O&45-7949/853.00 .W 0 1985 ergamon ergamon ressLtd. ressLtd.
Contprrrers & S~rwtw es Vol. 21. No. 6. pp. 1355-1359. 1985
Printedn PrintednGreatBritain. Great Britain.
EXACT EXACT
STIFF NE SS
MATRIX MATRIX FOR BEAMS FOUNDATION
MOSHE EISENBERGER
Faculty of Civil Engineering, Engineering,
(Received
Abstract-An exact element is required only a few elements a computer program
and
DAVID
28
ebruary
Technion-Israel Technion-Israel
Z.
ON ELASTIC
YANKELEVSKY
Institute of Technology,
Haifa 32000, 32000 , Israel
1984)
stiffness matrix of a beam element on o n elastic foundation is formulated. A single to exactly represent represent a continuous part of a beam on a Winkler foundation. Thus are sufficient for a typical problem solution. The stiffness matrix is assembled in and some numerical numerica l examples are presented.
INTRODUCTION The problem of beams on an elastic foundation has been treated by numerous numerous authors and closed form solutions of the differential equation have been proposed[l-41. Numerical methods, like the finite difference method, have been applied to solve this problem[5] and yield approximate solutions to the differential equation, which converge to the exact solution with increasing number of nodes. Such Su ch numerical techniques can easily be extended extend ed to handle nonlinear foundation foundation properties, variation of cross section di mensions, etc. Finite element solution of a beam on elastic foundation foundation may be obtained by discrete springs connected to structure nodes, thus roughly approximating the problem[6]. A more complicated model is the Winkler Winkler foundation foundation where the discrete springs are smeared to form a continuous continuous foundation[6-81. This is a common technique adopted in a finite element element formulation formulation of the problem. However, the shape functions functions which yield exact solution for a free beam, yield only only approximate solutions for a beam on elastic foundation, and accuracy is improved by increasing increasing the number number of elements. In this paper an exact stiffness matrix for a beam on a Winkler foundation is formulated. formul ated. Using this element, a single element is required requir ed between discontinuities (i.e. concentrated forces, abrupt change in section dimensions, dimensions, etc.) to yield exact solutions. Only Only a few elements are therefore required to exactly solve a typical problem and solution may be obtained on any small size microcomputer. Some examples show the agreement with exact solutions and comparisons with approximate methods. STIFFNESSMATRIXDERIVATION MATRIXDERIVATION The differential equation for the deflection curve of the beam supported on an elastic foundation in Fig. 1, is[l]
EI3 +
y
0.
The general general solution of eqn (1) can be written in terms of four functions (sign convention is shown in Fig. 1): Y(X) = YOFI(h)
+ - 8oFz(hx)
where F,(hr)
= cash AXcos AX,
Fz(Ax)
= t(cosh Ax sin AX + sinh hr cos LXX), L XX),
(3a)
(3b) F3@_x) = f sinh Ax sin AX, R(hr)
(3c)
= t(cosh Ax sin Xx - sinh hx cos Ax), (Ml
h=
4k
EI’
and yo, 00, MO, and Q. are the t he values at x = 0. 0 . The slope along the th e beam is given by Y’(X) = 4XYoF4(hx) + BOFl(hx)
The terms of the stiffness matrix, are defined as the holding holding actions at the ends ends of the beam, beam, due to unit translations translation s and rotations, rotatio ns, as shown in Fig. 2. From the expression in eqns (2) and (41, all 16 terms of the stiffness matrix can be found. As an example, the values Sll-S,4, for the case shown in Fig. 2(a), are computed. The values for y. and O. are 1 and 0, respectively, and thus deflection deflection and slope at
13.55
M.
35
EISENBERGER
nd D. Z.
YANKELEVSKY
‘k Fig. 1. Beam on elastic foundation.
I.
L
1
1
1
1
(d) Fig. 2. Member stiffnesses. =
can be written Y(L)
=
Fl(AL) + AI
J”(L)
where
as
F,(AL)S,,
= -4AF,(AL) +
&I
C
&I
= 0,
(W
- &IFz(hL)s,,
F3(AL)S,,
= 0.
II
A
k sinh2 AL
2A’
sin2 AL c
(6~)
M(x) = - $
yoF 3W
-
;
BoF dW
(5b) MoF~(hr)
+
QoF2W,
Va)
with two unknowns,
= If cos AL sinh AL + cos AL sin AL
s2, = -
sin2 AL.
The actions at x = can be found from the general expressions for M(x) and Q(x):
-
These are two equations and the terms are found as
sinh* AL
Q(x) = ; (6a) (6bl
~oF2(Ax )
+
4AMoF,(hr)
8oF~(Ax) - QoF, (hx).
(7b)
Substitution of the values for the shear and moment at x = 0 given in eqn (6) yields the actions at the
1357
Exact stiffness matrix right end (x = L): k sinh AL cos AL + cash AL sin AL
s3, = --
so, = ksinh A2
AL sin AL
=
sinh AL cash AL - sin AL cos AL c
S4I
(lla)
s24
=
s42,
(lib)
s34
=
s43,
(llc)
s44
=
s22.
(lid)
This stiffness matrix has been incorporated into a standard beam programI91. Only minor changes are required to extend the program’s capabilities.
are
EXAMPLES
1
Example The beam analyzed by Hetenyi (Ref. 1, p. 47) was solved using the computer program (Fig. 3). For this loading system, four elements are required to obtain the exact solution (elements Al?, BC, CD, and DE). The resulting ground pressure distribution and bending moment diagram are shown in the tigure, and these are identical with those given in Ref. 1.
(9b)
(9c)
s32
=
s
= _k_cosh AL sin AL
7
sinh AL cos AL
2A3
3
(94
SI3 =
S3I
s23
=
s32,
(10b)
s33
=
s,,,
(1Oc)
s43=
=
Pa)
s22 = 2A3
42
in Figs. 2(b)-2(d)
s21,
s4I
@aI
(8b)
In a similar way, actions found to be Sl2
,
Sl4
7
(W
-s21,
Example 2 For partially loaded beams on elastic foundation, Hetenyi proposes a solution in form of infinite
(1W
= 100 k?b?./in2 (689.5 30in
(0.762m)
LP=5000 (22.241 60
in
ebs KN)
L
/
48 in (I.219
KN/m2
m)
L2Oin (0.508m)
1.524 m )
O-
*O
2-20 4I-
/;
.
IO 12-
-60
/d
., *-.-.‘-*
40
/.dC
PRESSURE
DISTRIEUTIDN
ebsh21
-80
KN/m2
25000
-
30000
-
35000
-
BENDING
MOMENTS
- 400
4oDOo-
in-ebs
KN.m
Fig. 3. Example 1: Beam on elastic support.
M.
1358
EISENBERGERnd D. Z. YANKELEVSKY
Table I. Example 2: Deflections at x = 3.m a n d x =
q= KN/m
EJ=45000KNm2
4.m n
3m
Im
Im
1’
Lk=106 L-
L=lm
Im
KN/m2 4
Fig. 5. Example 3: Beam with both rigid and elastic supports.
trigonometric
series
(Ref.
0.300 0.488 0.448 0.509 0.509 0.498 0.499 0.498 0.497 0.497 0.497 0.497 0.497 0.497 0.497 0.497 0.497 0.497 0.497 0.497
2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Fig. 4. Example 2: Partially loaded beam.
YfEl~5000KNm2
Y(X = 3.m
1, p. 79):
Exa ct
sin
Y(x = 4.m)
622E -06 699E -06 927E -06 465E -06 465E -06 283E -06 782E -06 022E -06 307E-06 307E-06 572E -06 81OE -06 739E -06 918E -06 918E -06 826E-06 845606 813E-06 796E-06 7968-06
0.185 0.490 0.554 0.5 17 0.5 17 0.523 0.521 0.518 0.518 0.518 0.5 I8 0.518 0.518 0.518 0.518 0.518 0.518 0.518 0.518 0.518
0.497 817E-06
795E-06 109E -06 462E -06 047E -06 047E -06 958E-06 533E-06 687E-06 245E-06 245E-06 409E-06 794E-06 91OE -06 799606 799E-06 856E-06 826E-06 775E-06 764E-06 764E-06
0.518 787606
!!
Y=
These values agree with the exact solution given i Ref. 10.
(12) CONCLUSIONS
for in Fig. 4. The exact solution was obtained computer program, with the beam divided segments. The results are given in Table seen that three digits accuracy can be 3.m with eight terms for both points (x
The convergence
the beam using the into two 1. It is achieved
of this solution
and x = 4.m). Four digits accuracy
was tested
will require
An exact stiffness matrix for a beam on elastic foundation is directly formulated. An exact solution of a typical problem may be achieved by assembling a few elements. Nodes are placed at points of abrupt changes in loading, foundation or beam stiffness, and local supports. The efftciency and exact results are demonstrated through examples. This procedure can easily be implemented into any beam program.
12
terms. Example 3
A simply supported beam on elastic foundation (Fig. 5) was analyzed by Mohr[8] using a contact stiffness matrix. The same problem is solved by the proposed stiffness matrix formulation using one segment and the equations for the rotation at the left end +, and the deflection under the load v2 are
sinh h3C
cash
sinh
L
sin
The solution of these equations values for +, and v2:
sin A L
co s A L
I.
gives the exact
0.8366P x lo+,
(14a)
$,
0.1064P
(14bl
M. Hetenyi, Beams on Elastic Foundation. University of Michigan Press, Ann Arbor, Michigan (1964).
2. C. Miranda and K. Nair, Finite beams on elastic foundation. J. St. Div. ASCE 92, 131-142 (1966).
hsinh
h*(sinhALcoshXL
VT
x 10-5.
REFERENCES
sin AL
+ sinhLcosAL)
].rl]={ vz
-Or}.
(131
3. A. Dodge, Influence functions for beams on elastic foundations. J. Sr. Div. ASCE 90, 63-101 (1964). 4. B. Y. Ting, Finite beams on elastic foundation with restraints.-.I. Sr. Div. ASCE 108, 61 l-621 (1982). 5. F. W. Beaufait and P. W. Hoadlev. Analvsis ofelastic beams on nonlinear foundations.-Compur. Structures 12, 669-676 (1980).
Exact stiffness matrix 6. R. D. Cook, Concepts and Applications of Finite Element Analysis. Wiley, New York (1981). I. F. Miyahara and J. G. Ergatoudis, Matrix analysis of structure foundation interaction. J. St. Div. AXE 102, 251-265 (1976). 8. G.
A. Mohr, A contact stiffness matrix for finite ele-
1359
ment problems involving external elastic restraint. Comput. Struct. 12, 189-191 (1980). 9. W. Weaver and J. M. Gere, Matrix Analysis of Framed Structures. Van Nostrand Reinhold, New York (1980). 10. S. P. Timoshenko, Strength of Materials, Part 2. 3rd Edn. Van Nostrand Reinhold, New York (1956).
