Multiple three-dimensional equilibrium solutions for cantilever beams loaded by dead tip and uniform distributed loads

Pergamon

lnf. J Non-Linear

Mechanics. Vol. 31, No. 3, pp. 297-311, 1996 Copynghf 0 1996 Elsevm Science Ltd Printed in Great Bntain. All rights reserved OOZO-7462/96 $15.00 + 0.00

0020-7462(95)00070-4

MULTIPLE THREE-DIMENSIONAL EQUILIBRIUM SOLUTIONS FOR CANTILEVER BEAMS LOADED BY DEAD TIP AND UNIFORM DISTRIBUTED LOADS D. W. Raboud, Department

of Mechanical

M. G. Faulkner Engineering, Canada

(Received 8 Junuary

and A. W. Lipsett

University T6G 2G8

1995; in reoised

of Alberta,

form 16

Edmonton,

September

Alberta,

1995)

Abstract-The development of multiple solutions for orthotropic cantilever beams in a fully three-dimensional setting is investigated. The governing equations are solved using an iterative shooting procedure that converts the original boundary value problem into a sequence of initial value problems that converge to the desired solution. This method is well suited to finding multiple equilibrium solutions. Several classes of equilibrium configurations are described and illustrated including planar shapes, buckled planar shapes and fully three-dimensional configurations which appear far removed from the initial plane of loading. The solutions for the planar shapes and the buckled configurations compare favourably to previously published results. The development of the far-removed shapes is shown to be qualitatively similar to that of the planar shapes. The behaviour is shown to be highly dependant upon the aspect ratio of the cross-section. For certain aspect ratios it is shown, somewhat surprisingly, that out-of-plane equilibrium solutions can exist at loads below those required for multiple planar solutions. Copyright c> 1996 Elsevier Science Ltd. Keywords: multiple

solutions,

cantilever

beams, orthotropic

rods

1. INTRODUCTION

Even though Euler had considered the possibilities for various shapes of elastic rods over 250 years ago [ 11, there continues to be considerable activity directed towards understanding the emergence of the multiplicity of solutions. While much of the research has been confined to planar problems [2-51, there has also been discussion of out-of-plane buckling [6]. The present study is an attempt to further explore the possibilities for multiple solutions in a completely three-dimensional context and also to determine how the multiple planar solutions “fit” into the larger picture. For the case of a cantilever beam loaded by tip or uniform distributed loads, Navaee and Elling have detailed the development and geometry of planar multiple equilibrium shapes as the loading is increased [4,5]. It is interesting to note that in their experimental verification of these multiple shapes (which used orthotropic rods), it was difficult to maintain some of the planar shapes as the rod has a tendency to twist out of the plane. This suggests that there exists out-of-plane equilibrium shapes. The problem of the buckling of a deep cantilever beam (i.e. an orthotropic rod) due to tip loading has been considered by Hodges and Peters [6]. In particular, they consider the loads necessary for out-of-plane buckling for rectangular cross-section rods with various width to height (aspect) ratios. This analysis does not provide post-buckled shapes but considers only the loading necessary to sustain a perturbed out-of-plane shape. In the present study the rod is assumed to be able to undergo bending and twisting so that completely three-dimensional shapes are possible. The formulation of the problem is from an initial value viewpoint similar to the method described for planar problems by Lipsett et al. [7]. The rod is considered to have an appropriate number of segments and the solution is determined by solving one segment at a time. Force and geometric compatibility relations are used between the segments so that the complete rod can be analysed. The original problem is then solved by considering a sequence of initial value problems that

Contributed

by K. R. Rajagopal. 297

D. W. Raboud et al.

298

converge to the required boundary conditions. In what follows we will first consider the kinematics of the deformation and the equations of equilibrium. Following a description of the numerical procedure, the problems of a cantilever beam loaded by a tip load and a uniformly distributed load are considered. The solutions obtained are compared with the previous results for multiple solutions of planar rods as well as those obtained for out-of-plane buckling. In addition, the development of several classes of multiple solutions for fully three-dimensional cases is described and illustrated.

2. KINEMATICS,

CONSTITUTIVE

ASSUMPTIONS

AND

EQUILIBRIUM

EQUATIONS

In what follows a rod is defined to be a one-dimensional continuum which deforms through bending and twisting as it is assumed to be inextensible. A reformulation of the theory for such rods in a variational setting has been recently presented in detail by Steigmann and Faulkner [S] and their results will be referred to as required. A configuration of the rod is characterized by a set of coordinates and orthonormal basis {r(s), ei(s)> which define the location and orientation at any point on the rod in terms of the arc length parameter s. The vector e1 is a unit vector in the tangent direction of increasing arc length while e2 and e3 are unit vectors which define the orientation of the cross-section and are embedded in the material. For orthotropic rods, e2 and e3 are in the principal directions of the cross-section. This embedded orthonormal basis will be referred to as the material basis. The material basis differs from the Frenet basis in that the latter depends only on the shape of the centreline of the rod and does not take the orientation of the cross-sections into account. The material basis has the advantage of being uniquely determined even in the case when the rod remains straight. The rate of change of the material basis {ei} with respect to arc length is determined by the vector K (K = Kjej), that is ei =

K X ei

(1)

where the prime indicates differentiation with respect to arc length. The ICYcomponent is the twist per unit length along the rod, while rc2 and x3 are the components of curvature. In the present work, the rod’s undeformed configuration is assumed to be straight and prismatic. The equations of equilibrium are well known [l, 8,9] and are

F’+b=O, M’=Fxe,,

(2)

where F, M and b are the normal force resultant, moment resultant and body force per unit length, respectively. The rods being considered are assumed to have a quadratic strain energy function U, that is U = +[GJrc;

+ EIz&

+ E13&]

(3)

where El2 and EI, are the flexural rigidities about the principal e2 and e3 axes, respectively, and GJ is the torsional rigidity. This in turn implies that the moment can be expressed as

M = GJlclel

+ EI,lc,e,

+ E13K3e3.

(4)

Note that, since non-circular cross-sections will be considered, J is not the usual polar moment of inertia but must rather take into account the contribution due to warping of the cross-section [lo].

3. NUMERICAL

PROCEDURE

Consider a segment of the rod as shown in Fig. 1. Here (Ei} is a fixed global basis in which the rod problem is formulated. For example, if gravity loading is included it may act in the negative E, direction. {ei} is the embedded material basis which changes orientation along the length of the rod segment. At the start of the segment (s = 0) the values of {ei> are {es), which serve as a fixed basis in a particular segment of the rod. The basis {eo} will be referred to as the local basis.

Multiple three-dimensional

299

equilibrium solutions

Fig. 1. Arbitrary rod segment.

The orientation of the material basis in terms of the local basis is most conveniently expressed using Euler angles. For convenience, the set of Euler angles commonly referred to as the yaw ($), pitch (0) and roll ($) angles [11] are used which move the singularity away from the null rotation. In terms of these angles, the components of the material basis (ei} are el e2

=

e3

e? CR1 & , e3”

CeQ’ [R] =

(5)

Ces4

- se

S$SeCb

-

C*S4

S*SeS#

+

C*Cd

CeS*

CJISeC$

+

S*Sb

C*SeS+

-

SsC,#,

Ce CJI

y

(6)

where c and s represent cosine and sine, respectively. Therefore the Euler angles at the start of the segment are {O,O,0} (’i.e. a null rotation). The components of curvature and twist along the segment can then be expressed as rcl = *’ - sin04’, ic2 = cos9sin*#

+ COS+~‘,

rc3 = cos8cosIc/~’ + sin*W.

(7)

In terms of the local basis, the rod segment starts at the local coordinates (x, y, z) = (0,0,O).Since el = r’(s) is the unit tangent vector, the coordinates of the centreline of the rod in the local basis satisfy x’ = cos 8 cos f#&

y’ = cos 8 sin 4,

z’ = - sin 8.

(8)

The force vector F at the start of the segment can be expressed as F = F?e? I I

(9)

where the F? are the segment’s initial tension and shear components. For a dead uniform distributed load b = bi$ (bi are constants) acting on the rod, (2)r can be integrated to give F(s) = (FO - bis)eo.

(10)

However, to obtain the true tension and shear components, F needs to be expressed in the material basis so that F= (11) where

300

D. W. Raboud

Combining

et al.

(4) with (2)2 gives

which is combined

El,li,e;

+

F x e, = GJrcle; + El,lc2e;

+ GJK-;el -i- EI,K;e2

with (l), (7), and the dimensionless

p2!!

(13)

parameters

FiL2 Vi=F>

Ri = rCiL,

+ E131cje3,

biL3 Ili’m>

(14)

El ’

where EZ (no subscript) is the larger of El2 or E13 and L is the length of an individual segment. The result is a system of three second-order differential equations for the three Euler angles c’,Ac,

d;=

+ ~,I$, ’

AIce

(15) where

e’1=(R -

A)&&

r(SocJ,J8 + CfjS~&h + c&j)- v2,

+

c2 =(r -a)k17& +&S&d c,

=

(A -

and where the superposed (12) become

-C&~~~ + S&b)+

Vj,

r)eziz3 +ck(&, dot indicates

(16)

differentiation

with respect to p. Similarly,

(7) and

Iz, =lj-s,& R2 = C&i

+ c,e,

123

-

=

C&cj

s,d,

(17)

and [VI; jV21

=

[R]

(18)

iV3!

while (8) becomes

simply

Equations (14H19) completely describe the deformation of the rod segment in terms of the initial conditions. The forces, moments and geometry (position and orientation) at the end of the segment can be determined by the forces, moments and geometry at the start of the segment by direct numerical integration. The coordinates of material positions obtained along the rod are related to the global basis (Ei} to allow the specific position and orientation of the rod to be determined. Because it is significantly more computationally efficient when high precision is required, the Bulirsch-Stoer method, rather than the more common Runge-Kutta methods, is used to perform the integration [12]. The discussion above provides a method of solution for an individual segment of the rod. To solve the entire rod, which can be composed of a number of segments, the individual segments are assembled such that force and geometric compatibility are maintained. This is accomplished by using the values for the force and geometry obtained at the end of one segment as the starting values for the next. For example, the material basis {ei} at the end of the kth segment becomes the local basis (eo} for the kth + 1 segment. This procedure is continued from segment to segment until a solution is obtained for the entire rod. In this

Multiple

three-dimensional

equilibrium

301

solutions

way, the forces, moments and geometry at the end of the rod are determined by the forces, moments and geometry at the start of the rod. While it is possible to consider many rod problems using only one segment, it is often advantageous to use several segments. In cases of complex loading, changing material properties or complex initial geometry, the segmented rod is much easier to implement. In addition, and more importantly for the present work, the introduction of several segments along the rod allows the ability to reset all the Euler angles to zero and avoid the numerical problems caused by the singularity which occurs at 8 = i n/2. The solution as presented is an initial value approach in that the conditions must be completely specified at one end of the rod to obtain a solution. Most rod problems are actually two-point boundary value problems where some boundary conditions are known at each end of the rod. An iterative shooting procedure, similar to that described in detail for planar problems [7], is employed to solve these problems. This procedure requires that 12 initial conditions (force and moment components, position coordinates and Euler angles to specify orientation) be specified at the start of the rod. In general, not all of these will be known a priori . Some of these values will be unknown, but a corresponding number of conditions will be known at the other end of the rod. The unknowns must be initially estimated to start the numerical procedure. From these initial conditions the solution throughout the rod is obtained using the method described. However, as the conditions at the end of the rod will not in general agree with the corresponding known conditions at that end, a Newton-Raphson false position method is then used to iteratively improve the estimates of the unknown initial values until the required boundary conditions at the end of the rod are satisfied to within some specified tolerance. A more general implementation of this method, which encompasses initially curved rods, has been used previously for several problems including an initially straight circular rod bent and twisted into a helix, an initially straight rectangular rod deformed into a Mobius strip, an initially curved cantilever beam deformed out of the plane by dead tip and distributed loads as well as fully threedimensional deformations of an orthodontic retraction appliance with a complex initial geometry [13]. As discussed in ref. [13], there are several checks available which the numerical solutions obtained must satisfy. A generalization of the classical energy integral [l] is available which must remain constant as well as the necessary condition that the determined shape must satisfy static equilibrium. Both of these conditions were consistently satisfied for all of the numerical results presented in the following sections.

4.

MULTIPLE

SOLUTIONS

OF

AN ORTHOTROPIC

CANTILEVER

The problem considered is the deformation of a cantilever beam due to a dead tip load as illustrated in Fig. 2. The cross-section is rectangular with an aspect ratio a/b. In the undeformed configuration the longitudinal axis of the beam is aligned with the global El axis while the principal directions of the cross-section are in the E, and E, directions. The load P is applied in the negative E2 direction. If the shooting procedure is begun at the fixed end, all the initial values of the forces, moments and geometry are known except the moment components in the El and E3 directions (M, and MJ, respectively). (The corresponding known conditions at the free end of the beam are that the moment components are all zero.) If M, equals zero the deformation of the centreline remains in the El-E2 plane. For the case of planar deformations, it has been previously shown [2,4,5] that for a tip load the number of equilibrium shapes is a function of the load parameter

where P is the load magnitude and L is the length of the rod. The number of equilibrium shapes for various ranges of c[ is given in Table 1. When multiple shapes first develop (i.e. at M = 3.214) one additional shape is possible but this shape immediately splits into two as the load is increased above the bifurcation point. These bifurcation values and corresponding shapes can be reproduced using the above technique by setting Ml = 0 at the fixed end of

302

D. W. Raboud

et al.

1

2b i

Fig. 2. Cantilever

Table 1. Number

beam with dead tip load, P.

of planar equilibrium solutions of the load parameter a

Load parameter

as a function

Number of equilibrium configurations

0 < G(< 3.214 a = 3.214 3.214
1 2 3 4 5

1.0

-

(I=

I-pL2

I

E4

I

=5.0

I :II

3 o.5 s

2

z 40.0 &

------

---

_--

3

z U k -0.5

1

-LO]

IIIIIIIII,II/J111,,

-1.0

~,,,,,,,,,,,,,,,,,,,

-0.5

0.0

X Coordinate

Fig. 3. Planar

equilibrium

solutions

corresponding

0.5

1 .o

(n-r)

to a load G(= 5.

the beam to ensure planar deformations. Figure 3 shows the three resulting planar shapes for a l-m long beam corresponding to a load c1= 5 using the present technique. Note that these computed shapes show excellent agreement with previously published results. Hodges and Peters [6] used a perturbation approach and presented numerical results for the loads necessary to allow infinitesimal out-of-plane deformations for deep cantilevers as a function of the aspect ratio (a/b) of the cross-section. (Note that for the remainder of this

Multiple

three-dimensional

equilibrium

Tip Load Distributed

+ \

303

solutions

Load

0.8

g b

0.6

? Qz

0.4

0.0

0.2

0.4

Aspect Fig. 4. Dimensionless

buckling

loads

0.6

Ratio

0.8

1

(a/b)

as a function of aspect distributed loads.

ratio

for dead

tip and

uniform

work, this transition from a planar equilibrium state to an infinitesimally nearby out-ofplane solution will be referred to as buckling.) Using the present technique with M1 # 0 allows the determination of the buckling loads shown in Fig. 4 where the non-dimensional loading parameter is given by

P=

@I&i.

(21)

These results show excellent agreement with those of Hodges and Peters [6]. In particular, when the curve is extrapolated to an aspect ratio of zero, the dimensionless buckling load approaches /I,, = 4.0126. (22) As well, when the curve is extrapolated to the horizontal axis it is also seen that the load increases without limit as the aspect ratio approaches 0.607 indicating that at higher aspect ratios the rod does not have a perturbed out of the plane solution. Figure 4 also shows the dimensionless loads which provide nearby out-of-plane equilibrium shapes if the tip load is replaced with a uniformly distributed dead load acting in the same plane. Again, as the aspect ratio approaches zero, the loading parameter approaches y0 = 12.85

(23)

where (24) and wYis the magnitude of the distributed load per unit length. It can also be seen that the load rises indefinitely as the aspect ratio approaches 0.603. Hodges and Peters [6] did not consider buckling due to uniformly distributed loads. The results shown in Fig. 4 for the distributed load case were verified using an aluminium cantilever beam with an aspect ratio of 0.260 loaded under its own weight. By changing the length, it was found experimentally that the beam buckled out of the plane between a length of 3.38 and 3.40 m. This agrees closely with the predicted length of 3.398 m. Figure 4 illustrates that, relative to PO and yo, the buckling loads for the tip load and uniformly distributed loads at any aspect ratio are very close, but not identical, to each other. The results of Hodges and Peters [6], as well as those above in Fig. 4, indicate only the loads at which out-of-plane deformations begin through the buckling of a planar equilibrium shape. However, as has been previously discussed, more than one planar equilibrium

304

D. W. Raboud

et al.

Multiple Planar Solutions Begin Here (8=30.25)

0

20

10

6C

80

1 (10

P Fig. 5. MI components

required

to maintain

buckled equilibrium of li1.7.

configurations

for an aspect ratio

shape can exist for certain loads. The question naturally arises whether these other planar equilibrium shapes may also buckle out of the plane in a manner similar to the buckling behaviour discussed so far. To illustrate, consider a cantilever with a tip load and an aspect ratio of l/1.7. For this aspect ratio out-of-plane buckling occurs for PO/b = 0.248 or p = 16.18 (from Fig. 4) and increasing the load beyond this value results in the moment M1 increasing from zero as shown in Fig. 5 as shape 1. The twisting moment M, required to maintain the equilibrium configuration continues to increase as fi is increased, implying that the free end is moving further from the original plane of the rod. Also indicated is the point at which multiple planar equilibrium configurations begin to occur. For this case, this occurs at fi = 30.25 (which is equivalent to CI= 3.214 in Table 1) so that the out-of-plane buckled deformations are possible at lower loads than are the multiple planar solutions. For loads above B = 30.25 it was found that each of the new planar solutions can also buckle out of the plane. These are indicated in Fig. 5 which shows that above ,G = 59.76 (shape 2, Fig. 3) and p = 42.02 (shape 3, Fig. 3) the other two planar solutions exhibit buckled equilibrium configurations similar to the behaviour described above. As the load is further increased above these values, the moment component M1 necessary to maintain equilibrium also increases. To appreciate the deformed shapes of these rods, Fig. 6 illustrates the deformed shape of a l-m long rod with a load p = 64 for shape 2 while Fig. 7 shows shape 3 at a load of p = 44. In both these figures the deformed shapes are observed looking down the El axis (in the positive E, direction). Also, the cross-sections here, and in all such following figures, have been expanded by a factor of five for clarity. All of the above analysis only considers the development of out-of-plane shapes as the buckling of planar ones. A larger question is whether or not other three-dimensional equilibrium shapes exist which do not develop as perturbed planar ones. In terms of the moment component M 1,this may mean that its value is considerably above zero. Using the shooting procedure described above, larger values of Ml can be input to search for these shapes. Since this involves a two-parameter shooting problem (A4, and M3 are unknowns), the actual technique used was a more systematic approach involving the use of contour maps as described by Lipsett et al. 131. Figure 8 shows non-zero M1 values which result in equilibrium being satisfied. (The numbers shown beside each shape will be discussed later.) This figure indicates that the development of these new shapes is qualitatively similar to the development of multiple equilibrium shapes in the plane. At the critical load where B = 18.87, shown as point 1 in Fig. 8, a new equilibrium shape is possible (far removed from the planar solution). As the load is increased, this shape bifurcates into two others (shape

Multiple

three-dimensional

equilibrium

solutions

Fig. 6. Geometry

of buckled

shape 2, aspect

ratio = 1/1.7, B = 64.

Fig. 7. Geometry

of buckled

shape 3, aspect

ratio = 111.7, B = 44.

305

A and shape B) which represent distinct equilibrium configurations. It is somewhat surprising that this occurs at loads below those at which multiple planar solutions occur (B = 30.25) and yet the shapes taken by these equilibrium deformations look similar to these planar shapes. These new shapes occur at loads only slightly above those necessary for the perturbed buckling described previously. As an example, Fig. 9 shows the three views of the centreline of shape B at a load of p = 2.5 as well as an oblique view showing the effect of the twist. In the Er-EZ plane this shape appears similar to planar shape 2 shown in Fig. 3, even though there is a considerable twist and relatively large out-of-plane movement of the tip of the beam. Figure 10 shows the same oblique view for shape A at the same load j = 25. These shapes are fundamentally different from those which develop as buckled

et al.

D. W. Raboud

306

Shape A (2.14) Shape E (18.03) Shape C (10.65)

Shape F (18.94)

Shape B (6.64) Shape #l (-21.46) Shape #3 (13.13) Shape D (17.21)

/

I

Shape #2 (20.17)

O~,,,i,,,,,,‘,,,,,,,,,!,,,,,,,,, Fig. 8. MI components required and the associated

I

to maintain equilibrium configurations far removed from the plane potential energies (Nm) for an aspect ratio of l/1.7.

planar shapes as shown in Fig. 5 as well as in Fig. 8 as the dashed curves. These are not buckled shapes as they do not develop from planar shapes (i.e. no equilibrium path exists from a planar shape). As well as the two branches which develop from the point I, there are additional shapes which develop from points II and III shown in Fig. 8. These are again distinct equilibrium configurations. Where the branches seem to cross each other and the Ml components are equal, the M3 components of the moment (not shown, but necessary to maintain equilibrium) are different so that the shapes are all distinct. Note that point II (/3 = 25.93) also occurs at a load below that at which multiple planar equilibrium shapes exist. It is also of interest to consider the potential energies associated with the various shapes considered. Since a stable equilibrium state is a minimizer of the potential energy function, a look at the potential energies allows a global comparison of the shapes obtained even though we do not have any indication of the local stabilities of these configurations. A more complete stability analysis would require the generation of all possible kinematically admissible displacement fields which satisfy the required boundary conditions to ensure the energy is actually a minimum. This is difficult to do in the present context of using this initial value numerical solution. The potential energy E is defined as E (K, f) = S(K) - P(r)

(25)

where

s L

s(K)

=

U(K) ds

0

is the strain

energy stored along the rod [with

U(K)

P(r) = P- r(L)

given by equation

(3)] and (27)

is the load potential associated with the dead tip load P. Table 2 shows the potential energies for the various shapes considered in Fig. 8 for a steel rod (E = 200 GPa, v = 0.3) l-m long with a 2.0 x 3.4 mm rectangular cross-section at a load of /I = 64. (These values are also shown in Fig. 8.) At this load three planar shapes exist, for which the Ml component is zero, and each of these will have an associated buckled configuration as well (the dashed curves in Fig. 8). Shape 1 has the lowest absolute potential energy. The potential energies of the buckled shapes l-3 are all lower than those of their associated

Multiple three-dimensional

equilibrium solutions

307

planar shapes, although there is only a small difference in the energies. Of the shapes far removed from the plane, the branches AB from point I, which starts at the lowest load, has the lowest energy followed by branches CD from point II and EF from point III. It can also be seen that the top portions of these branches, namely shapes A, C and E, which are further removed from the plane, consistently have potential energies lower than the bottom portions of the corresponding branches. Also note that branches AB have potential energies substantially lower than either of the planar shapes 2 and 3 or their buckled configurations while the other branches have energies which are comparable to these shapes. It should be noted again that a solution with a low energy is not necessarily stable, and one with a high energy unstable, with respect to small perturbations. To this point only rods with aspect ratios less than unity (i.e. deep cantilevers) have been considered. When the aspect ratio is greater than unity, a similar behaviour is seen to occur.

-0.5

1, -0.4

I

I

I,

I

-0.3

I

I,,

I

-0.2

I

/,

,,,,

-0.1

,I / / -0.0

I,,,

/I

0.1

,(

0.2

,I,,

,

,,

,

,

,

,

,

,

/

,

,

,/

0.4

0.5

0

,,",,,,,,"rrl,','~,,,,,,",,,,,,,,,,,,,,,,,,,,, -0.4 -0.3 -0.2 -0.1 -0.0 0.1 0.2 0.3 0.4

0.5

c

X Coordinate

(aI

0.5

0.3

(m)

, J

0.4 -

I

/ I

0.3 3

0.2 4

s

0.1

-

z ;;3

0.0

Lr7-__

z 0

-0.1

-

-0.2

-

-0.3

-

-0.4

-

v N

1 / I 1

I I /

/

/ I -0.5

@I

X Coordinate Fig. 9(a) and (b).

(m)

308

D. W. Raboud

c.5

+r~-Trrrrm-~-,

-0

(c)

:, -0

4 -0

I)

I

3 -0.2

Z

-0.1

I

0.0

Coordinate

I,,

0.1

I

I,,

/

0 2

(m)

I,,

0.3

et al.

I

I

0.4

-

c.5

(4

Fig. 9. Geometry of shape B, aspect ratio = l/1.7, b = 25. (a) Centreline in E,-EZ (b) Centreline in E,-E, plane. (c) Centreline in Ej-Ez plane. (d) Oblique view of deformed

plane. shape.

PC Fig. 10. Oblique

view of shape A, aspect ratio = 111.7, p = 25.

Figure 11 shows the results, similar to Fig. 8, for a rod with an aspect ratio of 1.7. As can be seen there are again numerous equilibrium configurations possible. One major difference between these two figures is that for the deep cantilever there are no longer out-of-plane shapes which develop as buckled planar ones (the dashed curves in Fig. 8). This is expected for shape 1 since it was shown that this planar shape would not buckle above an aspect ratio of 0.607. It was also found that the planar shapes 2 and 3 would not buckle above aspect ratios of approximately 0.7 and 0.6, respectively. Therefore the only shapes indicated in Fig. 11 are those which are far removed from the planar solutions. As an example of a deformed configuration, Fig. 12 shows an oblique view of the deformed shape corresponding to branch A in Fig. 11 at a load of /I = 20. Figure 13 shows

--3 E

Multiple

three-dimensional

equilibrium

solutions

309

Table 2. Potential energies of various equilibrium configurations a deep cantilever (aspect ratio = l/1.7) at a load p = 64 Shape Planar

Potential (MI = 0)

Buckled

Far removed from the plane

energy

Shape 1 Shape 2 Shape 3

21.45 20.18 13.14

Shape 1 Shape 2 Shape 3

_ 21.46

Shape Shape Shape Shape Shape Shape

for

(N m)

20.17 13.13 2.14 6.64 10.65 17.21 18.03 18.94

A B C D E F

Shape A (8.38)

/ Shape C (17.54)

Shape E (28.87)

Shape D (20.17) Shape F (33.72) Shape B (14.71)

ol,,,,,,,,,,,,,,,,,,,,,,,,, 0 10 20

I,,,,,,,,,,,,] 30

40

50

60

70

80

P Fig. 11. MI components required to maintain equilibrium configurations far removed plane and the associated potential energies (Nm) for an aspect ratio of 1.7.

from the

the same view for the same rod loaded as a deep cantilever with the same absolute load (corresponding in this case to ,!I = 34). Note that the two shapes are somewhat similar for the same load despite being shallow and deep cantilevers, respectively. The potential energies for the various equilibrium configurations are also shown in Fig. 11, as well as in Table 3, corresponding to a load /I = 45. Note that this corresponds to a load of a = 8.688 [from (20)] so that five planar shapes will exist as shown in Table 1. The usual three planar shapes considered thus far (similar to those of Fig. 3) have the lowest potential energies, with shape 1 having by far the lowest. As mentioned previously there are no associated buckled configurations for these shapes at this aspect ratio. Of the shapes far removed from the plane, branches AB again have the lowest energies followed by branches CD and EF. All these shapes have potential energies significantly higher than the three planar ones. This is in contrast to the deep cantilever case shown in Fig. 8, where some of these shapes had lower potential energies than several of the planar shapes. As well, the two new planar shapes 4 and 5 (not shown) which exist at this higher load also have much higher potential energies than the other planar shapes.

310

D. W. Raboud

Fig. 12. Oblique

Fig. 13. Oblique

view of geometry

view of geometry

et al.

of shape A, aspect ratio = 1.7, b = 20.

of shape B, aspect ratio = l/1.7, /I = 34.

5. CONCLUDING

REMARKS

The existence of several classes of multiple equilibrium configurations for cantilever beams under dead tip loads in a fully three-dimensional setting was established. The most well known of these are the multiple planar shapes which have been detailed previously [2,4, 51. Another well-known class is the configurations which result as planar shapes buckle out of the plane. While this has been investigated previously for one particular planar shape [6] it has been shown here that other pIanar shapes can also buckle out of the

Multiple Table

three-dimensional

equilibrium

3. Potential energies of various equilibrium configurations a shallow cantilever (aspect ratio = 1.7) at a load /? = 45 Potential

Shape

for

energy (N m)

(M, = 0)

Shape Shape Shape Shape Shape

1 2 3 4 5

_ 29.60 1.58 - 1.32 26.14 23.08

Far removed from the plane

Shape Shape Shape Shape Shape Shape

A B C D E F

8.38 14.71 17.54 20.17 28.87 33.12

Planar

311

solutions

plane. The third class described here are the fully three-dimensional shapes which appear far removed from the plane. At some critical load one of these shapes appears and then immediately bifurcates into two shapes as the load is increased in a manner similar to the development of multiple planar shapes. Surprisingly, it was found that these new shapes could exist at loads below those required for multiple planar solutions. Further, it is not known whether the out-of-plane shapes suggested in Figs 8 and 11 are exhaustive, but the possibility of numerous equilibrium configurations is certainly shown. As a final note, it should be stressed that the results presented are highly dependent upon the aspect ratio used. This is especially true of the relative magnitudes of the loads required for the various shapes to occur. For example, while some of the shapes far removed from the plane were found to exist at loads below those required for multiple planar solutions, this was for a specific aspect ratio and will not be true for all aspect ratios. The aspect ratios were chosen to illustrate some of the different types of behaviour possible.

REFERENCES 1. A. E. H. Love, A Treatise on the Mathematical Theory of Elasticity. Dover, New York (1944). 2. M. G. Faulkner, A. W. Lipsett and V. Tam, On the use of a segmental shooting technique for multiple solutions of planar elastica problems. Comput. Meth. Appl. Mech. Engng 110, 221-236 (1993). 3. A. W. Lipsett, M. G. Faulkner and V. Tam, Multiple solutions for inextensible arches. Trans. CSME 17,1&15 (1993). 4. S. Navaee and R. E. Elling, Equilibrium configurations of cantilever beams subjected to inclined end loads. ASME .I. Appl. Mech. 59, 512-579 (1992). 5. S. Navaee and R. E. Elling, Large deflections of cantilever beams. Trans. CSME 15, 91-107 (1991). 6. D. H. Hodges and D. A. Peters, On the lateral buckling of uniform slender cantilever beams. Int. J. Solids Struct. 11, 1269-1280 (1975). I. A. W. Lipsett, M. G. Faulkner and K. El-Rayes, Large deformation analysis of orthodontic appliances. ASME J. Biomech. Engng 112, 29-37 (1990). 8. D. J. Steigmann and M. G. Faulkner, Variational theory for spatial rods. J. Elast. 33, l-26 (1993). 9. L. D. Landau and E. M. Lifshitz, Theory ofElasticity, 3rd Edn. Pergamon Press, Oxford (1986). 10. E. H. Dill, Kirchoff’s theory of rods. Arch. Hist. Exact Sci. 44, l-23 (1992). 11. H. Goldstein, Classical Mechanics, 2nd Edn. Addison-Wesley, MA (1980). 12. W. H. Press, S. A. Teukolsky, W. T. Vetterling and B. P. Flannery, Numerical Receipes, 2nd Edn. Cambridge University Press, Cambridge (1992). and A. W. Lipsett, A segmental approach for large three dimensional rod 13. D. Raboud, M. G. Faulkner deformations. Int. J. Solids Struct. 33, 1131-1156 (1996).