equilibrium configurations of cantilever columns under · pdf fileequilibrium configurations...

Canadian Journal of Basic and Applied Sciences

©PEARL publication, 2014

CJBAS Vol. 02(02), 46-63, 2014

ISSN 2292-3381

Equilibrium configurations of cantilever columns under a tip-concentrated

subtangential follower force

M. Mutyalarao a, D. Bharathi a, B. Nageswara Rao b

a Department of Engineering Mathematics, College of Engineering, Andhra University, Visakhapatnam - 530003,

India b Department of Mechanical Engineering, School of Civil and Mechanical Sciences, KL University, Green Fields,

Vaddeswaram – 522 502, India

Keywords: Abstract

Cantilever column,

Large-deflections,

Tip-concentrated load,

Tip-angle,

Subtangential,

parameter

Studies are made on the post-buckling behavior of a cantilever column subjected to a

tip-concentrated subtangential follower force. The formulation of the problem results

in non-linear ordinary differential equations amenable to numerical integration. A

relation is obtained for the applied load in terms of the subtangential parameter and

the tip-angle of the column. Also identified the range of the load parameter for which

one can get multiple solutions.

1. Introduction

One of the interesting and oldest problems of elastomechanics is the determination of

equilibrium shapes of a cantilever beam under a tip-concentrated load. Assuming that the curvature

of the beam is proportional to the bending moment, Euler provided a differential equation with an

infinite series solution for the deformed shape of the beam and also classified all equilibrium states

(see Antman [1]). The problem of nonlinear bending of cantilever beams has been studied by many

researchers [2-9].

Slender structural components, such as beams and columns, constitute basic parts of many

structures. For example, slender cantilever columns are used extensively as struts, carrying

compressive loads. In the analysis of such structures, nonlinearities arise due to large deformations

and material properties. The determination of elastic curve of a column is essential since it is often

required that not only stresses induced in the column should not exceed the allowable stress but also

the maximum deflection of the column should not be greater than a certain predetermined value

depending on the operating conditions of the column. Study of such structures under the action of

non-conservative forces is of particular importance in modern engineering practice.

Corresponding Author:

E-mail, [email protected] – Tel, (+91) 86456948 – Fax, (+91) 8645247249

mailto:[email protected]

M. Mutyalarao et al. - Can. J. Basic Appl. Sci. Vol. 02(02), 46-63, 2014

47

Timoshenko and Gere [10] have studied the Euler’s column and solved the problem in terms of

elliptical integrals. Chen [11] has proposed a new integral approach for large deflection cantilever

columns. Application of the Euler method in these studies is used mainly due to the external forces

which are acting on the body considered to be conservative. Bolotin [12] gives the extension of the

Euler method to the problem of the stability of a column under the action of a compressive force

(follower force) which after deformation rotates together with the tip of the column and at all times

remains tangential to its deformed axis. The stability of nonconservative systems has extensively

studied [13-24].

The presence of nonconservative loads makes the linearization of equation system

mathematically non-self-adjoint and the corresponding eigenvalue problem is ruled by a non-

symmetric matrix and can exhibit complex eigenvalues. Langthjem and Sugiyama [25] and

Elishakoff [26] have made excellent surveys on the static and dynamic stabilities of systems loaded

with follower forces. Finite element analysis has been carried out to examine the nonlinear stability

of Beck’s columns [27-31]. It has been known since the time of Euler that flexible beams can

assume multiple equilibrium solutions under a given load. The existence and emergence of these

solutions has received much attention in the literature [32-36]. An equally interesting question

concerns the stability of these multiple configurations.

The post-buckling analysis of a uniform cantilever column is performed under a tip-

concentrated subtangential follower force. A relation is obtained for the applied load in terms of the

subtangential parameter and the tip-angle of the column. There is a limitation on the load rotation

parameter in obtaining solution of the equations arrived from the static stability criterion. The

limitation can be overcome adopting the dynamic stability criterion. The deformed configurations

of the column are obtained by solving directly the resulting non-linear differential equations

through fourth-order Runge-Kutta integration scheme. The range of load parameter is arrived at

which multiple equilibrium configurations of the column exist.

Figure 1. A uniform cantilever column under a tip-concentrated subtangential follower load


48

2. Theoretical Formulation

The formulation of the problem is mainly based on an important relation of the flexural theory

(i.e.,ds

d

EI

M

1), the quantity

1 (the curvature of the deflected axis of the column)

characterizes the magnitude of bending deformation, which is proportional to bending moment, M

and inversely proportional to the product EI, called flexural rigidity of the column. Figure-1

represents a follower load P not always be tangent, but can be subtangential depending on a

parameter . Here 0 represents Euler’s column (unidirectional, vertical, tip-concentrated load

P) and 1 represents Beck’s column (tip-concentrated tangential load P). The differential

equation of the deflection curve is formulated here for the angle between the tangent to the bent

axis and the vertical, as a function of the length s of the curve measured from the tip of the column

as shown in Figure 1.

The moment-curvature relationship of a uniform cantilever column subjected to a tip-

concentrated subtangential follower load (P) is as follows [14, 17, and 18]

)()0(sin)()0(cos aa XXPYYPds

dEI

(1)

where

L

s

dsX )(cos)( (2)

L

s

dsY )(sin)( (3)

Here E is the Young’s modulus; I is the moment of inertia; L is the length of the column; is a

dummy variable; )0( is the tip-angle of the column; and is the subtangential parameter. At s

= 0, equations (2) and (3) give tip coordinates ( aa YX , ) of the column.

Differentiating equations (1) to (3) with respect to s, the following system of equations are

obtained:

0)0(sin2

2

Pds

dEI (4)

cosds

dX (5)

sinds

dY (6)

Boundary conditions for the differential equations (4) to (6) are:

At the tip of the column (s=0)


49

0ds

d (7)

At the root of the column (s=L)

0 , X=0, Y=0 (8)

The solution of equations (4) to (8) is obtained in terms of elliptic integrals for one equilibrium

deformed configuration of the column [17, 18]. In this paper, a direct procedure is followed to

obtain many equilibrium solutions. The details of which are presented below.

Equations (4) to (8) form a two-point boundary value problem. DefiningEI

PL2

;

L

s ;

L

Xx ; and

L

Yy ; equations (4) to (8) are written in non-dimensional form as

0)0(sin2

2

d

d (9)

0cos d

dx (10)

0sin d

dy (11)

0

d

d at 0 (12)

0 yx at 1 (13)

Here

L

Y

L

Xyx aa

aa ,),( represents the tip-coordinates of the column in the non-dimensional

form.

Multiplying equation (9) by 2

d

d and integrating, one can obtain

)}0()1cos{()}0(cos{2

2

d

d (14)

This satisfies )0( at 0 . Now writing 22 sin21)0(cos k and

221)0()1(cos k , equation (14) is transformed to

)sin1( 22

2

k

d

d

(15)

The boundary conditions for equation (15) are:

nn

n

2)12(11sin

11

1

at 0 (16)

2

)0(sin

1sin 1

0

k at 1 (17)


50

where

2

)0()1(sin

k . (18)

It should be noted that the condition (16) in Ref. [17] is for n =1(i.e., 1 =2

), which

corresponds to the first equilibrium configuration of the column. Integrating equations (15) to (17),

one can obtain a relation between the load parameter ( ) and the tip-angle ( )0( ) with

subtangential parameter ( ) as

2

22

1

0sin1

k

d (19)

By applying )0( 0 in equation (17) one can obtain a sequence of load parameters

1

),(n

n as

2

11

1sin

2121),(

nn

nfor

2

10 (20)

It should be noted that

1sin 1 does not exist for 1

2

1 and hence no static load is

possible to evaluate for the column [14, 17, 18]. In the sequence of load parameters, )1,( is the

linear critical load parameter for the specified subtangential parameter ( ). It is interesting to note

from equation (20) that the linear critical load parameter )1,( decreases with negative values of

. This is mainly due to the fact that the horizontal component of the force aids the bending of the

column. For2

10 , the horizontal component acts as a force opposing bending. As the

parameter increases upto the value of2

1 , the critical load parameter increases and finally at

2

1 attains the value 2 . Finite element analysis (FEA) results [27, 29, and 31] are found to be in

good agreement with the critical load parameter, )1,( given in equation (20). For2

1 , there is

no force able to maintain equilibrium of the column in its deflected position. To determine the

critical loads )1,( in this range, one has to follow the dynamic stability criterion in which straight

equilibrium configuration is the only equilibrium configuration [17]. Only one equilibrium

configuration is possible for the specified load parameter, )2,( . One can expect m ( 2 )

equilibrium configurations for the specified load parameter,


51

)]1,(),,([ mm for 2

10 (21)

)]12,(),12,([ mm for 2

1 (22)

However for2

1 , multiple equilibrium configurations are possible for the specified load

parameter ( ). It should be noted that for 1k , equation (19) gives the load parameter,

2

00

11

tansec

tansecln

(23)

which is undefined for all the values of 1 in equation (16). Using 1k in equation (18) one can

find the tip-angle,

)1()0(

(24)

Figure 2. Variation of tip-angle, )0( with the subtangential parameter ( ) at which the load parameter ( )

is undefined.

The load parameter (λ) is undefined for the specified tip-angle, )0( of equation (24) and the

solution of the problem becomes singular. Figure-2 shows the variation of )0( with the applicable

range of the subtangential parameter ( ) at which the load parameter is undefined. For the

specified subtangential parameter ( ) and tip-angle ( )0( ), load parameter ( ) can be evaluated

from equation (19) through numerical integration. Gauss-Legendre quadrature scheme available in

MATLAB is utilized for evaluation of the integral in equation (19). For the tip-angle, )0( and the

corresponding load parameter ( ), equation (9) is written into two first-order differential equations

and integrated from 0 to 1 utilizing the fourth-order Runge-Kutta method and obtained the

unknown

d

d at 1 . Using the determined value of

d

d at =1 as well as other conditions in

0 0.1 0.2 0.3 0.4 0.5180

200

220

240

260

280

300

320

340

360

rotation parameter ()

tip

-an

gle

,

(0)

(deg

)


52

equation (13): 0 yx at =1, the fourth-order non-linear differential equations (9) to (11)

were solved from 1 to 0 using the fourth-order Runge-Kutta method and obtained the

deformed configuration of the column. A uniform step size ( ) of 0.001 is considered to obtain

the numerical solution for the non-linear ordinary differential equations. The deformed shape of the

column including its tip-coordinates ( aa yx , ) are obtained at the end of the integration.

Replacing the constant angle ‘ ’ by ( 1 ) )0( in the mathematical formulation [37], one can

obtain resulting differential equation (4). By specifying the value of equivalent to )0( ( 1 ),

it is not possible to obtain the linear critical load parameter for the applicable range of the load

rotation parameter, <2

1. Because, specifying the tip-angle, )0( close to zero, the constant angle,

in Ref. [37], becomes zero for all the load rotation parameters ( ).

3. Numerical Results

Post-buckling behavior of a uniform cantilever column subjected to a tip-concentrated

subtangential load is examined. Many load parameters ( ) are obtained from equation (19) varying

n (= 1, 2, 3… etc.) for the specified tip-angle ( )0( ) and the subtangential parameter )2/1,0( .

Figure 3 and 4 show the variation of the load parameter ( ) with the tip-angle, )0( for the

subtangential parameter, β = 0.0 and 0.5. The asymptotic nature of the load parameter ( ) close to

the values of the tip-angle, )0( of Figure 2 can be seen clearly in Figures 3 and 4. Multiple

equilibrium solutions can be seen from these Figures 3 and 4 having different tip-angles ( )0( ) of

the cantilever column for a specified load parameter ( ) and the subtangential parameter ( ).

Figure 3. Variation of load parameter ( ) with tip-

angle, )0( of the cantilever column for the

subtangential parameter, β=0.0.

Figure 4. Variation of load parameter ( ) with tip-

angle, )0( of the cantilever column for the

subtangential parameter, β=0.5.

0 20 40 60 80 100 120 140 160 1800

50

100

150

200

250

300

tip-angle, (0)(deg)

loa

d p

ara

me

ter(

)

0 50 100 150 200 250 300 3500

50

100

150

200

250

300

tip-angle, (0)(deg)

loa

d p

ara

me

ter(

)


53

Figures 5 to 8 show the deformed configurations of the column for the specified tip-angle,

060)0( and subtangential parameter, β=0.0, 0.1, 0.3 and 0.5, respectively. These figures are

drawn from the first, second and third equilibrium solutions having different values of the load

parameter ( ). The deformation pattern for a specified tip-angle )0( is found to differ due to

different value of the subtangential parameter, β.

Figure 5. Deformed configurations of the cantilever

column for β = 0.0 and 060)0(

Figure 6. Deformed configuration of the cantilever


Figure 7. Deformed configuration of the cantilever


Figure 8. Deformed configurations of the cantilever

column for β=0.5 and 060)0(

Figure 9 depicts the variation of along the length of the column for three load parameters

corresponding to β=0.0 and β=0.5, respectively for specified tip-angle, 060)0(

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

y

x

12 3

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

y

x

21

3

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

y

x

1

2

3

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

y

x

2

31


54

Figure 9. Variation of along the length of the column for the three load parameters

(a) 060)0( , 0.0 ; (b)

060)0( , 5.0

For the case of first equilibrium solution, )(s varies monotonically from zero to the tip-angle (

)0( ) whereas the second and third equilibrium solutions, the trend of )(s variation is different

and its magnitude does not exceeds the tip-angle, )0( . For higher load parameters having the same

tip-angle, (s) variation along the axis of the column is highly non-linear. In order to obtain the

deformed configuration of the column from equations (2) and (3), one has to store large amount of

(s)-data with increasing the load parameter, as evidenced from Figures 8. To overcome this type

of problem numerically the present approach is for direct integration of non-linear differential

equations providing the required initial conditions. The first equilibrium solutions are found to be in

good agreement with those of elliptic-function solution [17]. It may be noted that the increase in

load parameter give rise to the increase in the number of configurations.

Table 1. Comparison of load-deflection data for buckled column ( 0.0 ).

Finite element analysis [31] Present analytical

solution

)0( (deg) ax ay ax ay

2.5044 19.8 0.970 0.218 2.5046 0.9703 0.2173 2.6228 40.3 0.880 0.426 2.6270 0.8795 0.4251 2.8424 59.8 0.744 0.591 2.8390 0.7426 0.5917 3.1903 79.5 0.564 0.716 3.1817 0.5643 0.7170 3.7455 99.4 0.356 0.789 3.7258 0.3556 0.7902 4.6486 119.2 0.132 0.802 4.6044 0.1323 0.8039 6.2697 139.5 0.101 0.751 6.2172 -0.1012 0.7525 9.9411 159.3 0.330 0.628 9.7359 -0.3318 0.6306

Table 1 gives a comparison of the finite element analysis (FEA) results [31] with the present

analytical results for =0. It should be noted that the tip-angle, )0( and tip-deflections are

obtained from FEA by specifying the load parameter ( ) and the subtangential parameter ( ). In

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-60

-40

-20

0

20

40

60

(= s/L)

(

deg

)

=2.841754

=25.575788

=71.043856

a)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

10

20

30

40

50

60

(= s/L)

(

deg

)

=10.216231

=91.946082

=255.405785

b)


55

the present analytical solution, the obtained tip-angle, )0( of FEA is considered for estimating the

load parameter ( ) and obtaining the deformed configuration of the column.

Table 2. Comparison of load-deflection data for the specified subtangential parameter, and tip-angle, )0( upto

600

Finite element analysis [31] Present analytical solution

)0( (deg) ax ay ax ay

1.0

2.8745 20.8 0.968 0.227 2.8669 0.9676 0.2262

3.2644 60.9 0.735 0.597 3.1733 0.7369 0.5949

2.0

3.3778 18.9 0.973 0.204 3.3529 0.9736 0.2037

3.8442 59.9 0.748 0.582 3.6203 0.7492 0.5807

3.0

4.1206 18.9 0.973 0.204 4.0804 0.9742 0.2002

4.7029 59.6 0.757 0.571 4.3180 0.7576 0.5684

5.3814 21.8 0.966 0.223 5.3245 0.9672 0.2227

6.1734 59.7 0.766 0.553 5.5402 0.7673 0.5511

497.0

9.0899 19.3 0.976 0.178 8.9536 0.9799 0.1435

10.6074 60.1 0.797 0.490 9.2348 0.8154 0.4022

FEA results in Table 2 for the specified subtangential parameter ( ) and the tip-angle, )0(

upto 600 are comparable with those obtained from the present analytical solutions. The discrepancy

in the FEA results may be in the formulation of the nonlinear problem or in the FE models (which

may need more refinement) of Ref. [31].

Tables 3 to 8 provide the load-deflection data of the buckled cantilever columns for the

specified subtangential parameter ( ). The elliptic integral solutions of Timoshenko and Gere [10]

for =0 and those of Ref. [17] for =0.1 to 0.5 are found to be in good agreement with the

present first equilibrium solutions in Tables 3 to 8. Results of the present analysis will be useful to

validate the geometric non-linear finite element solutions of cantilever columns.

Mutyalarao et al. [38] have examined the dynamic stability of cantilever columns under a tip-

concentrated subtangential follower force. A simple numerical iterative scheme is presented to

obtain the critical and post-critical loads of the columns for the specified tip-angle ( )0( ) and the

load rotation parameter ( ). The critical load parameter ( cr ) for the load rotation parameter ( )

was obtained by specifying the tip-angle, )0( =0.010. The critical load parameter ( cr ) is found to

vary with the load rotation parameter ( ). The critical load parameters obtained from the first two

4.0


56

eigencurves (i.e., load versus frequency curves) [38] are found to be in excellent agreement with the

present analysis results (i.e., )1,( and )2,( for2

1 ).

Table 3. Load-deflection data of the buckled cantilever columns ( 0.0 ).

tip-angle, )0( (deg) First equilibrium Second equilibrium Third equilibrium

ax ay ax ay ax ay

(0) 0 2.4674 1.0000 0.0000 22.2066 1.0000 0.0000 61.6850 1.0000 0.0000

10 2.4768 0.9924 0.1108 22.2914 0.9924 -

0.0369

61.9205 0.9924 0.0221

20 2.5054 0.9697 0.2194 22.5485 0.9697 -

0.0731

62.6348 0.9698 0.0439

30 2.5541 0.9324 0.3239 22.9865 0.9324 -

0.1080

63.8514 0.9326 0.0647

40 2.6245 0.8812 0.4222 23.6204 0.8812 -

0.1407

65.6121 0.8815 0.0844

50 2.7192 0.8170 0.5126 24.4727 0.8170 -

0.1708

67.9796 0.8176 0.1025

60 2.8418 0.7410 0.5932 25.5758 0.7411 -

0.1977

71.0439 0.7419 0.1187

70 2.9972 0.6546 0.6626 26.9749 0.6547 -

0.2208

74.9302 0.6560 0.1327

80 3.1925 0.5594 0.7195 28.7329 0.5595 -

0.2398

79.8136 0.5613 0.1441

90 3.4376 0.4569 0.7628 30.9383 0.4571 -

0.2542

85.9398 0.4596 0.1528

100 3.7465 0.3490 0.7915 33.7183 0.3493 -

0.2638

93.6619 0.3538 0.1586

110 4.1401 0.2372 0.8052 37.2606 0.2376 -

0.2683

103.5017 0.2390 0.1610

120 4.6506 0.1232 0.8032 41.8550 0.1239 -

0.2675

116.2640 0.1306 0.1605

130 5.3305 0.0082 0.7851 47.9745 0.0095 -

0.2614

133.2624 0.0085 0.1554

140 6.2728 -

0.1069

0.7504 56.4549 -

0.1042

-

0.2494

156.8193 -

0.1101

0.1483

150 7.6622 -

0.2223

0.6979 68.9596 -

0.2181

-

0.2316

191.5543 -

0.2266

0.1381

160 9.9438 -

0.3403

0.6246 89.4945 -

0.3427

-

0.2088

248.5960 -

0.3300

0.1219

170 14.682

2

-

0.4714

0.5200 132.140

2

-

0.4679

-

0.1743

367.0562 -

0.5092

0.0862

Table 4. Load-deflection data of the buckled cantilever columns ( 1.0 )


ax ay ax ay ax ay

(0) 0

2.8296 1.0000 0.0000 21.1696 1.0000 0.0000 63.4464 1.0000 0.0000

10 2.8381 0.9925 0.1099 21.2356 0.9941 -

0.0165

63.6416 0.9934 0.0369

20 2.8640 0.9701 0.2177 21.4352 0.9766 -

0.0331

64.2323 0.9739 0.0728

30 2.9079 0.9332 0.3213 21.7739 0.9477 -

0.0499

65.2343 0.9417 0.1068

40 2.9710 0.8825 0.4188 22.2610 0.9079 -

0.0670

66.6749 0.8975 0.1381

50 3.0550 0.8191 0.5084 22.9105 0.8578 -

0.0844

68.5948 0.8420 0.1658

60 3.1624 0.7442 0.5882 23.7418 0.7981 -

0.1022

71.0509 0.7761 0.1892

70 3.2964 0.6592 0.6568 24.7816 0.7297 -

0.1204

74.1206 0.7010 0.2076

80 3.4617 0.5656 0.7130 26.0656 0.6536 -

0.1390

77.9080 0.6182 0.2206

90 3.6640 0.4652 0.7557 27.6421 0.5710 -

0.1578

82.5537 0.5288 0.2278

100 3.9117 0.3599 0.7840 29.5772 0.4829 -

0.1768

88.2491 0.4339 0.2287

110 4.2160 0.2515 0.7974 31.9625 0.3906 -

0.1959

95.2595 0.3363 0.2236

120 4.5928 0.1419 0.7956 34.9280 0.2951 -

0.2147

103.9612 0.2322 0.2102

130 5.0654 0.0327 0.7783 38.6655 0.1977 -

0.2332

114.9068 0.1319 0.1918

140 5.6696 -

0.0743

0.7455 43.4689 0.0993 -

0.2510

128.9428 0.0266 0.1653

150 6.4631 -

0.1779

0.6973 49.8171 0.0006 -

0.2679

147.4436 -

0.0779

0.1334

160 7.5469 -

0.2771

0.6331 58.5549 -

0.0979

-

0.2833

172.8289 -

0.1748

0.0995

170 9.1209 -

0.3718

0.5519 71.3614 -

0.1976

-

0.2977

209.8940 -

0.2737

0.0536

180 11.6623 -

0.4635

0.4499 92.2790 -

0.3101

-

0.3147

270.1496 -

0.3755

0.0034

190 16.8353 -

0.5595

0.3143 135.5107 -

0.4144

-

0.3279

393.9067 -

0.5204

-

0.0745


57



ax ay ax ay ax ay

(0) 0

3.3251 1.0000 0.0000 19.8890 1.0000 0.0000 65.7180 1.0000 0.0000

10 3.3328 0.9926 0.1087 19.9384 0.9953 0.0045 65.8768 0.9941 0.0514

20 3.3563 0.9705 0.2153 20.0878 0.9813 0.0084 66.3566 0.9765 0.1014

30 3.3958 0.9342 0.3178 20.3402 0.9583 0.0110 67.1672 0.9476 0.1489

40 3.4523 0.8844 0.4141 20.7013 0.9265 0.0118 68.3261 0.9078 0.1927

50 3.5270 0.8220 0.5026 21.1791 0.8865 0.0104 69.8583 0.8579 0.2316

60 3.6214 0.7484 0.5814 21.7848 0.8389 0.0061 71.7987 0.7989 0.2646

70 3.7377 0.6650 0.6492 22.5333 0.7843 -

0.0013

74.1933 0.7317 0.2909

80 3.8788 0.5734 0.7046 23.4439 0.7237 -

0.0122

77.1025 0.6576 0.3097

90 4.0482 0.4755 0.7467 24.5420 0.6578 -

0.0268

80.6046 0.5778 0.3205

100 4.2507 0.3730 0.7746 25.8608 0.5876 -

0.0451

84.8020 0.4935 0.3228

110 4.4925 0.2680 0.7880 27.4441 0.5141 -

0.0673

89.8292 0.4063 0.3163

120 4.7819 0.1623 0.7866 29.3502 0.4382 -

0.0932

95.8652 0.3171 0.3006

130 5.1298 0.0581 0.7703 31.6589 0.3608 -

0.1226

103.153

1

0.2275 0.2762

140 5.5516 -

0.0430

0.7395 34.4811 0.2829 -

0.1554

112.030

2

0.1398 0.2431

150 6.0691 -

0.1391

0.6944 37.9766 0.2053 -

0.1911

122.979

5

0.0588 0.2036

160 6.7144 -

0.2287

0.6357 42.3839 0.1287 -

0.2294

136.719

8

-

0.0283

0.1529

170 7.5376 -

0.3103

0.5638 48.0789 0.0535 -

0.2699

154.376

6

-

0.1096

0.0968

180 8.6221 -

0.3830

0.4790 55.6955 -

0.0198

-

0.3123

177.838

6

-

0.1838

0.0341

190 10.120

7

-

0.4460

0.3808 66.4125 -

0.0925

-

0.3577

210.597

9

-

0.2507

-

0.0388 200 12.358

8

-

0.4990

0.2671 82.7742 -

0.1683

-

0.4068

260.146

2

-

0.2982

-

0.1103 210 16.233

1

-

0.5431

0.1308 111.902

3

-

0.2203

-

0.4471

347.317

6

-

0.3709

-

0.2013 220 26.386

1

-

0.5844

-

0.0622

191.383

1

-

0.4778

-

0.5200

581.176

9

-

0.4465

-

0.3170



ax ay ax ay ax ay

(0) 0

4.0550 1.0000 0.0000 18.2284 1.0000 0.0000 68.8384 1.0000 0.0000

10 4.0621 0.9928 0.1068 18.2635 0.9959 0.0263 68.9650 0.9944 0.0654

20 4.0835 0.9712 0.2116 18.3694 0.9838 0.0518 69.3468 0.9777 0.1293

30 4.1195 0.9357 0.3123 18.5477 0.9637 0.0753 69.9898 0.9502 0.1903

40 4.1706 0.8871 0.4070 18.8015 0.9362 0.0961 70.9044 0.9126 0.2469

50 4.2377 0.8263 0.4939 19.1352 0.9017 0.1134 72.1054 0.8655 0.2979

60 4.3217 0.7546 0.5713 19.5546 0.8608 0.1264 73.6131 0.8098 0.3421

70 4.4242 0.6734 0.6379 20.0675 0.8143 0.1344 75.4539 0.7468 0.3784

80 4.5468 0.5845 0.6924 20.6836 0.7629 0.1371 77.6612 0.6776 0.4061

90 4.6917 0.4895 0.7338 21.4154 0.7076 0.1338 80.2773 0.6035 0.4244

100 4.8618 0.3904 0.7615 22.2787 0.6494 0.1245 83.3557 0.5259 0.4330

110 5.0605 0.2892 0.7751 23.2933 0.5892 0.1089 86.9633 0.4463 0.4314

120 5.2921 0.1879 0.7743 24.4846 0.5280 0.0871 91.1851 0.3661 0.4198

130 5.5622 0.0884 0.7594 25.8850 0.4668 0.0592 96.1298 0.2861 0.3968

140 5.8778 -

0.0073

0.7307 27.5369 0.4065 0.0254 101.9377 0.2092 0.3655

150 6.2484 -

0.0974

0.6889 29.4961 0.3480 -

0.0139

108.7930 0.1360 0.3241

160 6.6861 -

0.1802

0.6349 31.8379 0.2921 -

0.0583

116.9424 0.0682 0.2738

170 7.2076 -

0.2542

0.5696 34.6662 0.2395 -

0.1071

126.7242 0.0083 0.2181

180 7.8363 -

0.3181

0.4941 38.1289 0.1908 -

0.1598

138.6162 -

0.0501

0.1507

190 8.6061 -

0.3706

0.4097 42.4448 0.1465 -

0.2158

153.3191 -

0.0988

0.0771

200 9.5687 -

0.4108

0.3173 47.9538 0.1069 -

0.2746

171.9122 -

0.1392

0.0060

210 10.8078 -

0.4379

0.2179 55.2194 0.0723 -

0.3355

196.1686 -

0.1688

-

0.0764 220 12.4723 -

0.4512

0.1116 65.2645 0.0424 -

0.3994

229.2764 -

0.1855

-

0.1604 230 14.8645 -

0.4497

-

0.0026

80.2149 0.0145 -

0.4677

277.7944 -

0.1833

-

0.2434 240 18.7530 -

0.4319

-

0.1299

105.6158 0.0049 -

0.5267

358.6438 -

0.1988

-

0.3560 250 27.3317 -

0.3925

-

0.2910

165.1976 -

0.0354

-

0.6373

543.3032 -

0.2388

-

0.5508


58

Table 7. Load-deflection data of the buckled cantilever columns ( 4.0 ).


ax ay ax ay ax ay

(0) 0

5.2924 1.0000 0.0000 15.8616 1.0000 0.0000 73.6801 1.0000 0.0000 10 5.2991 0.9931 0.1034 15.8843 0.9959 0.0500 73.7792 0.9944 0.0786 20 5.3194 0.9724 0.2048 15.9525 0.9835 0.0989 74.0778 0.9776 0.1556 30 5.3535 0.9384 0.3023 16.0673 0.9632 0.1454 74.5792 0.9500 0.2295 40 5.4017 0.8918 0.3940 16.2299 0.9354 0.1885 75.2893 0.9122 0.2985 50 5.4646 0.8337 0.4782 16.4424 0.9007 0.2271 76.2165 0.8651 0.3616 60 5.5428 0.7651 0.5532 16.7077 0.8598 0.2604 77.3720 0.8095 0.4172 70 5.6374 0.6876 0.6178 17.0290 0.8136 0.2875 78.7700 0.7468 0.4644 80 5.7493 0.6028 0.6708 17.4110 0.7631 0.3078 80.4284 0.6782 0.5022 90 5.8801 0.5124 0.7114 17.8589 0.7092 0.3207 82.3692 0.6051 0.5300

100 6.0313 0.4183 0.7388 18.3795 0.6532 0.3260 84.6192 0.5291 0.5472 110 6.2050 0.3225 0.7527 18.9809 0.5961 0.3234 87.2111 0.4517 0.5536 120 6.4037 0.2268 0.7531 19.6731 0.5392 0.3129 90.1845 0.3745 0.5491 130 6.6304 0.1333 0.7403 20.4683 0.4836 0.2947 93.5881 0.2993 0.5343 140 6.8886 0.0438 0.7146 21.3816 0.4303 0.2691 97.4812 0.2260 0.5073 150 7.1829 -

0.0400

0.6769 22.4317 0.3804 0.2366 101.9371 0.1585 0.4725 160 7.5187 -

0.1164

0.6282 23.6421 0.3349 0.1978 107.0470 0.0974 0.4292 170 7.9028 -

0.1840

0.5696 25.0426 0.2946 0.1534 112.9258 0.0421 0.3754 180 8.3440 -

0.2414

0.5025 26.6718 0.2602 0.1043 119.7199 -

0.0041

0.3161 190 8.8532 -

0.2877

0.4285 28.5798 0.2324 0.0513 127.6191 -

0.0396

0.2522 200 9.4449 -

0.3220

0.3491 30.8337 0.2115 -

0.0046

136.8740 -

0.0683

0.1808 210 10.1384 -

0.3436

0.2660 33.5254 0.1978 -

0.0624

147.8238 -

0.0840

0.1122 220 10.9602 -

0.3523

0.1808 36.7843 0.1916 -

0.1211

160.9405 -

0.0919

0.0353 230 11.9477 -

0.3479

0.0951 40.7995 0.1928 -

0.1800

176.9046 -

0.0898

-

0.0365 240 13.1568 -

0.3302

0.0104 45.8606 0.2015 -

0.2380

196.7430 -

0.0767

-

0.1051 250 14.6750 -

0.2992

-

0.0721

52.4387 0.2175 -

0.2944

222.1009 -

0.0526

-

0.1800 260 16.6515 -

0.2550

-

0.1516

61.3660 0.2411 -

0.3482

255.8370 -

0.0120

-

0.2440 270 19.3722 -

0.1967

-

0.2281

74.3014 0.2743 -

0.3989

303.5468 0.0379 -

0.3127 280 23.4987 -

0.1222

-

0.3033

95.2552 0.3218 -

0.4307

378.4959 0.0974 -

0.3817 290 31.2565 -

0.0232

-

0.3853

138.3885 0.3603 -

0.5184

526.4898 0.1646 -

0.4963



ax ay ax ay ax ay

(0) 0

9.8696 1.0000 0.0000 88.8264 1.0000 0.0000 246.7401 1.0000 0.0000 10 9.8790 0.9943 0.0870 88.9110 0.9943 0.0870 246.9752 0.9943 0.0870 20 9.9073 0.9773 0.1723 89.1656 0.9774 0.1723 247.6822 0.9774 0.1723 30 9.9547 0.9494 0.2544 89.5921 0.9495 0.2544 248.8669 0.9495 0.2544 40 10.021

6

0.9112 0.3317 90.1941 0.9114 0.3317 250.5391 0.9114 0.3317 50 10.108

5

0.8636 0.4027 90.9766 0.8638 0.4027 252.7127 0.8639 0.4027 60 10.216

2

0.8075 0.4662 91.9461 0.8078 0.4663 255.4058 0.8079 0.4663 70 10.345

7

0.7442 0.5211 93.1110 0.7446 0.5212 258.6416 0.7447 0.5213 80 10.497

9

0.6750 0.5664 94.4814 0.6756 0.5667 262.4484 0.6757 0.5668 90 10.674

4

0.6015 0.6015 96.0698 0.6022 0.6019 266.8604 0.6024 0.6022 100 10.876

7

0.5252 0.6259 97.8907 0.5259 0.6265 271.9185 0.5262 0.6268 110 11.106

8

0.4476 0.6393 99.9616 0.4485 0.6402 277.6711 0.4488 0.6407 120 11.367

0

0.3705 0.6417 102.303

2

0.3715 0.6431 284.1754 0.3719 0.6437 130 11.660

0

0.2954 0.6335 104.939

6

0.2966 0.6353 291.4989 0.2969 0.6363 140 11.988

8

0.2239 0.6152 107.899

6

0.2245 0.6164 299.7210 0.2249 0.6176 150 12.357

4

0.1574 0.5873 111.216

9

0.1579 0.5889 308.9357 0.1581 0.5898 160 12.770

2

0.0971 0.5509 114.931

6

0.0973 0.5529 319.2544 0.0978 0.5541 170 13.232

4

0.0444 0.5070 119.091

4

0.0443 0.5095 330.8095 0.0449 0.5103 180 13.750

4

0.0000 0.4570 123.753

3

-

0.0001

0.4588 343.7593 0.0001 0.4624 190 14.331

8

-

0.0352

0.4020 128.986

1

-

0.0353

0.4058 358.2948 -

0.0348

0.4057 200 14.985

9

-

0.0606

0.3437 134.873

1

-

0.0615

0.3457 374.6474 -

0.0610

0.3465 210 15.724

0

-

0.0760

0.2835 141.516

3

-

0.0771

0.2855 393.1010 -

0.0788

0.2901 220 16.560

3

-

0.0811

0.2229 149.042

4

-

0.0824

0.2271 414.0066 -

0.0829

0.2217 230 17.512

2

-

0.0762

0.1635 157.609

8

-

0.0772

0.1689 437.8049 -

0.0769

0.1650 240 18.602

2

-

0.0616

0.1067 167.420

2

-

0.0626

0.1065 465.0560 -

0.0626

0.1092 250 19.859

4

-

0.0377

0.0539 178.734

6

-

0.0378

0.0553 496.4850 -

0.0378

0.0531 260 21.322

0

-

0.0053

0.0063 191.897

9

-

0.0027

0.0054 533.0497 -

0.0039

0.0040 270 23.041

8

0.0348 -

0.0349

207.376

3

0.0323 -

0.0348

576.0453 0.0343 -

0.0356 280 25.091

1

0.0818 -

0.0687

225.819

8

0.0780 -

0.0671

627.2771 0.0775 -

0.0667 290 27.574

1

0.1346 -

0.0943

248.167

0

0.1326 -

0.0954

689.3528 0.1326 -

0.0869 300 30.648

7

0.1923 -

0.1111

275.838

3

0.1860 -

0.1092

766.2174 0.1911 -

0.1099 310 34.568

4

0.2540 -

0.1186

311.115

3

0.2481 -

0.1161

864.2092 0.2510 -

0.1183 320 39.775

4

0.3193 -

0.1165

357.978

2

0.3163 -

0.1211

994.3839 0.3169 -

0.1123 330 47.138

8

0.3883 -

0.1046

424.249

6

0.3900 -

0.1022

1178.4712 0.3905 -

0.1069 340 58.729

0

0.4628 -

0.0829

528.560

9

0.4641 -

0.0866

1468.2247 0.4645 -

0.0834 350 81.729

7

0.5507 -

0.0595

735.567

1

0.6835 -

0.1072

2043.2418 0.5488 -

0.0283


59

Concluding Remarks

Every nonlinear problem must be treated as a self-respecting entity and to be solved on its own

merit. This may be the reason why the literature shows several investigations on the nonlinear

problem of cantilever columns loaded by a subtangential follower force for the past seven decades.

Rao and Rao [14-23] have published extensively in this area in the late 1980s and early 1990s;

recently by the current authors [36-38]; and other researchers [46-50]. Many of these articles were

not discussed about the existence of multiple solutions for the nonlinear differential equations while

studying the post-buckling behavior of cantilever columns.

A simple numerical procedure is followed in this article to obtain the deformed configuration of

the column when it is subjected to a tip-concentrated subtangential follower load. Equation (20)

clearly indicates the limits on the subtangential parameter ( ) to apply static stability criterion for

buckling and post-buckling analysis of cantilever columns. Multiple equilibrium solutions are

found from the generated load versus tip-angle curves having different tip-angles ( )0( ) for the

specified load parameter (λ), which indicates non-uniqueness in the solution of the problem [51].

The present boundary value problem can also be solved using a symbolic computer algebra code

like Mathematica.

Future work is planned to analyze several test configurations (viz., tangentially loaded column,

cantilever column mounting solid rocket motor at free-end, free-free column under an end thrust

and post-buckling of simply supported stepped column) for investigating the buckling and post-

buckling behavior of columns. Other challenging tasks identified are the dynamic stability

characteristics of tapered truncated cantilever wedges and cones, deformations of beams/columns

considering both geometric and material nonlinearities and optimum design of beam/column

configurations for the specified load and allowable limits.

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equilibrium configurations of cantilever columns under · pdf fileequilibrium configurations...

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