Acta Mechanica 130, 67 79 (1998) ACTA MECHANICA �9 Springer-Verlag 1998

Exact solutions for extensible circular curved Timoshenko beams with nonhomogeneous elastic bo, undary conditions

S. M. Lin, Tainan, Taiwan

(Received March 18, 1997; revised April 4, 1997)

Summary. A generalized Green function of nth-order ordinary differential equation with forcing func- tion eomposed of the delta function and its derivatives is obtained. The generalized Green function can be easily and effectively applied to both the boundary value problems and the initial value problems. The generalized Green function is expressed in terms of n linearly independent normalized homogeneous solutions. It is the generalization of those given by Pan and Hohenstein, and Kanwal. Accordingly, the exact solution for static analysis of an extensible circular curved Timoshenko beam with general non- homogeneous elastic boundary conditions, subjected to any transverse, tangential and moment loads is obtained. The three coupled governing differential equations are uncoupled into one complete sixth-order ordinary differential characteristic equation in the tangential displacement. The explicit relations between the angle of rotation due to bending, the transverse displacement and the tangential displacement are obtained. The deflection curves due to a unit generalized displacement at nodal coordinate, and the exact element stiffness matrix are derived based on the solution for the general system. A finite element method can be developed based on the results for the dynamic analysis. Meanwhile, the stiffness locking pheno- mena accompanied in some other curved beam element methods does not exist in the proposed method.

1 Introduction

Many physical problems such as diffusion, heat conduction, vibration, wave propagation etc.,

can be described by a nth-order linear ordinary differential equation. Pan and Hohenstein [1]

offered a method of solution for a fourth-order ordinary differential equation with the forcing

terms which are composed of the delta function and its derivatives. Kanwal [2] and Yang and

Tan [3] found the matrix Green function of a matrix differential equation without the deriva-

tives of delta function. It is obviously difficult to calculate the fundamental matrix and its

inverse whose dimensions are large. Stakgold [4] discussed just some properties of Green's

function of an ordinary differential equation. So far, no generalized Green function of a nth-

order, linear ordinary differential equation with forcing terms which are composed of the

delta function and its derivatives, has been obtained.

Curved beams, which have importance in many practical applications such as aircraft structures, bridges, was a textbook subject for a long time. An interesting review of the subject can be found in the reviewer papers [5], [6]. The exact solutions for static analysis of exten-

sional circular curved Timoshenko beams with some special conditions has been obtained by using the Castigliano's theorem [7]. Fettahlioglu and Mayers [8] found the analytical solution for the displacement and stress results for specific Bernoulli-Euler beams. Gauthier and Jahs-

68 S.M. Lin

man [9] studied the micropolar elastic effect on the displacement of semicircular ring subjected to end shear force. Kardomateas [10] found the analytical solution for the state of stress in pure bending of cylinderically orthotropic curved beams. Reddy and Volpi [11] studied the static behavior of Timoshenko curved beams by using the method of finite element. Bagci [12] found the exact elasticity solutions of stresses and deflections in curved beams subjected to moment loading only or force loading only. Lee and Sin [13] and Yang and Sin [14] presented the curvature-based beam element for the analysis of Timoshenko beam to eliminate the stiff- ness locking phenomena. Bucalem and Bathe [15] studied the locking behavior of isoparame- tric curved beam finite elements. Lee and Lin [16] study the dynamic response of a straight Bernoulli-Euler beam with time dependent elastic boundary conditions by generalizing the method of Mindlin-Goodman [17]. There still is no study on the analysis of extensional circu- lar Timoshenko beams with general elastic nonhomogeneous elastic boundary conditions.

The purpose of this paper is to find the generalized Green function of a nth-order ordinary differential equation with forcing function composed of the delta function and its derivatives. The generalized Green function can be easily and effectively applied to both the boundary value problems and the initial value problems. The Green function is expressed in terms of the n linearly independent normalized homogeneous solutions. The method of solution presented here is more simple than those given by Pan and Hohenstein [1] and Yang and Tan [3]. It is the generalization of that given by Pan and Hohenstein [1] and Kanwal [2]. Accordingly, a systematic theoretical development of the static analysis of extensional circular curved Timo- shenko beams with general nonhomogeneous elastic boundary conditions is presented. The exact solution for the analysis is obtained. The three coupled governing differential equations are uncoupled into one complete sixth-order ordinary differential characteristic equation in the tangential displacement. The explicit relations between the transverse displacement, the tangential displacement, and the angle of rotation due to bending are established. The deflec- tion curves due to a unit generalized displacement at nodal coordinate, and the exact element stiffness matrix are derived based on the solution for the general system. Moreover, the consi- stent mass matrix can be constructed via the deflection curves. A finite element method can be developed based on the results for the dynamic analysis. Meanwhile, the stiffness locking phenomena accompanied in some other curved beam element methods does not exist in the proposed method.

2 General solution

Consider a nth-order ordinary differential equation

d~w dn- lw dw ~-l dip,i,(~) qn ~ - qn- l d~YF + " " + ql ~ + q~ = Z d~i

i=O

(0,1) (2.1)

where the coefficients {qi} are constants on the closure domain [0, 1], and the leading coeffi- cient % does not vanish anywhere on closure domain.

The general solution w(~) of Eq. (2.1) can be written as

w(~) = ~p(~) + ~ C~V~(~), (2.2) i = l

where wp(~) is the particular solution. {Ci} are the constants to be determined. {Vi(~)} are n linearly independent homogeneous solutions of Eq. (2.1), which satisfy the following norma-

Exact solutions for curved Timoshenko beams 69

lized condition

v,(o) �89

W(o) W(o)

v~ ' :~ -~ / (o ) ~(~-~)(o)

.v~< ~ ~)(o) ~4(~<)(o)

. . ~7n-1(0 )

.. v L , ( 0 )

.. vs

. . ~ f ~ ) ( o )

v,.(o)

v, ' (o)

v,,/,-~/(0)_ [i 0 o o] 1 . 0 O]

0 .. . 1 0

0 . o !

(2 .3 )

The particular solution wp(~) of Eq. (2.1) can be written as

n - 1 1

~(!~) = E f>(~)E~(~ - ~)do~, i=0 0

where {E~(~)} are the Green functions of the following equations, respectively

d" Ei d~-I E, + dEi dis qr,,--d~,V q- qn-l d~C ~- ...-}- q1 ~ q- qoEi : ~ , i = 0 , 1 , 2 , . . . , n - 1

in which 6 is the Dirac delta function. The Green functions {/~({)} can be written as

(2.4)

(2.5)

E0(~:) = G ( ~ ) , i

i = l , 2 , . . . , n - 1 ,

(2.6)

where {/)i (~)} are the Green functions of the following equations, respectively

d~"E~ + d~-is dE~ ~ dig ~-1 %--d=( y- % 1 d ~ 2 5 - + ' " + q l ~ + q ~ i = ~ + j _ ~ 0 q~-i+r d~J' i=0 ,1 ,2 , . . . , n - ! . ( 2 .7 )

The Green functions {Ei (~)} can be obtained

~%(~) = @(~)H(~), i : 0, 1, 2 , . . . , ~ - 1 (2.s)

wheJce H(() is the Heaviside function, if the function {Gi(~)} satisfy the following conditions

dJGid~j ~=0 = 0, j (= n - i - 1, (2.9)

d*~-i-tG~d~ i-1 ~=0 =--'qnl i = 0., 1, 2, . . . , n - 1 (2.10)

in the distributional sense. Letting the function Gi (~) be

j = l

and substi tuting Eq. (2.1 l) into Eqs. (2.8) (2.9), the coefficients {a<j} and the Green functions {Ei (~) } are obtained respectively

1 a h n i = - - q,7,

a i , j == O, j 7 s n -- i,

q.T~

i = O, 1, 2 , . . . , n - 1.

i = 0 1 . 2 , . . . , n - 1 .

(2.12)

(2.13)

70 S.M. Lin

Substituting Eq. (2.13) into Eqs. (2.2), (2.4), and (2.6), the particular solution and the general solution are obtained. Substituting the general solution into the associated boundary condi- tions, the coefficients {Ci} can be obtained, It should be noted that the generalized Green function can be easily and effectively applied to both the boundary value problems and the initial value problems. Based on the results the analysis of a curved Timoshenko beam is made as following.

3 Exact solutions

3.1 Extensional curved Timoshenko beams

Consider the static response of a extensional circular Timoshenko beam with general non- homogeneous boundary conditions, subjected to any transverse, tangential and moment loads, as shown in Fig. 1. In terms of the following dimensionless quantities,

k F1 F2 - L ' A = L -

A F4 F5 - L ' f s = L -

f3 = F3,

FI*L ~ F,2 *L 2 Fa*L f l* - - -- fa*-- E 1 ' A* ~:.r , E.r '

F4* L 2 Fs*L 2 F6* L f 4 * - E r ' 5 ' - E r ' 5 " Z _ r '

M(O)L 2 P(0)L a Q(0)L 3 (3.1) m ( ~ ) - E I ' P ( { ) - E I ' q ( ~ ) - E I '

U W L L ' w = , o L = ~ ,

KuLL 3 KwLL 3 KoLL /31 E ~ ' /32 - E I ' /33 E I '

Ku•L 3 K w R L a KoaL /34= E [ ' / 32 - 17,I ' / 36 - E I '

E I A L 2 0

# = ~ G A L 2 ~ ( - I ' ~ oe'

where W(O) and U(O) are the tangential and transverse displacements, respectively, g*(0) is the angle of rotation due to bending. P(O), Q(0) and M(O) are the applied transverse, tangential and moment loads, respectively. E, G, ~, ! and A denote Young's modulus, shear modulus, shear correction factor, moment of inertia and area per unit length, respectively. Kur, KwL and KOL, and KuR, KwR and Ko• are the transverse translational spring constants, the tan- gential translational spring constants and the rotational spring constants at 0 = 0 and 0 = a, respectively. R and L are the radius and the length of the circular beam, respectively. F1, F2, F3, FI*, F2* and F3*, and F4, Fs, F6, F4*, F5 ~ and F6*are the transverse displacements, the tangential displacements, the slopes, the shear forces, the tangential forces and the

Exact solutions for curved Timoshenko beams 7 t

moment loads at the left end and the right end of the beam, respectively. The coupled govern-

ing differential equations are

+ - _ =

~-d7 + aw - L~ - an = q(~), (3.3)

1 + aw - r -- m(~) (3.4) d~ 2 .

The associated boundary condition are

at{ = 0:

l ( d d ~ + a w g ' ) - 3 ' n u - 3 ' n f l - q q 2 f F , "/12 #

0'22r -- aZt -- Y21W = --')'21f2 -- 3'22f; ,

dg' "Y32 ~ - ~'3~ = -%s~f3 - %2fa*,

at{ = 1:

#t [du _ ) "/42 ~.~ , C~W -- e + /̂41U : "741f4 + ~/42f4",

- a u + O m w = % 1 f 5 + 7 ~ < f s * ,

dg'

where

"7~'~ = i = 1 , . . . , 6 "m -- ~ + / % , l + , & '

(3..5)

(3.6)

(3.7)

(3.8)

(3.9)

(3.~o)

(3.!1)

. 1( dfn i dSq)

( 3 . 1 2 . 1 )

(3.12.2)

U ~ 1 dSw 1 d3w

a:3<61 d{ 5 a<6t d~ 3

1 dw 1 62 d2p 1-- a ~ + ~p ~<6~ d~ 2

It should be noted that if the moment load m(~) is neglected and the beam is clamped at both ends, the coupled governing differential equations (3.2)-(3.4) and the boundary conditions (3.5) (3.10) become the same as those given by Reddy and Volpi [11]. Taking the method given by Lee and Lin [19], the angle of rotation due to bending, and the transverse displace- ment can be expressed in terms of the tangential displacement, respectively

72 S.M. Lin

P(0)

"TNZx.'" i / \ \ : . :

Fig. l. Geometry and coordinate system of curved beam with general nonhomogeneous elastic boundary conditions, subjected to transverse, tangential and moment loads

where

(~i = 1+O~ 2 , 8 + , (~2 = # @ ~ , & = l + 2 a 2 # + �9 (3.12.3)

Substituting the relations (3.12.1-3) into Eqs. (3.3) and (3.5)-(3.11), the complete sixth-order ordinary differential characteristic equation and the associated boundary conditions in the tangential displacement are obtained, respectively

daw d4w oz4 d~w dp (~ ) dSp d2q l d4q d~m dT+2~'~ + ~ = ~ - +" ~ ~'q+~Vdp ~ ~'~-~d~'

(3.13) at ~ = 0:

"}/11 d5w dp

dSw 2 daw ")/22 ~ g @ '-Y220~ ~ -- 'T21Gf2(~1 w = --G2~'1('-Y21/2 -t- "Y22f2*) ,

d5w (~ d4w (5 d3w d2w 2 dw %,&~- ~1 .~+ ~,..~- z~l~V + ~.,~ ~1~- ~'~1~ = --ct~1('~81/8 T ")'32/8"),

daw dSw d2w 2 ~ dw z l , ~ ~ ~ + ~11~ ~ ~ - ~ 1 ~ ~ ~ - ~1i~ ~ 1 - ~ = - ~ % ( ~ 1 / ~ + ~l , f~*) , (a.~4)

(3.15)

(3.16)

a t e = 1:

-%1 dSw 2 d4w �9 d3w oz4 d2w 2 dw "}'42 0~ d ~ - ~ d~ 5 - ~42(22 ~ - - ~42 ~ - + "~41C~ (~1 ~ - c~8(~1 (741 f4 + %2 f4 * ) (3.17)

dSw o2 daw 2~52 ~g- + ~52 ~ + "/51c~2(~1w = c~2~1 (751f5 + 752f.5'), (3.18)

dSw 5 d4w ~ d3w d2w ~ dw

(3.19)

Exact solutions for curved Timoshenko beams 73

The six exact independent normalized homogeneous solutions of Eq. (3.13) are

Vl (~) = 1,

�89 : ~,

2 5 § ~ cos a~,

~ sin a~,

3 sin c~ + V~ (~) -- Ct 4 20z3 20~4 COS 0~(,

(3.20)

which satisfy the following normalized condition

-v~(o) v~(o) �89 v~(o) ~(0) v~(o) vs v~'(o) v3'(o) �88 vj(o) v~'(o) v~"(o) v~"(o) vj'(o) vg'(o) vj'(o) vo"(o) Vltt' (0) v2tH (0) v3m (0) v4m (0) v5m (0) v6m (0) ~(4/(0) v~/4/(o) v3<4/(0) vS~/(o) vs<~/(o) v~/4/(o) ~/5/(0) �89 ~(~/(0) v55/(0) �89 v~<~/(o)

The,' general solution of Eq. (3.13) is

6 ~(~) : ~(~) + E cy4~),

i=1

L 1 0 0 0 0 Oil 0 1 0 0 0 0 o = 0 0 1 0 0

0 0 0 1 0 " 0 0 0 0 1 0 0 0 0 0

(3.2/)

(3.22)

where the particular solution Wp(~) can be obtained from Eqs. (2.4), (2.6), and (2.13). Substi-

tuting the general solution (3.22) into the boundary conditions (3.14)-(3.19), the associated coefficients {Ci} are obtained

CL

- b36 b26 b16

= 73262 "~22

b35 b34 b33 b31 b31 b25 b24 b23 b22 b21 b15 b14 hi3 hi2 bll --~31~2 "~32~3 731~3 "~32Ct2~1 --7310Z2~I 0 ~/22a 2 0 0 -721a26!

--'-/120:2 "/11 OZ2 ?120/4 --7110~261 0 r

-1

a5 a4 a3 a2 al

(3.23)

where

al := -- r d~ 5 712a 2 d c4 + " / l la 2 d~. 3

d%(O) -- ~11810Z 2 ~ - - / ] -- Ofi61s

d%,(O) d%p(O) a 2 = - 722 d( ~ ~-722a 2 d~ 3

d5%(0) a 3 : = - 7a~2 d~ 5

+ 73251a 2 dwp(O) d~

yno? d~ 2

"]/21610~2Wp(0)) -- 0~261s ,

d%(O) d%p(O) d%;(O) ~'a~2 ~ - + 73253 d~ 3 %1~3 d~ 2

~/31(~10!2Wp(0)) -- Ct51s

(3.24.1)

74 S.M. Lin

_ "/41 dSwp(1) 2 d4wp(1) 2 d3wp(1) w.(

/ dSwp(1) a2 d3wp(1) ~_ %181a2wp(1)) + a2~les, ~ = - ~,%~ ~ + "Y~ dr

d~(~) - %2613 2 76161o}wp(1) + o~61c6 ; d~

bli 74~ dSV/(1) 2 d4V~(1) 2 d3V~(1) d2V/(1) - - 7420 ~ d~- 7410z ~ )~42 ~ d~2

d~V~(~) d3~(~) b2/= %2 ~ -+ %2a2 d~3 ~- %13261V/(1) '

dsV~(1) ~ d4V//(1) (5 d3Ki(1) ~ d2V//(1) ~ 2 dVi(1) v3~ = % ~ ~ K - + %~ ~ ~ - + % ~ ~ - + % ~ ~ - + %~1~

+%15~a2V~(1), i = 1 , . . . , 6

,~41510~ 2 dwp(1) + 3351e4 ' d~

d~(1) - - ~4151a2 d~

(3.24.1)

(3.24.2)

in which

~ 1 = 1 + a 2 # + 52 = # + ~ ,

ei = 7ilfi § 7i2.fi*, i = 1, . . . , 5, 6

63 = 1 + 2 3 2 # + , (3.25)

3.2 Inextensional curved Timoshenko beams

For inextensional beams the effect of centerline extensionsibility is neglected. By letting C --+ ~c in Eq. (3.13), the governing characteristic differential equation is obtained:

dew 2 d4w 4 d2w dp d3p 2 2 d2q d2m (3.26) + 2~ ~+~ ~ = ~ - ~ V - - ~ q+~ ,~_~3~_~ ~2 d~6

The relations between the angle of rotation due to bending, the transverse displacement and

the tangential displacement are obtained via Eqs. (3.12. l-3), respectively

1 + 1) [" d4w "1 2 - d ~ - d2w --(a ctp~dp)}__ - ~2(~2~ J ~ + am + 2~ ~) + ~ ( 1 + s - ~ u 2 q + ~ . ~ + ,

(3.27)

1 dw _ (3.28)

a d~

The corresponding general solution can be obtained via Eq. (3.22) by letting < ~ oc.

4 Def lect ion curves and exact e lement stiffness matrix

To establish the element stiffness matrix, the deflection curves due to a unit generalized displa- cement at one of the nodal coordinates are to be determined. Moreover, the consistent mass matrix can be constructed via the curves. A finite element method can be successfully devel- oped for the dynamic analysis [18].

Exact solutions for curved Timoshenko beams 75

Letting "m = 1, and ~i~ = 3~j* = 0, i = 1, 2 , . . . , 6, and f~, = 1 while all other nodal coordi-

nates are maintained at zero generalized displacement, n = 1, 2 , . . . , 6, respectively, six kinds

of tile boundary conditions are obtained as following:

ease 1: u(0) = 1, w(0) = k~(0) = u(1) = w(1) = k~(1) = 0;

case 2: w(0) = 1, u(0) = k~(0) = u(1) = w(1) = g'(1) = 0;

case 3: ~P(0) = I , g(0) = w(0) = u(1) = w(1) = LR(1) ---- 0; (4.1.1 -- 6) c~e 4: ~(I) : 1, "4o) = ~(o) -- ~(o) = ~(I) = ~(1) = o; ease 5: w(1) : 1, u(0) = w(0) = tP(0) = u(1) -- ~P(1) = 0;

case 6: kO(1) : i, u(O) = w(O) = g'(O) = u(1) : w(1) : O.

Neglecting the external forces and moments p(~), q(~) and m(~), and substituting one of the

six kinds of conditions into the general solution (3.22), the corresponding displacement w(~)

is obtained

6 ~'(~) : E @<(~), (4.2)

i=1

where the corresponding coefficients are listed in the Appendix. Substituting w(~) into the

relations (3.12.1-3), respectively, the displacements u(~) and the angle due to bending ~P(~)

are obtained. Letting the external forces and moment be zero and based on Eq. (3.3) and the relations

(3.1:!.1-3), the dimensionless normal force N(~), the bending moment M(#) and the shear

force Q(~) can be expressed in terms of the tangential displacement w, respectively

(dw ) 1 dSw 1 daw (4.3.1) X(~) = COg ~-- a~ -- <2261 de 5 } 61 d~ 3 '

Ce (d'u + _@) _ 1 dSw F 1 d4w (4,3.2) Q(~) , ~ ~6~ dp ~ d~4 '

F dSw d3w 2 dwl ~//{(~) -- + s 61 ~]"

The relation between the generalized forces and displacements at the nodal coordinate is writ-

ten in matrix notat ion as

I N(o) l F ~(o) ] O(o) | ~(o) | n~(o)/ / w(o) / N ( 1 ) | = [k~J]6• / u(1) 1" (4.4) Q(1)| /~(1)/ M(1)j L~'(~)J

Substituting the six kinds of deflection curves (4.2) due to a unit generalized displacement at one of the nodal coordinates into Eqs. (4.3.1-3), the stiffness matrix [kij] can be obtained

6

~lj =: E i=1

k2j =: L i=1 6

]~3j =: E i=1

61 <22 d ~ + d~3 ],

<,(7 j) [ ] d~ d4V~(O)] O~61 [ OZ 2 d~ 6 -1- d~ 4 ] '

r d V~(0) W~(0) 2~ dV~(0)l

76

]~4j = E Ti'(7-J) i=1 61

6

k5j : E - - i=1

6

]~6j = E - - - i=i

1 d~V~(1) d3V~(1)l

Tid7-j) F 1 d6 V/(1) d4V/(1)~

T~,t7-j/ [6 d5~(1) d3V~(1)

S.M. Lin

(4.5)

5 V e r i f i c a t i o n

To verify the previous analysis, the following example is presented.

Example." Consider the deformat ion of a cantilever extensional Timoshenko beam sub-

jected to a unit transverse load at the right end of the beam. The corresponding coefficients

are

~11 =721 =~31 =~42 : ~ 5 2 ='~62 = 1, ~/12 =~/22 =~32 = ~ / 4 1 = 7 5 1 :~/61 = 0 ,

p(~) = q(~) = m(~) = f l = f l * = f2 = f2* = f3 = f3* = f4 = f5 = fs* = f6 = 5* = O, (5.1)

f4* = i, a = 7r/2.

The homogeneous solutions (3.20) have been obtained. The par t icular solution is zero.

Accordingly, the corresponding general solution in the tangential displacement can be ob-

tained via Eq. (3.22). Substi tut ing the general solut ion into Eqs. (3.12.1-3), the solutions in

transverse displacement and the angle of ro ta t ion due to bending are obtained, respec-

tively. The solution in transverse displacement determined by using Cast igl iano's second

Table 1. The transverse and longitudinal displacements and the angle of rotation due to bending of cantilever extensional Timoshenko beam subjected to a unit transverse load at the right end [c~ = ~r/2, ~ = 100,

= 0.011

w ~ u u

0 0 0 0 0 0.1 4).000 82 0.062 56 0.003 33 0.003 33 0.2 -0.000 58 0.12465 0.013 14 0.013 14 0.3 0.00173 0.18575 0.02896 0.02896 0.4 0.007 02 0.245 36 0.050 00 0.050 00 0.5 0.01604 0.30291 0.075 18 0.075 t8 0.6 0.02938 0.35780 0.10322 0.10322 0.7 0.04737 0.40935 0.13263 0.13263 0.8 0.070 12 0.456 86 0.161 79 0.161 79 0.9 0.09748 0.49957 0.18902 0.18902 1.0 0.12901 0.53666 0.21264 0.21264

* Present study ** By using Castigliano's second theorem

Exact solutions for curved Timoshenko beams 77

theorem is

(5.2)

It is shown in Table 1 that the numerical results given by the present analysis are the same as those determined by using Castigliano's second theorem.

6 Conclusion

In this paper, a generalized Green function of a nth-order ordinary differential equation with forcing function composed of the delta function and its derivatives is obtained. The gener- alized Green function can be easily and effectively applied to both the boundary value prob- lems and the initial value problems. It is the generalization of those given by Pan and Hohen- stein, and Kanwal. The exact solution for static analysis of an extensible circular curved Timoshenko beam with nonhomogeneous elastic boundary conditions is obtained based on the generalized Green function. The explicit relations between the angle of rotation due to bending, the transverse displacement and the tangential displacement of the beam, the deflec- tion curves due to a unit generalized displacement at nodal coordinate, and the exact element stiffness matrix are obtained. A finite element method can be developed based on the results for tlhe dynamic analysis. Meanwhile, the stiffness locking phenomena accompanied in some other curved beam element methods does not exist in the proposed method.

Acknowledgement

The support of the National Science Council of Taiwan, R. O. C., is gratefully acknowledged (Grant number: Nsc87-2212-El 68-004).

Appendix: The coefficients of deflection curves

Case 1." The corresponding solution w(~) is

6 ~(~) = E T~V~(~),

i=1

where

[~jj =

"ba6 ba5 b34 ba3 ba 526 525 b~4 b23 b22 b16 bl5 514 bla 512 o -62 o -6a o 0 0 0 0 0 1 c~ 2

_~ o T o - ~

631 b21 bll -a261

0~2~1

0

-1

78 S.M. Lin

in which

b l i -

b2i = ~261Vi(1),

d~v~(1) d2v~(1)

I d5~(1) a2 daV~(1) ( d~ 5 d~ a ~- 51OZ 2 d~(1) ,

i = 1 , . . . , 6

(}) 63 : i+2~ 2 #+ .

Case 2." The corresponding solution w(~) is 6

~(~) : E T~sV~(~). i=l

where the coefficients are the same as those of case 1.

Case 3. The corresponding solution w(~) is 6

~(~) : E T~aV~(~), i=l

where the coefficients are the same as those of case 1.

Case 4." The corresponding solution w(~) is 6

~(~) : E T ~ ( ~ ) , /=1

where the coefficients are the same as those of case 1.

Case 5." The corresponding solution w(~) is 6

~(~) : E TJ~(~), {=i

where the coefficients are the same as those of case 1.

Case 6." The corresponding solution w(~) is 6

~(~) : E T~V~(~), i=1

where the coefficients are the same as those of Case 1.

References

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J. Appl. Mech. Trans. ASME 59, 1009-1014 (1992). [4] Stakgold, I.: Green's functions and boundary value problems. New York: John Wiley, 1979. [5] Laura, P. P. A., Maurizi, M. J.: Recent research on vibrations of archtype structures. Shock Vib.

Dig. 19, 6~9 (1987). [6] Chidamparam, P., Leissa, A. W.: Vibrations of planar curved beams, rings, and arches. Appl. Mech.

Rev. ASME 46, 467-483 (1993).

Exact solutions for curved Timoshenko beams 79

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Authors' address: S. M. Lin, Associate Prof. of Mechanical Engineering Department, Kung Shan Institute of Technology, Tainan, Taiwan 710, Republic of China