To explore the feasibility of using buckled columns to provide a soft support system for simulating a free-free boundary condition in dynamic testing, the nonlinear load-shortening behavior of initially imperfect, eccentrically loaded slender columns is analyzed. Load-shortening curves are obtained for various combinations of load eccentricity and uniform initial curvature and are compared, for reference purposes, with the limiting case of the classical elastica. Results for numerous combinations of initial curvature and load eccentricity show that, over a wide range of shortening, an axially loaded slender column exhibits load-deflection compliance which is of the same order as that of a straight but otherwise identical cantilever beam under lateral tip loading

Fichter, W. B.

Pinson, Mark W.

English

NASA Technical Reports Server

NASA Technical Memorandum 101643
LOAD-SHORTENING BEHAVIOR OF AN
INITIALLY CURVED ECCENTRICALLY
LOADED COLUMN
W. B. Fichter and Mark W. Pinson
December 1989
NA. A
National Aeronautics and
Space Administration
Langley Research Center
Hampton. Virginia 23665
(NASA-T_-IOlo4 3)
OF AN INITIALLY
COLUM_ (NASA)
LOAD-.SHORT_NING BFHAVIOR
CURVED FCCENTRICALLY LCAOEO
22 p CSCL 20K
G3/j9
N ,o0-16293
Uncl,_s
0256749
https://ntrs.nasa.gov/search.jsp?R=19900006977 2020-03-20T00:20:30+00:00Z
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=
LOAD-SHORTENING BEHAVIOR OF AN INITIALLY
CURVED ECCENTRICALLY LOADED COLUMN
W.B. Fichtcr
and
Mark W. Pinson
SUMMARY
To explore the feasibility of using buckled columns to provide a soft
support system for simulating a free-free boundary condition in dynamic
testing, the nonlinear load-shortening behavior of initially imperfect,
eccentrically loaded slender columns is analyzed. Load-shortening curves
are obtained for various combinations of load eccentricity and uniform
initial curvature and are compared, for reference purposes, with the
limiting case of the classical elastica. Results for numerous combinations of
initial curvature and load eccentricity show that, over a wide range of
shortening, an axially loaded slender column exhibits load-deflection
compliance which is of the same order as that of a straight but otherwise
identical cantilever beam under lateral tip loading.
LIST OF SYMBOLS
A
C
E
I
L
M
P
R
a
k
P
q
r
s
area of cross section
constant of integration
Young's modulus
cross-sectional moment of inertia
length of column
moment
applied vertical force
radius of curvature
moment arm for applied vertical force
force-stiffness parameter
parameter
normalized vertical force
parameter
arc-length coordinate
tX
Y
A
0
K
integration variable
vertical coordinate
horizontal coordinate
column tip rotation
column shortening
column cross-section rotation
column curvature
2
Subscripts 0 and e denote initial stifle and Euler buckling value,
respectively.
INTRODUCTION
In many structural dynamics tests it is desirable to minimize the
interaction between a structure and its support system. One frequently
used scheme involves suspending the structure on long, flexible cords
when, for example, free vibration frequencies are of interest. Such an
arrangement is rrrost satisfactory when the suspension cords are long
enough to minimize side loads and the natural frequencies of the
suspension system are separated sufficiently from those of the test article.
Another possible approach is to mount the structure on
compression springs, if the springs can be made to have sufficiently
low stiffness while supporting the weight of the test article. One device
which might be able to satisfy the combined flexibility and load-bearing
requirements, and simultaneously avoid the need for spacious overhead
support capability, is the buckled column. After buckling, the column
continues to support its buckling load, while over a considerable deflection
range the slope of its load-shortening curve (i.e., its tangent stiffness) may
be low enough to satisfy test requirements.
To investigate the feasibility of such an approach, the shortening due to
the large lateral deflections of an eccentrically loaded, slender elastic
column with initial curvature is examined. Both the initial curvature and
the load eccentricity serve to preclude buckling instability which would be
troublesome in a support system. For reference, the load-shortening
curves for various combinations of initial imperfection and load
eccentricity are compared with results from reference 1 for the classical
elastica problem.
3
ANALYSIS
Governing Equations
A sketch of a cantilevered, initially curved, inextensional column, before
and during eccentric loading at the free end, is shown in figure 1. For
simplicity, the initial imperfection shape is assumed to be a circular arc of
radius R 0. Hence, the initial tip rotation 70 is given by 70 =_I.,_ ,where L is
R0
the length of the column. The vertical force P ia applied at a normal
distance a from the column centroid, and so as to cause tip rotation of the
same sense as the initial curvature. During deformation, the moment arm
of length a is assumed to rotate with the end of the column. Equating the
internal moment at a distance s measured along the column centerline
from the tip, to the moment due to the applied force P yields
-EI0c-_Co)=P(y+acosT) (1)
where the initial curvature is r0=--L- 7is the angle of tip rotation relative
Ro '
to the vertical, y is the lateral displacement relative to the tip, _¢ is the total
curvature, and E1 is the column bending stiffness.
In terms of 0, the angle of rotation of the tangent at s, _¢=d0
ds
equation (1) can be written as
Thus,
do ___1_ k2(y + a cos 7)
ds Ro (2)
where k2= P The boundary conditions areE1 "
and
0(0) = Y (and y = 0)
0(L)=0
(3)
(4)
Differentiation of equation (2) gives
d20 =_
ds 2 ds
or
d20 =_ k2sin0
ds 2
4
(5)
Multiplication by dO and integration yields
ds
d_s! = 2k2 cos 0 + C (6)
Applying boundary condition (3) and using equation (2) in
equation (5) gives
C = {R_o+ kZa cos T_ - 2k2 cos _/
Thus,
d0_s-4(_ +_ oos_)_,k_(co,_o_o) (7)
where the negative sign is consistent with the fact that 0(s) is a decreasing
function of s.
Employing the identity cos 0= 1-2sin2_, integrating over the
length of the column, and using boundary condition (4) yields
I' 00
L= _/(R_0 + k2a cosy)2+ 4k2(sin_ _sin_ )
(8)
With the definition p = sin_ , and the change of variable pt = sin2_, equation
(8) becomes
f'L = 2p (It
4( 1 k2a cos _t)2
_+ +4k_p_ 1-4i-_p_1-ff_
(9)
5where
r = n2e q cos
_o + 4L +/t2p2q
With the Euler buckling force for a
perfect column denoted by Pe =x2EI, then k 2 n2
4L2 = 4---_' where
q =-P-- is the normalized applied force. Finally, equation (9) can be written
Pe
as:
(10)
which is the governing relation between the applied force and the initial
and final tip rotations.
To obtain the relation between applied force and the column
shortening, which is defined herein as the vertical tip displacement in
excess of that due to the initial curvature, an expression for vertical
displacement is needed. (Note that in the present analysis the axial
compressive compliance of the column has been neglected.) Since _d#__=cos 0,
ds
where x is measured downward from the tip of the column, then
dx = cos0ds = cos0d-0-_d0, so that use of equation (7) gives
dO
dx = - cos 0 dO
_/(R_o + kZa c°s Y)2+ 4k2{ sin2-T2 "sin2OI
(11)
With the definitions and change of variable employed earlier, integration
of equation (11) over the length of the column yields
= 4rL f' 'V_-r2t 2dt_ L
xt _-"_J0 _ (12)
where xt is the height of the deformed column, and equation (10) has
been used to simplify the result. The column shortening due solely to
initial curvature is L-_; hence, the net shortening due to the vertical
%
tip force is A =L-xt-(L-_1 or% J
A=lq sin% 4r (l_dt
L Vo  ¢q]o (13)
To obtain the relation between applied force and net column shortening
for specified values of initial curvature and force eccentricity, equations
(10) and (13) must be solved simultaneously.
Computations
A numerical approach was used in solving equations (10) and (13).
Equation (10) was rewritten as
_-_qI' 1 =0 (14)f(V,Vo,a,q) = ,qr__ p2t25/-__dr r2t2
which is the form required by routines for solving nonlinear equations.
Before this equation could be solved, however, a numerical problem in
evaluating the integrals in equations (13) and (14) required resolution.
When r is very close to one, both integrals are nearly singular, giving rise
to numerical integration inaccuracy. To circumvent this problem a change
of variable was made, and asymptotic expansions were developed to
evaluate the integrals on the subinterval 0.99 < t < 1.0. The integrals and
their replacements are
1 = 1+
dt
---- lr-{J1 J12}
II = 1-_]-_p_2_ 1- r2t2 -
where
,T
Jil =
7r2! ' 4 r 2
J12= + 72__(1__28 l p2- 9p4 5p6_/(-i _ T)732r 2 8r 4+2r61 •
and
where
I2 =/'/1 "f_- p2t2dt = ]r-(J21+ J22)
.Io
iVJ21 = _ _ u2 du
- r2
and where, for the present computations, T = 0.99.
For prescribed values of a, _'o and ?, equation (14) can be solved for the
normalized force q. The integrals Jll and J21 were evaluated using a
standard mathematical library routine based on Simpson's rule. Equation
(14) was solved by use of an available mathematical library routine which
first bounds the root and then uses interpolation to obtain a converged
value. An error criterion for f of 10 -4 was used in equation (14). After
equation (14) was solved, _- was calculated from equation (13), again using
L
the routine based on Simpson's rule. A complete load-shortening curve
was generated by incrementing 7 and repeating the solution process for
numerous prescribed values of a and _'o"
RESULTS AND DISCUSSION
The results are presented in figure 2 as plots of q(_) for various
,,Ik,,d
combinations of initial curvature and load eccentricity. For reference, the
8
elastica solution, which the present solution approaches from below as load
eccentricity and initial curvature vanish, is also shown in each figure.
Figure 2(a) shows that for zero load eccentricity, the present results
are barely distinguishable from the elastica solution when initial curvature
is very small (T_= 0.0001). As initial curvature is increased, the deviation
from the elastica solution becomes more pronounced, particularly at small
values of column shortening. However, for #-- greater than about 0.05, all of
L
the load-shortening curves are essentially parallel to the elastica curve,
indicating that the tangent stiffness of all the "buckled" columns is about
equal to that of the elastica.
For progressively larger dimensionless load eccentricity, _ , the relative
importance of initial curvature is seen to diminish. In figure 2(f), for
example, note the small range in the results for initial curvature values
from 0.0001 to 0.03. Hence, a value of 0.05 for dimensionless load
eccentricity would ensure smooth nonlinear load-shortening behavior, with
little chance of troublesome column instability. However, the slope of the
curve is somewhat greater than that of the elastica in the A-range of
L
interest, say, 0.1 to 0.3, indicating a tangent stiffness which is above the
minimum achievable.
Figures 2(b) through 2(e) are for values of load eccentricity in the
range 0.002 < a< 0.02. As expected, 311 of these results lie in the envelope
L
defined by the curves for load eccentricity, a of zero and 0.05, and show
L'
in the A-range of interest a gradual but discernable trend toward higher
L
slope (thus, higher tangent stiffness) with increasing load eccentricity.
Thus, the desire for the lowest practicable tangent stiffness would require
that load eccentricity be kept reasonably small.
The results also show that the effects of initial curvature and load
eccentricity are essentially the same. Thus, the relatively soft
"postbuckling" behavior of the nearly perfect slender column can be
obtained by imposing reasonable values of either initial curvature or load
eccentricity. In most situations it would appear that, of these two
9
perturbing influences, load eccentricity would be easier to quantify and
control.
An appreciation of the magnitude of the tangent stiffness of the bent
column can be gained by comparing the slope of the load-shortening curve
for bending with that of the prebuckling load-shortening curve for the
straight column under axial compression. For illustrative purposes, the
two curves are combined in the sketch below to represent a continuous
prebuckling and postbuckling load-shortening curve for a perfect column.
The slope of the postbuckling curve is exaggerated for easier visualization.
P
A
Note that
tanct =(dP)bdA = _--_LP--)2tan[3<<(dP)c =tan[3"dA
From the elastica solution (ref. 1), it can be shown (See Appendix) that
the initial postbuckling slope of the load-shortening curve is, in
10
dq =1, which can be rewritten as (d__)=xZEI,
dimensionless terms, _ b
,L,,
where the subscript b denotes bending. (As can be seen in figure 2, this is
a close approximation to the present results for a large range of column
shortening.) The corresponding expression for prebuckling axial
compression is /dPl = ,_-_-, where the c denotes compression.
_ /c 1-'dA
the two tangent stiffness measures is
The ratio of
dal E _ _2I rc2/ t2
(d_-)c 8L2---A - _ ,L-] '
where P is the radius of gyration of the column cross section. For slender
columns, this ratio is very small. However, to put this result in
perspective, note that the initial postbuckling slope is of the same order of
magnitude as the slope of the load-deflection curve for the tip-loaded
straight cantilever beam of length L, i.e.,
This means, of course, that a buckled-column support system would
simulate free-free specimen support conditions about as well as would a
system employing cantilever beams. Hence, although the slope of the load-
shortening curve for the buckled column is very small in comparison with
the perfect column's prebuckling slope, it is not negligible, particularly
when compared with the initial postbuckling slope of the plot of load as a
function of lateral tip deflection, which is zero.
CONCLUDING REMARKS
In this paper, the load-shortening behavior of an eccentrically loaded,
initially imperfect slender column has been analyzed. The results indicate
that such columns can be designed to behave as relatively soft springs
11
over a wide range of column shortening, for reasonably broad ranges of
initial curvature and load eccentricity. Both initial curvature and load
eccentricity are seen to promote similar nonlinear behavior. Because load
eccentricity is probably easier to impose with accuracy, it is likely to be
the preferred parameter to specify in order to produce and control the soft
"postbuckling" behavior of the slender column. If the tangent stiffness of a
"buckled" column is low enough to satisfy test requirements, then a
specimen support system that incorporates eccentrically loaded slender
columns may present a useful alternative to a system requiring very long
suspension cords. It should be noted, however, that the tangent stiffness
of the deformed column is of the same order as that of an initially straight,
tip-loaded cantilever beam and, thus, may not be low enough to satisfy
some test requirements.
REFERENCES
1. Timoshenko, Stephen P. and Gere, James M.: Theory of Elastic Stability,
second edition, McGraw-Hill Book Company, New York, 1961.
2. Abramowitz, Milton; and Stegun, Irene, Eds.: Handbook of Mathematical
FUnctions, National Bureau of Standards Applied Mathematics Series 55,
1968
APPENDIX
The Initial Postbuckling Slope of a Perfect Inextensional Column
From reference 1, the two equations governing the load-shortening
behavior of the elastica are, in dimensionless terms,
and
_L = k-_F-(p)- 1 (A1)
kL = K(p) (A2)
where K(p) and E(p) are complete elliptic integrals of the first and second
kind, respectively, p=sin_, and k= Xfq-. Letting _. =_, (A1) and (A2)2 L
become
and
I+K - 2E(p)
K(p)
12
(A3)
_/-q-= K(p)
2
Differentiating both equations with respect to 7_,
and
dp _ K2
dp dp}
(A4)
(A5)
dX
Inserting (A5) into (A6) gives
dq _ 4 _r-q- d_K_d__EP
dpd_.
K_K
d q = 2¢q-_. dp
d_. _ BdK_ KdE
dp dp
For load only slightly above the buckling load (q=l), p=sin tx
2
(A6)
(A7)
is very small.
Hence, K(p) and E(p) have the valid power series
(A8)
and
E(P)=2_(1 P2 )4 64_-p4 +"" (A9)
Substitution of (A8) and (A9) into (A7) yields
d q _ "(q-(1 + O(p2))
dK 2
13
As q_l+, p_0; therefore,
from above.
as the buckling load is approached
14
x T
i11]!1
Figure 1. Coordinate system and nomenclature.
q1.2
1.0
0.8
0.6
0.4
0.2
0.0 I I g
15
0.0 0.1 0.2 0.3
A
L
(a) a= 0
L
Figure 2.- Load-shortening curves for an initially curved, eccentrically loaded column.
0.4
q1.2
1.0
0.8
0.6
0.4
0.2-
I I I
0.1 0.2 0.3
A
L
1 6
o•
0.4
(b) a = 0.002
L
Figure 2.- Continued.
q1.2
1.0
0.8
0.6
0.4'
0.2
0.0
i=.=
Elastica . .. '.. "'..'"" ""
• • • • °
= •
: _ = 0.0001
\ ro=O.O 
• To= 0.02
To= 0.03
I I I
17
°•
0.0 0.1 0.2 0.3
A
L
(c) a = 0.005
L
0.4
Figure 2.- Continued.
1o0'
18
q
0.0
0.0
Figure 2.- Continued•
!
0.1
I
0.2
A
L
(d) a = 0.01
L
!
0.3 0.4
q1.2
1.0
0.8
0.6
0.4
0.2
0.0
°°
• ° ° •
• • o •
J • • •
• " °° ° °
.." "-".'_ 70= 0.0001
0.0
!
0.1
(e)
!
0.2
A
L
a = 0.02
L
0.3
-•
19
0.4
Figure 2.- Continued.
q1.2
1.0
0.8
0.6
0.4
0.2
0.0
i
0.03
' I I I
0.0
Figure 2.- Concluded.
0.1 0.2 0.3
A
L
(f) a- = 0.05
L
2O
0.4
Report Documentation Page
1. Report No
NASA TM-I01643
2 Government Accession No.
4. Title and Subtitle
Load-Shortening Behavior of an Initially Curved
Eccentrically Loaded Column
7. Author(s)
W. B. Fichter and Mark W. Pinson"
9 Pedorming Organization Name and Address
Langley Research Center
Hampton, VA 23665-5225
12. Sponsoring Agency Name and Address
National Aeronautics and Space Administration
Washington, DC 20546-0001
3. Recipient's Catalog No.
5. Report Date
December t989
6. Performing Organization Code
8. Performing Organization Report No
10. Work Unit No.
585-02-31-01
11. Contract or Grant No.
13. Type of Report and Period Covered
Technical Memorandum
14. Sponsoring ,_gency Code
15. Supplementary Notes
16. Abstract
To explore the feasibility of using buckled columns to provide a soft support
system for simulating a free-free boundary condition in dynamic testing, the
nonlinear load-shortening behavior of initially imperfect, eccentrically loaded
slender columns is analyzed. Load-shortening curves are obtained for various
combinations of load eccentricity and uniform initial curvature and are compared,
for reference purposes, with the limiting case of the classical elastica. Results
for numerous combinations of initial curvature and load eccentricity show that,
over a wide range of shortening, an axially loaded slender column exhibits
load-deflection compliance which is of the same order as that of a straight but
otherwise identical cantilever beam under lateral tip loading.
17, Key Words (Sug_sted by Authorls))
Column buckling
Nonlinear ana]ysis
Elastica
18. Distribution Statement
Unclassified - Unlimited
Subject Category - 39
19. Securi_ Cla_if. (of this report)
Unclassified
20. SecuriW Cla_if. (of this pa_}
Unclassified
NASA FORM 1626 OCT 86
21. No. of pages
21
22. Price
A0 3



Load-shortening behavior of an initially curved eccentrically loaded column

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