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Buckling behaviour of uniformly and non-uniformly pretwisted beams
Computers
Pergamon
00457949(94)00357-2
TECHNICAL
& Strucrures Vol. 54, No. 3. pp. 551-556, 1995 Copyright 0 1995 Elsevier Science Ltd Printed in Great Britain. All rights reserved 004s7949/95 $9.50 + 0.00
NOTE
BUCKLING BEHAVIOUR OF UNIFORMLY AND NON-UNIFORMLY PRETWISTED BEAMS N. Madhusudbana,t TDepartment
B. P. Gauthamz and N. Canesants
of Mechanical Engineering, and IMachine Dynamics Laboratory, Department Mechanics, Indian Institute of Technology, Madras-600 036, India (Received
of Applied
15 July 1993)
Abstract-The buckling behaviour of continuously, discontinuously and oppositely pretwisted beams under a variety of combinations of twist angle and cross-section has been analysed. Both uniformly and non-uniformly twisted beams are considered. The twisted rod is modelled as a beam with varying bending strength depending on the twist. A Cartesian global coordinate system is used with a rotating local coordinate system in deriving the governing system of differential equations. These are then solved using the Galerkin finite element formulation with the Hermite polynomials as interpolating functions. The various combinations of parameters are compared and the characteristic behaviour of each family indicated. The choice of combinations in order to maximize the buckling load have been considered.
1. INTRODUCTION
Pretwisted beams are widely used as structural elements. A beam is considered pretwisted if, in the stress-free state, the cross-section of the beam rotates relative to the end crosssection along the length of the beam. The blades of propellers, turbines and fans and drill bits are usually modelled as pretwisted beams. Pretwisted thin-walled members are being increasingly used in the construction of bridges. Pretwist introduces coupling between the bending strengths of the beam about the principal axes, significantly influencing the buckling load. The review paper by Rosen [l] gives a detailed account of the work that has been carried out on pretwisted beams. A lot of work is available in the literature in relation to finding the buckling loads and the natural frequency of vibration of pretwisted beams [2,3]. Nixdorff and Brass analysed the buckling under axial compression. Fisher-Fay investigated the stability of continuously and discontinuously pretwisted rods under axial compression. Tebedge [5] analysed the buckling of pretwisted rods having thin cross-sections using finite element stability analysis based on a non-linear forrnulation. Rosen et al. [4] used their model based on a principal curvature transformation, and specialized generalised coordinates to investigate the stability of pretwisted beams. Magrab and Gilsinn [6] modelled drill bits and fluted cutters as twisted beams and used a Cartesian coordinate system to investigate the axial buckling of the drill bit, considering it to be clamped at both the ends. The governing system of differential equations are in a Cartesian coordinate system similar to [6], which has been extended to consider non-linear twist. The equations are solved using the Galerkin finite element formulation with Hermite polynomials as interpolating functions. The solution procedure used was found to be quite efficient. Any complicated twist can also be analysed with the present model. The purpose of this paper is to analyse the behaviour of continuously, discontinuously and oppositely pretwisted beams clamped at both ends. A comparison between the § Author
to whom
correspondence
should
various combinations of twist and cross-section with respect to their buckling load has been carried out. An optimal combination is also discussed. 2. FORMULATION
Figure 1 shows a pretwisted beam. A Cartesian coordinate system (x, y, z) is used. The angle 6 defines the local rotation between the X, y coordinates and the cross-sectional principal directions, y. 1. The coupled lateral buckling equations [5] can be written as
Fig. 1. Pretwisted
be addressed. 551
beam.
Technical
552
where U and V are the transverse displacements of the beam in the x-y and the y-z planes, respectively, n = z/L
(34 g(rl) = 4sinP8,
Note and
are displacements and slopes at nodes I and element. The two equations can be combined giving
+ 28’51
2 of each
B0 LLijl
h(?/) = A, - E, cos[28, + 2/I’<]
(3c)
4[J,l-
&[K,l
A, = 0.5(1 + (la) B, = 0.5( 1 - a,)
The two matrices are now rearranged to permit global assembly giving the stiffness matrix and the mass matrix of the element. These matrices are computed numerically using Gaussian four point integration. The matrices for the entire structure are assembled and the boundary conditions imposed. The eigen value problem was solved using a standard non-symmetric matrix solver. Though the present formulation results in a non-symmetric matrix, requiring a nonsymmetric matrix solver, it was found to have many advantages due to its convergence properties and versatility.
0 =?X
B = 8, + P’5, Be+
I =
8,
B’h,,
+
where E is the Young’s modulus of the beam, 0, , O2are the twists of the end cross-sections at nodes 1 and 2 relative to the global reference. p’ represents the rate of twist in that element. The twist is considered uniform in each element. The subscript e refers to the element and h, refers to the length of the element. The variational formulation of the Galerkin finite element method was used to solve the above set of governing differential equations. The Hermite polynomials were used as interpolating functions (cp,, ‘p,). The above set of differential equations yield the following system of equations
Ao[J,,l{VI + ~oK,,l{ VI + ~o[Lt,l{~I) + wf,,l{ V) = 0 W-4 4JJ,l{~~ - ~oW,,lI~~ + &[L,,l{ V} + S[M,,]{ U} = 0, (4b) where
Jz, =
s h*d$
d2p
_Ldl,
i,j=l,..,,
o dt2 dt2
4
(54
h,
K, =
cos(2& + 2/I’<) g
$+
d&
s0
i,j=l,...,4
L,, =
sin(2p, + 28’5) 2
i,j = 1
1..,>
M,, =
(5b)
s
hed2q, --,dt, o dt2
{V,={K.(“,fl);
{U) ={&($
3
dt,
(5c)
4 i,j=l,_..,
v’?r$),>
u2_(gx>
4
(54
3. RESULTS AND DISCUSSION The convergence characteristics of the model were found to be very good even for large amount of twist, for twists of less than 60” per element the results were within 1% error. Some of the results were also compared with the results from analysis of the COSMOS/M finite element package using 192 solid elements and are given in Table 1. The results were within 2% error. For the present study the twisted beam is modeled with 16 elements which gives nearly converged results. The results obtained by the above solution procedure for various cross-sections and combinations of twist were plotted. The results presented are for a beam clamped at both l/J a, represents the relative bending ends. The parameter strengths of the beams cross-sections about the two principal axes. If S, is the critical buckling load about the x-axis and S, the critical buckling load about the y-axis of an untwisted beam, then from eqn (2) we find that S,/S, = l/a,. For the case of a rectangular cross-section I/a, reduces to h/b, where h, b are defined in Fig. 1 when a, = 1, S = 47r2. b represents the twist about the Z-axis. In Fig. 2 the buckling coefficient is plotted as a function of a = l/duo for various values of /I. Figure 3 gives the variation of the normalized buckling load with twist. These results are for uniformly twisted beam. It is seen that there is a significant increase in the buckling load with twist. At all twists the weaker section, taken around the y-axis, is the one that governs the buckling of the twisted beam. It was found that there is a rapid drop in the buckling coefficient S as a is increased from unity, as seen in Fig. 2, but from Fig. 3 as a increases the normalised buckling coefficient, S/S, (So is the buckling coefficient for a non-twisted beam of the same cross-section) increases for a given value of twist. This indicates that the coupling of the stronger cross-section results in a significant increase in S, in the case of low buckling load beams, i.e. beams with higher a. The Table
1. Validation
of results for bucklinn
a
Twist (8)
Present study
2 4 6
180 90 180
16.49 4.95 8.83
coefficient
COSMOS/M 16.51 4.86 8.87
Technical
30.00
.L? .I! r Q) s
70.00
.-F 22 2 m
10.00
6.00
6.00
4.00
1/(oo)~‘2 Fig. 2(a). Buckling
coefficient
as a function
of a,, and B.
40.00
00
2.00
4.00
6.00
6.00
I /(Q)“l
Fig. 2(b). Buckling
coefficient as a function (high).
553
buckling load is dependent on the mode shape of buckling which is influenced by the twist, this explains why the buckling load does not increase steadily with twist. Further it was found from Fig. 3 that as B increases the norrnahsed buckling coefficient shows some characteristic behaviour for all values of G[. It has an initial peak at approximately p = 100” then drops a bit and later increases to reach a maximum at /? 2 200” and again drops, from here on it oscillates the peak rising a bit with every oscillation till it reaches a somewhat steady value at /I z 1200”. The value never reaches the previous maximum attained at B z 200”. The peak shows some characteristic behaviour of its own. It occurs at approximately /I = 240” for a = 2, fi = 225” for u = 3, )Cl= 210” for a = 4 and at /I = 180” for a = 6, i.e. there is a direct proportional decrease of 15” from 240” for every unit increase in a, above a = 2. Hence it can be concluded that for a particular cross-section there is a definite amount of twist for which the beam can have the maximum buckling load. The buckling loads of discontinuously twisted beams are found to be significantly different, here the beam is twisted over a portion of the entire length in one or more parts. The buckling load was also found to be dependent on the position of the twist. This must be the case because the position of the twist results in a different buckling mode shape. The variation of the normalized buckling coefficient S/S,, with X, the normalized distance between the midpoint of the twist section and one end, is shown in Fig. 4. The beam is twisted by 90” over one eighth of the beam length. As expected the buckling load is maximum when the twist section is at the center. For higher values of a there is a significant variation as the twist is displaced away from the centre. Figure 5 gives the behaviour of a similar beam twisted by 180” over a length of 3L/8. This was found to have a minimum at X z 0.26 in addition to the maximum at X = 0.5. Further for /I = 225” (Fig. 6), the maximum again occurred at X = 0.5 but no minimum was found. In contrast to the previous case there is a steady increase as the twist position proceeds towards the center from the least value at the ends. The case of opposite twist was also considered. The variation of S/S, with X, the normalized distance between one end and mid-point of the twist portion, for a twist of (- 45”, 45”) over one eighth of the entire length is shown in Fig. 7. A significant difference was noticed when compared to a (45”, 45”) twist, a second maximum was found at
m -;I
Note
of a,, and p
2.50
0.50
0.00
200.00
Total Fig. 3. The variation
600.00
600.00
400.00
Twist
of (S/S,)
1000
00
1200.00
(Degrees)
with twist for various
values of a.
1400
00
Technical
554
Note
3.00
2.50
080
Fig.
4. Variation of S/S,, with position of twist. twist = 90” over a length L/8.)
(Total
X z 0.5 and a minimum at X z 0.25. Further the buckling load was higher for the case when the twist was at the end than when at the center, however the buckling loads for the case of opposite twist were lower than that for unidirectional twist. Figure 8 gives the variation of S/S,, with X for a twist of (- 90”, 90”) successfully over a length L/4. A prominent dip was noticed when the twist section was at X x 0.26. The tip values were nearly equal to the centre twist values, but, these were also lower than the corresponding equivalent of 180” unidirectional twist, though the difference was lower than that for the 90” case. Thus it may be concluded that only uniformly twisted beams have the expected buckling relationships whereas discontinuously twisted and non-uniform twisted beams may not have a maximum when the twist portion is positioned at the centre. Further unidirectional twist must always be preferred to opposite twist if the sole aim is to increase the buckling load. The buckling load was found to be dependent on the length over which it is to be twisted (L’). In the results given W = L’/L. The twist is considered to be symmetric about the centre of the beam. The dependence of S/S, for a 90” twist spread over a normalized length W for various values of I is shown in
0.30
0.40
0 50
6. Variation of S/S, with twist position. twist = 225” over a length 5L/16.)
0.60
(Total
Fig. 9. The variation with a was found to be characteristically different. For a > 4 it was seen to show approximately the same behaviour. For a = 2 a maximum occurs at W z 0.64 and a minimum at W z 0.24, whereas for a = 3 the maximum was at W 5 0.25 and the minimum at W N 0.60, for a = 4 the normalized buckling coefficient dropped from a maximum at W z 0.1 to a minimum at W z 0.74. Further the maximum values attained were found to be significantly higher than those for uniform twist of the same amount over the entire length. For a (45”, -45”) the variation of buckling coefficient with length of twist is shown in Fig. 10. This shows a steady increase to a maximum from a minimum value. This behaviour is quite different from the previous one which had a minimum also. This may be due to the symmetric opposite twist balancing for the vacillation of just unidirectional twist, forcing the buckling load to steadily increase. Some variety of combinations have been considered and presented in Table 2. It was always found that an opposite twist reduces the buckling load for any particular combination of twist position and total twist.
1 20
I”\,
1.50
0.20
X Fig.
1 b0 CY
-““““~I~~~~‘~“~I~“~‘,,~‘I”‘~‘~(“I’~
0 10
c
6
N-
1 40
3 1.15 :
1 10 c
s0
1.30
0
'c
120
u) 1.05 ; F
1.10
100
1 00
090 O.lU
0.20
0.30
0.40
0 50
X Fig.
5. Variation of S/S, with location of twist. twist = 180” over a length 3L/8.)
:
0 60
(Total
Fig.
X
7. Variation of S/S, with twist position. [Twist = (45”, -45”) over a length L/8.]
Technical
Note
1.50
555
2.80
?
I
(x=6
i
1.40
\
z’40: L cr=4
i 1.30 0
:
k
2.00
I
0
1.20
a=3
? t Lo 1.60 =
0.00
--s
a=
0.20
0.40
0.60
2
0.80
1.00
1.20
W Fig. 8. Variation of buckling coefficient with position twist. [Twist = (45”, 45”, -45”, -45”) over L/4.]
of
Fig.
9. Variation of S/S,, with length twist = 90” spread over normalised 4.
A case where a discontinuous beam is made up of two portions of twist symmetric about the centre was also considered. The dependence of S/S, with Y, the normalized length between the two twist portions, for various values of a is shown in Fig. 11. Y = 0 represents the case when the beam is singly discontinuous. It was found that the maximum buckling load occurs at Y z 0.22 for 01= 2, at YxO.l6fora=3,at Yx0,22fora=4andat YxO.24 for a = 6. These maxima are significantly higher than the maxima attained with uniform twist along the entire length. Another case wherein the two twist portions are oppositely twisted relative to each other was also considered. The behaviour was significantly different as shown in Fig. 12. The maximum was found to occur at Y x 0.38, this occurred at the same point for all values of a. Further it was found that the maxima attained with opposite twist was lower than the maxima attained with unidirectional twist for al! values of a. Table 2. Buckling
coefficient
of twist. (Total length W.)
CONCLUSIONS
The present study gives the characteristic behaviour of certain families of pretwisted rods. The formulation developed can be efficiently used in the analysis of complicated forms of twist. The maximum normalized buckling load of various families has been compared. Though the maximum buckling load for any combination is found to be greatly dependent on a, certain general results can be used without much error to achieve maximum buckling loads. The twist of nearly 225” was found to be the best possible total twist to achieve maximum buckling loads. The unidirectional twist was always found to have a higher buckling load than an equal amount of any oppositely twisted combination, the discontinuously twisted rods were found to have a higher buckling load than those twisted uniformly along the entire length. Further it can be concluded that within the cases considered here, the buckling load is the maximum achieved for the case when the total twist of 225” is in two portions separated by X z 0.20 and unidirectional.
for some twist combinations,
for various
values of a
Case No.
Element number 1 2 3 4 5 6 I
8 9 10 11 12 13 14 15 16 Buckling coejicient a=2 a=3 a=4 a=6
1
2
3
4
0 45 45 0 0 45 45 0 0 45 45 0 0 45 45 0
0 28.1 28.1 0 0 28.1 28.1 0 0 28.1 28.1 0 0 28.1 28.1 0
0
0
15 15 0 0 15 15 0 0 15 15 0 0 15 15 0
45 45 0 0 -45 -45 0 0 45 45 0 0 -45 -45 0
11.7 3.1 1.4 ^_ 0.2
17.29 5.21 2.40 .~ 0.360
16.29 4.95 2.37 0.425
11.13 2.87 1.29 0.209
5 0
28.1 28.1 0 0 -28.1 -281 0 0 28.1 28.1 0 0 -28.1 -28.1 0 10.40 2.63 1.17 0.360
6 0
45 -45 0 0 45 -45 0 0 45 -45 0 0 45 -45 0 10.33 2.61 1.16 0.209
I 0
28.1 -28.1 0 0 28.1 -28.1 0 0 28.1 -28.1 0 0 28.1 -28.1 0 10.04 2.52 1.12 0.360
8
28.1 28.1 28.1 28.1 28.1 28.1 28.1 28.1 -28.1 -28.1 -28.1 -28.1 -28.1 -28.1 -28.1 -28.1 11.42 3.02 1.36 0.200
9
14 14 14 14 14. 14 14 14 -14 -14 -14 -14 -14 -14 -14 - 14 19.26 5.99 2.76 0.416
in 0
45 -45 45 0 0 45 -22.5 -22.5 45 0 0 45 -45 45 0 16.69 5.38 2.58 0.202
Technical
556
Note
2.00 a=4
0 ? tn 1.50
a=3
[1:.:: /
c(-2
1.00
k
Fig.
10. Variation of S/S, [Twist = (45”, -45”)
with W for various over a length IF.1
a.
Fig. 12. Variation of normalized buckling coefficient with Y. [Two parts of 112.5”, oppositely twisted over (3L/l6).]
2.80
REFERENCES
1.60
0.80 i 0.00
0.40
0.20
0.60
Y Fig. 11. Variation of normalized buckling coefficient with Y. [Total twist = 225”, 112.5” is over a length (3L/l6).]
Structural and dynamic behaviour of 1. A. Rosen, pretwisted rods and beams. Appl. Mech. Rev. 44, 483-515 (1991). 2. W. Carnegie, Vibrations of pretwisted cantilever blading. Proc. Inst. Mech. Engrs 173, 343-362 (1959). Coupled bending-bending vibrations of 3. B. Dawson, pretwisted cantilever blading treated by the Rayleigh-Ritz method. J. Mech. Engng Sci. 10, 381-388 (1968). 4. A. Rosen, R. G. Loewy and M. B. Mathew, Nonlinear analysis of pretwisted rods using ‘principal curvature transformation’. Part I: theoretical derivation. AIAA Jnl 25, 470478 (1987). 5. N. Tebedge, Buckling of pretwisted columns. J. Sfrucr. Engng 108, 11241139 (1982). 6. E. B. Magrab and D. E. Gilsinn, Buckling loads and natural frequencies of drill bits and fluted cutters. J. Engng Indust. 106, 196204 (1984).
Pergamon
00457949(94)00357-2
TECHNICAL
& Strucrures Vol. 54, No. 3. pp. 551-556, 1995 Copyright 0 1995 Elsevier Science Ltd Printed in Great Britain. All rights reserved 004s7949/95 $9.50 + 0.00
NOTE
BUCKLING BEHAVIOUR OF UNIFORMLY AND NON-UNIFORMLY PRETWISTED BEAMS N. Madhusudbana,t TDepartment
B. P. Gauthamz and N. Canesants
of Mechanical Engineering, and IMachine Dynamics Laboratory, Department Mechanics, Indian Institute of Technology, Madras-600 036, India (Received
of Applied
15 July 1993)
Abstract-The buckling behaviour of continuously, discontinuously and oppositely pretwisted beams under a variety of combinations of twist angle and cross-section has been analysed. Both uniformly and non-uniformly twisted beams are considered. The twisted rod is modelled as a beam with varying bending strength depending on the twist. A Cartesian global coordinate system is used with a rotating local coordinate system in deriving the governing system of differential equations. These are then solved using the Galerkin finite element formulation with the Hermite polynomials as interpolating functions. The various combinations of parameters are compared and the characteristic behaviour of each family indicated. The choice of combinations in order to maximize the buckling load have been considered.
1. INTRODUCTION
Pretwisted beams are widely used as structural elements. A beam is considered pretwisted if, in the stress-free state, the cross-section of the beam rotates relative to the end crosssection along the length of the beam. The blades of propellers, turbines and fans and drill bits are usually modelled as pretwisted beams. Pretwisted thin-walled members are being increasingly used in the construction of bridges. Pretwist introduces coupling between the bending strengths of the beam about the principal axes, significantly influencing the buckling load. The review paper by Rosen [l] gives a detailed account of the work that has been carried out on pretwisted beams. A lot of work is available in the literature in relation to finding the buckling loads and the natural frequency of vibration of pretwisted beams [2,3]. Nixdorff and Brass analysed the buckling under axial compression. Fisher-Fay investigated the stability of continuously and discontinuously pretwisted rods under axial compression. Tebedge [5] analysed the buckling of pretwisted rods having thin cross-sections using finite element stability analysis based on a non-linear forrnulation. Rosen et al. [4] used their model based on a principal curvature transformation, and specialized generalised coordinates to investigate the stability of pretwisted beams. Magrab and Gilsinn [6] modelled drill bits and fluted cutters as twisted beams and used a Cartesian coordinate system to investigate the axial buckling of the drill bit, considering it to be clamped at both the ends. The governing system of differential equations are in a Cartesian coordinate system similar to [6], which has been extended to consider non-linear twist. The equations are solved using the Galerkin finite element formulation with Hermite polynomials as interpolating functions. The solution procedure used was found to be quite efficient. Any complicated twist can also be analysed with the present model. The purpose of this paper is to analyse the behaviour of continuously, discontinuously and oppositely pretwisted beams clamped at both ends. A comparison between the § Author
to whom
correspondence
should
various combinations of twist and cross-section with respect to their buckling load has been carried out. An optimal combination is also discussed. 2. FORMULATION
Figure 1 shows a pretwisted beam. A Cartesian coordinate system (x, y, z) is used. The angle 6 defines the local rotation between the X, y coordinates and the cross-sectional principal directions, y. 1. The coupled lateral buckling equations [5] can be written as
Fig. 1. Pretwisted
be addressed. 551
beam.
Technical
552
where U and V are the transverse displacements of the beam in the x-y and the y-z planes, respectively, n = z/L
(34 g(rl) = 4sinP8,
Note and
are displacements and slopes at nodes I and element. The two equations can be combined giving
+ 28’51
2 of each
B0 LLijl
h(?/) = A, - E, cos[28, + 2/I’<]
(3c)
4[J,l-
&[K,l
A, = 0.5(1 + (la) B, = 0.5( 1 - a,)
The two matrices are now rearranged to permit global assembly giving the stiffness matrix and the mass matrix of the element. These matrices are computed numerically using Gaussian four point integration. The matrices for the entire structure are assembled and the boundary conditions imposed. The eigen value problem was solved using a standard non-symmetric matrix solver. Though the present formulation results in a non-symmetric matrix, requiring a nonsymmetric matrix solver, it was found to have many advantages due to its convergence properties and versatility.
0 =?X
B = 8, + P’5, Be+
I =
8,
B’h,,
+
where E is the Young’s modulus of the beam, 0, , O2are the twists of the end cross-sections at nodes 1 and 2 relative to the global reference. p’ represents the rate of twist in that element. The twist is considered uniform in each element. The subscript e refers to the element and h, refers to the length of the element. The variational formulation of the Galerkin finite element method was used to solve the above set of governing differential equations. The Hermite polynomials were used as interpolating functions (cp,, ‘p,). The above set of differential equations yield the following system of equations
Ao[J,,l{VI + ~oK,,l{ VI + ~o[Lt,l{~I) + wf,,l{ V) = 0 W-4 4JJ,l{~~ - ~oW,,lI~~ + &[L,,l{ V} + S[M,,]{ U} = 0, (4b) where
Jz, =
s h*d$
d2p
_Ldl,
i,j=l,..,,
o dt2 dt2
4
(54
h,
K, =
cos(2& + 2/I’<) g
$+
d&
s0
i,j=l,...,4
L,, =
sin(2p, + 28’5) 2
i,j = 1
1..,>
M,, =
(5b)
s
hed2q, --,dt, o dt2
{V,={K.(“,fl);
{U) ={&($
3
dt,
(5c)
4 i,j=l,_..,
v’?r$),>
u2_(gx>
4
(54
3. RESULTS AND DISCUSSION The convergence characteristics of the model were found to be very good even for large amount of twist, for twists of less than 60” per element the results were within 1% error. Some of the results were also compared with the results from analysis of the COSMOS/M finite element package using 192 solid elements and are given in Table 1. The results were within 2% error. For the present study the twisted beam is modeled with 16 elements which gives nearly converged results. The results obtained by the above solution procedure for various cross-sections and combinations of twist were plotted. The results presented are for a beam clamped at both l/J a, represents the relative bending ends. The parameter strengths of the beams cross-sections about the two principal axes. If S, is the critical buckling load about the x-axis and S, the critical buckling load about the y-axis of an untwisted beam, then from eqn (2) we find that S,/S, = l/a,. For the case of a rectangular cross-section I/a, reduces to h/b, where h, b are defined in Fig. 1 when a, = 1, S = 47r2. b represents the twist about the Z-axis. In Fig. 2 the buckling coefficient is plotted as a function of a = l/duo for various values of /I. Figure 3 gives the variation of the normalized buckling load with twist. These results are for uniformly twisted beam. It is seen that there is a significant increase in the buckling load with twist. At all twists the weaker section, taken around the y-axis, is the one that governs the buckling of the twisted beam. It was found that there is a rapid drop in the buckling coefficient S as a is increased from unity, as seen in Fig. 2, but from Fig. 3 as a increases the normalised buckling coefficient, S/S, (So is the buckling coefficient for a non-twisted beam of the same cross-section) increases for a given value of twist. This indicates that the coupling of the stronger cross-section results in a significant increase in S, in the case of low buckling load beams, i.e. beams with higher a. The Table
1. Validation
of results for bucklinn
a
Twist (8)
Present study
2 4 6
180 90 180
16.49 4.95 8.83
coefficient
COSMOS/M 16.51 4.86 8.87
Technical
30.00
.L? .I! r Q) s
70.00
.-F 22 2 m
10.00
6.00
6.00
4.00
1/(oo)~‘2 Fig. 2(a). Buckling
coefficient
as a function
of a,, and B.
40.00
00
2.00
4.00
6.00
6.00
I /(Q)“l
Fig. 2(b). Buckling
coefficient as a function (high).
553
buckling load is dependent on the mode shape of buckling which is influenced by the twist, this explains why the buckling load does not increase steadily with twist. Further it was found from Fig. 3 that as B increases the norrnahsed buckling coefficient shows some characteristic behaviour for all values of G[. It has an initial peak at approximately p = 100” then drops a bit and later increases to reach a maximum at /? 2 200” and again drops, from here on it oscillates the peak rising a bit with every oscillation till it reaches a somewhat steady value at /I z 1200”. The value never reaches the previous maximum attained at B z 200”. The peak shows some characteristic behaviour of its own. It occurs at approximately /I = 240” for a = 2, fi = 225” for u = 3, )Cl= 210” for a = 4 and at /I = 180” for a = 6, i.e. there is a direct proportional decrease of 15” from 240” for every unit increase in a, above a = 2. Hence it can be concluded that for a particular cross-section there is a definite amount of twist for which the beam can have the maximum buckling load. The buckling loads of discontinuously twisted beams are found to be significantly different, here the beam is twisted over a portion of the entire length in one or more parts. The buckling load was also found to be dependent on the position of the twist. This must be the case because the position of the twist results in a different buckling mode shape. The variation of the normalized buckling coefficient S/S,, with X, the normalized distance between the midpoint of the twist section and one end, is shown in Fig. 4. The beam is twisted by 90” over one eighth of the beam length. As expected the buckling load is maximum when the twist section is at the center. For higher values of a there is a significant variation as the twist is displaced away from the centre. Figure 5 gives the behaviour of a similar beam twisted by 180” over a length of 3L/8. This was found to have a minimum at X z 0.26 in addition to the maximum at X = 0.5. Further for /I = 225” (Fig. 6), the maximum again occurred at X = 0.5 but no minimum was found. In contrast to the previous case there is a steady increase as the twist position proceeds towards the center from the least value at the ends. The case of opposite twist was also considered. The variation of S/S, with X, the normalized distance between one end and mid-point of the twist portion, for a twist of (- 45”, 45”) over one eighth of the entire length is shown in Fig. 7. A significant difference was noticed when compared to a (45”, 45”) twist, a second maximum was found at
m -;I
Note
of a,, and p
2.50
0.50
0.00
200.00
Total Fig. 3. The variation
600.00
600.00
400.00
Twist
of (S/S,)
1000
00
1200.00
(Degrees)
with twist for various
values of a.
1400
00
Technical
554
Note
3.00
2.50
080
Fig.
4. Variation of S/S,, with position of twist. twist = 90” over a length L/8.)
(Total
X z 0.5 and a minimum at X z 0.25. Further the buckling load was higher for the case when the twist was at the end than when at the center, however the buckling loads for the case of opposite twist were lower than that for unidirectional twist. Figure 8 gives the variation of S/S,, with X for a twist of (- 90”, 90”) successfully over a length L/4. A prominent dip was noticed when the twist section was at X x 0.26. The tip values were nearly equal to the centre twist values, but, these were also lower than the corresponding equivalent of 180” unidirectional twist, though the difference was lower than that for the 90” case. Thus it may be concluded that only uniformly twisted beams have the expected buckling relationships whereas discontinuously twisted and non-uniform twisted beams may not have a maximum when the twist portion is positioned at the centre. Further unidirectional twist must always be preferred to opposite twist if the sole aim is to increase the buckling load. The buckling load was found to be dependent on the length over which it is to be twisted (L’). In the results given W = L’/L. The twist is considered to be symmetric about the centre of the beam. The dependence of S/S, for a 90” twist spread over a normalized length W for various values of I is shown in
0.30
0.40
0 50
6. Variation of S/S, with twist position. twist = 225” over a length 5L/16.)
0.60
(Total
Fig. 9. The variation with a was found to be characteristically different. For a > 4 it was seen to show approximately the same behaviour. For a = 2 a maximum occurs at W z 0.64 and a minimum at W z 0.24, whereas for a = 3 the maximum was at W 5 0.25 and the minimum at W N 0.60, for a = 4 the normalized buckling coefficient dropped from a maximum at W z 0.1 to a minimum at W z 0.74. Further the maximum values attained were found to be significantly higher than those for uniform twist of the same amount over the entire length. For a (45”, -45”) the variation of buckling coefficient with length of twist is shown in Fig. 10. This shows a steady increase to a maximum from a minimum value. This behaviour is quite different from the previous one which had a minimum also. This may be due to the symmetric opposite twist balancing for the vacillation of just unidirectional twist, forcing the buckling load to steadily increase. Some variety of combinations have been considered and presented in Table 2. It was always found that an opposite twist reduces the buckling load for any particular combination of twist position and total twist.
1 20
I”\,
1.50
0.20
X Fig.
1 b0 CY
-““““~I~~~~‘~“~I~“~‘,,~‘I”‘~‘~(“I’~
0 10
c
6
N-
1 40
3 1.15 :
1 10 c
s0
1.30
0
'c
120
u) 1.05 ; F
1.10
100
1 00
090 O.lU
0.20
0.30
0.40
0 50
X Fig.
5. Variation of S/S, with location of twist. twist = 180” over a length 3L/8.)
:
0 60
(Total
Fig.
X
7. Variation of S/S, with twist position. [Twist = (45”, -45”) over a length L/8.]
Technical
Note
1.50
555
2.80
?
I
(x=6
i
1.40
\
z’40: L cr=4
i 1.30 0
:
k
2.00
I
0
1.20
a=3
? t Lo 1.60 =
0.00
--s
a=
0.20
0.40
0.60
2
0.80
1.00
1.20
W Fig. 8. Variation of buckling coefficient with position twist. [Twist = (45”, 45”, -45”, -45”) over L/4.]
of
Fig.
9. Variation of S/S,, with length twist = 90” spread over normalised 4.
A case where a discontinuous beam is made up of two portions of twist symmetric about the centre was also considered. The dependence of S/S, with Y, the normalized length between the two twist portions, for various values of a is shown in Fig. 11. Y = 0 represents the case when the beam is singly discontinuous. It was found that the maximum buckling load occurs at Y z 0.22 for 01= 2, at YxO.l6fora=3,at Yx0,22fora=4andat YxO.24 for a = 6. These maxima are significantly higher than the maxima attained with uniform twist along the entire length. Another case wherein the two twist portions are oppositely twisted relative to each other was also considered. The behaviour was significantly different as shown in Fig. 12. The maximum was found to occur at Y x 0.38, this occurred at the same point for all values of a. Further it was found that the maxima attained with opposite twist was lower than the maxima attained with unidirectional twist for al! values of a. Table 2. Buckling
coefficient
of twist. (Total length W.)
CONCLUSIONS
The present study gives the characteristic behaviour of certain families of pretwisted rods. The formulation developed can be efficiently used in the analysis of complicated forms of twist. The maximum normalized buckling load of various families has been compared. Though the maximum buckling load for any combination is found to be greatly dependent on a, certain general results can be used without much error to achieve maximum buckling loads. The twist of nearly 225” was found to be the best possible total twist to achieve maximum buckling loads. The unidirectional twist was always found to have a higher buckling load than an equal amount of any oppositely twisted combination, the discontinuously twisted rods were found to have a higher buckling load than those twisted uniformly along the entire length. Further it can be concluded that within the cases considered here, the buckling load is the maximum achieved for the case when the total twist of 225” is in two portions separated by X z 0.20 and unidirectional.
for some twist combinations,
for various
values of a
Case No.
Element number 1 2 3 4 5 6 I
8 9 10 11 12 13 14 15 16 Buckling coejicient a=2 a=3 a=4 a=6
1
2
3
4
0 45 45 0 0 45 45 0 0 45 45 0 0 45 45 0
0 28.1 28.1 0 0 28.1 28.1 0 0 28.1 28.1 0 0 28.1 28.1 0
0
0
15 15 0 0 15 15 0 0 15 15 0 0 15 15 0
45 45 0 0 -45 -45 0 0 45 45 0 0 -45 -45 0
11.7 3.1 1.4 ^_ 0.2
17.29 5.21 2.40 .~ 0.360
16.29 4.95 2.37 0.425
11.13 2.87 1.29 0.209
5 0
28.1 28.1 0 0 -28.1 -281 0 0 28.1 28.1 0 0 -28.1 -28.1 0 10.40 2.63 1.17 0.360
6 0
45 -45 0 0 45 -45 0 0 45 -45 0 0 45 -45 0 10.33 2.61 1.16 0.209
I 0
28.1 -28.1 0 0 28.1 -28.1 0 0 28.1 -28.1 0 0 28.1 -28.1 0 10.04 2.52 1.12 0.360
8
28.1 28.1 28.1 28.1 28.1 28.1 28.1 28.1 -28.1 -28.1 -28.1 -28.1 -28.1 -28.1 -28.1 -28.1 11.42 3.02 1.36 0.200
9
14 14 14 14 14. 14 14 14 -14 -14 -14 -14 -14 -14 -14 - 14 19.26 5.99 2.76 0.416
in 0
45 -45 45 0 0 45 -22.5 -22.5 45 0 0 45 -45 45 0 16.69 5.38 2.58 0.202
Technical
556
Note
2.00 a=4
0 ? tn 1.50
a=3
[1:.:: /
c(-2
1.00
k
Fig.
10. Variation of S/S, [Twist = (45”, -45”)
with W for various over a length IF.1
a.
Fig. 12. Variation of normalized buckling coefficient with Y. [Two parts of 112.5”, oppositely twisted over (3L/l6).]
2.80
REFERENCES
1.60
0.80 i 0.00
0.40
0.20
0.60
Y Fig. 11. Variation of normalized buckling coefficient with Y. [Total twist = 225”, 112.5” is over a length (3L/l6).]
Structural and dynamic behaviour of 1. A. Rosen, pretwisted rods and beams. Appl. Mech. Rev. 44, 483-515 (1991). 2. W. Carnegie, Vibrations of pretwisted cantilever blading. Proc. Inst. Mech. Engrs 173, 343-362 (1959). Coupled bending-bending vibrations of 3. B. Dawson, pretwisted cantilever blading treated by the Rayleigh-Ritz method. J. Mech. Engng Sci. 10, 381-388 (1968). 4. A. Rosen, R. G. Loewy and M. B. Mathew, Nonlinear analysis of pretwisted rods using ‘principal curvature transformation’. Part I: theoretical derivation. AIAA Jnl 25, 470478 (1987). 5. N. Tebedge, Buckling of pretwisted columns. J. Sfrucr. Engng 108, 11241139 (1982). 6. E. B. Magrab and D. E. Gilsinn, Buckling loads and natural frequencies of drill bits and fluted cutters. J. Engng Indust. 106, 196204 (1984).