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Large elastic deformations in thin cantilever rods due to concentrated loadings
Im J Non Lmtar ~ h t h a m ~ Vol 16 No 1 pp 53 6~ 1981 Printed in Great Britain
002(~7462 81 010053 1 1 5 0 2 0 0 0 ~)1981 Pergamon Press Ltd
LARGE ELASTIC D E F O R M A T I O N S IN THIN CANTILEVER RODS D U E TO C O N C E N T R A T E D LOADINGS D E PANAYOTOUNAKOS Department of Mathematics The National Technical Umversitv of Athens, Athens Greece and P S THEOCARIS Department of Theoretical and Apphed Mechanics, The National Techmcal University of Athens, Athens. Greece (Re~elted 2 November 19791 Abstract In this paper the problem of large elasttc deformations in cantilever thin rods subjected to concentrated loads is considered Taking mto account the lncompresslbihty assumption of the center line and the equations relating the internal moments with the curvatures and torsion of the rod before and after the deformation, the non-linear equihbrlum system, composed of six coupled differential equations of hrst order is transformed to a new system of higher order The cases of geometries of mitmlly curved rods and their cross-sections were investigated, for which the higher order system of equahons may be deconpled and solved in a closed form Several apphcattons of thm curved cantilever rods were made and the potentiahties of the method were shown with these examples
1 INTRODUCTION Costello, in a recent paper [3], has considered the problem of pure bending of a hehcal s p r i g , which can be submitted to large deformations Also, an exact solution for the lmear system of the e q m h b r l u m differential equations was presented, when the cross-section of the spring has a kinetic symmetry, so that the principal m o m e n t s and the torsional m o m e n t of inertia are equal (I~ = 12 = 13) This solution was then used as a first a p p r o x i m a t i o n in Plcard's m e t h o d for a kinetic s y m m e t r y of the rod, where only the pnnclpal m o m e n t s of inertia are equal In this case the previous system becomes non-linear In a paper by the authors the closed-form solution of the non-linear system describing the foregoing problem was given [4] If the spring is not permitted to expand radially by some type of cylindrical constraint, the spring stiffness increases This problem was solved (neglecting friction) t h r o u g h an exact solution by Blanco and Costello [7] Also, Love [1] gave a closed-form solution of large deflections of a helical spring subjected to an axial torsional m o m e n t by m a k i n g the assumption that a kinetic s y m m e t r y analogous to that o f [ 3 ] is valid Finally, P a n a y o t o u n a k o s and Theocarls [6] solved t h r o u g h an exact solution the problem of the non-linear elastic analysis of a circular cantilever b e a m subjected to a co-planar concentrated force The present paper deals with the p r o b l e m of large elastic deformations of a cantilever thin rod, subjected to a concentrated load at its free end This p r o b l e m is associated with a system of six coupled non-linear differential equations of first order T a k i n g into a c c o u n t the lncompresslblhty assumption of the center-line of the rod a n d the equations relating the internal m o m e n t s with the curvatures KI(S) ~c2(S) and torsion z(S) after the deformation, the previous system, t h r o u g h several analytical treatments, leads to a new strongly nonlinear system of three equations with respect to ~a(S), K2(S) and z(S) The aim of this paper is to examine in what types of initial curves of the rod-axis and under what geometrical conditions of their cross-sections this system can be further decoupled and solved in a closed form Finally, several applications were examined in cases of rods accepting a closed-form solution and the results were c o m p a r e d with those given by [1] and [6] 53
54
D E PANAYOTOUNAKOSand P S THEOCARIS 2 ANALYSIS
Consider a cantilever skew-curved than rod AB of uniform cross-section, with one of its ends A fully fixed, subjected to a terminal concentrated force P = (P~, P2, P3) at its free end B (Fig 1) Let us denote xo(So), So(So), zo(So) the two curvatures and the torsion of the rod before deformation and x 1(S), x2(S), T(S) the same quantities after deformation All previous functions are related with the right-handed principal system of inertia 123 of the crosssection
Fig 1 Geometry and sign convention
The mcompresslblhty assumption of the center-line of the rod may be expressed by the relation dS0 = dS where So, S are the arc lengths before and after the deformation respectively Finally, consider the Fr6net tnhedron (t, n, b) at an arhatrary point of the deformed rod, where t = (tl, t2, t3) is the umt tangent vector pointing to the direction of the increasing S, n = (nl, n2, n3) is the umt normal vector pointing to the center of curvature and b = (bl, b2, b3) is the umt banormal vector, b as defined m such a way that the corresponding Fr6net trihedron to be a right-handed system It should be noticed that the principal system of inertia 123 does not coincide with the corresponding Fr6net trihedron The generahsed equlhbnum chfferenual equations of the thin rod w~th respect to the prinopal system are [1] N'~ - N 2 z + 7 k 2 + X = 0
(1 1)
N'2-TxI+NlZ+Y=O T'--N1K2+N2K 1 + Z = 0
(12)
G'I-G2z+Htc2-N2+K1 =0 G'z--Hxx+Giz+N1-K2=O H ' - Gl1¢2 + G2tc I + O = 0
(2 1) (22)
(1)
(1 3) (2)
(2 3)
where X, Y, Z, K~, K2 and O are the components of the force and couple-resultants per umt length along the rod and Nx, N2, T, Gi, G 2 and H are the components of the force and coupleresultants acting on a genenc cross-section Also, primes denote dlfferentiat~on w~th respect to S The bending and twisting couples G1, Gz and H are related to the final curvatures x~(S), Kz(S) and the final torsion z(S) by Gl=A0cl-Xo),
Gz=B(Ic2-~o) ,
H=C(x-Zo)
(2a)
where A = EI 1, B = EI 2, C = GI3, E, G denote the moduh of elasticity and shear respectively, 11, 12 are the moments of lnertm about the principal axes 1 and 2 of the cross-section,
whereas 13 represents the moment of inertia of the cross-section about the 3 axis (torsional moment of inertia) The quantmes X, Y, Z, K~, K 2 and ® are equal to zero because of the concentrated load P = (Px, P2, P3) Thus, the non-linear differential system of (1) and (2) becomes
Large elastic deformatmns m thin cantileverrods due to concentrated loadmgs N'~ - N 2 t +
55 (3 1)
T/¢2 = 0
N ~ - 7k 1 + N i t = 0
(3 2)
T' - N ~ K 2 + N21cx = 0
(3 3)
G't - GEt + H/c2 = N2
(4 1) (4 2)
G'2 - H K a + G l t = --N1
(3)
(4)
(4 3)
H'-GI~2+GEKx=O
Equations (4) through relations (3) yield Ax'~ + F x 2 t + B~ot - C t o K 2 - - Ax~
= N2
(5 1)
Bx'2 + E x t t + C t o x * - A x o t - B # ~ ' o
= -N,
(5 2)
Cz' - - D t c l K
2+
(5)
(5 3)
A~Co~C2 -- Bg, oX 1 - Cz'o = 0
in which D=A-B,
E=A-C,
(6)
F=C-B=D-E
For the decouphng of the non-hnear system (3) and (5) the following analytical treatments are used Differentiating (5) we have A~c~+
F(K2"C )' +
B(~ot ~
-
-
C(ZoK2)' --
Ar~ = N~
B ~ + EOq t~ + C(to~x)' -- AOCot~ -- Bg~) = - N'x
(6a)
B~c~ + E ( r 1z)" + C(tolC 1)" -- A0Cot) " -- Bg~ = - N~
Now multiplying (3) with ix(S), ~c2(S), t(S) respectively, adding the resulting relations and using the first two equations of (6a) together with (3 3) and (5), the following non-linear differential equation of the second order may be derived, as A ( - - K 2 K ~ --KoKz'[2+K1K'I'~--KI"CK'o)+A[--KI(KO'~Y+K2K'~]
+ B[~q K~ -~%(#oz) ' + ~c2t~c~ - x2t~ ~ + xltzt~o] - B~q #3 +
CEKI('~oK 1)t --j-K2("CoK2f ]
"~ EEKI(K 1"~)' "3vK2"C2K1]-~"F [ -- x2(~c2t)' + ~qt2K2] = 0
(7)
Moreover, through differentiation of (3 1) and using relations (3 3), (5) and (6a) we have
+ A [ - K~t~o + K~(~ot)' - ~ 2(~otY' - ~ t ~ 3 v
t ,v
+ ~[~2~ - ~ + ~' + ~,~o 3 ~t -it -m + B [ - ~2~:o + ~'2~o - ~2~Co]
- ~,
~ 22- ~ = t '
+ ~t~'~o - ~:~o
+ ~t]
+ ~:~(~o~)'] (8)
+ C [ - ~ ( t o ~ 1)' + ~2(t o~ 0" - t o t ' ~ + t o~2~ht - ~2t(to~2)']
+ e[~,~-
~ ( ~ , ~ ) ' + ~2(~i~)"] + F [ ~ i ~ t
+ ~'
- ~ :
+ ~:~(~)'] = 0
So, the decouphng of the non-hnear system of (3) and (5) leads to a strongly non-linear system of higher order with respect to xl, K2, z composed of(5 3), (7) and (8) The scope of this Investigation is not to find numerical solutions of this system, but to examine under what analytical equations of initial skew-curved thin rods and under what kinetic conditions of their cross-sections this system would be further decoupled and solved in a closed form 2 1 Strazght rod wtth mttml constant torsmn
Consider now the cantilever straight rod AB (to = ~o = 0) of initial torsion t o = constant A kinetic symmetry of its cross-section is supposed such that I 1 = I 2 , which leads to A=B,
D=0, E= -F
56
D E PANAYOTOUNAKOSand P S THEOCARIS
T h r o u g h the foregoing assumptions equatmns (53), (7) and (8) become respectively z = c 1 = const
(91)
/£ 1/£2 "~/£2Kal r'-----'~'(/£1/£~--/£2/£~)
(92)
/£2/£~--/£tl/£t2=Z/£2(/£1/£tl'q-/£2/£'2)'q-,~(/£2/£~t--/£P2/£~)
(93)
(9)
where A
2= -
(94)
q(A-F)+Cz o
and cl is an integration constant Based on relation (93) and taking mto account that it is valid
\/£~I
/£~
/£~
\/£21
the following equation results
/£~/= ~,1.(/£1 +/£:0' +,'1. \/£-~/ from the integration of wtuch we have x,1 _2/£ 2_ ._ ~2/£z(/£x2+/£2)+c2/£21
(10)
where c2 is an integration constant Also, the second equation of (9) through relation
(/£ ~/£~)' - (/£2/£i)' =/£~/£~ -/£2/£7 can be eqmvalently written as (/£2 +/£2), = 22(/£1/£~ _/£2/£'!)' from the lntegratmn of wbach we have /£2 +/£2 = 2;t(/£1x~ --/£2/£i)+ c3
(11)
where c3 is again an integration constant Thus, (9 1), (10) and (11) constitute a new, strongly non-linear, differential system of second order, written under the form 17=C1 (121)
/£'1 --/~/£~ /£2
2(/£2+/£ 2) C2 ~'-~'~
(122)
c3 1 + ~2 _ 22~' =/£~ where
(12)
(123)
( =/£2//£1
(13)
Finally, t h r o u g h &vismn of (122), (123) and taking into account relations (13), we find that 2c2 -- 2c3 (1 + ~2 _ 22~') = Ka 2 x ~2C3 /£2/£2
_ 22~. (14)
The combination of (14) with the relations
1//£2 =f/Ca (1//£2)' =f'/c3 /£i//£~ = - f ' / 2 c 3 /£i //£ , = - f ' /2f ,, 3 /£1//£1 = --f"/2c3 + 3f'z/4f c3
(15)
Large elastic deformationsm thm cantdever rods due to concentrated loadmgs
57
where f = 1 + (2 _ 22(' yields the following final strongly non-hnear differentml equation of third order with respect to ( [822( ' - 4 2 ( ( 2 + 1)]~'" + 24).(('(" - 1222((")2 + 3(22cq -~2)(( 2 + 1)(' + 6(2 + 20c2)(~') 2 - 24~.(~')3 -- 30q( 4 - 6 0 ~ 1 ( 2 - 30q = 0
(16)
in which 0~1 - -
2(2c2 - 2 c 3 ) 32
Ct2= 4 ( __22C2 + 222C3 + 1) Assuming that, if S vamshes, H vamshes too, the third of relatmns (2a) leads to 'C(0) = "1S= C 1 = Z O
(17)
~.= - 1/2%
(18)
and because of (9 4) we have
The only unknown constant coefficients of(16) are the integration constants c2 and c3 Since the centroldal axis of the rod is a umform skew-curve, the function ( (x 14= 0) is continuously &fferentlable, thus, we can ask for a solution of(16) under the form (= ~
0c.S"
(19)
n=O
Substituting relation (19) into (16) the coefficients 0tn may be found including, m ad&tmn, the constants of lntegratmn c2 and Ca For the determlnatmn of c2, c3 and of the coefficients 0~n we make the following boundary conditions (i) F r o m (2a), (13) and for S = 0 we have 4(0) :/£2(0)//£
1(0) :
K~(0)/KT~I(0) :
--
P1/P2 = cto
(n) Also, based on the first two equations of (6a), (3), (4) and (5) and for S = 0 the following relations may be derived ~'~(0) = - ( 1 / A ) P 2 x'2(O)=(1/A)PI
x'~(0) = (2P 1/A)zo
(20)
~c'~(O) = (2P2/A)z o
After the definmon of the quantities c2, c3, ~, and consequently of the functmns xl, x2 and z, the force and couple resultants can be readily derwed through (2a), (3) and (4) 2 2 Straight thin rod wzthout m m a l torslon
An interesting special case of the previous general problem is that of a cantilever straight thin rod AB with a kinetic symmetry of its cross-section 11 = 12 and without lmtlal torsion (% = 0) Assume also that the rod is subjected to a co-planar force P = (P cos 0, P sin 0, 0) = (P1, P2, 0) at its free end B, where 0 IS the angle between the tangent and the direction of the force P after the deformation, the curled bar characterlses any quantmes at the free end (origin of the arc measurement) It IS obvious that the following is vahd
Xo=Xo=Z0=0,
x l=z=0
(21)
58
D E PANAYOTOUNAKOSand P S THEOCARIS
Since z o = ~c~=0, the coefficient 1/2 becomes zero, thus, (9 2) is an identity whereas (9 3) leads to the following strongly non-linear differential equation of the third order . . .- . .x2~:2 . 3 , =0 ~:2~:2 + ~:2~:2
(22)
The closed-form solution of the last equatmn has already been given in [6] Making use of the foregoing general solution written under the form
/£2(S)=o92 dn
u
K~= - 0
(23)
where
u=o92(S +e3)/2, dn u = ( 1 - p Z
pZ=(o92_o9~)/o9~< 1,
smZq~)l/2,
o9~ = cl -(c~ + c2) '/~
x2(O)=O K'z(O)=(1/A)P sin 0 ~(0)--- -(1/A)lc2(O)P cos 0
(24)
Based on (23) we have (25 1)
tc~(S)-- - ( o 9 2 p 2 / 2 ) sn u cn u
(25)
x'~(S)=(o92/4)[(2-O2)- 2 x~2o9(7)-l~:2(S)
(25 2)
Through the formula hm ~:~(S)_ ~:"t0) 2t = s-o K2(S) K~(0)
-(1/A)P
cos 0
the boundary conditions (24) can be expressed as (26 1)
dnul~= o = 0
(1/A)P sin 0
-(o92p2/2) sn uis=o cn u[~=o = o922(2 _ p 2 ) =
-(4/A)P
cos 0
(26 2)
(26)
(26 3)
From (26 1) one may conclude that sn ul~=o = _+ 1,
cn uis=o =0,
p~ 1
(27)
By now, through the combination of relation (27) with (26 2) and (26 3), we infer that 0=re,
sn uis=o = - 1
Finally, (26 3) leads to
o922=(4/A)P
(28)
Consequently, based on relation (23), the function u at the free end is derived by u]s= o~ -
f(rU2, re/2) =
-
where F is the complete elliptic integral of the first kind So, one may easily determine the integration constant c3 and therefore the curvature function of the deformed rod It can be noticed here, that Love [1] gave analogous to (23) formula in the elastwa problem (p p 402-405)
Large elastic deformations m thin cantilever rods due to concentrated loadmgs
59
2 3 Skew-curved thin rod In this section the problem of the non-hnear analysis of a skew curved thin rod, subjected to a terminal concentrated force, will be examined, the centroldal axis of the rod is considered as a uniform curve and the principal system of inertia of every cross secUon coincides with the corresponding Fr6net trlhedron before and after the deformation In this case, because of relations KO:KI~0,
/~0 = KO
K2=K
Love's generalized equations (2a), (3) and (4) expressed with respect to the Fr6net trlhedron are given by Gb = Afro--Xo),
(29)
H, = C(z - % )
N~,- ZNb+ xTt= 0
(30 1)
Ng+zN.
=0
(30 2) (30)
T't--xN.
=0
(30 3)
G',, --'CGb+ xH, = Nb
(31 1)
Gg+zG.
= --N.
(31 2) (31)
H;-xG,,
=0
(31 3)
Equation (31 1) combined with (31 2) and relations (29) becomes N.-
C'c /£ (z - - % ) ' - A(x -- Xo)'
(32)
Also, (31 1) combined with (31 3) and relations (29) is written as Nb = C
+ Cx(z-%)-Az(x-xo)
(33)
Finally, based on relations (30 1), (30 2), (32) and (33), equations (30 2) and (30 3) lead to the following strongly non-linear differential equations of the third order + C[~c(z - % ) ] --A[z(K --Xo) ] = C - - (z K
o) + AT(K-- ~Co)'
t~L ~ J
J
+ C[ 1 "cz(K-~Co)]'+ CEz('c-Zo)]'- A[~K~(x --Xo)] = -Cz(z-%)'-A t~(x-Ko)' which after successive dlfferentmtlons can be written as
2z~..( z - % ) ' +
=2
[
. z,
2 z (z_%).+
(3_< \ ~ca2
~c" ~:
2z'x' xz _~z+K 2
(
--(K'--KT0)"-~---(/£--/£0)"-~('~2--K2)(/£--KT0)'q'-'C 2Z' - - -x'z K K
(~-Zo)'+ ~'(~-~o)
)
(X-Xo)
1
(34)
where 2 = A/C Equations (34) constitute a strongly non-hnear system of third order ordinary differential equations with respect to ~:, r that can be integrated numerically by applying Runge-Kutta's
60
D E PANAYOTOUNAKOSand P S THEOCARIS
m e t h o d under suitable mitml conditions But, in some special cases of initial curves for the rod axis the previous system can be solved in a closed-form These cases will be investigated m the following
2 3 1 Skew-curved thin rod deformed to a czrcle We try to d e t e r m m e a cantilever skew-curved rod subjected to a given concentrated terminal force P = ( P . P., Pb) which is d e f o r m e d to a circle of k n o w n curvature K = 1/R after loading In this case it is vahd that K = constant,
z= 0
(35)
So, equations (34) lead to "C~ -~- K2T~) = 0,
(36)
K~ -[- K2K~) = 0
E q u a t i o n s (36) are linear ordinary &fferentlal equations with constant coefficients whose general integrals are respecUvely To(S)~---c 1 s i n KS--}-c 2 c o s KS + c 3
(37) K o t S } = ~ 1 s i n K S q - ~ 2 c o s K-~-~ 3
where c,, }, (t = 1, 2, 3) are constant of integration D e t e r m i n a t i o n of c,, ~, can be achieved t h r o u g h the b o u n d a r y conditions, taking into a c c o u n t that the rod is unlvocally dlfined, since the curvature ~cis k n o w n and it has one fixed end So, based on relations (29) to (33) a n d for S = 0 we have xo(O) = K,
AK~)(0)= - - P . ,
%(o)=o,
~ o(0) = 0,
(38)
K
z ~ ( 0 ) = ~ Pb,
~(o) =
A
Pt
Finally, from (37) a n d (38) the following relations result zo(S)= x~--~(1-cos KS) (39) Ko(S)=c-71~ [P,(1 - - c o s t c S ) - P , sin KS] + K
Since n o w the initial curvature xo(S) a n d torsion zo(S ) of the rod are known, all the internal forces a n d m o m e n t s can be found by using (29), (30) and (31)
2 3 2 Planar curved rod deformed to a hehx W e determine n o w a cantilever planar curved rod subjected to an u n k n o w n concentrated loading P = (Pt, P., Pb), M = (M,, M., Mb) at its free end such that the rod is deformed to a helix of k n o w n curvature x a n d torsion z In this case it is valid that r o = O,
z = k n o w n constant,
K = k n o w n constant
(40)
T h e first of equations (34) leads to -2zx b =0 from which one can result that Ko = constant O n the other hand, the second of equations (34) leads to G - ( z2 - K 2 ) ~ = 0
(41)
Large elastic deformations in thin cantdever rods due to concentrated loadmgs
61
whach, because of the fact that Ko = constant, becomes an identity Thus, it may be concluded that the initial planar curve should be a circle F r o m (31 3) and (32), combined with relations (29), one m a y also result that the loading P and M must be such that the quantatIes G. and N . vanish Also, relations (33), (31 1) combined with (32), (29) and (41) give the following two equations, wtuch must be valid for a generac cross-section of the helax
Nb= C~cz-Az(x--~:o),
T,=-~ Nb
(42)
It should be noticed now, that the ratio Nb/T~= tan 6 is constant, where 6 is the angle of the helix It will be proved that the only loading, satisfying the equations G . = 0 , N . = 0 and (42), is a concentrated force P and a concentrated m o m e n t M, whose vectors are on the axes of the helix Indeed. at as obvious that the effect of the m o m e n t M causes G. = 0, N . = 0 at any cross-section Also, Nb = T, = 0 Based on the equations CK"c = K M t
Az(X--Ko)=zMb x M t - z M b = X M cos 6 - z M san 6=zM tan 6 cos 6 - z M sin 6 = 0 relations (42) yield
Nb=tCMt--'rMb=O,
T
Tt=- Nb=O K
On the other hand, the force P, intersected with the radius p of the cylinder, gives G. = 0 and obviously N . = 0 Finally, because the relations T , = P c o s 6 and N ~ = P s I n 6 are valid, equations (42) are verified In this point, at should be noticed that an analogous result is included in Love's book, where the case was examined to determine the loading conditions of the rod wtuch deforme an initial known helix to another known helix ([-1, pp 4 1 3 ~ 1 4 ] ) The determination of the constant curvature Ko was obtained by using relations (29) So, for S = 0 , we have
A(~C-Ko)=Gb=M sin 6+Pp cos 6 Now, the curvature function ~co results in 1
~o = K - ~ [ m sin 6 + Pp cos 6]
(43)
Finally, the loading acting on the free end of the initial circular rod consists of the four generahsed forces Mcosf+Ppsan6,
Psin6,
Msinf+Ppcos6,
Pcos6
with respect to the axis of t and b respectively
2 3 3 Planar curved rod deformed to a cwcle We determine a cantilever planar-curved rod subjected to an unknown concentrated force P = (P. P., Pb) of known direction at its free end such that the rod to be deformed to a circle of a given curvature ~: = 1/R In this case it is valid that z = z o = 0,
~c= known constant
(44)
The first of equations (34) is an identity, whale the second leads to the differential equation ~c~+ K2~c~= 0
(45)
62
D E PANAYOTOUNAKOSand P S THEOCARIS
The general Integral of this equation is K0(S)=c 1 s i n l£S"]-C2 COS
(46)
I£S"~-C 3
where c, 0 = 1, 2, 3) are integration constants It is obvious that relation (30 1) gives Nb = 0 for any cross-section of the circle So, the component Pb of the force P vamshes and therefore P is a co-planar force Now taking into account that the circle is umvocally defined, since the curvature ~ is known and xt has one fixed end, the angle 0 between the tangent at the fixed end and the direction of the force P is also known The boundary conditions are 1
~Co(0) = x,
~'0(0)= -- ~ P.,
~c
~(0) = ~ Pt
(47)
From the solution of the previous system the unknown curvature x 0 results in x0(S) = ~
[P,(1 - c o s xS) - P . sin xS-] + ~c
(48)
Moreover, for the fixed end of the rod the following relations are vahd Crb= A [x - tCo~)] = P,r. - P.r,
-0 = 0~) = -
f2
1%(S)dS
(49)
where r = (r. r., 0) denotes the known posmon vector from the fixed end to the free end of the circle with respect to the Fr6net trmhedron t, h,/) Now, from the solution of the last system one may determine the components P, and P. as ~:2A(O + xS)(xr, + sin ~ ) Pt =
( x r . + 1 - c o s xsx1 - c o s ~ ) + ( x r , +
sin xSXsm ~ - ~ )
K.2ACO+ ~-S)(tcrnq- 1 - c o s ~ ) P" - (xr. + 1 - cos ~ ) ( 1 - cos xS) + (xr, + sin ~ ) ( s m ~ - ~ )
3
(50)
CONCLUSIONS
In ttus paper relationships yielding the large elastic deformations of a thin incompressible rod subjected to a concentrated general type of load are established F r o m this investigation the following results were derived (1) The transformation of the general non-linear differential system composed of six coupled equations governing the equlhbrlum of a skew-curved element, to a strongly non-hnear system of three equations with respect to the curvature and torsion function (2) A closed-form solution of the three equation system under statable boundary conditions corresponding to a straight thin rod with mltml torsion and kinetic symmetry of its cross-section such that the principal moments of inertia are equal (3) The decouphng of the non-hnear system with respect to the Fr6net trlhedron accompanying a skew-arc element and its closed-form solution m several cases of cantilever curved rods where the curvature and torsion functions before and after deformation, as well as the concentrated loading are considered either as given known in advance, or unknown quantities
REFERENCES E H Love, A Treattse on the Mathematical Theory oJ Elastlclt~ Dozer Pubhcatlons New York (19441 Tlmoshenko and J Gere, Theory ofElastw Stabdlty McGraw-Hill, New York (1961) G Costello, Large deflections ofhehcal spring due to bending J En 9 Mech Dlv 103, 481 (1977) E P a n a y o t o u n a k o s and P S T h e o c a n s N o n h n e a r analysis of cantilever rods due to terminal m o m e n t s lny 4rchtt 49 73 11980) 5 J Kennedy and A S Aggarwal Effect of weight on large deflections of arches J Era! ~4et h DIt. 97 637 (1971) 6 D E P a n a y o t o u n a k o s and P S T h e o c a n s N o n h n e a r elastic analysts of circular beams loaded b~ co-planar concentrated forces Z A M M (in press)
1 2 3 4
A S A D
Large elastic deformations m thin cantilever rods due to concentrated loadmgs
63
7 J A Blanco and G A Costello, Cylindrical constraint of helical springs J appl Mesh 41, 1138 (1974) 8 G B Sinclair, The non-linear bending of a cantilever beam with shear and longitudinal deformations Int J Non-Lmear Mesh 14 I l l (1979) 9 R Schmldt and D A DaDeppo Non-hnear theory of arches and beams with shear deformation J appl Mesh 39, 1144 (1972) 10 A E Green and W Zerna Theoretical Elasttc~t~ Oxford University Press (1954) 1l A E Green and J E Adk~ns, Larqe Ela~tt~ DeJorrnatton~ Clarendon Press, Oxford (1970) 12 H T Davis lntrodu¢tton to Nonhnear Dtflerenttal and Integral Equatton~ Do~er Publications, New York (1970) 13 l S Gradshteyn and I M Ruzhik, Tables o/lnteqrals Ser~e~and Products Academic Press~ New York (19651 R~sum~
Dans le p r e s e n t a r t i c l e on c o n s l d e r e le problEme des grands d e f o r m a t i o n s e l a s t t q u e s d'une p o u t r e mince a c a n t i l e v e r , soumls ~ des f o r c e s concentr~es En t e n a n t compte de la cond l t J o n d ' l n c o m p r e s s i b i l t t ~ de la l i g n e c e n t r a l e t de l ' ~ q u a tton reliant les moments internes avec les courbures et la torsion de la poutre avant et apr~s la deformation, le syst~me non llneatre de l'~qullJbre, qul est compos~ de six ~quatlons dlfferent0~lles couples du premier ordre, est transform~ a un nouveau syst~me d'ordre plus ~l~v~ Les cas des g~ometries des poutres tnttlallement encurv~ et de leur section transversale a ~t~ ~xamln~ Dans ces cas le system d'ordre ~l~v~ peut etre decoupl~ et resolu de facon que nous pouvons fournlr une solution ferm~e On a effectu~ plur0eures appllcatlons de poutres minces et la pulssence de la m~thode est demontr~e
Zusammenfassung
In d l e s e r A r b e l t das Problem yon grossen e l a s t u s c h e n DeformatJonen dunner be~se~tlg g e l a g e r t e n Stabe u n t e r k o n z e n t r t e r t e r Belastungen wurde u n t e r s u c h t Die Inkompessibll0tatsannahme f u r d i e N u l l - L i n i e und d i e g l e l c h u n g e n , d i e innere Momente mlt den Stabkrummungen u n d - t o r s l o n e n , vor und nach der Deformation v e r k n u p f e n , wurden b e t r a c h t e t und das n t c h t l t n e a r e g l e l c h g e w t c h t s s y s t e m , zusammengesetzt aus sechs gekoppelten D J f f e r e n t i a l g l e t c h u n g e n e r s t e r Ordnung, wurde zu elnem neuen System h o h e r e r Ordung transformJert Die geometrten der a n f a n g l , c h gekrunmten Stabe und i h r e Querschmltte, f u r welche d i e glelchungssysteme h o h e r e r Ordnung e n t k o p p e t t und in g e s c h l o s s e n e r Form g e l o s t werden konnten, wurden u n t e r s u c h t Mehrere Anwendungen von d l e s e n gekrummten e l n s e l t l g g e l a g e r t e n Stabe wurden g e b l l d e t und mJt dlesen B e l s p l e l e n wurden d i e MoglJchkelten der Methode g e z e t g t
002(~7462 81 010053 1 1 5 0 2 0 0 0 ~)1981 Pergamon Press Ltd
LARGE ELASTIC D E F O R M A T I O N S IN THIN CANTILEVER RODS D U E TO C O N C E N T R A T E D LOADINGS D E PANAYOTOUNAKOS Department of Mathematics The National Technical Umversitv of Athens, Athens Greece and P S THEOCARIS Department of Theoretical and Apphed Mechanics, The National Techmcal University of Athens, Athens. Greece (Re~elted 2 November 19791 Abstract In this paper the problem of large elasttc deformations in cantilever thin rods subjected to concentrated loads is considered Taking mto account the lncompresslbihty assumption of the center line and the equations relating the internal moments with the curvatures and torsion of the rod before and after the deformation, the non-linear equihbrlum system, composed of six coupled differential equations of hrst order is transformed to a new system of higher order The cases of geometries of mitmlly curved rods and their cross-sections were investigated, for which the higher order system of equahons may be deconpled and solved in a closed form Several apphcattons of thm curved cantilever rods were made and the potentiahties of the method were shown with these examples
1 INTRODUCTION Costello, in a recent paper [3], has considered the problem of pure bending of a hehcal s p r i g , which can be submitted to large deformations Also, an exact solution for the lmear system of the e q m h b r l u m differential equations was presented, when the cross-section of the spring has a kinetic symmetry, so that the principal m o m e n t s and the torsional m o m e n t of inertia are equal (I~ = 12 = 13) This solution was then used as a first a p p r o x i m a t i o n in Plcard's m e t h o d for a kinetic s y m m e t r y of the rod, where only the pnnclpal m o m e n t s of inertia are equal In this case the previous system becomes non-linear In a paper by the authors the closed-form solution of the non-linear system describing the foregoing problem was given [4] If the spring is not permitted to expand radially by some type of cylindrical constraint, the spring stiffness increases This problem was solved (neglecting friction) t h r o u g h an exact solution by Blanco and Costello [7] Also, Love [1] gave a closed-form solution of large deflections of a helical spring subjected to an axial torsional m o m e n t by m a k i n g the assumption that a kinetic s y m m e t r y analogous to that o f [ 3 ] is valid Finally, P a n a y o t o u n a k o s and Theocarls [6] solved t h r o u g h an exact solution the problem of the non-linear elastic analysis of a circular cantilever b e a m subjected to a co-planar concentrated force The present paper deals with the p r o b l e m of large elastic deformations of a cantilever thin rod, subjected to a concentrated load at its free end This p r o b l e m is associated with a system of six coupled non-linear differential equations of first order T a k i n g into a c c o u n t the lncompresslblhty assumption of the center-line of the rod a n d the equations relating the internal m o m e n t s with the curvatures KI(S) ~c2(S) and torsion z(S) after the deformation, the previous system, t h r o u g h several analytical treatments, leads to a new strongly nonlinear system of three equations with respect to ~a(S), K2(S) and z(S) The aim of this paper is to examine in what types of initial curves of the rod-axis and under what geometrical conditions of their cross-sections this system can be further decoupled and solved in a closed form Finally, several applications were examined in cases of rods accepting a closed-form solution and the results were c o m p a r e d with those given by [1] and [6] 53
54
D E PANAYOTOUNAKOSand P S THEOCARIS 2 ANALYSIS
Consider a cantilever skew-curved than rod AB of uniform cross-section, with one of its ends A fully fixed, subjected to a terminal concentrated force P = (P~, P2, P3) at its free end B (Fig 1) Let us denote xo(So), So(So), zo(So) the two curvatures and the torsion of the rod before deformation and x 1(S), x2(S), T(S) the same quantities after deformation All previous functions are related with the right-handed principal system of inertia 123 of the crosssection
Fig 1 Geometry and sign convention
The mcompresslblhty assumption of the center-line of the rod may be expressed by the relation dS0 = dS where So, S are the arc lengths before and after the deformation respectively Finally, consider the Fr6net tnhedron (t, n, b) at an arhatrary point of the deformed rod, where t = (tl, t2, t3) is the umt tangent vector pointing to the direction of the increasing S, n = (nl, n2, n3) is the umt normal vector pointing to the center of curvature and b = (bl, b2, b3) is the umt banormal vector, b as defined m such a way that the corresponding Fr6net trihedron to be a right-handed system It should be noticed that the principal system of inertia 123 does not coincide with the corresponding Fr6net trihedron The generahsed equlhbnum chfferenual equations of the thin rod w~th respect to the prinopal system are [1] N'~ - N 2 z + 7 k 2 + X = 0
(1 1)
N'2-TxI+NlZ+Y=O T'--N1K2+N2K 1 + Z = 0
(12)
G'I-G2z+Htc2-N2+K1 =0 G'z--Hxx+Giz+N1-K2=O H ' - Gl1¢2 + G2tc I + O = 0
(2 1) (22)
(1)
(1 3) (2)
(2 3)
where X, Y, Z, K~, K2 and O are the components of the force and couple-resultants per umt length along the rod and Nx, N2, T, Gi, G 2 and H are the components of the force and coupleresultants acting on a genenc cross-section Also, primes denote dlfferentiat~on w~th respect to S The bending and twisting couples G1, Gz and H are related to the final curvatures x~(S), Kz(S) and the final torsion z(S) by Gl=A0cl-Xo),
Gz=B(Ic2-~o) ,
H=C(x-Zo)
(2a)
where A = EI 1, B = EI 2, C = GI3, E, G denote the moduh of elasticity and shear respectively, 11, 12 are the moments of lnertm about the principal axes 1 and 2 of the cross-section,
whereas 13 represents the moment of inertia of the cross-section about the 3 axis (torsional moment of inertia) The quantmes X, Y, Z, K~, K 2 and ® are equal to zero because of the concentrated load P = (Px, P2, P3) Thus, the non-linear differential system of (1) and (2) becomes
Large elastic deformatmns m thin cantileverrods due to concentrated loadmgs N'~ - N 2 t +
55 (3 1)
T/¢2 = 0
N ~ - 7k 1 + N i t = 0
(3 2)
T' - N ~ K 2 + N21cx = 0
(3 3)
G't - GEt + H/c2 = N2
(4 1) (4 2)
G'2 - H K a + G l t = --N1
(3)
(4)
(4 3)
H'-GI~2+GEKx=O
Equations (4) through relations (3) yield Ax'~ + F x 2 t + B~ot - C t o K 2 - - Ax~
= N2
(5 1)
Bx'2 + E x t t + C t o x * - A x o t - B # ~ ' o
= -N,
(5 2)
Cz' - - D t c l K
2+
(5)
(5 3)
A~Co~C2 -- Bg, oX 1 - Cz'o = 0
in which D=A-B,
E=A-C,
(6)
F=C-B=D-E
For the decouphng of the non-hnear system (3) and (5) the following analytical treatments are used Differentiating (5) we have A~c~+
F(K2"C )' +
B(~ot ~
-
-
C(ZoK2)' --
Ar~ = N~
B ~ + EOq t~ + C(to~x)' -- AOCot~ -- Bg~) = - N'x
(6a)
B~c~ + E ( r 1z)" + C(tolC 1)" -- A0Cot) " -- Bg~ = - N~
Now multiplying (3) with ix(S), ~c2(S), t(S) respectively, adding the resulting relations and using the first two equations of (6a) together with (3 3) and (5), the following non-linear differential equation of the second order may be derived, as A ( - - K 2 K ~ --KoKz'[2+K1K'I'~--KI"CK'o)+A[--KI(KO'~Y+K2K'~]
+ B[~q K~ -~%(#oz) ' + ~c2t~c~ - x2t~ ~ + xltzt~o] - B~q #3 +
CEKI('~oK 1)t --j-K2("CoK2f ]
"~ EEKI(K 1"~)' "3vK2"C2K1]-~"F [ -- x2(~c2t)' + ~qt2K2] = 0
(7)
Moreover, through differentiation of (3 1) and using relations (3 3), (5) and (6a) we have
+ A [ - K~t~o + K~(~ot)' - ~ 2(~otY' - ~ t ~ 3 v
t ,v
+ ~[~2~ - ~ + ~' + ~,~o 3 ~t -it -m + B [ - ~2~:o + ~'2~o - ~2~Co]
- ~,
~ 22- ~ = t '
+ ~t~'~o - ~:~o
+ ~t]
+ ~:~(~o~)'] (8)
+ C [ - ~ ( t o ~ 1)' + ~2(t o~ 0" - t o t ' ~ + t o~2~ht - ~2t(to~2)']
+ e[~,~-
~ ( ~ , ~ ) ' + ~2(~i~)"] + F [ ~ i ~ t
+ ~'
- ~ :
+ ~:~(~)'] = 0
So, the decouphng of the non-hnear system of (3) and (5) leads to a strongly non-linear system of higher order with respect to xl, K2, z composed of(5 3), (7) and (8) The scope of this Investigation is not to find numerical solutions of this system, but to examine under what analytical equations of initial skew-curved thin rods and under what kinetic conditions of their cross-sections this system would be further decoupled and solved in a closed form 2 1 Strazght rod wtth mttml constant torsmn
Consider now the cantilever straight rod AB (to = ~o = 0) of initial torsion t o = constant A kinetic symmetry of its cross-section is supposed such that I 1 = I 2 , which leads to A=B,
D=0, E= -F
56
D E PANAYOTOUNAKOSand P S THEOCARIS
T h r o u g h the foregoing assumptions equatmns (53), (7) and (8) become respectively z = c 1 = const
(91)
/£ 1/£2 "~/£2Kal r'-----'~'(/£1/£~--/£2/£~)
(92)
/£2/£~--/£tl/£t2=Z/£2(/£1/£tl'q-/£2/£'2)'q-,~(/£2/£~t--/£P2/£~)
(93)
(9)
where A
2= -
(94)
q(A-F)+Cz o
and cl is an integration constant Based on relation (93) and taking mto account that it is valid
\/£~I
/£~
/£~
\/£21
the following equation results
/£~/= ~,1.(/£1 +/£:0' +,'1. \/£-~/ from the integration of wtuch we have x,1 _2/£ 2_ ._ ~2/£z(/£x2+/£2)+c2/£21
(10)
where c2 is an integration constant Also, the second equation of (9) through relation
(/£ ~/£~)' - (/£2/£i)' =/£~/£~ -/£2/£7 can be eqmvalently written as (/£2 +/£2), = 22(/£1/£~ _/£2/£'!)' from the lntegratmn of wbach we have /£2 +/£2 = 2;t(/£1x~ --/£2/£i)+ c3
(11)
where c3 is again an integration constant Thus, (9 1), (10) and (11) constitute a new, strongly non-linear, differential system of second order, written under the form 17=C1 (121)
/£'1 --/~/£~ /£2
2(/£2+/£ 2) C2 ~'-~'~
(122)
c3 1 + ~2 _ 22~' =/£~ where
(12)
(123)
( =/£2//£1
(13)
Finally, t h r o u g h &vismn of (122), (123) and taking into account relations (13), we find that 2c2 -- 2c3 (1 + ~2 _ 22~') = Ka 2 x ~2C3 /£2/£2
_ 22~. (14)
The combination of (14) with the relations
1//£2 =f/Ca (1//£2)' =f'/c3 /£i//£~ = - f ' / 2 c 3 /£i //£ , = - f ' /2f ,, 3 /£1//£1 = --f"/2c3 + 3f'z/4f c3
(15)
Large elastic deformationsm thm cantdever rods due to concentrated loadmgs
57
where f = 1 + (2 _ 22(' yields the following final strongly non-hnear differentml equation of third order with respect to ( [822( ' - 4 2 ( ( 2 + 1)]~'" + 24).(('(" - 1222((")2 + 3(22cq -~2)(( 2 + 1)(' + 6(2 + 20c2)(~') 2 - 24~.(~')3 -- 30q( 4 - 6 0 ~ 1 ( 2 - 30q = 0
(16)
in which 0~1 - -
2(2c2 - 2 c 3 ) 32
Ct2= 4 ( __22C2 + 222C3 + 1) Assuming that, if S vamshes, H vamshes too, the third of relatmns (2a) leads to 'C(0) = "1S= C 1 = Z O
(17)
~.= - 1/2%
(18)
and because of (9 4) we have
The only unknown constant coefficients of(16) are the integration constants c2 and c3 Since the centroldal axis of the rod is a umform skew-curve, the function ( (x 14= 0) is continuously &fferentlable, thus, we can ask for a solution of(16) under the form (= ~
0c.S"
(19)
n=O
Substituting relation (19) into (16) the coefficients 0tn may be found including, m ad&tmn, the constants of lntegratmn c2 and Ca For the determlnatmn of c2, c3 and of the coefficients 0~n we make the following boundary conditions (i) F r o m (2a), (13) and for S = 0 we have 4(0) :/£2(0)//£
1(0) :
K~(0)/KT~I(0) :
--
P1/P2 = cto
(n) Also, based on the first two equations of (6a), (3), (4) and (5) and for S = 0 the following relations may be derived ~'~(0) = - ( 1 / A ) P 2 x'2(O)=(1/A)PI
x'~(0) = (2P 1/A)zo
(20)
~c'~(O) = (2P2/A)z o
After the definmon of the quantities c2, c3, ~, and consequently of the functmns xl, x2 and z, the force and couple resultants can be readily derwed through (2a), (3) and (4) 2 2 Straight thin rod wzthout m m a l torslon
An interesting special case of the previous general problem is that of a cantilever straight thin rod AB with a kinetic symmetry of its cross-section 11 = 12 and without lmtlal torsion (% = 0) Assume also that the rod is subjected to a co-planar force P = (P cos 0, P sin 0, 0) = (P1, P2, 0) at its free end B, where 0 IS the angle between the tangent and the direction of the force P after the deformation, the curled bar characterlses any quantmes at the free end (origin of the arc measurement) It IS obvious that the following is vahd
Xo=Xo=Z0=0,
x l=z=0
(21)
58
D E PANAYOTOUNAKOSand P S THEOCARIS
Since z o = ~c~=0, the coefficient 1/2 becomes zero, thus, (9 2) is an identity whereas (9 3) leads to the following strongly non-linear differential equation of the third order . . .- . .x2~:2 . 3 , =0 ~:2~:2 + ~:2~:2
(22)
The closed-form solution of the last equatmn has already been given in [6] Making use of the foregoing general solution written under the form
/£2(S)=o92 dn
u
K~= - 0
(23)
where
u=o92(S +e3)/2, dn u = ( 1 - p Z
pZ=(o92_o9~)/o9~< 1,
smZq~)l/2,
o9~ = cl -(c~ + c2) '/~
x2(O)=O K'z(O)=(1/A)P sin 0 ~(0)--- -(1/A)lc2(O)P cos 0
(24)
Based on (23) we have (25 1)
tc~(S)-- - ( o 9 2 p 2 / 2 ) sn u cn u
(25)
x'~(S)=(o92/4)[(2-O2)- 2 x~2o9(7)-l~:2(S)
(25 2)
Through the formula hm ~:~(S)_ ~:"t0) 2t = s-o K2(S) K~(0)
-(1/A)P
cos 0
the boundary conditions (24) can be expressed as (26 1)
dnul~= o = 0
(1/A)P sin 0
-(o92p2/2) sn uis=o cn u[~=o = o922(2 _ p 2 ) =
-(4/A)P
cos 0
(26 2)
(26)
(26 3)
From (26 1) one may conclude that sn ul~=o = _+ 1,
cn uis=o =0,
p~ 1
(27)
By now, through the combination of relation (27) with (26 2) and (26 3), we infer that 0=re,
sn uis=o = - 1
Finally, (26 3) leads to
o922=(4/A)P
(28)
Consequently, based on relation (23), the function u at the free end is derived by u]s= o~ -
f(rU2, re/2) =
-
where F is the complete elliptic integral of the first kind So, one may easily determine the integration constant c3 and therefore the curvature function of the deformed rod It can be noticed here, that Love [1] gave analogous to (23) formula in the elastwa problem (p p 402-405)
Large elastic deformations m thin cantilever rods due to concentrated loadmgs
59
2 3 Skew-curved thin rod In this section the problem of the non-hnear analysis of a skew curved thin rod, subjected to a terminal concentrated force, will be examined, the centroldal axis of the rod is considered as a uniform curve and the principal system of inertia of every cross secUon coincides with the corresponding Fr6net trlhedron before and after the deformation In this case, because of relations KO:KI~0,
/~0 = KO
K2=K
Love's generalized equations (2a), (3) and (4) expressed with respect to the Fr6net trlhedron are given by Gb = Afro--Xo),
(29)
H, = C(z - % )
N~,- ZNb+ xTt= 0
(30 1)
Ng+zN.
=0
(30 2) (30)
T't--xN.
=0
(30 3)
G',, --'CGb+ xH, = Nb
(31 1)
Gg+zG.
= --N.
(31 2) (31)
H;-xG,,
=0
(31 3)
Equation (31 1) combined with (31 2) and relations (29) becomes N.-
C'c /£ (z - - % ) ' - A(x -- Xo)'
(32)
Also, (31 1) combined with (31 3) and relations (29) is written as Nb = C
+ Cx(z-%)-Az(x-xo)
(33)
Finally, based on relations (30 1), (30 2), (32) and (33), equations (30 2) and (30 3) lead to the following strongly non-linear differential equations of the third order + C[~c(z - % ) ] --A[z(K --Xo) ] = C - - (z K
o) + AT(K-- ~Co)'
t~L ~ J
J
+ C[ 1 "cz(K-~Co)]'+ CEz('c-Zo)]'- A[~K~(x --Xo)] = -Cz(z-%)'-A t~(x-Ko)' which after successive dlfferentmtlons can be written as
2z~..( z - % ) ' +
=2
[
. z,
2 z (z_%).+
(3_< \ ~ca2
~c" ~:
2z'x' xz _~z+K 2
(
--(K'--KT0)"-~---(/£--/£0)"-~('~2--K2)(/£--KT0)'q'-'C 2Z' - - -x'z K K
(~-Zo)'+ ~'(~-~o)
)
(X-Xo)
1
(34)
where 2 = A/C Equations (34) constitute a strongly non-hnear system of third order ordinary differential equations with respect to ~:, r that can be integrated numerically by applying Runge-Kutta's
60
D E PANAYOTOUNAKOSand P S THEOCARIS
m e t h o d under suitable mitml conditions But, in some special cases of initial curves for the rod axis the previous system can be solved in a closed-form These cases will be investigated m the following
2 3 1 Skew-curved thin rod deformed to a czrcle We try to d e t e r m m e a cantilever skew-curved rod subjected to a given concentrated terminal force P = ( P . P., Pb) which is d e f o r m e d to a circle of k n o w n curvature K = 1/R after loading In this case it is vahd that K = constant,
z= 0
(35)
So, equations (34) lead to "C~ -~- K2T~) = 0,
(36)
K~ -[- K2K~) = 0
E q u a t i o n s (36) are linear ordinary &fferentlal equations with constant coefficients whose general integrals are respecUvely To(S)~---c 1 s i n KS--}-c 2 c o s KS + c 3
(37) K o t S } = ~ 1 s i n K S q - ~ 2 c o s K-~-~ 3
where c,, }, (t = 1, 2, 3) are constant of integration D e t e r m i n a t i o n of c,, ~, can be achieved t h r o u g h the b o u n d a r y conditions, taking into a c c o u n t that the rod is unlvocally dlfined, since the curvature ~cis k n o w n and it has one fixed end So, based on relations (29) to (33) a n d for S = 0 we have xo(O) = K,
AK~)(0)= - - P . ,
%(o)=o,
~ o(0) = 0,
(38)
K
z ~ ( 0 ) = ~ Pb,
~(o) =
A
Pt
Finally, from (37) a n d (38) the following relations result zo(S)= x~--~(1-cos KS) (39) Ko(S)=c-71~ [P,(1 - - c o s t c S ) - P , sin KS] + K
Since n o w the initial curvature xo(S) a n d torsion zo(S ) of the rod are known, all the internal forces a n d m o m e n t s can be found by using (29), (30) and (31)
2 3 2 Planar curved rod deformed to a hehx W e determine n o w a cantilever planar curved rod subjected to an u n k n o w n concentrated loading P = (Pt, P., Pb), M = (M,, M., Mb) at its free end such that the rod is deformed to a helix of k n o w n curvature x a n d torsion z In this case it is valid that r o = O,
z = k n o w n constant,
K = k n o w n constant
(40)
T h e first of equations (34) leads to -2zx b =0 from which one can result that Ko = constant O n the other hand, the second of equations (34) leads to G - ( z2 - K 2 ) ~ = 0
(41)
Large elastic deformations in thin cantdever rods due to concentrated loadmgs
61
whach, because of the fact that Ko = constant, becomes an identity Thus, it may be concluded that the initial planar curve should be a circle F r o m (31 3) and (32), combined with relations (29), one m a y also result that the loading P and M must be such that the quantatIes G. and N . vanish Also, relations (33), (31 1) combined with (32), (29) and (41) give the following two equations, wtuch must be valid for a generac cross-section of the helax
Nb= C~cz-Az(x--~:o),
T,=-~ Nb
(42)
It should be noticed now, that the ratio Nb/T~= tan 6 is constant, where 6 is the angle of the helix It will be proved that the only loading, satisfying the equations G . = 0 , N . = 0 and (42), is a concentrated force P and a concentrated m o m e n t M, whose vectors are on the axes of the helix Indeed. at as obvious that the effect of the m o m e n t M causes G. = 0, N . = 0 at any cross-section Also, Nb = T, = 0 Based on the equations CK"c = K M t
Az(X--Ko)=zMb x M t - z M b = X M cos 6 - z M san 6=zM tan 6 cos 6 - z M sin 6 = 0 relations (42) yield
Nb=tCMt--'rMb=O,
T
Tt=- Nb=O K
On the other hand, the force P, intersected with the radius p of the cylinder, gives G. = 0 and obviously N . = 0 Finally, because the relations T , = P c o s 6 and N ~ = P s I n 6 are valid, equations (42) are verified In this point, at should be noticed that an analogous result is included in Love's book, where the case was examined to determine the loading conditions of the rod wtuch deforme an initial known helix to another known helix ([-1, pp 4 1 3 ~ 1 4 ] ) The determination of the constant curvature Ko was obtained by using relations (29) So, for S = 0 , we have
A(~C-Ko)=Gb=M sin 6+Pp cos 6 Now, the curvature function ~co results in 1
~o = K - ~ [ m sin 6 + Pp cos 6]
(43)
Finally, the loading acting on the free end of the initial circular rod consists of the four generahsed forces Mcosf+Ppsan6,
Psin6,
Msinf+Ppcos6,
Pcos6
with respect to the axis of t and b respectively
2 3 3 Planar curved rod deformed to a cwcle We determine a cantilever planar-curved rod subjected to an unknown concentrated force P = (P. P., Pb) of known direction at its free end such that the rod to be deformed to a circle of a given curvature ~: = 1/R In this case it is valid that z = z o = 0,
~c= known constant
(44)
The first of equations (34) is an identity, whale the second leads to the differential equation ~c~+ K2~c~= 0
(45)
62
D E PANAYOTOUNAKOSand P S THEOCARIS
The general Integral of this equation is K0(S)=c 1 s i n l£S"]-C2 COS
(46)
I£S"~-C 3
where c, 0 = 1, 2, 3) are integration constants It is obvious that relation (30 1) gives Nb = 0 for any cross-section of the circle So, the component Pb of the force P vamshes and therefore P is a co-planar force Now taking into account that the circle is umvocally defined, since the curvature ~ is known and xt has one fixed end, the angle 0 between the tangent at the fixed end and the direction of the force P is also known The boundary conditions are 1
~Co(0) = x,
~'0(0)= -- ~ P.,
~c
~(0) = ~ Pt
(47)
From the solution of the previous system the unknown curvature x 0 results in x0(S) = ~
[P,(1 - c o s xS) - P . sin xS-] + ~c
(48)
Moreover, for the fixed end of the rod the following relations are vahd Crb= A [x - tCo~)] = P,r. - P.r,
-0 = 0~) = -
f2
1%(S)dS
(49)
where r = (r. r., 0) denotes the known posmon vector from the fixed end to the free end of the circle with respect to the Fr6net trmhedron t, h,/) Now, from the solution of the last system one may determine the components P, and P. as ~:2A(O + xS)(xr, + sin ~ ) Pt =
( x r . + 1 - c o s xsx1 - c o s ~ ) + ( x r , +
sin xSXsm ~ - ~ )
K.2ACO+ ~-S)(tcrnq- 1 - c o s ~ ) P" - (xr. + 1 - cos ~ ) ( 1 - cos xS) + (xr, + sin ~ ) ( s m ~ - ~ )
3
(50)
CONCLUSIONS
In ttus paper relationships yielding the large elastic deformations of a thin incompressible rod subjected to a concentrated general type of load are established F r o m this investigation the following results were derived (1) The transformation of the general non-linear differential system composed of six coupled equations governing the equlhbrlum of a skew-curved element, to a strongly non-hnear system of three equations with respect to the curvature and torsion function (2) A closed-form solution of the three equation system under statable boundary conditions corresponding to a straight thin rod with mltml torsion and kinetic symmetry of its cross-section such that the principal moments of inertia are equal (3) The decouphng of the non-hnear system with respect to the Fr6net trlhedron accompanying a skew-arc element and its closed-form solution m several cases of cantilever curved rods where the curvature and torsion functions before and after deformation, as well as the concentrated loading are considered either as given known in advance, or unknown quantities
REFERENCES E H Love, A Treattse on the Mathematical Theory oJ Elastlclt~ Dozer Pubhcatlons New York (19441 Tlmoshenko and J Gere, Theory ofElastw Stabdlty McGraw-Hill, New York (1961) G Costello, Large deflections ofhehcal spring due to bending J En 9 Mech Dlv 103, 481 (1977) E P a n a y o t o u n a k o s and P S T h e o c a n s N o n h n e a r analysis of cantilever rods due to terminal m o m e n t s lny 4rchtt 49 73 11980) 5 J Kennedy and A S Aggarwal Effect of weight on large deflections of arches J Era! ~4et h DIt. 97 637 (1971) 6 D E P a n a y o t o u n a k o s and P S T h e o c a n s N o n h n e a r elastic analysts of circular beams loaded b~ co-planar concentrated forces Z A M M (in press)
1 2 3 4
A S A D
Large elastic deformations m thin cantilever rods due to concentrated loadmgs
63
7 J A Blanco and G A Costello, Cylindrical constraint of helical springs J appl Mesh 41, 1138 (1974) 8 G B Sinclair, The non-linear bending of a cantilever beam with shear and longitudinal deformations Int J Non-Lmear Mesh 14 I l l (1979) 9 R Schmldt and D A DaDeppo Non-hnear theory of arches and beams with shear deformation J appl Mesh 39, 1144 (1972) 10 A E Green and W Zerna Theoretical Elasttc~t~ Oxford University Press (1954) 1l A E Green and J E Adk~ns, Larqe Ela~tt~ DeJorrnatton~ Clarendon Press, Oxford (1970) 12 H T Davis lntrodu¢tton to Nonhnear Dtflerenttal and Integral Equatton~ Do~er Publications, New York (1970) 13 l S Gradshteyn and I M Ruzhik, Tables o/lnteqrals Ser~e~and Products Academic Press~ New York (19651 R~sum~
Dans le p r e s e n t a r t i c l e on c o n s l d e r e le problEme des grands d e f o r m a t i o n s e l a s t t q u e s d'une p o u t r e mince a c a n t i l e v e r , soumls ~ des f o r c e s concentr~es En t e n a n t compte de la cond l t J o n d ' l n c o m p r e s s i b i l t t ~ de la l i g n e c e n t r a l e t de l ' ~ q u a tton reliant les moments internes avec les courbures et la torsion de la poutre avant et apr~s la deformation, le syst~me non llneatre de l'~qullJbre, qul est compos~ de six ~quatlons dlfferent0~lles couples du premier ordre, est transform~ a un nouveau syst~me d'ordre plus ~l~v~ Les cas des g~ometries des poutres tnttlallement encurv~ et de leur section transversale a ~t~ ~xamln~ Dans ces cas le system d'ordre ~l~v~ peut etre decoupl~ et resolu de facon que nous pouvons fournlr une solution ferm~e On a effectu~ plur0eures appllcatlons de poutres minces et la pulssence de la m~thode est demontr~e
Zusammenfassung
In d l e s e r A r b e l t das Problem yon grossen e l a s t u s c h e n DeformatJonen dunner be~se~tlg g e l a g e r t e n Stabe u n t e r k o n z e n t r t e r t e r Belastungen wurde u n t e r s u c h t Die Inkompessibll0tatsannahme f u r d i e N u l l - L i n i e und d i e g l e l c h u n g e n , d i e innere Momente mlt den Stabkrummungen u n d - t o r s l o n e n , vor und nach der Deformation v e r k n u p f e n , wurden b e t r a c h t e t und das n t c h t l t n e a r e g l e l c h g e w t c h t s s y s t e m , zusammengesetzt aus sechs gekoppelten D J f f e r e n t i a l g l e t c h u n g e n e r s t e r Ordnung, wurde zu elnem neuen System h o h e r e r Ordung transformJert Die geometrten der a n f a n g l , c h gekrunmten Stabe und i h r e Querschmltte, f u r welche d i e glelchungssysteme h o h e r e r Ordnung e n t k o p p e t t und in g e s c h l o s s e n e r Form g e l o s t werden konnten, wurden u n t e r s u c h t Mehrere Anwendungen von d l e s e n gekrummten e l n s e l t l g g e l a g e r t e n Stabe wurden g e b l l d e t und mJt dlesen B e l s p l e l e n wurden d i e MoglJchkelten der Methode g e z e t g t