Buckling Loads For Variable Cross-Section Members With Variable Axial Forces
Buckling Loads For Variable Cross-Section Members With Variable Axial Forces
Buckling Loads For Variable Cross-Section Members With Variable Axial Forces
MOSHEEISENBERGER~
Department of Civil Engineering. Carnegie Mellon University. Pittsburgh. PA 15213. C.S.4
Abstract-This uork gives exact solutions for the buckling toads varidbk cross-section coiumns, loaded by variabie axial I&P. for severai booundary conditions. Both the cross-section bending sti%ess and the axial luad can vary along the column as polynomiaf expressions. The proposed sob&on is based on a new method &hat enabitz one to get the stiffness matrix for &he member including the effects of the axial loading. The buckting load is found as the load that makes the determinant of the stiffness matrix equal zero. Severai examples are given and compared to published
of
results to demonstrate the accuracy and flexibility of the method. New exact results are given for several other CUCS.
INTRODUCTION
Columns
with non-uniform
in engineering.
requirements.
columns were derived in the past. The cases that were treated
in the past and solved in this work can be divided into three subsets as Fotlotrs.
Gallagher
approximate
finite clcmcnt solution for monomial 1%~ (IWX) used improved vorsians of variable stiffness columns. Lately, Smith
vari;ltion of the tlcxuritl stitfncss. Mcich (1952) prcsentcd cxnct solutions for simpic monomial stillncss variations. of the Ra)Icigh Iremonger (1980) method fscrt (1983) and Elishakofland to obtain approximate solutions
for the buckling load using the cncrgy method, but these itrc
Timoshenko
(1932) presented exact solutions for mono(cantilever or symmetrically loaded added the solutions for three more cases of distributed axial force.
conditions
(1931) prescntcd exact solutions for monoElishakofJ solutions for two sets of boundary
lo&.
of both axial and stitJnncss load for simple boundary conditions. variation ofstilfncss
ilnci
and Pcllcgrini (1957) presented exact and approximate conditions and monomial
ilXi:ll
When using the finite element method for the case of vnriahle properties of rhc crosssection along the cofumn. it is common practice to USC some equivaicnt moment
solutions arc approximate. and improvement can be nchicvcd by using larger number of elements. This involves much work in the preparation of data. and results in larger finite element models for solution. Recently. Eiscnberger and Reich (1959b) presented an approximate finite element
solution that can be used far variable cross-section members. Later, Eisenberger and Reich
Institute
of Tcfhnology. 135
Technion
City 32OW
Israel
YI 2i.2.1
I?6
\l.
EISEVHFR(;EK
(I9YYa) used this method for all the three subsets that were presented above. and _cot excellent results for several known cases. In this paper an exact method for the stability of the stiffness matrix for variable cross-section (1989). Here. the stitTnrss matrix analysis ofcolumns with variable tlcxural rigidity and variable axial load along their length is introduced. It is based on the derivation members that was presented in Eisenberger functions of member is derived for general polynomial
properties including the effect of the variable axial load. The method is based on the wellknown power series solution stiffness to diKerential equations with variable coefficients. However, has been used to form the member, rather than solve for a particular set of procedure of the direct stiffness matrix is that
LW
this is the first time. to the authors knowledge. that this solution matrix of a variable cross-section This for the structure boundary conditions. on the solution
stitTness matrix is the exact stiffness matrix. and from that point is as for the well-known The advantage of having a stiffness
it can be combined with existing tinite element codes directly. and that all the well established procedures that were developed for the finite element method are valid here too. The of the stiffness conditions examples). method in the solution at the same ease (as shown in all the combinations as well as asscmblics in space. where variable cross-section analysis enables one to treat all combinations of boundary that were solved in the possible members arc desirable
application to large structures for weight reduction). The results of the stability
The diIti.rcntial
of a tapcrcd mcmbcr
equation rc;ids
whcrc /(_v) is the moment of inertia along the the axial force. /(.Y) is the distributed The solution the beam is not gcnerillly ilvail;lble.
bWln.
)I
Using the linitc clement technique. it is possible to derive the terms in the stifTncss matrix. We assume that the shape functions for the element are polynomials and we have to find the appropriate coctiicicnts. It is widely known that exact terms will result. if one uses the solution of the JitTcrential equation as the shape functions, for the derivation of the terms in the stiffness matrix. In this work exact shape functions arc used. to derive the exact stilincss cocllicients. These shape functions arc exact or up to ;L preset value set by the analyst. WC take the cocllicicnts bcil m in cqn (I) iis the following polynomial variation along the
LIP to
f?(x) = i
R,,r
A
(2)
N(x)
= f:
iv,.\.I
P,.\-.
(3)
P(x)
c
0
the number
of terms
in each series.
This
I37
representation
is very general.
and many
functions <
can be represented
(6)
with
(7)
(8)
(9)
infinite
power
series
\I(<) =
I
c
-
IV,<.
(10)
back into eqn (6) we have
,I
Calculating
and substituting
the expressions
+?)(i-k+
I ,
I)(i-k+2)rk,ztt,
_k+$
,F;,k~,,2(k+I)(i-k+I)(i-~~2)(i-k+3)r,+,l~,-k+)5'
i (i-k+ I)(i-k +?)(i-k+3)(i--k+4)r,\r, k+,< = i
r-
+ i
p,i.
(I 1)
, . I, k a I,
i
k .I,
(k+I)(i-k+l)ffk+Ibr,
.kcl-
(i-k+I)(i-k+2)ffkbr,
k+:
k - 0
c
k=
(k+ I)(k+?)(i-k+
I)(i-k+2)fk+2)(.t
.k+?
+ C (i-k+ k =I,
i)(i-k+2)(i-k+3)(i-k+4)rk~~,_k+d
= pi
(12)
13x
Or
EISENBERGER
+ k~o(i-L+i)(i-kf2)n,ir,.L,~x
c
k z 0
(k+ i)(k+?)(i-k-+-
i)(i-X-+7)rl,+2)(,_kcI
C l(fi+
k = 0
i)(i-kf
i)(i-X-+2)fi-X-+3)r,+,r~,_,+~
,$,
(i-x-+
I)(i-~+?)(i-~+3)(i-ii+?)rkl~,_k+~].
(1%
The terms for IL, + 4 tend to 0 as i --+ cc. Now we have all the W, coefficients except for the first four. that should be found using the boundary conditions. as degrees of freedom in the fo~uiation of the beam element. At < = 0 we have $10= W(0) and It1 = W(0) so the tirst two terms arc rcadiiy known from the boundary conditions. The terms II: and )1-j are found as follows: first four, and wt fan write All the H,Sarc linearly dependent on the (15) (14) For this case we choose the lateral deflection and rotation at the two ends
W(I) = t: W, = C,~W,,fC,W,
i-CJH?-tCJWj+
c;,,r,
, . f, w(I) = i
iI
(16)
07)
The 10 C coctiicients (c,,. C, , c?, C,, C;,. C;. Ci, C,. C,,,, and C,I,,) are expressible in terms ofail the coellicicnts in I+(;), n(T) and p(r). C,, for example, is the value of ~(1) when IV,, = I and II*, = II- = II) = pi = 0 calculated from eqn (IO) using the recurrence formula in eqn
as foiiows :
Irk
(18)
kbbqk = if
jJ kw,
k-4
(1%
both with iv& [from cqn (13)j based on II, = I. We,, = pk = 0; i, k = 0. I, 2.. . . , T_. and
(20)
i; I-0
c;, =
w( I) =
kf4 kw,
x
(21)
both with !lk [from eqn (13)f based on W, = 0; i = 0, I, 2. 3, and using the values p, for the particular loading. Then, knowing ail the terms in eqns (I 8)-(3 1). the values of wO and w,
members
13Y
IS)]. and the boundary conditions at x = L(< = 1) we can solve eqns (16) and (17) and find the unknowns H?and )vj. Thus, for any given variable polynomial functions [eqns (7)-(9)] we can find all the coefficients Itiin eqn (13). The terms in the stiffness matrix can be found as in the finite element method using the following expression
[eqns (14)-(
I
S=
F(<)El(j)F(<) d<
(22)
are the second derivatives of the basis functions. The four basis functions F
are found using eqns (IO). (14)-( 15) and (16)-( 17) for an
(1)
(2)
=
=
0 ;
= \t(l) = 0;
(3)
(4)
The shnpc functions that are found using this tcchniquc have the special property that they arc the exact solution for the diffcrcntial equation. The word exact in the previous This is so since the
GISCS I:
scntcncc stands for as exact as WC can get on a digital computer. the contribution
\vils
calculation of the C cocflicicnts is stopped according to a preset criteria : it could bc until of the next clcmcnt is less than an arbitrary IO Ix)
or
chosen
as
until the C
WIUCS
convcrgc complctcly (for the accuracy of the [ruthcr than by cqn (22)]. as follows: the
computer). In this work, the terms in the stifrncss matrix arc found in a simpler and faster way using the propcrtics of the shape functions due to unit translation or rotation. terms in the stiftncss matrix arc defined as the holding actions at both ends of the beam, at each of the four dcgrccs of freedom, one at a time.
Thus, corresponding to the four sets of boundary conditions above there arc four solutions W,;i = 1, 2, 3.4 for ~$5) which are found using eqns (IO), (13) and (16)-(21). Then. the holding actions will be :
r(0) d W, w=Ld+~d:~+-7 h
= 69 W,.,+2L 1 dr(0) d' W, n(0) d W,
L
r(0)
dc
(23)
r(0) d W,
wo)
r(1) d' W,
-Ld-~d_--7---
= - F
dI
4
(24)
V(l)=
n(l) L.
dWi d5
M. EISEUEKGER
(17)
(29)
S(4, i) =
r$ k-2 2 x-(x--
l)Ct;.,
where IV;,, are calculated using the r,r coeflicients. Then the buckling load for variable cross-section members, or frames with such members, can be found as the axial loads N(X) in the members, that cause the determinant of the corresponding stiffness matrix to become zero. This is done using r? routine that converges on the values of the axial load that satisfy this criteria. The procedure was incorporated examples. At this point, before going into examples, an ova-all discussion and conlp~lrisoll of the proposed method with the finite clcmcnt mcthod is prcsentod : onccan look at the proccdurc suggested in this work as an addition to the linitc element method, as one dcvcloping a methodology to derive shape functions solutions that yield the exact stifrncss matrix. When using the However. it will take scvcral in order to upply ;Ln error estimate that will Using the proposed method this is not From the computational matrix to dcrivc the exact stillness finite clement method, one can convcrgc to the solution. with increasing number ofelcmcnts yield a very good, but still approximate solution. into ;I regular beam analysis program and demonstntcd in the following
as outlined in this work. But, when this is viewed in comparison to assembling the stiffness matrix for 20 or 50 clcments, and the fitct that the size of the eigcnvulue problem that results in the stability analysis, is much smaller, more than otTsets the longer derivation time. As an cxamplc, for a fixed-free of the buckling column with variable cross-section, and variabte axial load
load (as shown in an cxamplc in the next section). Icads to a 40 by 40 the power of the new method is demonstrated 1984) compared to the result in the
eiyenvalue problem, compared with a 2 by 2 matrix for the proposed method. In the examples that follow, solution of many cases where exact solutions were not available. Also, some comparisons in (Swenson,
that were made to the wrong values [(Bert, 1952)] are pointed out.
EXAMPLES
The method was first checked for the classic Euler buckling casts for columns. For all the c;1scs. the method yiddcd the exact theoretic;ll solutions, using only one element for the whole member. The exanplcs in this section arc divided according to the three cases that were presented in the introduction. (i) C~~ri~~hi~~~~surf~l sI(f~iw.s.swith cwisfwt trxid iwtl
Consider the column that was solved by Swenson (1952) and later by Bert (1984) and Elishnkoff and Bert (1988). The mcmbcr moment of inertia is given as
Buckling loads for variable cross-section members Table I. Values of f in eqn (32) for members with variable flexural stiffness and constant axial loads Buckling load Example I Swenson (1952) 3.1176968 4.14183;;6 14.51 lllY540 29.~S96~~~6 Example 2 Bleich ( 1951) 3.836376918 6.731865407 20.79Z!X8J56 42.10917612~ 42.10)176122 81.923363881
141
Boundary conditions
9478X-Wb
;7:393956;36
simple supports at both ends (and the reported nondimensional buckling load iii 13= NL
El,,
are given in Table conditions. for the fixed-fixed I. Also results are given for five more combinations
(2)
of bounddry
In all the examples only one element was used to find the critical load. except case where two elements were used (but only two degrees of freedom). using the npproximatc method in Eisrn-
It should also be noted that all the results that are presented in this paper were checked against the converged values that were obtained bcrgcr and Reich (lW%). Another cxamplc
is the column that was solved by Blcich (1952) and latcr by Bert
For this example. the moment of inertia along the
f(5) = I,,( I
The values of the normali& of boundary conditions. buckling
+i,:.
(33)
Blcich solved exrtctly for the hinged - hinged cusc and obtained the cast are exactly the same. This is only due to the particular from it, this no longer holds.
s;lmc value. It should be noted here that for this special member the buckling loads for the fixed -hinged and hinged-fixed variation (ii) in cross-section properties, and for small deviation
loud along the member that was solved by Dinnik Grrc (I 96 I) on p. I3 I. The variation
of the distributed
I/(X) = c/#y
(3)
where the subscript h indicates the values at the base of the column. Then. the critical loads
are given as
L IFI
EI
Table 2. Values of ,)I in eqn (3.5) for members with constant flexural stiflhcss and variable axial loads Boundary conditions Upper end Free f lineed Fix&i Hinged Fixed Lower end Fixed Hinged Fixed Fixed p=o 7.837347 1X.56X725 30.00942 I 52.500663 74.628569 /!=I 16.100953 23.238937 36 7686 78:98%!~9 107.s732t2 Buckling loads p=2 27.256905 X674598 41.916950 I~.~~055 139.54143-t P 3 p=4 58.24450 32.703955 50.4607 IY 158.89539-i 205.0370~1 p=S 78.07591 I 35.630368 54.431 I3 I9O.O-W53 240.67I I3
Approximate-FE
3 elements 11.90513 2Y.58 1692 SX.17.i81 65.Y26002 I IY.0136Y
Weak
md
5 elements
13.557930 2Y.257630 47.7173 XL;?7 171) Il7.775740
IO elements
13.503883 29.21 I988 Jh.Ul7JI X3.YJ571 I Il7.610867
20 elements
13.865669 29.200512 16.257SX X4.34X360 ll7.6179IX
50 elements
13.854X4 19.197909 16.203(650 X4.1663RO I I7.630016
Timoshenko
for the free-fixed case. In Table 1 the Table 2 contains also values for the and Gcre (1961) for II = 4.5. and these combinations are given in Table but 18.53 for the
exact values of HI that ivcre calculated using the proposed method are shown, and they agree lvith the values in Timoshenko free-fixed case that were not given by Timoshenko Reich ( IYYYa). 2. Frisch-Fay Four
values agree with the con\,ergcd approximate values that were reported by Eisenbergcr and more cases of boundary conditions distributed axial force. The appro~itn~ltions.
for three more cases of boundary conditions, values that he gave were:
2nd 74.65 for the fixed -fixed cast. It hinged hinged cast; 51.40 for the hinged 3ixcd cast, . ,
can hc seen that thcsc arc still calculation that he pcrli)rmcd. probably due to the accuracy of the as his method is csact. All other cases appear hcrc, apparliar the higher values of /I in Table 1 indicate that the
51, ~*nilicant
is so. ;is tlic load is conccntratcd niorc in the loivcr part of the ;incl Lhc tipper hall 01 the column is hardly loadccl, so that its 1hCsh:ipc at the top.
;is p is incrciscd.
that arc available for this cast are those given by Timoshenko
and
However.
in alI thcsc cases, the moment of inertia at the top of the column Such casts with zero stillness, cannot be solved Therclorc, for this case, the results will be in this work.
was taken as zero, which is not realistic. using the mcthotl prcscntctl
using the finite element method, the member Orxural rigidity and the axial are taken as constant all along the element, as the value at the mid length point of the element. As an example, the column in the first example that was solved by Swenson (1952). but with uniformly distributed load along the member, will be used here. The load is taken in such a rvuy that the maximum axial load
is at the stronger end. In Table 3 the rcsulls arc given ror tivc combinations of boundary and compared with the results trom the approximate solution using 2, 5, IO, 20
conditions.
and 50 elcmcnts along the mcmbcr. It is seen that the appro?timatc results converge to the exact results for all the c;lscs. Thcrc are two problems with the well-known finite element solution in thcsc casts: the first is that the relative errors arc not known and several runs are needed to Knd if the solution is within some error criteria. The second is that the convergence is for some cases conscrvativc (i.e. the exact buckling load is below the Cnitc elcmcnt solution) and in other cases it is nonconservative estimate. Overall. the computer time for the more exact tinitc clcmcnt solutions (20 and 50 clcmcnts) was longer than the time for the exact solution ;IS prcscntcd in this Lvork. There is also the guarantee that only one solution is ncctlcd and th:tt it will yield the exact solution, lvhen using the proposed method.
Another exam& is that or the sway bucklin, 0 or the frame in Fig. I. The frame is composed of four tapered members with linc;lrly varying moment of inertia. bvith end values
I43
Fig. I. Example
as shown. Utilizing
symmetry of the problem. only two members were used for the solution
DISCUSSIOS The
AND
SUMMARY
method that was presented in this work is hased on the solution of the difTerential variation of the cross-section properties. Then, the results for is
the buckling loads arc exact. The application of diKerent sets of boundary conditions
strni_chtfortvnrtt{ as in the standard stitfncss method of analysis. The first advantage of the method is that it gives exact values for the buckling load (rather than approximate in other methods). ~~)Iiip;irir~~ this method to the tinitc clement method or the finite difliircncc
used
method points out the second ndvantagc ofthc method : only one clement is nccdcd for the solution. Thus, the results ilrc computed much faster. The method wu also
to
In this work, exact buckling loads (up to the accuracy of the computer) for variable cross-section mcmhers with variahlc ilxi;tl loads arc given. These were derived using a new elcmcnt based method that enables one to find the stiffness matrix for mcmbcrs with any polynomial variation of the cross-section and axiiil Ionding. In the cxnmplcs, for the three it is shown that the method gives exact results clusscs that wcrc listed in the introduction, for various combinations complex structures.
compared to known buckling loads. Many new exact values for buckling loads are given
of boundary
can be incorporilte~~ into regul:tr frame programs to yield exact buckIin~ loads for more
REFERENCES Bert, C. (1984). Improved tLvhniquc for rstimating buckling loads. J. Enytry dfrch. Die. 110, 1655-1665. Blcich, F. f 19.57). ~~~~~~lj~f~ S~rr~~~~~t ~~.,~~~fu~ Sfrrrtturrs. McGraw flill, New York. Dinnik. A. (1932). Design ofcolumns of varying cross-sation. Tru~.r. ASMES4, 165-171. Elhcnbcrger, M. (19X9). Enact static and dynamic stiKncss matrices for variable cross scction mcmbcrs. In 30111 S/rucfrtrcs, D~~~rtrn~ics. trntl .\/orrritr/s Cotr/:. pp. 1(52-X%. Mobile. Alabama AIAA, U.S.A. Eisenbcrgcr. M. and Reich, Y. ( ISYYa). Buckling of variable cross-sation columns. In Srvel S~rucrurrs. Proc. Slrttcrures Conyr.. pp. 443 451. San Francisco. ASCE, New York. Eiscnbcrger, M. and Reich. Y. (198Yb). Static. vibration, and stability analysis of non-uniform barns. Cun~pur. Srrut-r. 31. 567-573. EIishakolf. I. and Bert. C. (19%). Comparison of Rayleighs noninteger-power method with Rayleigh-Ritz method. Cotttp. .tlerlr, .4&, .\Icch. &qtrq 67, 297.-30Y. Ehshakoli. I. and Pcllqrini. F. ( 1987). Application olbesscl and lommel functions. and the undctcrmincd multiplier Galcrkin mcthod bcrsion. for rtahility of non-uniform columns. 1. Sorcnd Vibr. 115. IX?-186. Frlsch-pay. R. (1966). On the stability of ;Lstrut under uniformly distributed axial forces. Irrr. 3. .Soli~f.s Slrucrures 2. 361 -369. Gallagher. R. and Lee. C. (t970). Matrix dynamic and instability annlysis with non-uniform clcmcnts. faf. /. *Vtmlr~r. sl<~th. EfaqlJ{q 2. x5 275. Ircmonger, M. ( 19%)). Finite diffcrcncc buckling aniilysis of non-uniform columns. Cr,ntpuf. Sfruc~. 12,7Jl-748. Smith. W. f 1988). Analytical solutions for tapered column buckling. Contpltf. S/rucl. 2% 677-681. Swenson. G. (1052). Analysis of nonuniform columns and beams hy n simple d.c. network analyzer. J. Aerorruur. Sci. 19, 73 -776. Timoshenko, S. and Gcrc. J. ( IYhl ). 7%cr~r~ O/E/mfic Smhili!~. _n d edn. McGraw Hill. New York.