal internal resonances in 3- DOF systems subjected to wide-band random excitation. J.
Sound
Vib. 1989, 131, 305-321 Muscolino, G. Stochastic analysis of linear structures subjected to multicorrelated filtered noises. Eng.
Struct.
1986, 8, 119-126 Branstetter, L. and Thomas, L. Dynamic response moments of random parametered structures with random excitation. Pro-
ceedings of the Third Conference in Dynamic Response of Struc- fures,
March 31-April 2, 1986, pp.
668-675
Vanmarcke, E. and Grigoriu, M. Stochastic finite analysis of simple beams. J. Eng.
Mech.
1983, 109, 1203-1214 Spanos, P. D. and Ghanem, R. Stochastic finite element ex- pansion for random media. J.
Eng. Mech.
1989,115, 1035-1053 Mahmoud, A. and El Tawil, M. On stochastic analysis of beams.
Model. Simulation Control 1990, 33,
1-8 Pipes, L. and Harvill, L.
Applied Mafhematicsfor Engineering Physicists,
3rd ed. McGraw-Hill, New York, 1970 Morensen, R. E.
Random Signals and Systems.
John Wiley, New York, 1987
334
Appl.
Math. Modelling, 1992, Vol. 16, June

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