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Beams on generalized foundations: supplementary element matrices
EngineedngStructures,Vol.
ELSEVIER
PIh S0141-0296(97)00179-4
19, No. 11, pp. 910-920, 1997 © 1997 Elsevier Science Ltd All rights reserved. Printed in Great Britain 0141-0296/97 $17.00+ 0.00
Beams on generalized foundations: supplementary element matrices B. N. Alemdar and P. Giilkan* Department of Civil Engineering, Middle East Technical University, 06531 Ankara, Turkey (Received August 1996; revised version accepted December 1996)
A general analytical solution for the shape functions of a linear beam segment supported on a two-parameter elastic foundation is derived. The solution is not restricted to a particular range of magnitudes of the foundation parameters. The closed form shape functions are utilized to derive analytic expressions for work equivalent nodal forces for arbitrary transverse loads and coefficients of the consistent mass and geometrical stiffness matrices. Each work equivalent nodal force and each coefficient of the element matrices is compared with its conventional counterpart for an ordinary beam segment supported by no foundation. © 1997 Elsevier Science Ltd. Keywords:two-parameter elastic foundation, work equivalent nodal force, consistent mass matrix, consistent geometric stiffness matrix
1.
Introduction
special case of a beam segment supported by a singleparameter Winkler foundation, which in turn, is a subset of the more general two-parameter case. This follows directly from the governing differential equation. The derivation of the exact shape functions for a beam segment supported by a two-parameter elastic foundation must precede all other calculations. Once exact onedimensional shape functions have been derived, it becomes possible to manipulate them to derive exact expressions for coefficients of the element stiffness matrix, or the consistent geometric and mass matrices for beam segments as well as work-equivalent joint forces for a variety of concentrated or distributed loads. We call these the supplementary element matrices. They correspond to the quantities which can be derived from cubic Hermitian polynomials for uniform beam segments. Their expressions are lengthy and cumbersome, but this is outweighed by virtue of the fact that once they are made part of a structural analysis software, they enable exact analyses to be performed with fewer elements. One aim of this paper is to display in graphical form the analytic expressions for the element matrices, noting the smooth transition from the complex Pasternak solution to the familiar beam cubic displacement-based expressions for each distinct coefficient. A brief note on implementation is given before a sample problem is solved.
The need for a solution for stresses in beams, plates and shells in contact with, or wrapped by, elastic or viscoelastic media is commonly encountered in many diverse problems ~-4 such as buried pipelines, piles, foundation elements, reinforcing filaments in composite materials, and even endodontics. The beam theory can be traced to Winkler's original work on stresses in railway tracks where the idea of representing the soil underneath tracks by a series of closely packed, but otherwise unconnected springs leads to a solution for the internal forces 5. This type of foundation is characterized by a single distributed spring stiffness called the Winkler foundation parameter. Subsequent generalizations j,6, have interpreted the possible physical interaction between these springs through what will be designated in the following as the Pasternak parameter, enabling applications of wider range. More recent studies 7-9 have focused on the derivation of elements of the stiffness matrix for beam segments, but given the lengthy mathematical expressions the coefficients have usually been restricted to a particular range of the relative magnitudes of the Winkler and the Pasternak parameters. This limitation can be removed with little mathematical inconvenience, as will be demonstrated in the following. Geometric stiffness and mass matrices for beam segments embedded in this more general foundation have also found use in stability and vibration problems. The development below will reconfirm that a beam segment not supported by any foundation is a
Notation A B
B
* Present address: School o f Civil Engineering, Purdue University, W e s t Lafayette, IN 4 7 9 0 7 - 1 2 8 4 , U S A .
910
kJEI k/E1 column vector of transverse displacement functions
S u p p l e m e n t a r y e l e m e n t m a t r i c e s : B. N. A l e m d a r a n d P. GOIkan
C ci H
E1 ke
kg k kt L m
m(x) N P P p
q(x) S U W
column vector containing constants gration ci integration constant, i = 1 - 4 matrix defined iin equation (8) flexural stiffness beam element stiffness matrix consistent geometric stiffness matrix Winkler foundation parameter Pasternak foundation parameter beam segment length consistent beam mass matrix distributed moment row vector of siaape functions N~ vector of work equivalent nodal loads axial force AL transverse load
of inte-
d4w
w Z
6/A 2
ol
A2~
w(x) = c, cos13x coshax + -I- C3
+
X/8- )t 2
which
correspond
A
4 k ~/4EI
/x
2.
sin13x coshctx + c 4 sin13x sinhax
(2a)
C3e-4q~x
+
(2b)
c4xe -4@x
+ c3 cosh13x sinhax + c4 sinh13x sinhax
to
PL ~/1 - z or pL x/z - 1, respectively
kfl(4El)
C2 COS13X sinhax
w(x) = cl cosh13x coshax + c2 sinh13x coshax J
3
(1)
W(X) = Ci e4q'~x + c2xe 4q'~x
=p
or
+ kw = q(x)
Equation ( 1 ) is a broad statement which contains, as special cases, both a beam on a Winkler-type elastic foundation (k~ = 0) and a conventional beam (both k and k~ equal to zero). It is to be expected that any solution based on equation (1) should mask either of these two eventualities. This fact was not lost on researchers from the earliest stages of finite element development when interpolation functions dealt primarily with one-dimensional continua. Defining A = kj/EI and B =k/E1 the homogenous solution of the fourth-order equation (1) can be computed as any of the expressions in equation ( 2 a - c ) depending on which of the conditions A < 2qB, A = 2qB or A > 2~/B, respectively, holds
4~/(k/EI)
Xj)t2 -- 8
d2w
E l dx 4 - k, ~
strain energy functional column matrix of displacement boundary conditions transverse beam displacement
13
911
(2c)
In equations (2) the auxiliary variables are defined by a = ~/3f + 3
(3)
]3 = ~/~2 -- 3 in equation (2a)
distributed mass dimensionless coordinate x/L
Two-parameter foundation
An infinitesimal part of a beam supported by a two-parameter elastic foundation which terminates at the ends of the beam is shown in Figure 1. For the problem at hand both of these parameters are positive constants. The Winkler parameter, denoted by k, represents the distributed translational resistance of the foundation (F/L2), and the Pasternak parameter, symbolized by k~, accounts for its distributed rotational stiffness (F.L/L). Other interpretations for the second constant are possible ~. The governing differential equation of the beam segment in Figure I for transverse displacement w is derived from vertical equilibrium
(4)
A=4
(5)
and
3-4EI
w(x) = c] sinAx sinhAx + c2 sinAx coshAx + c3 cosAx sinhAx + c4 cosAx coshAx
(t I k|C
v
(6)
To date, most solutions for beams supported by generalized Pasternak-type foundations have considered the case in equation (2a) for which A < 2~/B7"8a°, although on physical grounds an unqualified justification for this restricq
'v' .._ jt3
dw
dx
InCo~l:r f s s ~ t e
H~(~/dx)dx
l) )
kl(
t 1' I' I'J~wI' I' f 1't ¢lx
Figure 1 Beam segment on a generalized foundation
k
When k~ approaches zero, all three expressions for w(x) in equations (2) reduce to
F& w2
ELL
13 = ~ j 3 - A2 in equation (2c)
V'~dV/dxlalw := dw d w ant 2
dx
Supplementary element matrices: B. N. Alemdar and P. GOIkan
912
tion is debatable, as noted also by Razaqpur and Shah 9. It is among our objectives in this study to remove this artificial restriction, and also to demonstrate that while the mathematical form of the solution varies in the three ranges the transition from one to the next is smooth, as would be expected in heuristic reasoning. We also note that for most real physical parameters the mathematically exact statement A = 2 ~ I B is not achieved, so that the solution for equation(2b) should be extractable from either equation (2a) or (2c). We note that equations (2) may be written conveniently in matrix form
where
s =
4~Eik
(12)
For A > 2qB NI = coshooc cosh/3x + [/3cosh/3x(-/3sinh2aL - asinh2/3L) sinh~x + acosh~x (/3sinh2aL + asinh2/3L) sinh/3x + a/3 ( - cosh2aL + cosh2/3L) sinhax sinh/3x)]/
(7)
w =BrC
where C is a column vector containing the four constants of integration c~, i = 1-4, and B is a column vector which contains any one of the three functions in equations (2) arranged appropriately. The four geometric boundary conditions which are the two transverse displacements w~ and w2 and the two rotations d w ~ / d x and d w 2 / d x at each end of the segment are utilized to express the integration constants. Inserting the four generalized displacements into a column matrix W = {w~, d w l / d x , w2, d w 2 / d x } r for the appropriate relative magnitudes of A and B, we obtain (8)
W=HC
When C is solved symbolically from equation (8) and substituted into equation (7) we find (9)
w = BrH-IW = N W
(a2 _/32 +/32cosh2a L _ aZcosh2/3L)
(13)
We can obtain shape functions for beams on Winklertype foundations if k~ approaches zero in any of equations (10), ( t 1 ) and (13). In all instances N~ becomes N~ = [cosAx coshh(2L - x) + cosh(2L - x) coshhx - 2coshx coshhx + sinAx sinhh(2L - x) - sinh(2L - x) sinhhx]/ ( - 2 + cos2Lh + cosh2Lh)
(14)
If we continue this process by allowing h also to approach zero in equation (14), we obtain the cubic Hermitian polynomial of the first kind Ni = 1 -
(15)
3~ 2 - 2~ 3
+ a 2 cos/3(2L - x ) coshax
where ~ = x / L . This provides a check on the accuracy of the computations in that we can successively reduce equations (10), (11) and (13) to equation (15) by letting both k and kl approach zero. This is a common theme for most derivations in this paper. Except for differences in notation the shape functions for beams on Winkler or Pasternak foundations have been k n o w n 7a2 o r independently rediscovered ~3. It is instructive to illustrate the variation of the shape function as a function of k or k~, and to compare them with the corresponding Hermitian polynomials. This can be done either by fixing k and letting k~ vary in a two-dimensional plot, or by describing their interaction in the form of a three-dimensional surface graph. For either purpose it is convenient to introduce two new nondimensional variables p and z such that
- a 2 cos/3x coshax
For A < 2~/B
The row vector N includes four shape functions each of which corresponds to a unit value of the generalized displacements. Taking the inverse of matrix H and premultiplying it by matrix B r is an arduous task. Computer software ~ was therefore used to perform all symbolic calculations implied in e q u a t i o n s ( 7 ) - ( 9 ) to obtain shape functions for each set of solutions stated in equation (2) for the different ranges of the Winkler and Pasternak parameters. To give a brief idea, we present the first shape function N~ that represents unit vertical displacement at left end while all remaining displacements are set equal to zero. For A < 2~/B Ni = [/32 cos/3x cosha(2L - x)
_/32 cos/3x coshax + a/3 sin/3x sinha(2L - x) a/3 sin/3(2L - x) sinhocr]/
-
a=~p N/l+z
(_a2 _/32 + a2 cos2/3L +/32 cosh2aL)
/3 = p ~ , v / i - z
(16)
/ 3 = {p ( z -
(17)
(10) For A > 2qB
For A = 2~/B e (2L-~)~ x s (-1 + e 2L~ + 2 L s ) - - s e - 2 L s + 2 L s 2)
a = { p (1 + z
1
+ e (2L+x)s x ( s
N~ = ( 1 - 2e 2L" + e 4L' - 4e2~LZs 2) e (2L+x)~
(--1 + e - 2 b -
+ e (2L-~)~
+
(--1 + e 2Ls +
where
2Ls - 2L2s2) 2Ls
-
Z
=
6/h 2 and p = AL
(18)
2 L 2 s 2)
( I - 2e 2z'' + e 4L~ - - 4e2L~Les z)
(11)
The eight adjacent surface plot frames of F i g u r e 2 show
Supplementary element matrices: B. N. Alemdar and P. GOIkan Xl,
913
II1, p-O.1
Ip~O.I
1
l -
Nlo
N1,
p-1.0
:1 0.7! O. 0.2
~
0
1 N1,
p"S.0
Nlo
~S
l NX,
p-lO.O
!il,
~10
0.3'
O.
0
0.:
0~
1
0
z
ks 0"6 0.8--"~~ " 1 Figure2 Surface plots fo,r N~ the variation of N1 through a range of values for z at four distinct values o f p = 0.1, 1, 5 and 10. The horizontal axis denoted by ks is the nondimensional axial coordinate = x/L. The four stacked frames on the left side of the figure correspond to z < 1, or equivalently to A < 2x/B, and those on the fight side to z > 1. The shapes N3 are simply the symmetric mirror images of these functions. It can be observed that firstly, for small k (approximately characterized by 0 ~
z) exhibits an increasingly dominant presence only for large values of k (characterized by p). This is, of course, not surprising because this parameter is, in effect, a reflection of the degree of interaction between adjacent Winkler elements. With these elements missing or weakly represented, interaction between them becomes moot. And, finally, the case for z = 0 is the Winkler foundation solution. In Figure 3 we show in similar fashion the variation of N2 which corresponds to a unit rotation induced at the left end of the beam segment. The functions N4 are antisymmetric mirror images of these functions. In all frames in Figures 2 and 3 the vertical planes of termination at z = 1 of the frames on the left match precisely the planes of initiation at z = 1 of the frames on the right, so that the smooth transition required by heuristic reasoning and the physical reality is in fact achieved.
Supplementary element matrices: B. N. Alemdar and P. GOIkan
914 N2,
p-0.1
N2,
Ip'0,1
N2,
p- 1.0
N2,
1 1)'1.0
;0
z
1 N2,
N2,
p"5.0
~S.O
0.15 0.1 0.0!
~0.
l U N2o
1
p-10.0
N2,
p'10
0 0.1 0.0'
8z
10
Figure3
3.
Surface plots f o r N 2
Element matrices
3.1. Element stiffness matrix The 4 x 4 element stiffness matrix ke which relates the nodal forces to nodal displacements (see Figure 1) can be obtained from the minimization of the strain energy functional U
the coefficients of the matrix N are different. Again, as k~ and k are allowed to vanish, the coefficients tend to either the Winkler or the customary beam solutions, and in the general case continuity exists along transitional planes. For example, when the Pasternak parameter vanishes the coefficient k,, becomes k,, =
f d2NT d2N f dN T dN ke = EI dx 2 dx 2 dx + k, dx ~ dx +
Ndx
4EIA3(sin2LA + sinh2LA) - 2 + cos2LA + cosh2LA
(20)
It is checked that equation (20) tends to 12EI/L 3 if k is also allowed to vanish, which constitutes another assurance. (19)
In this equation the integration is carried out over the entire length of the segment, and for each range of the solution
3.2. Work equivalent nodal forces The closed form expressions of the shape functions can be utilized to derive expressions for the vector P of work equivalent nodal loads for an arbitrarily distributed load q(x)
Supplementary element matrices: B. N. Alemdar and P. GOIkan ( P = J Nrq(x)dx
915
For A = 2x/B (21) 2qo(-1 + eLs) 2 UFI = UF2 =
The integration must be performed over the range of x where the distributed load is defined. In the case of a distributed moment m ( x ) equation (21) is modified
'r
P=
fdN ~
s(-1 + e 2L~ + 2Lse Ls)
qo(-1 + e 2z~- 2Lse Ls) UMI = - U M 2 - s2(_ 1 + e2~ + 2LseLS )
(25) (26)
where s is given by equation (12). (22)
m(x)dx
For A > 2~/B
In F i g u r e s 4 and 5 equation (21) is illustrated for a uniformly distributed load of intensity qo, and for a triangular load having an intensity of zero at x = 0 and of qo at x = L, respectively. In both sets of plots the horizontal axes are the variables p and z, and the vertical axis is normalized with respect to the magrfitude of the fixed end force corresponding to the case of the ordinary beam. Again, the value of z determines which set of interpolation functions should be substituted into equation (21). Denoting by UF] = UF2 the fixed end shear forces, and by UM] = - U M 2 the fixed end moments for uniform load qo, symbolic integration of equation (21) yields For A < 2x/B 2q,,c~fl( c o s ~ L - coshaL) UF, = UF= = ( a 2 +/32)(_asin/3 L _/3sinhc~L)
2a~qo[-/3sinh2Lo~ - asinh(a -/3)L +/3sinh(a -/3)L - asinh2/3L + ~sinh(c~+ fl)L +/3sinh(a +/3)L] UFI = UF2 - (a: -/32)(-0? +/32 _/32cosh2aL + c~Zcosh2/3L) (27) qo[-o? -/32 +/32cosh2c~L + 2ot/3cosh(ot-/3)L
+ o?cosh2/3L- 2oq3cosh(ot+/3)L]
UMI = -UM2 - (o? -/3~)(o? -/32 +/32cosh2aL - otZcosh2/3L)(28)
It is verified from Figure 4 that when k and k, both vanish the fixed end shear forces tend to (l/2)qoL, and the fixed end moments become (1/12)qoL 2, respectively. The closed form expressions for the triangular load are too involved to be included here, but their normalized values also tend in Figure 5 to unity for no foundation stiffness. Again, for z = 0, the display is for the Winkler foundation. 3.3
(23)
Consistent mass matrix
The 4 x 4 consistent mass matrix m for a uniform segment with mass per unit length/x is given by
qo( asin[3L - ]3sinhaL) UM1 = - U M 2 -- (Or2 +/32)(_c~sini3L _/3sinhaL)
m = t x f N r N dx
(24)
(29)
UF1
UF1
1 0° 0 o
O.I O. O.
5 uN1
Figure4
Normalized fixed end forces f o r u n i f o r m load
uM1
Supplementary element matrices: B. N. Alemdar and P. GOIkan
916
Tn
TFI
0.8 O.I d
5 'I'M1
1, 0.8 0.(
0.8 O.
o.,~
O.d
0
~USmm~
~
$ T~
S
o.:,~/ " , ~ , m u w ~ r / o . , :v..._ ~ l i l m r / o . , -
l ~
~llLltW/*.4
-
Figure 5
°%" o~
,
"
4 --,~v5o
Normalized fixed end forces for triangular load
In Figure 6 the distinct coefficients of m are displayed in normalized form where the constants of normalization are the corresponding entries for the conventional beam
156 m'
~lKl~i~Y" ~ , 0
~L
=4201
22L 54
I-13L
22L 4L 2 13L
54 13L 156
-22L I
-3L 2 -22L
4L21
The consistent geometric stiffness matrix kx for a constant compressive axial force of P is given by
f
-13L I -3L 2 I
3.4. Consistent geometric stiffness matrix
(30)
dNr~
The terms contained in equation(31) have also been obtained in closed form. They are presented in Figure 7 in normalized form where the coefficients which normalize the corresponding entries are the distinct members of the consistent geometric stiffness matrix for the conventional beam
Supplementary element matrices: 8. N. Alemdar and P. G(~lkan
917 mll
$
',,12
m12
5 U
m13
m13
$ m14
I 0.7.' 0.! 0.2
B
1 0.75 0.' 0.2!
z
5
5 U
-72
m22
m2.4
~24
I 0.7.~ O.I
0.2
1 0.75 0..' 0.2,'
0,75
0
2
$ 5
Figure 6
Consistent mass matrix
Supplementary element matrices: B. N. Alemdar and P. G(Jlkan
918 kgll
kgll
2 1.7: 1.2
5 kg12
kgl2
kg$3
kg13
S
5
5u kg14
kg14
kg22
kg22
k924
kg24
1
0.~
m z
Figure 7 Consistent g e o m e t r i c stiffness matrix
Supplementary element matrices: B. N. A l e m d a r and P. GOIkan
p
ii
3L
k~. = 3OL -36
4L 2
-3L _t 2
36 112 -3L -L 2 36
-3L
given to modelling the appropriate force boundary conditions at the free ends because the shear force at a free end does not vanish unless the rotation there also vanishes. One way of arriving at a satisfactory solution is to introduce concentrated translational springs at such termination points and to assign them a stiffness equal to ~/kk~ (see Eisenberger and Bielak3). Another approach would be to consider a virtual extension of the beam beyond its physical end, and to assign this extension very small section properties. The issue then becomes: how long should this extension be? If the problem domain with the beam segment is defined as 0 < - x - < L then for x < 0 and x > L , equation ( 1) becomes
(32)
-3L
We note again that equations (30) and (32) are simply the outcomes of equations (29) and (31), respectively, for p = z = 0. When only z = 0 is considered, the solution is for a Winkler foundation. 4.
d2wi
- kl ~
+ kwl = 0
(33)
where foundation deflection beyond the beam ends has been denoted by wj. The solution of equation (33) is
Implementation
In principle the beam element described in this paper can be built into any existing static and dynamic structural analysis software. Specifying only the Winkler parameter k, or both k and the Pasternak parameter kj, transforms only the element stiffness, mass or geometric stiffness matrices to increasingly unfamiliar expressions but these are the source equations from which the special cases can be cloned by successive simplificatiorts. For stress analysis, equivalent nodal forces (represented for two typical cases in F i g u r e s 4 and 5) must be used as an alternative to using many short beam segments which etfectively reduce p = )tL. Displacements and forces calculated at joints are then exact, in the same way as ordinary frame analysis yields exact values at the joints on the basis of Hermitian polynomials. Closed form expressions for coefficients of the stiffness matrix in equation(19), the fixed-end forces in equation(21) for both uniform and triangular loads, the consistent mass matrix in equation (29), and the consistent geometric stiffness matrix in equation (31 ) are too involved to be reproduced here fully (but see Reference 14, a web site for possible downloading.) When a finite-length beam is supported by a longer twoparameter foundation, special consideration needs to be
Table 1
919
w l ( x ) = cle - ~ x )
(34)
+ c2 e ~k'ff~7-~l)(x)
We now argue that for x ---* - ~ , w~ ---, 0 so that cj = 0, and for x = 0 the condition c2 = w~(0) must hold. This implies that for x < 0 (35)
w l ( x ) = w ( O ) e "1~7~)~x)
and for x > L w , ( x ) = w ( L ) e "~k/k,) (c - x~
(36)
In either case, the virtual length of the beam segment should be extended so that w~(x) becomes a small fraction of either w(0) or w ( L ) . If this ratio is taken as 0.001, then the length of the foundation required beyond a free end can be calculated to be 6 . 9 { ( k ~ / k ) . A single element is sufficient for this representation. We demonstrate an application through the device of computing the vibration frequencies of a cantilever beam on a two-parameter foundation. This problem is emulated after Eisenberger and Bielak 3. The results are summarized
Frequencies of a cantilever beam E= 1 I= 1 L = 1 /x = 1 (consistent units) Frequency oJ2, rad2/s 2
0
Extension b e y o n d end 0.5L
1.0L
Reference 3
Foundation p a r a m e t e r s
Mode
k= 1 kN/m 2 kl = 4.9348 kN
1 2 3
5.83 25.39 64.74
6.00 25.54 64.88
6.13 25.58 64.89
6.45 25.57 64.79
k = 100 k~ = 4.9348
1 2 3
11.53 27.67 65.50
14.13 29.36 66.51
14.18 29.41 66.54
14.06 29.02 66.19
k = 10 000 kl = 4.9348
1 2 3
100.23 103.24 119.23
101.34 108.97 127.14
101.34 108.97 127.14
101.22 107.67 123.81
k = 1000000 kl = 4.9348
1 2 3
1054.56 1059.62 1060.33
1059.58 1060.33 1064.28
1059.58 1060.33 1064.28
1000.15 1001.29 1005.04
Supplementary element matrices: B. N. Alemdar and P. GOIkan
920
in Table 1 where onl~¢ the case where k = 1, kj = 4.9348 corresponds to A > 2~/B, and the three remaining cases fall into the solution range of A < 2"4B. The frequencies determined with the consistent mass matrix shown graphically in Figure 6 show very good agreement with those of Reference 3 where the same problem is solved by affixing to the free end a concentrated spring to account for the correct force boundary condition. The table indicates the calculated frequencies when a part of the foundation beyond the free end equal to 0, 0.5 and 1.0 times the beam's own length has been included in the finite element model. We note that for A > 2x/B none of these virtual extensions satisfies the above criterion, whereas in the remaining cases for A < 2X/B extensions beyond the beam end of both 0.5L and 1.0L do. This is verified by the fact that the corresponding frequencies are virtually the same. For k = 1.0 x l06 all three frequencies tend to the same value, but larger differences exist between these and the values reported by Eisenberger and Bielak 3. As we do not know the details of that solution and the eigensolution routine which was employed, we will refrain from commenting on the causes of this difference.
into the element routines of frame analysis software, a broad spectrum of problems can be handled in a routine way. A web site ~4 has been prepared from which ASCII files containing the closed form expressions for element matrices for the full range of foundation parameters can be extracted.
References 1 2 3
4
5 6
7
5.
Conclusions
Many diverse problems involving beams on generalized elastic foundations may be solved routinely with the use of the exact shape functions illustrated in Figures 2 and 3. This exactness is in the same sense as the cubic displacement functions are exact for a uniform beam segment. Through appropriate manipulations of these functions exact expressions for the coefficients of the element matrices, i.e., stiffness, work-equivalent fixed end forces, consistent mass and geometric stiffness, can be generated. It has been demonstrated that the corresponding matrices for beams on Winkler-type foundations or for beams supported by no foundation can be derived as special solutions of this general case. The expressions for these coefficients are typically, very involved, but once they have been incorporated
8
9
10
11 12 13
14
Kerr, A. D. 'Elastic and viscoelastic foundation models', J. Appl. Mech., ASME 1964, 31 (3), 491-498 Scott, R. F. Foundation analysis, Prentice Hall, Englewood Cliffs, NJ, 1981 Eisenberger, M. and Bielak, J. 'Remarks on two papers dealing with the response of beams on two-parameter elastic foundations', Earthquake Engng Struct. Dyn. 1989, 18 (7), 1077-1078 Tekkaya, A. E. and Aydin, A. K. 'Analysis of fixed partial dentures for diminishing periodontal support', Proc. XX. Int. Finite Element Method Congress, Baden-Baden, Germany, November 1991, pp 517-531 Hetenyi, M. Beams on elastic foundations, University of Michigan Press, Ann Arbor, 1946 Pasternak, P. L. 'On a new method of analysis of an elastic foundation by means of two foundation constants', (in Russian) (Gosudartsvennoe Izdateltsvo Literaturi po Stroitelstvu Arkhitekture) Moscow, USSR, 1954 (cited in Reference 1) Zhaohua, F. and Cook, R. D. 'Beam elements on two-parameter elastic foundations', J. Engng Mech. Div., ASCE 1983, 109 (6), 13901402 Williams, F. W. and Kennedy, D. 'Exact dynamics member stiffness for a beam on an elastic foundation', Earthquake Engng Struct. Dyn. 1987, 15 (2), 133-136 Razaqpur, A. G. and Shah, K. R. 'Exact analysis of beams on twoparameter elastic foundations', Int. J. Solids Struct. 1991, 27 (4), 435-454 Chiwanga, M. and Valsangkar, A. J. 'Generalized beam element on two parameter elastic foundation', J. Struct. Engng, ASCE 1988, 114 (6), 1414-1427 Wolfram, S. Mathematica: a system for doing mathematics by computer, Addison-Wesley, Redwood City, CA, 1991 Wang, C. K. Intermediate structural analysis, McGraw Hill, New York, 1983 Eisenberger, M. and Yankelevsky, D. Z. 'Exact stiffness matrix for beams on elastic foundation', Comput. Struct. 1985, 21 (6), 13551359 http://www.metu.edu.tr/-www62/dept/alemdar/tez.tar.gz/
ELSEVIER
PIh S0141-0296(97)00179-4
19, No. 11, pp. 910-920, 1997 © 1997 Elsevier Science Ltd All rights reserved. Printed in Great Britain 0141-0296/97 $17.00+ 0.00
Beams on generalized foundations: supplementary element matrices B. N. Alemdar and P. Giilkan* Department of Civil Engineering, Middle East Technical University, 06531 Ankara, Turkey (Received August 1996; revised version accepted December 1996)
A general analytical solution for the shape functions of a linear beam segment supported on a two-parameter elastic foundation is derived. The solution is not restricted to a particular range of magnitudes of the foundation parameters. The closed form shape functions are utilized to derive analytic expressions for work equivalent nodal forces for arbitrary transverse loads and coefficients of the consistent mass and geometrical stiffness matrices. Each work equivalent nodal force and each coefficient of the element matrices is compared with its conventional counterpart for an ordinary beam segment supported by no foundation. © 1997 Elsevier Science Ltd. Keywords:two-parameter elastic foundation, work equivalent nodal force, consistent mass matrix, consistent geometric stiffness matrix
1.
Introduction
special case of a beam segment supported by a singleparameter Winkler foundation, which in turn, is a subset of the more general two-parameter case. This follows directly from the governing differential equation. The derivation of the exact shape functions for a beam segment supported by a two-parameter elastic foundation must precede all other calculations. Once exact onedimensional shape functions have been derived, it becomes possible to manipulate them to derive exact expressions for coefficients of the element stiffness matrix, or the consistent geometric and mass matrices for beam segments as well as work-equivalent joint forces for a variety of concentrated or distributed loads. We call these the supplementary element matrices. They correspond to the quantities which can be derived from cubic Hermitian polynomials for uniform beam segments. Their expressions are lengthy and cumbersome, but this is outweighed by virtue of the fact that once they are made part of a structural analysis software, they enable exact analyses to be performed with fewer elements. One aim of this paper is to display in graphical form the analytic expressions for the element matrices, noting the smooth transition from the complex Pasternak solution to the familiar beam cubic displacement-based expressions for each distinct coefficient. A brief note on implementation is given before a sample problem is solved.
The need for a solution for stresses in beams, plates and shells in contact with, or wrapped by, elastic or viscoelastic media is commonly encountered in many diverse problems ~-4 such as buried pipelines, piles, foundation elements, reinforcing filaments in composite materials, and even endodontics. The beam theory can be traced to Winkler's original work on stresses in railway tracks where the idea of representing the soil underneath tracks by a series of closely packed, but otherwise unconnected springs leads to a solution for the internal forces 5. This type of foundation is characterized by a single distributed spring stiffness called the Winkler foundation parameter. Subsequent generalizations j,6, have interpreted the possible physical interaction between these springs through what will be designated in the following as the Pasternak parameter, enabling applications of wider range. More recent studies 7-9 have focused on the derivation of elements of the stiffness matrix for beam segments, but given the lengthy mathematical expressions the coefficients have usually been restricted to a particular range of the relative magnitudes of the Winkler and the Pasternak parameters. This limitation can be removed with little mathematical inconvenience, as will be demonstrated in the following. Geometric stiffness and mass matrices for beam segments embedded in this more general foundation have also found use in stability and vibration problems. The development below will reconfirm that a beam segment not supported by any foundation is a
Notation A B
B
* Present address: School o f Civil Engineering, Purdue University, W e s t Lafayette, IN 4 7 9 0 7 - 1 2 8 4 , U S A .
910
kJEI k/E1 column vector of transverse displacement functions
S u p p l e m e n t a r y e l e m e n t m a t r i c e s : B. N. A l e m d a r a n d P. GOIkan
C ci H
E1 ke
kg k kt L m
m(x) N P P p
q(x) S U W
column vector containing constants gration ci integration constant, i = 1 - 4 matrix defined iin equation (8) flexural stiffness beam element stiffness matrix consistent geometric stiffness matrix Winkler foundation parameter Pasternak foundation parameter beam segment length consistent beam mass matrix distributed moment row vector of siaape functions N~ vector of work equivalent nodal loads axial force AL transverse load
of inte-
d4w
w Z
6/A 2
ol
A2~
w(x) = c, cos13x coshax + -I- C3
+
X/8- )t 2
which
correspond
A
4 k ~/4EI
/x
2.
sin13x coshctx + c 4 sin13x sinhax
(2a)
C3e-4q~x
+
(2b)
c4xe -4@x
+ c3 cosh13x sinhax + c4 sinh13x sinhax
to
PL ~/1 - z or pL x/z - 1, respectively
kfl(4El)
C2 COS13X sinhax
w(x) = cl cosh13x coshax + c2 sinh13x coshax J
3
(1)
W(X) = Ci e4q'~x + c2xe 4q'~x
=p
or
+ kw = q(x)
Equation ( 1 ) is a broad statement which contains, as special cases, both a beam on a Winkler-type elastic foundation (k~ = 0) and a conventional beam (both k and k~ equal to zero). It is to be expected that any solution based on equation (1) should mask either of these two eventualities. This fact was not lost on researchers from the earliest stages of finite element development when interpolation functions dealt primarily with one-dimensional continua. Defining A = kj/EI and B =k/E1 the homogenous solution of the fourth-order equation (1) can be computed as any of the expressions in equation ( 2 a - c ) depending on which of the conditions A < 2qB, A = 2qB or A > 2~/B, respectively, holds
4~/(k/EI)
Xj)t2 -- 8
d2w
E l dx 4 - k, ~
strain energy functional column matrix of displacement boundary conditions transverse beam displacement
13
911
(2c)
In equations (2) the auxiliary variables are defined by a = ~/3f + 3
(3)
]3 = ~/~2 -- 3 in equation (2a)
distributed mass dimensionless coordinate x/L
Two-parameter foundation
An infinitesimal part of a beam supported by a two-parameter elastic foundation which terminates at the ends of the beam is shown in Figure 1. For the problem at hand both of these parameters are positive constants. The Winkler parameter, denoted by k, represents the distributed translational resistance of the foundation (F/L2), and the Pasternak parameter, symbolized by k~, accounts for its distributed rotational stiffness (F.L/L). Other interpretations for the second constant are possible ~. The governing differential equation of the beam segment in Figure I for transverse displacement w is derived from vertical equilibrium
(4)
A=4
(5)
and
3-4EI
w(x) = c] sinAx sinhAx + c2 sinAx coshAx + c3 cosAx sinhAx + c4 cosAx coshAx
(t I k|C
v
(6)
To date, most solutions for beams supported by generalized Pasternak-type foundations have considered the case in equation (2a) for which A < 2~/B7"8a°, although on physical grounds an unqualified justification for this restricq
'v' .._ jt3
dw
dx
InCo~l:r f s s ~ t e
H~(~/dx)dx
l) )
kl(
t 1' I' I'J~wI' I' f 1't ¢lx
Figure 1 Beam segment on a generalized foundation
k
When k~ approaches zero, all three expressions for w(x) in equations (2) reduce to
F& w2
ELL
13 = ~ j 3 - A2 in equation (2c)
V'~dV/dxlalw := dw d w ant 2
dx
Supplementary element matrices: B. N. Alemdar and P. GOIkan
912
tion is debatable, as noted also by Razaqpur and Shah 9. It is among our objectives in this study to remove this artificial restriction, and also to demonstrate that while the mathematical form of the solution varies in the three ranges the transition from one to the next is smooth, as would be expected in heuristic reasoning. We also note that for most real physical parameters the mathematically exact statement A = 2 ~ I B is not achieved, so that the solution for equation(2b) should be extractable from either equation (2a) or (2c). We note that equations (2) may be written conveniently in matrix form
where
s =
4~Eik
(12)
For A > 2qB NI = coshooc cosh/3x + [/3cosh/3x(-/3sinh2aL - asinh2/3L) sinh~x + acosh~x (/3sinh2aL + asinh2/3L) sinh/3x + a/3 ( - cosh2aL + cosh2/3L) sinhax sinh/3x)]/
(7)
w =BrC
where C is a column vector containing the four constants of integration c~, i = 1-4, and B is a column vector which contains any one of the three functions in equations (2) arranged appropriately. The four geometric boundary conditions which are the two transverse displacements w~ and w2 and the two rotations d w ~ / d x and d w 2 / d x at each end of the segment are utilized to express the integration constants. Inserting the four generalized displacements into a column matrix W = {w~, d w l / d x , w2, d w 2 / d x } r for the appropriate relative magnitudes of A and B, we obtain (8)
W=HC
When C is solved symbolically from equation (8) and substituted into equation (7) we find (9)
w = BrH-IW = N W
(a2 _/32 +/32cosh2a L _ aZcosh2/3L)
(13)
We can obtain shape functions for beams on Winklertype foundations if k~ approaches zero in any of equations (10), ( t 1 ) and (13). In all instances N~ becomes N~ = [cosAx coshh(2L - x) + cosh(2L - x) coshhx - 2coshx coshhx + sinAx sinhh(2L - x) - sinh(2L - x) sinhhx]/ ( - 2 + cos2Lh + cosh2Lh)
(14)
If we continue this process by allowing h also to approach zero in equation (14), we obtain the cubic Hermitian polynomial of the first kind Ni = 1 -
(15)
3~ 2 - 2~ 3
+ a 2 cos/3(2L - x ) coshax
where ~ = x / L . This provides a check on the accuracy of the computations in that we can successively reduce equations (10), (11) and (13) to equation (15) by letting both k and kl approach zero. This is a common theme for most derivations in this paper. Except for differences in notation the shape functions for beams on Winkler or Pasternak foundations have been k n o w n 7a2 o r independently rediscovered ~3. It is instructive to illustrate the variation of the shape function as a function of k or k~, and to compare them with the corresponding Hermitian polynomials. This can be done either by fixing k and letting k~ vary in a two-dimensional plot, or by describing their interaction in the form of a three-dimensional surface graph. For either purpose it is convenient to introduce two new nondimensional variables p and z such that
- a 2 cos/3x coshax
For A < 2~/B
The row vector N includes four shape functions each of which corresponds to a unit value of the generalized displacements. Taking the inverse of matrix H and premultiplying it by matrix B r is an arduous task. Computer software ~ was therefore used to perform all symbolic calculations implied in e q u a t i o n s ( 7 ) - ( 9 ) to obtain shape functions for each set of solutions stated in equation (2) for the different ranges of the Winkler and Pasternak parameters. To give a brief idea, we present the first shape function N~ that represents unit vertical displacement at left end while all remaining displacements are set equal to zero. For A < 2~/B Ni = [/32 cos/3x cosha(2L - x)
_/32 cos/3x coshax + a/3 sin/3x sinha(2L - x) a/3 sin/3(2L - x) sinhocr]/
-
a=~p N/l+z
(_a2 _/32 + a2 cos2/3L +/32 cosh2aL)
/3 = p ~ , v / i - z
(16)
/ 3 = {p ( z -
(17)
(10) For A > 2qB
For A = 2~/B e (2L-~)~ x s (-1 + e 2L~ + 2 L s ) - - s e - 2 L s + 2 L s 2)
a = { p (1 + z
1
+ e (2L+x)s x ( s
N~ = ( 1 - 2e 2L" + e 4L' - 4e2~LZs 2) e (2L+x)~
(--1 + e - 2 b -
+ e (2L-~)~
+
(--1 + e 2Ls +
where
2Ls - 2L2s2) 2Ls
-
Z
=
6/h 2 and p = AL
(18)
2 L 2 s 2)
( I - 2e 2z'' + e 4L~ - - 4e2L~Les z)
(11)
The eight adjacent surface plot frames of F i g u r e 2 show
Supplementary element matrices: B. N. Alemdar and P. GOIkan Xl,
913
II1, p-O.1
Ip~O.I
1
l -
Nlo
N1,
p-1.0
:1 0.7! O. 0.2
~
0
1 N1,
p"S.0
Nlo
~S
l NX,
p-lO.O
!il,
~10
0.3'
O.
0
0.:
0~
1
0
z
ks 0"6 0.8--"~~ " 1 Figure2 Surface plots fo,r N~ the variation of N1 through a range of values for z at four distinct values o f p = 0.1, 1, 5 and 10. The horizontal axis denoted by ks is the nondimensional axial coordinate = x/L. The four stacked frames on the left side of the figure correspond to z < 1, or equivalently to A < 2x/B, and those on the fight side to z > 1. The shapes N3 are simply the symmetric mirror images of these functions. It can be observed that firstly, for small k (approximately characterized by 0 ~
z) exhibits an increasingly dominant presence only for large values of k (characterized by p). This is, of course, not surprising because this parameter is, in effect, a reflection of the degree of interaction between adjacent Winkler elements. With these elements missing or weakly represented, interaction between them becomes moot. And, finally, the case for z = 0 is the Winkler foundation solution. In Figure 3 we show in similar fashion the variation of N2 which corresponds to a unit rotation induced at the left end of the beam segment. The functions N4 are antisymmetric mirror images of these functions. In all frames in Figures 2 and 3 the vertical planes of termination at z = 1 of the frames on the left match precisely the planes of initiation at z = 1 of the frames on the right, so that the smooth transition required by heuristic reasoning and the physical reality is in fact achieved.
Supplementary element matrices: B. N. Alemdar and P. GOIkan
914 N2,
p-0.1
N2,
Ip'0,1
N2,
p- 1.0
N2,
1 1)'1.0
;0
z
1 N2,
N2,
p"5.0
~S.O
0.15 0.1 0.0!
~0.
l U N2o
1
p-10.0
N2,
p'10
0 0.1 0.0'
8z
10
Figure3
3.
Surface plots f o r N 2
Element matrices
3.1. Element stiffness matrix The 4 x 4 element stiffness matrix ke which relates the nodal forces to nodal displacements (see Figure 1) can be obtained from the minimization of the strain energy functional U
the coefficients of the matrix N are different. Again, as k~ and k are allowed to vanish, the coefficients tend to either the Winkler or the customary beam solutions, and in the general case continuity exists along transitional planes. For example, when the Pasternak parameter vanishes the coefficient k,, becomes k,, =
f d2NT d2N f dN T dN ke = EI dx 2 dx 2 dx + k, dx ~ dx +
Ndx
4EIA3(sin2LA + sinh2LA) - 2 + cos2LA + cosh2LA
(20)
It is checked that equation (20) tends to 12EI/L 3 if k is also allowed to vanish, which constitutes another assurance. (19)
In this equation the integration is carried out over the entire length of the segment, and for each range of the solution
3.2. Work equivalent nodal forces The closed form expressions of the shape functions can be utilized to derive expressions for the vector P of work equivalent nodal loads for an arbitrarily distributed load q(x)
Supplementary element matrices: B. N. Alemdar and P. GOIkan ( P = J Nrq(x)dx
915
For A = 2x/B (21) 2qo(-1 + eLs) 2 UFI = UF2 =
The integration must be performed over the range of x where the distributed load is defined. In the case of a distributed moment m ( x ) equation (21) is modified
'r
P=
fdN ~
s(-1 + e 2L~ + 2Lse Ls)
qo(-1 + e 2z~- 2Lse Ls) UMI = - U M 2 - s2(_ 1 + e2~ + 2LseLS )
(25) (26)
where s is given by equation (12). (22)
m(x)dx
For A > 2~/B
In F i g u r e s 4 and 5 equation (21) is illustrated for a uniformly distributed load of intensity qo, and for a triangular load having an intensity of zero at x = 0 and of qo at x = L, respectively. In both sets of plots the horizontal axes are the variables p and z, and the vertical axis is normalized with respect to the magrfitude of the fixed end force corresponding to the case of the ordinary beam. Again, the value of z determines which set of interpolation functions should be substituted into equation (21). Denoting by UF] = UF2 the fixed end shear forces, and by UM] = - U M 2 the fixed end moments for uniform load qo, symbolic integration of equation (21) yields For A < 2x/B 2q,,c~fl( c o s ~ L - coshaL) UF, = UF= = ( a 2 +/32)(_asin/3 L _/3sinhc~L)
2a~qo[-/3sinh2Lo~ - asinh(a -/3)L +/3sinh(a -/3)L - asinh2/3L + ~sinh(c~+ fl)L +/3sinh(a +/3)L] UFI = UF2 - (a: -/32)(-0? +/32 _/32cosh2aL + c~Zcosh2/3L) (27) qo[-o? -/32 +/32cosh2c~L + 2ot/3cosh(ot-/3)L
+ o?cosh2/3L- 2oq3cosh(ot+/3)L]
UMI = -UM2 - (o? -/3~)(o? -/32 +/32cosh2aL - otZcosh2/3L)(28)
It is verified from Figure 4 that when k and k, both vanish the fixed end shear forces tend to (l/2)qoL, and the fixed end moments become (1/12)qoL 2, respectively. The closed form expressions for the triangular load are too involved to be included here, but their normalized values also tend in Figure 5 to unity for no foundation stiffness. Again, for z = 0, the display is for the Winkler foundation. 3.3
(23)
Consistent mass matrix
The 4 x 4 consistent mass matrix m for a uniform segment with mass per unit length/x is given by
qo( asin[3L - ]3sinhaL) UM1 = - U M 2 -- (Or2 +/32)(_c~sini3L _/3sinhaL)
m = t x f N r N dx
(24)
(29)
UF1
UF1
1 0° 0 o
O.I O. O.
5 uN1
Figure4
Normalized fixed end forces f o r u n i f o r m load
uM1
Supplementary element matrices: B. N. Alemdar and P. GOIkan
916
Tn
TFI
0.8 O.I d
5 'I'M1
1, 0.8 0.(
0.8 O.
o.,~
O.d
0
~USmm~
~
$ T~
S
o.:,~/ " , ~ , m u w ~ r / o . , :v..._ ~ l i l m r / o . , -
l ~
~llLltW/*.4
-
Figure 5
°%" o~
,
"
4 --,~v5o
Normalized fixed end forces for triangular load
In Figure 6 the distinct coefficients of m are displayed in normalized form where the constants of normalization are the corresponding entries for the conventional beam
156 m'
~lKl~i~Y" ~ , 0
~L
=4201
22L 54
I-13L
22L 4L 2 13L
54 13L 156
-22L I
-3L 2 -22L
4L21
The consistent geometric stiffness matrix kx for a constant compressive axial force of P is given by
f
-13L I -3L 2 I
3.4. Consistent geometric stiffness matrix
(30)
dNr~
The terms contained in equation(31) have also been obtained in closed form. They are presented in Figure 7 in normalized form where the coefficients which normalize the corresponding entries are the distinct members of the consistent geometric stiffness matrix for the conventional beam
Supplementary element matrices: 8. N. Alemdar and P. G(~lkan
917 mll
$
',,12
m12
5 U
m13
m13
$ m14
I 0.7.' 0.! 0.2
B
1 0.75 0.' 0.2!
z
5
5 U
-72
m22
m2.4
~24
I 0.7.~ O.I
0.2
1 0.75 0..' 0.2,'
0,75
0
2
$ 5
Figure 6
Consistent mass matrix
Supplementary element matrices: B. N. Alemdar and P. G(Jlkan
918 kgll
kgll
2 1.7: 1.2
5 kg12
kgl2
kg$3
kg13
S
5
5u kg14
kg14
kg22
kg22
k924
kg24
1
0.~
m z
Figure 7 Consistent g e o m e t r i c stiffness matrix
Supplementary element matrices: B. N. A l e m d a r and P. GOIkan
p
ii
3L
k~. = 3OL -36
4L 2
-3L _t 2
36 112 -3L -L 2 36
-3L
given to modelling the appropriate force boundary conditions at the free ends because the shear force at a free end does not vanish unless the rotation there also vanishes. One way of arriving at a satisfactory solution is to introduce concentrated translational springs at such termination points and to assign them a stiffness equal to ~/kk~ (see Eisenberger and Bielak3). Another approach would be to consider a virtual extension of the beam beyond its physical end, and to assign this extension very small section properties. The issue then becomes: how long should this extension be? If the problem domain with the beam segment is defined as 0 < - x - < L then for x < 0 and x > L , equation ( 1) becomes
(32)
-3L
We note again that equations (30) and (32) are simply the outcomes of equations (29) and (31), respectively, for p = z = 0. When only z = 0 is considered, the solution is for a Winkler foundation. 4.
d2wi
- kl ~
+ kwl = 0
(33)
where foundation deflection beyond the beam ends has been denoted by wj. The solution of equation (33) is
Implementation
In principle the beam element described in this paper can be built into any existing static and dynamic structural analysis software. Specifying only the Winkler parameter k, or both k and the Pasternak parameter kj, transforms only the element stiffness, mass or geometric stiffness matrices to increasingly unfamiliar expressions but these are the source equations from which the special cases can be cloned by successive simplificatiorts. For stress analysis, equivalent nodal forces (represented for two typical cases in F i g u r e s 4 and 5) must be used as an alternative to using many short beam segments which etfectively reduce p = )tL. Displacements and forces calculated at joints are then exact, in the same way as ordinary frame analysis yields exact values at the joints on the basis of Hermitian polynomials. Closed form expressions for coefficients of the stiffness matrix in equation(19), the fixed-end forces in equation(21) for both uniform and triangular loads, the consistent mass matrix in equation (29), and the consistent geometric stiffness matrix in equation (31 ) are too involved to be reproduced here fully (but see Reference 14, a web site for possible downloading.) When a finite-length beam is supported by a longer twoparameter foundation, special consideration needs to be
Table 1
919
w l ( x ) = cle - ~ x )
(34)
+ c2 e ~k'ff~7-~l)(x)
We now argue that for x ---* - ~ , w~ ---, 0 so that cj = 0, and for x = 0 the condition c2 = w~(0) must hold. This implies that for x < 0 (35)
w l ( x ) = w ( O ) e "1~7~)~x)
and for x > L w , ( x ) = w ( L ) e "~k/k,) (c - x~
(36)
In either case, the virtual length of the beam segment should be extended so that w~(x) becomes a small fraction of either w(0) or w ( L ) . If this ratio is taken as 0.001, then the length of the foundation required beyond a free end can be calculated to be 6 . 9 { ( k ~ / k ) . A single element is sufficient for this representation. We demonstrate an application through the device of computing the vibration frequencies of a cantilever beam on a two-parameter foundation. This problem is emulated after Eisenberger and Bielak 3. The results are summarized
Frequencies of a cantilever beam E= 1 I= 1 L = 1 /x = 1 (consistent units) Frequency oJ2, rad2/s 2
0
Extension b e y o n d end 0.5L
1.0L
Reference 3
Foundation p a r a m e t e r s
Mode
k= 1 kN/m 2 kl = 4.9348 kN
1 2 3
5.83 25.39 64.74
6.00 25.54 64.88
6.13 25.58 64.89
6.45 25.57 64.79
k = 100 k~ = 4.9348
1 2 3
11.53 27.67 65.50
14.13 29.36 66.51
14.18 29.41 66.54
14.06 29.02 66.19
k = 10 000 kl = 4.9348
1 2 3
100.23 103.24 119.23
101.34 108.97 127.14
101.34 108.97 127.14
101.22 107.67 123.81
k = 1000000 kl = 4.9348
1 2 3
1054.56 1059.62 1060.33
1059.58 1060.33 1064.28
1059.58 1060.33 1064.28
1000.15 1001.29 1005.04
Supplementary element matrices: B. N. Alemdar and P. GOIkan
920
in Table 1 where onl~¢ the case where k = 1, kj = 4.9348 corresponds to A > 2~/B, and the three remaining cases fall into the solution range of A < 2"4B. The frequencies determined with the consistent mass matrix shown graphically in Figure 6 show very good agreement with those of Reference 3 where the same problem is solved by affixing to the free end a concentrated spring to account for the correct force boundary condition. The table indicates the calculated frequencies when a part of the foundation beyond the free end equal to 0, 0.5 and 1.0 times the beam's own length has been included in the finite element model. We note that for A > 2x/B none of these virtual extensions satisfies the above criterion, whereas in the remaining cases for A < 2X/B extensions beyond the beam end of both 0.5L and 1.0L do. This is verified by the fact that the corresponding frequencies are virtually the same. For k = 1.0 x l06 all three frequencies tend to the same value, but larger differences exist between these and the values reported by Eisenberger and Bielak 3. As we do not know the details of that solution and the eigensolution routine which was employed, we will refrain from commenting on the causes of this difference.
into the element routines of frame analysis software, a broad spectrum of problems can be handled in a routine way. A web site ~4 has been prepared from which ASCII files containing the closed form expressions for element matrices for the full range of foundation parameters can be extracted.
References 1 2 3
4
5 6
7
5.
Conclusions
Many diverse problems involving beams on generalized elastic foundations may be solved routinely with the use of the exact shape functions illustrated in Figures 2 and 3. This exactness is in the same sense as the cubic displacement functions are exact for a uniform beam segment. Through appropriate manipulations of these functions exact expressions for the coefficients of the element matrices, i.e., stiffness, work-equivalent fixed end forces, consistent mass and geometric stiffness, can be generated. It has been demonstrated that the corresponding matrices for beams on Winkler-type foundations or for beams supported by no foundation can be derived as special solutions of this general case. The expressions for these coefficients are typically, very involved, but once they have been incorporated
8
9
10
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