Deflection of nonuniform beams resting on a nonlinear elastic foundation

Compums & Srrucrures Vol. 51. No. 5. pp. 513-519, 1994 Copyright Q 1994 Elsevier Science Ltd Printed in Great Britain. All rights rewved oLM5-7949/94-$7.00 + 0.00

Pergamon

DEFLECTION OF NONUNIFORM BEAMS RESTING A NONLINEAR ELASTIC FOUNDATION Y. H. Kuot and S. Y.

ON

LEE$,$

tAeronautica1 Research Laboratory, Taichung, Taiwan 400, Republic of China IMechanical Engineering Department, National Cheng Kung University, Tainan, Taiwan 701, Republic of China (Received 4 November 1992)

Abstract-The static deflection of a general elastically end restrained non-uniform beam resting on a non-linear elastic foundation subjected to axial and transverse forces, governed by a non-linear fourth order non-homogeneous ordinary differential equation with variable coefficients, is examined. By using the method of perturbation, the governing differential equation is transformed into a set of self-adjoint linear fourth order ordinary differential equations with variable coefficients. It is shown that the deflection of the beam can be expressed in terms of the fundamental solutions of these linear ordinary differential equations. Especially if the coefficients of the linear fourth order ordinary differential equations are in an arbitrarily polynomial form, then the exact solution for the static deflection of the beam can be obtained.

I. INTRODUCIION

The problem of beams on a linear elastic foundation has been studied by many investigators [l-12]. The closed-form solutions for the static and dynamic response of a uniform beam resting on a linear elastic foundation can be found in several references [l-3]. The static, dynamic and elastic stability analysis of a beam resting on a non-linear elastic foundation was presented by Beaufait and Hoadley [4], Massalas [5 and 61 and Hui [7], respectively. Recently, the exact and semi-exact analysis of a non-uniform beam with general elastic end restraints was given by Lee et al. [12-141. From the existing literature, it is seen that very few works [4] are related to the static analysis of non-uniform beams resting on a non-linear elastic foundation and the exact solution of the problem is still not available. In this paper, we generalize the previous analysis given by Lee et al. [12-151 and study the static deflection of a general elastically end restrained non-uniform beam resting on a non-linear elastic foundation subjected to axial and transverse forces, governed by a non-linear fourth order non-homogeneous ordinary differential equation with variable coefficients. By using the method of perturbation [ 161, the governing differential equation is transformed into a set of self-adjoint linear fourth order ordinary differential equations with variable coefficients. Finally, the deflection §Author to whom all the correspondence should be addressed. 513

of the beam is expressed in terms of the fundamental solutions of these linear ordinary differential If the coefficients of the linear equations. fourth order ordinary differential equations are in an arbitrarily polynomial form, then by employing the closed-form fundamental solutions developed by Lee and Kuo [15] the exact solution for the static deflection of the beam can be obtained. 2. ANALYSIS

For a general elastically end restrained nonuniform Bernoulli-Euler beam of finite length 1, resting on a non-homogeneous, small non-linear elastic foundation subjected to axial tensile and transverse forces, as shown in Fig. 1, the static flexural deflection r(X) satisfies the ordinary differential equation

+k(X)V((X)+k*(c,X,

v)=p(X),

XE(O,I),

(1)

where E(X), I(X) and k(X) denote the Young’s modulus, moment of inertia and the Winkler’s linear elastic foundation modulus respectively. k*(~, X, 0 is a non-linear function due to the non-linear elastic foundation and 6 is a small perturbation parameter. N(X) and p(X) are arbitrary axial tensile and transverse distributed forces, respectively. If the beam is subjected to an axial end force NO, then N(X) is a constant function

Y. H. Kuo and S. Y. Las

514

N(X) = Ni. Instead, of the beam N(X) = &g(q) d+ The associated boundary conditions are

where E,,ZO= E(O)Z(O) and E,Z,= E(Z)Z(Z) are the bending rigidities of the beam at X = 0 and X = 1, respectively. The governing differential equation can be rewritten in the following non-dimensional form

~[C(X)~]+~[R(x) + Q@>J’+ Sk x, VI = f’(x),

and

x E (0, 11, (5)

and the associated boundary conditions become at

x =0:

v” - /IoLV’= 0,

V”‘+G’(0)V”+R(O)?“+~TLV=O,

where KoL and KTL and KORand KTR are the rotational spring and translational spring constants at the leftend and right-end of the beam, respectively. Introduce the following non-dimensional auantities x=-,

G(x)=

1

-w)W)

2

00

and at

x=1:

v” + /!?@a V’ = 0, v” + Yav” - 6, Y’ - ,&a v = 0,

(7)

where primes indicate differentiation with respect to variable x. It is well known that if the coefficients functions in eqn (5) are arbitrary, the closed-form solution is not available. However, if these coefficients are in arbitrarily polynomial forms, then the closed-form solution can be obtained. To simplify the analysis, one considers the problem with the non-linear term S(c,x, V) =cI’*. By using the method of perturbation [16], one can seek the solution for small E and let

x

Ez

(6)

- N(X)P R(x)=r, 00

V(x;E)=

Q(x)=%,

Y,(x)+cY,(x)+c2Y*(x)

00 +c3Y3(x)+... k*(q S(E, x, V) = EoZo

where Yi(x), i = 0, 1,2, . . . are the solutions corresponding to ci and independent of E. One then substitutes this expansion into eqn (5), expands for small 6, and collects coefficients of each power of L. Since these equations must hold for all values of 6, each coefficient of .Cmust vanish independently

’

P (W3

w=r,

(8)

x, p)I3

00

s,,=$k 00

s.,=~, I I pn.+, 00 s,,=$ I I G’(x) YR= G(x)

6 = NW*

(4)

R44

+----I

x=,’

’

Fig. 1. Geometry and coordinate system of a general elastically end restrained non-uniform beam.

515

Deflection of nonuniform beams resting on a nonlinear elastic foundation

because sequences of 6 are linearly Consequently, one has

independent. =

Coefficient of co:

J

[l

0

0

01

0 i 0

1

0

0

1

0 0

0

0

0

9 (14)

1

then the genera1 solution can be derived via the method of variation of parameters and shown to take the following concise form Coefficient of L’:

Y,(x)=

G~,(x)+C*~*(x)+

+

+ Q(x)Y,= -Y;

GV,(x)+

V,(x)V4(5)1

'I[v,wv,(x)-

s0

(10) -[V,(OV,(x)-

Coefficient of 6’:

x =0:

where C, , C,, C, and C., are integration constants and will be determined from the specified boundary conditions. The algebraic details of the derivation of the general solution are similar to those given by Lee et al. [14]. After substituting the general solution into the specified boundary conditions, the static deflection of the beam is shown to be

Yy = &_Yj = 0, Y,(x)

Y~+G’(0)Y;+R(O)Y;+&Yi=Oo,

- ~2’,(x)f’4(5)1

(11)

After substituting eqn (8) into the boundary conditions, eqns (6) and (7), expanding for small E and collecting coefficients of like powers of 6, one has the associated boundary conditions for Yi, i=o,1,2,... at

~*(x)~,(r)l

+ G’@)[V,(~)V~(X)

+ Q(x)Y, = -2Y, Y,

GV,(x)

(12)

=

5’ 0

Ktx, W(tY d5,

and

at

x=1:

r; + /IORr; = 0, + Y~+y,Y;-6,Y(-B,,Yi=o.

(13)

It can be observed that the governing differential equations and the associated boundary conditions for Yi, i=o,1,2 ,... are in the same form. It is the governing differential equation for the static deflection of a general elastically restrained non-uniform Bernoulli-Euler beam resting on a non-homogeneous linear elastic foundation, subjected to axial and tensile forces. If V,(x), V*(x), V3(x) and V4(x) of eqn (5) are the four linearly independent fundamental solutions of the governing equation (9) and satisfy the following normalization conditions at the origin of coordinate system V,(O)

V*(O)

V,(O)

V4(0)

V(O)

WO)

K(O)

C(O)

v;(o)

Vi(O)

v;(o)

Vi(O)

V;"(O)

V;'(O)

VT(O)

V;'(O)

1

s’

Kztx, 5)05) dL

(16)

x

where P(x) is the force function defined in eqn (4) and K(x, c) is the Green’s function of the system, 4(x,5) and K2(x, r) are subfunctions of K(x,t). They are symmetric, i.e., K,(x, 5)= K2(<, x).The Green’s function for the beam with general elastically end restrained boundary conditions is given in the Appendix. The Green’s function for the beam with typical boundary conditions can be easily obtained from that for the beam with general elastically end restrained boundary conditions by taking a suitable limiting procedure. For example, by letting /lO~+c.o, 0, one can obtain the and B"R=BTR= BTL+03 Green’s function for the cantilever beam and it is expressed as

4(x, 0 = I-a,P’,(x)V&)

- NO)~,(x)~,(S)I

- v,(x)v,tS)

+ %W3(X)V‘l(<)

Y. H. Kuo and S. Y. LEE

516 + vd(x)vJ(r) -‘3’(‘3)~4(~)~4(01

-a,

-

>

a4 ~4(X)~4(01/~,

v,(x)v,(S)

(17)

and

where e,fand h are integers representing the number of terms in the series, then a power series representation of the normalized fundamental solutions can be constructed by employing the method of Frobenius. These exact normalized fundamental solutions were recently developed by Lee and Kuo [15]. They are in the form of

K,(x, 0 = K, (CTx), V,(x) = f b,,ixi, a, = [V;(l)V;(l) +

4K(1)KU)

uz= [V;(l)V;‘(l)

for

V,(x): b,. = 1, b,,, = bl,, = bl., = 0,

for

V,(x): bz,, = 1, b,,, = b2,2= b*,, = 0,

for

Vs(x): b3.2= f, b,,. = b,., = b,., = 0,

for

V4(x): b4,) = b, b4,0= b4,, = b4,2= 0,

- v;(l)V;(l)],

4 = [Vi(l)VY(l)

- Vr(l)V;(l)]

-i- WW)v;(l) a, = [V;(l)Vy(l)

(20)

where

- ~;(l)~;(l)I,

- VY(l)Vb’(l)]

+ UW)V(l)

n = 1,2,3,4,

i=O

- V;“(l)Vl;(l)]

(21)

- V;(l)V;(l)J, and

- V;“(l)V;(l)]

-1

+ s,[V;(l)Vg(l)

hi+4 = (i + l)(i + 2)(i +

- Vf(l)V;(l)]. (18)

After obtaining the solution Y,,(x), the transverse force on the right-hand side of the differential equation (10) can be determined. Following a similar procedure for the solution Y,(X), the solution Y,(X) can be obtained and Yi(x), i = 2,3,. . . can be solved successively. Finally, after substituting these solutions back to eqn (8), the static deflection of the beam is determined and expressed in terms of the fundamental solutions of the linear differential equations (9)-( 11). 3. NORMALIZED

FUNDAMENTAL SOLUTIONS

1 gixi,

$ r,x’, i=O

+ 4)!

(i _j)!

gibn,i-i+4

i 2(j + l)(i -j +C

+ 3)!

j=O

(i -j)!

i

(j+l)(j+2)(i-j+2)!

+

&+ Ibn,i-jC3

gj+2

(i -j)!

j=O

+(i-_j+ l)(i-j+2)r,

C (i

-I- 1W -i

1

b,.i_j+2

+ l)rj+lbn.i-j+

I

j=O

+

i j=O

1

qjbtt,i-j n=l,2,3,4;

i>,O.

(22)

After substituting these fundamental solutions in the corresponding Green’s function, the exact solution for the static deflection of the non-uniform beam is obtained. 4. VERIFICATION AND EXAMPLE

Example 1

i=O

Q(x) = i

(i -j

j_,

Two examples are given to verify and illustrate the previous analysis.

,=”

R(x) =

i

{

+

Since the governing differential equations (9)-( 11) for Y,, i=O,1,2 ,... are in the same form and their coefficients are the same, hence the fundamental solutions for the differential equations (9)-(11) are the same. It is well known that if the coefficients of the differential equation (9) are arbitrary functions, then, in general, the exact closed-form fundamental solutions of the differential equation are not available. However, if the coefficients of the differential equation (9) are given in the following polynomial variation form G(x) =

x

3)(i + 4)go

ax’,

(19)

To show the convergence and efficiency of the proposed method, one first considers a beam resting on a linear elastic foundation. Table 1 shows the deflection and bending moment of a uniform cantilever beam, resting on the variable two-

517

Deflection of nonuniform beams resting on a nonlinear elastic foundation Table I. Deflection and bending moment of a cantilever beam resting on a linear two-parameter elastic foundation E(X)I(X) = 500 kN . m*, k, (A’)= lOO(9- A’*)kN,

J

k(X) = SOO(9- X2)kN/m’, p(X) = 106(X - 3) kN and I = 3 m. Moment (N.m) at X=2m Deflection (cm) at X = 3 m

4.70 20 1.23 25 30 1.33 40 1.33 60 1.33 1.33 t t’Ihe results obtained from Eisenberger and Clastornik [8]. parameter elastic foundation subjected to a concentration force at the tip of the beam. The two parameters of the elastic foundation are k(X) and k,(X). Here, k,(X) is the shear modulus of the two-parameter foundation. When k, (X) = 0, this foundation model reduces to the Winkler model. It is well known that the mathematical model for the beam resting on a two-parameter elastic foundation 0

e E &

-2.5

38 8 G

-5

a

is equivalent to that of a beam resting on the Winkler elastic foundation with an axial tensile forces, N(X) = k,(X) (Eisenberger and Clastornik [8]). Hence, the previous analysis can be applied to this particular problem. The results obtained by the present analysis are compared with those given by Eisenberger and Clastornik [8]. It can be observed that both results are almost the same when the number of terms (J) in each fundamental solution equals 40 and the difference between both results is within 0.17% when the number of terms employed is 30 only. Example

-----Y.

0

+ CY,+ CZYZ

0.5 X

2

Now consider the static deflection of a cantilever beam with constant width and linearly varied depth resting on a small non-linear elastic foundation subjected to distributed axial compressive and transverse forces. The function /c*(c, A’, v given in eqn (1) is assumed in the form of &V*(X).The beam is fixed at X = 0 and the material properties and the axial and transverse forces are given as

-------u.+cu, -7.5

3884 3470 4028 4035 4035 4035

1 E(X)I(X)

= 500

kN m2,

kN,

k(X)=500[1.6-2:+6)

kN/m2,

Pw)=~w[($J--($)~] N/m, c = 250 kN/m3, I=lm.

0

0.5

1

X Fig. 2. Deflection and bending moment of a tapered cantilever beam resting on a small non-linear eiastic foundation: (a) deflection versus A’; (b) bending moment versus X.

(23)

The convergence of the deflection and bending moment distribution of the method of analysis is plotted in Fig. 2 and the solution is listed in Table 2. The differences in deflection at X = 1 m and bending moment at X =0 between and Y,+cY, Y, + t Y, + L*Y2are less than 0.5%. The results show the method of analysis converges rapidly.

518

Y. H. KIJO and S. Y. LEE Table 2. Deflection and bending of a tapered cantilever beam resting on a small non-linear elastic foundation

Position (m)

Bending moment (kN . m)

Deflection (cm)

yo

x

Y,+tY,

Y,+tY,

+c2y2

0

0

0

0

0.1

0.0817 0.3392 0.7868

0.0793 0.3293 0.7637 I .3885 2.1995 3.1811 4.305 1 5.5330 6.8208 8.1188

0.0797 0.3309 0.7676

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1.o

1.4306 2.2664 3.2780 4.4366 5.7025 7.0302 8.3794

5.

YO 78.1336 75.5932 69.5178 60.5111 49.4228 37.3119 25.3699 14.7982 6.6464

1.3956 2.2108 3.1975 4.3273 5.5616 6.8561 8.1629

CONCLUSION

In this paper, the static deflection of a general elasticity end restrained non-uniform beam resting on a non-linear elastic foundation subjected to axial and transverse forces, governed by a non-linear fourth order non-homogeneous ordinary differential equation with variable coefficients, is examined. By using the method of perturbation, the governing differential equation is transformed into a set of self-adjoint linear fourth order ordinary differential equation with variable coefficients. It is shown that if the coefficients of the linear fourth order ordinary differential equations are in an arbitrarily polynomial form, then the exact solution for the static deflection of the beam can be obtained. The solutions are exact UP to the accuracy of the computer or a preset value set by the analyst.

REFERENCES M. Hetenyi, Beams on Elastic Foundation. The University of Michigan Press, Ann Arbor, Mich. (1946). C. Miranda and K. Nair, Finite beams on elastic foundation. ASCE .I. Struct. Div. 92, 131-142 (1966). B. Y. Ting, Finite beams on elastic foundation with restraints. ASCE. J. Struct. Div. 108, 611621 (1982). F. W. Beaufait and P. W. Hoadley, Analysis of elastic beams on nonlinear foundations. Cornput. Struct. 12, 669676 (1980). 5. C. Massalas, Fundamental frequency of vibration of a beam on a non-linear elastic foundation. J. Sound Vib. 54, 613-615 (1977).

6. C. Massalas, Comments on the Fundamental frequency of vibration of a beam on a non-linear elastic foundation. J. Sound Vib. 58, 455-458 (1978). 7. D. Hui, Postbuckling behavior of infinite beams on elastic foundation using Koiter’s improved theory. Inr. J. Non-Linear Mech. 23, 113-123 (1988). 8. M. Eisenberger and J. Clastornik, Beams on variable two-parameter elastic foundation. ASCE J. Engng Mech. 113, 14541466 (1987). 9. D. Z. Yankelevsky and M. Eisenberger, Analysis of a beam column on elastic foundation. Cornput. Struct. 23, 351-356 (1986). 10. D. Karamanlidis and V. Prakash, Exact transfer and stiffness matrices for a beam/column resting on a twoparameter foundation. Comp. Meth. Appl. Mech. Engng 72, 77-89 (1989).

11. S. N. Sirosh, A. Ghali and A. G. Razaqpur, A general finite element for beams or beam-columns with or

1.6340 0

Y,+cY,

Yo+cY,+c2Y2

75.8512 73.3743 67.4662 58.7137 47.9425 36.1818 24.5897 14.3334 6.4315 1.5789 0

76.2364 73.7488 67.8125 59.0171 48.1923 36.3726 24.7214 14.4119 6.4678 1.5882 0

without an elastic foundation. Inr. J. Num. Meth. Engng 28, 1061-1076 (1989). 12. S. Y. Lee and H. Y. Ke, Free vibrations of a non-uniform beam with general elastically boundary conditions. J. Sound Vib. 136, 425-437 (1990). 13. S. Y. Lee, H. Y. Ke and Y. H. Kuo, Exact static deflection of a non-uniform Bernoulli-Euler beam with general elastic end restraints. Comput. Struct. 36,91-97 (1990).

14. S. Y. Lee, H. Y. Ke and Y. H. Kuo, Analysis of non-uniform beam vibration. J. Sound Vib. 142, 15-29 (1990).

15. S. Y. Lee and Y. H. Kuo, Exact solutions for the analysis of general elastically restrained non-uniform beams. ASME J. Appl. Mech. 59, 2055212 (1991). 16. A. H. Nayfeh, Perturbation Methods. John Wiley, New York (1973).

APPENDIX: GREEN’SFUNCTION

OF NON-UNIFORM BEAMS WITH GENERAL ELASTICALLY END RESTRAINED BOUNDARY CONDITIONS

JG(x, 5) = &Ju,Kw,(t)

-

V,(x)V,(S)

+

WO)V,(xP’2W

+

W’)V,(x)V,W

-

W)G’@)V&)J’,(S)I

+ ~,SoJ- Vdx)V,(S) + G’(O)V,(x)V,(t) + ‘W)V,(x)V&)

- G’2(0)V,(x)V,G)1

-a,BTLV~(X)v4(5)+a2V,(X)V,(S)

-

azBTLw,(X)Vm

+

-

G’(O)V,(x)V2(t)

+

WP(O)V,(x)V,(Ol

+

~~PoL&L[-

+

G’(O)V,(xV’,(S)

-

V,(x)V,(S)

R@)V&)V,(t;)

V&)f’dS)

+

-

G’2(0)Vq(x)V~(t;)1

-a,[V,(x)V,W+

V*(x)V,W

+

W’)V2(x)V2W

-

-

R@‘)V,(x)V,(5)

+

R2@)V&)V&N

-

G’V’)V,(x)VdO

W’%V&)V2(0

R(W’(O)V2(x)V,(5)

+%BoJ- V,(x)V,(t;) - V,(x)V,(t;)

Deflection of nonunifo~

beams resting on a nonlinear elastic foundation

519