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Spatial buckling of arches—A finite element analysis
0045-7949/90 53.00 + 0.00 c’ 1990 Pergamon Press plc
Con~puws & StrucrurcsVol. 34. No. 4. pp. 565-576. 1990 Printed in Great Britain.
SPATIAL
BUCKLING OF ARCHES-A ELEMENT ANALYSIS G.
KARAMI,~
FINITE
M. FARSHADS and M. R. BANAN
Departments of tMechanica1 Engineering and SCivil Engineering, University of Shiraz, Shiraz, Iran (Received 13 Januury 1989) Abstract-A finite element (FE) formulation is presented for the linear buckling analysis of curved beams and arches on elastic and/or point rigid foundations. The element shape function adapted herein embodies the rigid body modes as well as the deformation modes. With 12 degrees of freedom (six at each end), the master element can represent all possible general shapes of spatial curved beams and arches. To develop the element shape function, the kinematical relations are solved in terms of perturbations in the strain and the curvature change to yield the perturbations in a rigid body as well as the deformation modes. Specific examples of in-plane and out-of-plane stability problems of arches are investigated and the corresponding buckling loads and buckling modes are obtained.
1.
INTRODUCTION
The application of finite element (FE) method to the stability analysis of curved beams has been the
subject of considerable research interest in recent years. A variety of approaches to the problem has been adapted. However, comparatively few of these approaches are capable of dealing with curved elements of general geometry on elastic or rigid foundations. These approaches have mostly been limited to particular hinds of geometry such as straight beams or frames [l-4]. In several cases, variational methods have been employed. These studies have been limited to very special cases of rod buckling such as symmetrical, nonsymmetrical, in-plane or out-of plane buckling [S-9]. Most of the available formulations do not include elastic supports specially designed for general elements [3]. In the framework of FE analysis, curved beams and arched bodies can be investigated by replacing the elements by an assembly of straight beams. The degree of convergence of the solution obtained by this approximation is fairly well established. It is advisable, however, that the approximate deformation mode converges easily to the natural ones if the shape functions are in terms kinematical relations which precisely represent the deformation as well as the rigid body modes. Individual elements can be developed using either the linear or the nonlinear theory of elastic spatial rods, even though the linear theory finds many more applications in engineering analysis [8,9]. Two distinct methods can be employed in FE stability analysis; one is based on nonlinear theory of spatial curved beams and plotting of the load deformation variation [lo]. The second approach is based on the formation of the geometric stiffness matrix wherein either the stiffness method or the flexibility method can be utilized [l, 2,8].
In this paper, an FE formulation for linear stability theory of curved beams and arches is presented. The
general master-element developed can represent any geometrical shape or curved beams. With six degrees of freedom at each end the element cross-section may be variable along the length of the element. It can also rest on elastic and/or rigid point supports of linear or torsional type, each of which may be concentrated or be distributed along the length of the element in the local and/or global coordinate system. Also, the external applied loadings can be of a concentrated or uniformly distributed type. Both the force and the moment type distributed loadings can be expressed in either the local and/or global coordinate system. Moreover, the element can, in general, have six types of structural hinges of bending, twisting, axial, or shear type at each end. In what follows, the linear stability theory for curved beams is presented, followed by the derivation of the shape function and the element stiffness matrices. Several comparative examples are presented to show the accuracy of the formulations.
2. LINEAR THEORY AND LINEAR STABILITYTHEORY OF RLk3nC SPA~AL CURVED ELEMENTS
2.1. Linear theory of curved elements Let al a2a3 be the natural coordinates of the center line of the rod, with the a3 axis lying along the normal vector of the center line. Also, let xy be the principal axes of the rod cross-section making an angle C#J with the natural axes a, a2. Also, let the parameter s designate the coordinate along aj , as shown in Fig. 1. A brief summary of the classical (linear) theory for elastic spatial rods is now presented [l l-131. 2.1.1. The kinematical relations. In local principal axes, the strain-displacement {E}-(U), and the curvature-angle of rotations {k}-(O) relations can be 565
G. KARAMI et
al.
Fig. 1. Global and natural coordinate systems.
written as
twist for the curved beam elements is zero. In the natural coordinate system, the kinematical relations are written in the form
{sL} = {u’}’ + [G‘]{uLl~+ [J} {+} {k‘) = (@) + [CL1 {@},
(1)
where
(2) where
The superscript L implies that the associated quantity is expressed in the local (xy#,) coordinate system. The curvature matrix [G‘] and the matrix [J] are as follows: 0
1
1 0
VI= - i 01 00
00
in which K, = K sin 4, KY = K cos 4, K, = 0, where K is the curvature. Note that in the curvature matrix, [CL], the terms due to K, do not appear, as the local
PI =
0
o
o o ,
[ -K
0
K
1
0
0
where (u, v, w) are the displacements of the centerline along a,azaj, {e} is the rotation vector of the rod cross-section, {E} is the strain vector and {k} is the curvature-twist variation vector. Note that ( )’ = d( )/d.s. To clarify the symbolism used herein, the subscripts xy3 and 123 will be used to represent the variables in the xya3 and ala2a3 coordinate systems, respectively. 2.1.2. Equilibrium equations. In the local principal coordinate system the equilibrium equations can be
Spatial buckling of arches
567
In the natural coordinate system, the constitutive relations can be written as
written as IQ”)’ + [G”l{Q”) + @=‘I= (0)
{ML}’+ [G?{M‘} + [Jl{Q”} + {EL}= IO}, (3) where (6)
(7) Also in the natural coordinate system the equilibrium equations are written in the following form: in which
{QI’ + [Gl{Q) + W = to)
64,S,,S2,~1,~29~12)
PJ’ + [GIWI + Vl{Q) + (6) = (Oh
rrr
(4)
=
where
(1,a2,a1,a:,a:,ala2)daldap. JJJ
Also, z,=z,+z,=z,+z,,
where (Z,, I,,) =
in which {F} is the vector of distributed external forces, and {E} is the distributed moment vector along the length of an element of the rod, {Q} is the internal force vector and {M} is the internal moment vector at a typical section of the curved element. In the above formulations, it is assumed that M3 = M,, + M32 and k, = k32= k,, . 2.1.3. Matrix form of one-dimensional constitutive relations. The one-dimensional constitutive relations expressed in the local principal coordinate system are
(5)
{ML1= VlP% where
PI=
VI =
GA 0 i 0
1 1
0 GA 0
0 0 EA
EI,
0
0
0
EI,
0
0
G13
[ 0
.
In the above equations, modului of the body.
(y2,x2) dx dy. E and G are the elastic
2.2. Linear stability theory of curved elements Using the linear elastic theory for curved elements and arches, and utilizing the equilibrium method of stability analysis, we may now derive the genera1 stability (bifurcation) equations for curved elements. To follow the related procedure, we first assume an equilibrium state, to be called a critical state. Then, with a slight perturbation around the critical configuration, another equilibrium state, to be called a perturbed state, would be sought in the initial equilibrium neighborhood (see Fig. 2). If the equilibrium of the critical state is stable, then the perturbation field would identically vanish. On the other hand, a nonvanishing perturbed solution is indicative of bifurcation in the equilibrium path, i.e. the onset of the instability phenomenon. We now write a set of governing equations (kinematical, equilibrium, and constitutive relations) for the critical state. These governing equations plus the following governing equations in terms of perturbations make the stability theory governing equations for curved elements in natural coordinate
G. KARAMIet al. A
Critical state
Undeformed
Fig. 2. Different states in stability analysis.
systems. Note that the perturbation values are the values at the perturbed state subtracted from the values at the critical state [13, 151. 2.2.1. Kinematical relations in terms of perturbations. Let {u*} and {e*} designate the perturbations in the displacements and the rotation angles. The kinematical relations, in terms of perturbations, can be written in the local coordinate system as
[Q:+ e:Qo, + v*‘Qm - e:M,,,]’ + F: = 0 Q:’ - (K*w* + Ku*‘)Qos - KQ: + K0;Qo2 -K*e:M02+(Ke:
-K*O:)M,,,+F;=O
M:’ - (K*w* + Ku*‘)M,,, + KM:, - Q: + E: = 0 M:‘-
(K*w+ + Ku*‘)M,,+
Q: + Ej’=O
{E*} = {u*}‘+ [G]{u*} + [J}{e*} [Mj’ - (u *’ + Kw *)M,, - u*‘MO*]’ {k*} = {tP}‘+ [G]{L’*},
(8) -KM:+(u*‘+Kw+)Q,,-v*‘Q,,+E$=O,
(9)
where (&*I _1:3.
tu*) ={E}
where {F*} and {E*} are the perturbations in the distributed force vectors. Using the above equations, the perturbations in the applied distributed load vectors would vanish as constant direction type loading is assumed in application [ 13, 151. 2.2.3. Matrix form of constitutive relations in terms of perturbations. The one-dimensional constitutive equations in terms of perturbations are written as
2.2.2. Equilibrium equations in terms of perturbations. Let {M,,} and {Q,,} represent the forces and moments at the critical state; let {MI} and {Q’} be the associated perturbations, i.e. {Q}(at perturbed state) = {Q,} + {Q ‘} (10)
{M}(at perturbed state) = {M,} + {M*}. Hence, the equilibrium equations, in terms of perturbations and the values at the critical state, can be written as [12,13]
IQ?- e:Q~+(U+‘+Kw*)Q#J3+Ke:MOZ -(et’-Ke:)M,,yfKQ:+F:=o
The matrices [H,] and [H,] are defined in eqns (6) and (7). Note that Mf = M$, + M&. The formulations presented in eqns (&o-(O) form a basis for FE formulations and the stability analysis of curved elements. These relations will be utilized subsequently.
Spatial buckling of arches 3. THE FINITE ELEMENT FORMULATION
To develop a general FE scheme for stability analysis of curved beams and arches, the weighted residual method is employed herein. In order to do so, the equilibrium equations (9) are multiplied by a test function in matrix form and then integrated over the length of the element to implement the boundary conditions. The test function to be used is the same as the adapted shape functions [16] embodying the rigid body motion of the curved elements. Due to the similarities between the governing stability equations and the governing linear equations of the curved elements, these shape functions have been used for the static analysis as well as the stability analysis. It should be noted here that in order to form the element geometric stiffness matrix one has to have the internal loading at the critical state. Therefore, before one can perform the stability analysis the governing equations for the critical state must have been solved previously. Thus, an FE static analysis is performed for this purpose.
569
where {A } and {B} are column vectors of integration constants. In eqns (12) and (13), the first terms are the homogenous solutions which represent the rigid body modes. The second terms are the particular solutions corresponding to deformation modes. Utilizing the definitions a = [K: + K$]“’ {K)T=[Kx
The general are [16, 181
solutions
KY
to
{O*‘} = (l/a’{K}{K}‘+
01,
eqns
(12)
and
l/a2[GL]?Gt]
+ l/a[GLlsinas){A}
+ (l/a2{K}{K}T
+ Va*[G”l)(d) + (l/a4[GL]qGL] + s/a*[G‘l + l/a2~K}{K)~~2/2){e,2}
3.1. The shape function To develop the shape functions we set the strain and the change in curvature in the kinematical relations in terms of perturbations equal to zero. These equations for a linearly varying curvature change, a constant twist curvature, a constant axial strain and zero shear strain in the local principal coordinate system (xya,) may be expressed symbolically as
(13)
{u’“} = lIa2({K}{K}T+
(14)
[GL]TIGL]co~as
+ a[G%os m)(B)
+ l/a2({K}{K}r8
+ [GLl)h3} + Va4{K}(K}?J][GL]
x (-cosas{AI
+s{dj
+~*/2{e,,])
{fl*L}’ - [GL]{e*L} = {d} + s{e12} {Use}' -[GL]{u@}
-[J]{e*“} = {e3},
(11)
where
-I- 1/a’{K}{K}~J](a
sinus(d)
+&I)
+ lla4[Jl{K}{K}T({d}
+ &)I
+ ~/~4~G‘l~Jl{~}{K}~~{A}
+ s(d) + (s*/2 - l/a2){e,2}).
Formally, the solution to eqns (11) may be written as [16-H]
(15)
Equations (14) and (15) contain 12 constants of integration, three for each of the matrices {A}, {B}, {d) and also {e}. They enable us to develop a general element with 12 degrees of freedom. The perturbation field in the local coordinate system can be written as
{O*‘} = (exp[G‘]s){A} + (exp[GL]s)
x (exp - [G%)({d) +&H s
b
in which (c} are the parameters in both eqns (14) and (15), i.e.
{u”“) = (exp[GL]s){B} + (exp[GL]s) x
(exp - [GLls)([Jl{~*“) + @,}I h s
(16)
(12)
(13)
{c}={Al,A2,A3,BI,B2,B
39
d II d2, d3, e 39 e 2, e 1 }.
l/a* K,
-l/a*
KY
l/u* K,s
-llu4K;
l/u4 K, KY
K,
l/u* K,,s
- l/u4 K, K,,
l/u4 K;
-l/u*
l/a* K:s
l/a* K,K,,s
l/a* K,,
l/a* K, K,,s
l/a* Kzs
0
-1/u*K,cosas
I/a’K,K,sinas
l/u’K:sinas
K,cosus
-l/u*
l/u3 Kz sin us
cos as
I/u3KxKysinas
I/a KY sin as
- l/a K, sin as
l/u K, sin us
l/a* (K; + Kf cos us)
I/u*K,K,(l
-cosu.s)
- l/a KY sin as
-cosuS)
l/a*K,K,(l
l/a*(K;+K;cosas)
The elements of a 6 x 12 matrix [C] are derived as
0
0
l/u* (K$ + Kf cos as)
-cosas)
0
l/a* K,
l/a* Kg
0
0
0
l/a* K:s
0
l/a* K,
l/a* K,s
0
0
0
0
- u2s2/2))
l/a* [a2s2/2 - us sin as - l]K,
I/a” (1 - cos as)K$
-u2s2/2)
K,
l/a4 [a2s2/2 - as sin as
- l/u4 (1 - cos as)K,K,s
l/u4 (1 - cos as)K;s
-l/u*
l/a4 (Ki + K:a*s*/2)
-l/a4(KxK,(l
cOSl7.Y
l/aK,sinas
-l/aK,sinas
l/a4 (1 - cos as)K,K,s
l/u* K,,s
- l/a4 (K,K,,(l
l/u4 (Kf, + Kfa*s*/2)
- 1/a K, sin us
l/a* KY
l/u KY sin as
0
l/u*K,K,,(l
0
0
l/u*(K~+ K;G;osas)
0
0
571
Spatial buckling of arches [C] is used as the basis for computation of the shape functions and element stiffness matrices. The final form of the shape functions is derived by expressing unknown perturbations in terms of the nodal degrees of freedom at each end, i.e. The matrix
The perturbations within the element are then written in terms of the perturbations at the ends = [C(s)][R]{o*L)e= [N]{u*~}~. (18) [N] is the shape function as well as the test function matrix. {u *L}eis the perturbation of nodal degrees of freedom vector. The matrices [Nj and {u*~}~ have been expressed in the local coordinate system. Multiplying eqns (9) by [N]‘, using integration by parts, and utilizing eqn (18), we obtain
P,l=
P21=
By the symbol (E.E.) we mean the LHS of equilibrium eqns (9). Due to similarities between the governing equations in terms of perturbations and those for the critical state, and since we are using the same shape functions for perturbations as well as for the critical deformations, the matrix [K’J in eqn (19) is the generalized stiffness matrix of the element. In equilibrium eqns (9), the terms which are independent of the external forces will form the element stiffness matrix [Kr, while the load-dependent terms constitute the so-called geometric stiffness matrix [J&r. Matrices [O,], [02], and [03] have been derived as
0
0
0
0
0
0
0
0
0
0
0
0
0
0
-KM,,
-Qoz
0
0
$ 01
02
KQo,
0
0
Qo,
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
;
0
0
-MO,
0
-MO,
-Mo2
Qos
0
-MO* 0
0 0
Qo,
lo
O
0
1
0
0
0
0
-K2Mo,
0
0
0
0
0
-K2M02
0
0
0
0
KQoz
0
0
0
0
0
0
0
0
0
0
0
0
P31=
; 0
-K2M,,, -K2M,,, _. __ KQo,
L
(19)
in which
0
K;
1
([lul’+ [KJ){u*‘}e = {PIL)e,
0
- K*Qo,
G.
572
KARAMI et
We observe that the matrices [Dr] and [D,] are symmetrical and we also note that the last term in parenthesis in eqn (21) is the transpose of the first term. Therefore, for the conservative systems, [&r is also a symmetrical matrix. The load vector {P*}e is formed through integration by parts and it represents the perturbation in nodal concentrated loads. Since we have assumed that the applied loads are of constant-direction type, the assembled form of this vector would be a zero vector. After assembling the elemental equations, and by implementation of the geometrical boundary conditions, we will have a homogenous system of linear equations in the following form:
al.
+
{~*“yL%l{~*“}) CL%
(24)
in which the first term is due to the internal strain energy and the second term is the energy due to elastic supports. The matrices [A] and [I] are given in eqns (6) and (7). Z, , ZY, and ZSare the second moments of area with respect to the x, y and a3 axes. The matrices [S,] and [.S,J are the linear and rotational foundation stiffness of the Winkler type and are defined as
[f& and [KGISysare the overall system stiffness and geometric stiffness matrices, respectively. {U*}, is the vector of the nodal degrees of freedom in terms of perturbations. Now, by assuming that the geometric stiffness is linearly proportional to the applied loading, this may be stated mathematically as [4] Hence, eqn (24) can be written in the following condensed matrix form 1 is a single parameter characterizing the applied loading and which determines the proportionate critical buckling load, i.e. the critical buckling load is d times the applied load that was used to obtain the geometric stiffness. Equation (23) constitutes a characteristic eigenvalue problem. The parameter 1 corresponding to non-zero solutions of this problem is the eigenvalues while the family of non-zero solutions comprise the eigenvectors of the problem. Physically, the eigenvalues yield the critical (buckling) loads of the system and the corresponding solutions give rise to the buckling modes of the generally curved element.
u*’ = f{~““}r([Dl’ + [KSr){o*L}‘,
(25)
where [D]’ is the element stiffness matrix due to elastic deformation, [KS]e is the element stiffness matrix due to elastic supports, expressed in the local principal coordinate system. These matrices are as follows:
PI;2 x12 WI;,
=
mwll~l
(26)
x12= WI
[a
(27)
>
where
4. ELEMENT MATRICES
It is very convenient here to generate the element stiffness matrix from the perturbations in the elemental energy [16, 18-201. However, since the critical internal forces at any section of the element can be calculated using the matrix equilibrium equations, it is easier to develop the geometric element stiffness matrix by starting with the governing differential equations and employing the Galerkin method as was done previously in eqn (21). 4.1. Element stiffness matrix, [K] The strain and the curvature changes can be expressed in terms of shape functions. Thus, the element internal energy, U*e, may be calculated easily in the form
The matrix [KA 1, in eqn (26), depends on the shape function and the variation of the rod cross-section with respect to s. For the shape function employed herein, we have the strain energy as ;
+ so’(in*L}7al{&*L] {k*L}~z]{k*L})ds
s
o’Cie)Vl{e)
+w
++Mlw~
++NdJ
=i{cYt~~l{cJ = f{v*“}T[R]~KA][R]{u*~}
U” =
‘~{c*L)?A]{s*L} + {k*~}~f]{k*‘})d.r s0
=f{~*“}r[Dl’{~*~}.
(28)
Spatial buckling of arches
513
Hence, the matrix [K4] can be written in the following form:
(29) where
IN =
[~~Cl[~l,
where the matrix [Z’JaX6 is the transformation matrix from the xy axes to the u,az axes. Also, the internal forces at each section (needed for calculations of the matrices [D,], [DJ and [Q]) must also be first transformed into the natural coordinate system. If
EIJ
0
0
0
E~~12/2
0
0
EIJ
0
0
0
EI, 12/2
0
0
cr,r
0
0
0
0
0
0
EAI
0
0
E&l212
0
0
0
EIJ3/3
0
0
E&l’/2
0
0
0
EZ,,i3/3
WI =
Recalling eqn (25) the element stiffness matrix is [kr]e= [DY + [D1’.
(30)
In calculating the elements of the matrix [KS]c the shape function is expressed in the local principal coordinate system. Thus, if the spring stiffness matrix is expressed in a constant coordinate system it must be transformed into the local coordinate system. 4.2. The perturbation load vector (P*>
relation (33) is inserted into eqn (21), we obtain I vG1’ = FIT
-
1s 0
([~l~([~,l[yl+
P'l*@UEYl
P2lIn
- P,IrlxlN
ds WI,
1
(34)
in which VI = P-WI
Let (PEf’ represent the vector of element nodal forces due to the applied distributed forces across the element, then the element and internal forces vector (Ps} can be calculated according to fP*L)e = [K]e(v*t}6 - (P&Lil”,
(33)
(31)
in which the vector {Pe:“) is determined as
(F*L) and {E*L) are the ~~urbations in the distributed forces and moments along the length of the element and are expressed in the local coordinate system. If they are expressed in the global system, they must also be transformed.
WI = VI [Cl’ + I17’w3
In the special case of a straight beam the matrix [T’j is an identity matrix; therefore, for such cases relations (34) would become WI = [Cl,
WI = w.
(35)
The elastic support is assumed to be of a constant directional type. Otherwise, its influence must be incorporated into [&]e. Such is also the case for the type of defo~ation-dependent loadings such as the follower-type load [ 13, 151. 5. ILLUSTRATIVE
EXAMPLES
4.3. The geometrical stiffness matrix, [IC,p
5.1. In-plane buckling of a simply supported semi-circular arch
Once the internal forces at the ends of the element are evaluated, the matrix ~~~b~urn equations can be used to find the internal forces at any section along the length of the element. The forces arising from elastic supports can be calculated using the spring stiffness multiplied by the appropriate deformation field. To develop the element geometric stiffness matrix, the set of eqns (21) are employed. Again, since the shape function matrix [N] is expressed in the local coordinate system, it must first be transformed to the natural coordinate system, i.e.
A semi-circular arch under a constant direction pressure load is considered, as shown in Fig. 3. The cross-sectional properties are taken to be E = 207 GPa, G = 95 GPa, R = 5 m, A = 0.070686 m2, Z, = 10e6, Z, = 0.3976 x 10-4, Z, = 0.7952 x 10m4m4. In the present example out-of plane buckling is restrained. A curved element as well the available straight beam elements are use to discretize the arch. To show the rate of convergence by increasing the number of elements, as well as to check and compare the accuracy of the two kinds of elements used, the
G. KARAMI
574
et al.
P d
T
Sm
Q
1 I I I I I xc ::.: :.:.:.:.:.
Fig. 3. A simply-supported semi-circular arch.
6m
Y
results for the critical buckling loads are plotted in Fig. 4. As we observe, the results for the curved elements converge very rapidly, while the convergence of straight prismatic elements is not satisfactory. The accuracy in comparison with analytical solutions [21,22] is very satisfactory. The buckling modes are shown in Fig. 5. 5.2. An archedframe
t
under its own weight
In this example, the stability of an arched frame under its own weight is considered, as shown in
.I +-?-?=in=mmq
L
I-
Fig.
6.
lOm-
1 J
i
A simply-supported arched frame under its own
weight.
Fig. 6. The assumed cross-sectional properties are E = 207 GPa, G = 95 GPa, A = 0.7068 m2, Z, = Z,,= 10e3 m4, Z, = 21,. Altogether, six curved elements are used to model the problem. The critical
Mode II
n 0 Prismatic elements + * Curved element
0
r
I
Mode I %
Number of elements Fig. 4. The in-plane buckling critical loads of modes 1 and 2 for a simply-supported semi-circular arch.
Fig. 5. The in-plane buckling modes of the semi-circular arch.
575
Spatial buckling of arches
Fig. 7. The first two buckling modes of the arched frame.
Fig. 11. The first mode of buckling for the spatial arched structure.
T
Sm
6m
1 i--'OmI Fig. 8. An arched frame with partial elastic foundation.
buckling distributed loads corresponding to the first and the second modes are found to be qCrl= 0.47826 x IO6N/m and qer2= 0.281936 x 10’ N/m, respectively. The first and the second modes of buckling are also plotted in Fig. 7. 5.3. An arched frame with partial elastic foundations The previous example is considered again. Now, the sides and bottom are assumed to rest on partial elastic foundation (Fig. 8). The geometrical properties are the same as in example 2, except for Z, and I,,, which are assumed to be equal to 0.5 m4. Also, the axial stiffness of the elastic support is assumed to be 0.25 x lo9 N/m2. Using six curved elements to model the problem, the first critical buckling load is found to’ be q_, = 0.23635 x lo9 N/m. The first mode of buckling is shown in Fig. 9; the deformed buckled state is also shown. This example may be considered as a satisfactory model for a semi-buried tunnel and/or framework. 5.4. A spatial arched structure
Mode I
Deformed state
Fig. 9. The first mode and the deformed state of the arched frame on partial elastic foundation.
As the final example, a spatial arched structure as shown in Fig. 10 under a concentrated loading P is considered. The cross-sectional properties are the same as in example 2. Twelve curved elements are used to model the problem. The first and the second mode critical buckling loads are to be
Fig. 10. A spatial arched structure.
G. KARAMI et al.
516
Perl = 0.83759 x log and Pw2= 0.10742 x 1O’N. The first mode of buckling is shown in Fig. 11. This example may he considered as a model of a dome of revolution.
6. CONCLUSION
A master curved element to be used with FE modeling for the linear stability analysis of spatial curved elements and arches has been presented. The illustrating examples show that in the modeling of structural problems, even with a minimum number of such elements, the results are quite satisfactory.
REFERENCES 1. M. A. Bradford and N. S. Trahair, Lateral stability of beams on seats. J. struct. Engng, ASCE 109,2212-2215 (1983). 2. E. S. Awodalla, Elastic critical loads on multistory rigid frames. J. struct. Engng, ASCE 109, 1091-1106 (1983). 3. N. Tebedge and L. Tall, Linear stability analysis of beam-columns. J. struct. Div., ASCE 99, 24392457 (1973). 4. C. P. Johnson and K. M. Will, Beam buckling by finite element procedure. J. struct. Div., ASCE 100, 669-685 (1974). 5. W. J. Austin, In-plane bending and buckling of arches. J. struct. Div., AsCE 97, 1575-1592 (1971). 6. W. J. Austin and T. J. Ross, Elastic buckling of arches under symmetrical loading. J. struct. Div., ASCE 102, 1085-ld95 (1976). 7. C. Hong Yoo and P. A. Pfeiffer, Elastic stability of curved members. J. struct. Engng, ASCE 109, 2922-2940(1983). 8. R. K. Wen and J. Lange, Curved beam element for
arch buckling analysis. J. struct. Engng, ASCE 107, 20522069 (1981). 9. D. L. C. Lo and E. F. Mansur, Dynamic buckling of shallow arches. J. struct. Div., ASCE 102, 901917 (1976). 10. A. B: Sabir and A. C. Lock, Large deflection, geometrically nonlinear finite element analysis. ht. J. nu~h. Sci. 15, 3747 (1973). 11. V. V. Novozhilov, Foundation of The Nonlinear Theory of Elasticity, 3rd Edn. Graylock Press, Rochester, NY (1963). 12. M. Farshad, G. Karami and M. R. Banan, Nonlinear theory for elastic spatial rods. In?. J. Solids Struct. (submitted). 13. M. Farshad, G. Karami and M. R. Banan, A nonlinear stability analysis of spatial rods (submitted). 14. B. Tabarrok, A. M. Sinclair, M. Farshad and H. Yi, On the dynamics of spatially curved and twisted rods-a finite element formulation. J. Sound Vibr. 123, 315-326 (1988). 15. M. Farshad and B. Tabarrok, On stability equations of spatial rods-a technical theory. Truns CSME 11, 229-235 (1987). 16. M. R. Banan. G. Karami and M. Farshad. Finite element analysis of curved beams on elastic foundations. Cot&t. Struct. 32, 45-53 (1989). 17. L. A. Pines. Matrix Methods for Enaineers. Prentice Hall, Englewood Cliffs, NJ (1973). 18. B. Tabarrok, M. Farshad and H. Yi, Finite element formulation of spatially curved and twisted rods. Comput. Meth. appl. Mech. Engng 70, 275-299 (1988). 19. R. H. Gallagher,Finite Element Analysis Fundamentals. Prentic*Hall. Endewood Cliffs, NJ (1975). 20, M. Farshad, G. Karami and M. R. Banan, A theoretical and numerical finite element analysis of spatial rod systems. Comput. Meth. appl. Mech. Engng 73, 11l-132 (1989). 21. G. J. Simitses, An Introduction to Elastic Stability of Structures. Prentio+Hall, Englewood Cliffs, NJ (1976). 22. M. Farshad, On lateral-torsional instability of arches subjected to motion-dependent loading. J. Sound Vibr. 53, 165-171 (1977).
Con~puws & StrucrurcsVol. 34. No. 4. pp. 565-576. 1990 Printed in Great Britain.
SPATIAL
BUCKLING OF ARCHES-A ELEMENT ANALYSIS G.
KARAMI,~
FINITE
M. FARSHADS and M. R. BANAN
Departments of tMechanica1 Engineering and SCivil Engineering, University of Shiraz, Shiraz, Iran (Received 13 Januury 1989) Abstract-A finite element (FE) formulation is presented for the linear buckling analysis of curved beams and arches on elastic and/or point rigid foundations. The element shape function adapted herein embodies the rigid body modes as well as the deformation modes. With 12 degrees of freedom (six at each end), the master element can represent all possible general shapes of spatial curved beams and arches. To develop the element shape function, the kinematical relations are solved in terms of perturbations in the strain and the curvature change to yield the perturbations in a rigid body as well as the deformation modes. Specific examples of in-plane and out-of-plane stability problems of arches are investigated and the corresponding buckling loads and buckling modes are obtained.
1.
INTRODUCTION
The application of finite element (FE) method to the stability analysis of curved beams has been the
subject of considerable research interest in recent years. A variety of approaches to the problem has been adapted. However, comparatively few of these approaches are capable of dealing with curved elements of general geometry on elastic or rigid foundations. These approaches have mostly been limited to particular hinds of geometry such as straight beams or frames [l-4]. In several cases, variational methods have been employed. These studies have been limited to very special cases of rod buckling such as symmetrical, nonsymmetrical, in-plane or out-of plane buckling [S-9]. Most of the available formulations do not include elastic supports specially designed for general elements [3]. In the framework of FE analysis, curved beams and arched bodies can be investigated by replacing the elements by an assembly of straight beams. The degree of convergence of the solution obtained by this approximation is fairly well established. It is advisable, however, that the approximate deformation mode converges easily to the natural ones if the shape functions are in terms kinematical relations which precisely represent the deformation as well as the rigid body modes. Individual elements can be developed using either the linear or the nonlinear theory of elastic spatial rods, even though the linear theory finds many more applications in engineering analysis [8,9]. Two distinct methods can be employed in FE stability analysis; one is based on nonlinear theory of spatial curved beams and plotting of the load deformation variation [lo]. The second approach is based on the formation of the geometric stiffness matrix wherein either the stiffness method or the flexibility method can be utilized [l, 2,8].
In this paper, an FE formulation for linear stability theory of curved beams and arches is presented. The
general master-element developed can represent any geometrical shape or curved beams. With six degrees of freedom at each end the element cross-section may be variable along the length of the element. It can also rest on elastic and/or rigid point supports of linear or torsional type, each of which may be concentrated or be distributed along the length of the element in the local and/or global coordinate system. Also, the external applied loadings can be of a concentrated or uniformly distributed type. Both the force and the moment type distributed loadings can be expressed in either the local and/or global coordinate system. Moreover, the element can, in general, have six types of structural hinges of bending, twisting, axial, or shear type at each end. In what follows, the linear stability theory for curved beams is presented, followed by the derivation of the shape function and the element stiffness matrices. Several comparative examples are presented to show the accuracy of the formulations.
2. LINEAR THEORY AND LINEAR STABILITYTHEORY OF RLk3nC SPA~AL CURVED ELEMENTS
2.1. Linear theory of curved elements Let al a2a3 be the natural coordinates of the center line of the rod, with the a3 axis lying along the normal vector of the center line. Also, let xy be the principal axes of the rod cross-section making an angle C#J with the natural axes a, a2. Also, let the parameter s designate the coordinate along aj , as shown in Fig. 1. A brief summary of the classical (linear) theory for elastic spatial rods is now presented [l l-131. 2.1.1. The kinematical relations. In local principal axes, the strain-displacement {E}-(U), and the curvature-angle of rotations {k}-(O) relations can be 565
G. KARAMI et
al.
Fig. 1. Global and natural coordinate systems.
written as
twist for the curved beam elements is zero. In the natural coordinate system, the kinematical relations are written in the form
{sL} = {u’}’ + [G‘]{uLl~+ [J} {+} {k‘) = (@) + [CL1 {@},
(1)
where
(2) where
The superscript L implies that the associated quantity is expressed in the local (xy#,) coordinate system. The curvature matrix [G‘] and the matrix [J] are as follows: 0
1
1 0
VI= - i 01 00
00
in which K, = K sin 4, KY = K cos 4, K, = 0, where K is the curvature. Note that in the curvature matrix, [CL], the terms due to K, do not appear, as the local
PI =
0
o
o o ,
[ -K
0
K
1
0
0
where (u, v, w) are the displacements of the centerline along a,azaj, {e} is the rotation vector of the rod cross-section, {E} is the strain vector and {k} is the curvature-twist variation vector. Note that ( )’ = d( )/d.s. To clarify the symbolism used herein, the subscripts xy3 and 123 will be used to represent the variables in the xya3 and ala2a3 coordinate systems, respectively. 2.1.2. Equilibrium equations. In the local principal coordinate system the equilibrium equations can be
Spatial buckling of arches
567
In the natural coordinate system, the constitutive relations can be written as
written as IQ”)’ + [G”l{Q”) + @=‘I= (0)
{ML}’+ [G?{M‘} + [Jl{Q”} + {EL}= IO}, (3) where (6)
(7) Also in the natural coordinate system the equilibrium equations are written in the following form: in which
{QI’ + [Gl{Q) + W = to)
64,S,,S2,~1,~29~12)
PJ’ + [GIWI + Vl{Q) + (6) = (Oh
rrr
(4)
=
where
(1,a2,a1,a:,a:,ala2)daldap. JJJ
Also, z,=z,+z,=z,+z,,
where (Z,, I,,) =
in which {F} is the vector of distributed external forces, and {E} is the distributed moment vector along the length of an element of the rod, {Q} is the internal force vector and {M} is the internal moment vector at a typical section of the curved element. In the above formulations, it is assumed that M3 = M,, + M32 and k, = k32= k,, . 2.1.3. Matrix form of one-dimensional constitutive relations. The one-dimensional constitutive relations expressed in the local principal coordinate system are
(5)
{ML1= VlP% where
PI=
VI =
GA 0 i 0
1 1
0 GA 0
0 0 EA
EI,
0
0
0
EI,
0
0
G13
[ 0
.
In the above equations, modului of the body.
(y2,x2) dx dy. E and G are the elastic
2.2. Linear stability theory of curved elements Using the linear elastic theory for curved elements and arches, and utilizing the equilibrium method of stability analysis, we may now derive the genera1 stability (bifurcation) equations for curved elements. To follow the related procedure, we first assume an equilibrium state, to be called a critical state. Then, with a slight perturbation around the critical configuration, another equilibrium state, to be called a perturbed state, would be sought in the initial equilibrium neighborhood (see Fig. 2). If the equilibrium of the critical state is stable, then the perturbation field would identically vanish. On the other hand, a nonvanishing perturbed solution is indicative of bifurcation in the equilibrium path, i.e. the onset of the instability phenomenon. We now write a set of governing equations (kinematical, equilibrium, and constitutive relations) for the critical state. These governing equations plus the following governing equations in terms of perturbations make the stability theory governing equations for curved elements in natural coordinate
G. KARAMIet al. A
Critical state
Undeformed
Fig. 2. Different states in stability analysis.
systems. Note that the perturbation values are the values at the perturbed state subtracted from the values at the critical state [13, 151. 2.2.1. Kinematical relations in terms of perturbations. Let {u*} and {e*} designate the perturbations in the displacements and the rotation angles. The kinematical relations, in terms of perturbations, can be written in the local coordinate system as
[Q:+ e:Qo, + v*‘Qm - e:M,,,]’ + F: = 0 Q:’ - (K*w* + Ku*‘)Qos - KQ: + K0;Qo2 -K*e:M02+(Ke:
-K*O:)M,,,+F;=O
M:’ - (K*w* + Ku*‘)M,,, + KM:, - Q: + E: = 0 M:‘-
(K*w+ + Ku*‘)M,,+
Q: + Ej’=O
{E*} = {u*}‘+ [G]{u*} + [J}{e*} [Mj’ - (u *’ + Kw *)M,, - u*‘MO*]’ {k*} = {tP}‘+ [G]{L’*},
(8) -KM:+(u*‘+Kw+)Q,,-v*‘Q,,+E$=O,
(9)
where (&*I _1:3.
tu*) ={E}
where {F*} and {E*} are the perturbations in the distributed force vectors. Using the above equations, the perturbations in the applied distributed load vectors would vanish as constant direction type loading is assumed in application [ 13, 151. 2.2.3. Matrix form of constitutive relations in terms of perturbations. The one-dimensional constitutive equations in terms of perturbations are written as
2.2.2. Equilibrium equations in terms of perturbations. Let {M,,} and {Q,,} represent the forces and moments at the critical state; let {MI} and {Q’} be the associated perturbations, i.e. {Q}(at perturbed state) = {Q,} + {Q ‘} (10)
{M}(at perturbed state) = {M,} + {M*}. Hence, the equilibrium equations, in terms of perturbations and the values at the critical state, can be written as [12,13]
IQ?- e:Q~+(U+‘+Kw*)Q#J3+Ke:MOZ -(et’-Ke:)M,,yfKQ:+F:=o
The matrices [H,] and [H,] are defined in eqns (6) and (7). Note that Mf = M$, + M&. The formulations presented in eqns (&o-(O) form a basis for FE formulations and the stability analysis of curved elements. These relations will be utilized subsequently.
Spatial buckling of arches 3. THE FINITE ELEMENT FORMULATION
To develop a general FE scheme for stability analysis of curved beams and arches, the weighted residual method is employed herein. In order to do so, the equilibrium equations (9) are multiplied by a test function in matrix form and then integrated over the length of the element to implement the boundary conditions. The test function to be used is the same as the adapted shape functions [16] embodying the rigid body motion of the curved elements. Due to the similarities between the governing stability equations and the governing linear equations of the curved elements, these shape functions have been used for the static analysis as well as the stability analysis. It should be noted here that in order to form the element geometric stiffness matrix one has to have the internal loading at the critical state. Therefore, before one can perform the stability analysis the governing equations for the critical state must have been solved previously. Thus, an FE static analysis is performed for this purpose.
569
where {A } and {B} are column vectors of integration constants. In eqns (12) and (13), the first terms are the homogenous solutions which represent the rigid body modes. The second terms are the particular solutions corresponding to deformation modes. Utilizing the definitions a = [K: + K$]“’ {K)T=[Kx
The general are [16, 181
solutions
KY
to
{O*‘} = (l/a’{K}{K}‘+
01,
eqns
(12)
and
l/a2[GL]?Gt]
+ l/a[GLlsinas){A}
+ (l/a2{K}{K}T
+ Va*[G”l)(d) + (l/a4[GL]qGL] + s/a*[G‘l + l/a2~K}{K)~~2/2){e,2}
3.1. The shape function To develop the shape functions we set the strain and the change in curvature in the kinematical relations in terms of perturbations equal to zero. These equations for a linearly varying curvature change, a constant twist curvature, a constant axial strain and zero shear strain in the local principal coordinate system (xya,) may be expressed symbolically as
(13)
{u’“} = lIa2({K}{K}T+
(14)
[GL]TIGL]co~as
+ a[G%os m)(B)
+ l/a2({K}{K}r8
+ [GLl)h3} + Va4{K}(K}?J][GL]
x (-cosas{AI
+s{dj
+~*/2{e,,])
{fl*L}’ - [GL]{e*L} = {d} + s{e12} {Use}' -[GL]{u@}
-[J]{e*“} = {e3},
(11)
where
-I- 1/a’{K}{K}~J](a
sinus(d)
+&I)
+ lla4[Jl{K}{K}T({d}
+ &)I
+ ~/~4~G‘l~Jl{~}{K}~~{A}
+ s(d) + (s*/2 - l/a2){e,2}).
Formally, the solution to eqns (11) may be written as [16-H]
(15)
Equations (14) and (15) contain 12 constants of integration, three for each of the matrices {A}, {B}, {d) and also {e}. They enable us to develop a general element with 12 degrees of freedom. The perturbation field in the local coordinate system can be written as
{O*‘} = (exp[G‘]s){A} + (exp[GL]s)
x (exp - [G%)({d) +&H s
b
in which (c} are the parameters in both eqns (14) and (15), i.e.
{u”“) = (exp[GL]s){B} + (exp[GL]s) x
(exp - [GLls)([Jl{~*“) + @,}I h s
(16)
(12)
(13)
{c}={Al,A2,A3,BI,B2,B
39
d II d2, d3, e 39 e 2, e 1 }.
l/a* K,
-l/a*
KY
l/u* K,s
-llu4K;
l/u4 K, KY
K,
l/u* K,,s
- l/u4 K, K,,
l/u4 K;
-l/u*
l/a* K:s
l/a* K,K,,s
l/a* K,,
l/a* K, K,,s
l/a* Kzs
0
-1/u*K,cosas
I/a’K,K,sinas
l/u’K:sinas
K,cosus
-l/u*
l/u3 Kz sin us
cos as
I/u3KxKysinas
I/a KY sin as
- l/a K, sin as
l/u K, sin us
l/a* (K; + Kf cos us)
I/u*K,K,(l
-cosu.s)
- l/a KY sin as
-cosuS)
l/a*K,K,(l
l/a*(K;+K;cosas)
The elements of a 6 x 12 matrix [C] are derived as
0
0
l/u* (K$ + Kf cos as)
-cosas)
0
l/a* K,
l/a* Kg
0
0
0
l/a* K:s
0
l/a* K,
l/a* K,s
0
0
0
0
- u2s2/2))
l/a* [a2s2/2 - us sin as - l]K,
I/a” (1 - cos as)K$
-u2s2/2)
K,
l/a4 [a2s2/2 - as sin as
- l/u4 (1 - cos as)K,K,s
l/u4 (1 - cos as)K;s
-l/u*
l/a4 (Ki + K:a*s*/2)
-l/a4(KxK,(l
cOSl7.Y
l/aK,sinas
-l/aK,sinas
l/a4 (1 - cos as)K,K,s
l/u* K,,s
- l/a4 (K,K,,(l
l/u4 (Kf, + Kfa*s*/2)
- 1/a K, sin us
l/a* KY
l/u KY sin as
0
l/u*K,K,,(l
0
0
l/u*(K~+ K;G;osas)
0
0
571
Spatial buckling of arches [C] is used as the basis for computation of the shape functions and element stiffness matrices. The final form of the shape functions is derived by expressing unknown perturbations in terms of the nodal degrees of freedom at each end, i.e. The matrix
The perturbations within the element are then written in terms of the perturbations at the ends = [C(s)][R]{o*L)e= [N]{u*~}~. (18) [N] is the shape function as well as the test function matrix. {u *L}eis the perturbation of nodal degrees of freedom vector. The matrices [Nj and {u*~}~ have been expressed in the local coordinate system. Multiplying eqns (9) by [N]‘, using integration by parts, and utilizing eqn (18), we obtain
P,l=
P21=
By the symbol (E.E.) we mean the LHS of equilibrium eqns (9). Due to similarities between the governing equations in terms of perturbations and those for the critical state, and since we are using the same shape functions for perturbations as well as for the critical deformations, the matrix [K’J in eqn (19) is the generalized stiffness matrix of the element. In equilibrium eqns (9), the terms which are independent of the external forces will form the element stiffness matrix [Kr, while the load-dependent terms constitute the so-called geometric stiffness matrix [J&r. Matrices [O,], [02], and [03] have been derived as
0
0
0
0
0
0
0
0
0
0
0
0
0
0
-KM,,
-Qoz
0
0
$ 01
02
KQo,
0
0
Qo,
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
;
0
0
-MO,
0
-MO,
-Mo2
Qos
0
-MO* 0
0 0
Qo,
lo
O
0
1
0
0
0
0
-K2Mo,
0
0
0
0
0
-K2M02
0
0
0
0
KQoz
0
0
0
0
0
0
0
0
0
0
0
0
P31=
; 0
-K2M,,, -K2M,,, _. __ KQo,
L
(19)
in which
0
K;
1
([lul’+ [KJ){u*‘}e = {PIL)e,
0
- K*Qo,
G.
572
KARAMI et
We observe that the matrices [Dr] and [D,] are symmetrical and we also note that the last term in parenthesis in eqn (21) is the transpose of the first term. Therefore, for the conservative systems, [&r is also a symmetrical matrix. The load vector {P*}e is formed through integration by parts and it represents the perturbation in nodal concentrated loads. Since we have assumed that the applied loads are of constant-direction type, the assembled form of this vector would be a zero vector. After assembling the elemental equations, and by implementation of the geometrical boundary conditions, we will have a homogenous system of linear equations in the following form:
al.
+
{~*“yL%l{~*“}) CL%
(24)
in which the first term is due to the internal strain energy and the second term is the energy due to elastic supports. The matrices [A] and [I] are given in eqns (6) and (7). Z, , ZY, and ZSare the second moments of area with respect to the x, y and a3 axes. The matrices [S,] and [.S,J are the linear and rotational foundation stiffness of the Winkler type and are defined as
[f& and [KGISysare the overall system stiffness and geometric stiffness matrices, respectively. {U*}, is the vector of the nodal degrees of freedom in terms of perturbations. Now, by assuming that the geometric stiffness is linearly proportional to the applied loading, this may be stated mathematically as [4] Hence, eqn (24) can be written in the following condensed matrix form 1 is a single parameter characterizing the applied loading and which determines the proportionate critical buckling load, i.e. the critical buckling load is d times the applied load that was used to obtain the geometric stiffness. Equation (23) constitutes a characteristic eigenvalue problem. The parameter 1 corresponding to non-zero solutions of this problem is the eigenvalues while the family of non-zero solutions comprise the eigenvectors of the problem. Physically, the eigenvalues yield the critical (buckling) loads of the system and the corresponding solutions give rise to the buckling modes of the generally curved element.
u*’ = f{~““}r([Dl’ + [KSr){o*L}‘,
(25)
where [D]’ is the element stiffness matrix due to elastic deformation, [KS]e is the element stiffness matrix due to elastic supports, expressed in the local principal coordinate system. These matrices are as follows:
PI;2 x12 WI;,
=
mwll~l
(26)
x12= WI
[a
(27)
>
where
4. ELEMENT MATRICES
It is very convenient here to generate the element stiffness matrix from the perturbations in the elemental energy [16, 18-201. However, since the critical internal forces at any section of the element can be calculated using the matrix equilibrium equations, it is easier to develop the geometric element stiffness matrix by starting with the governing differential equations and employing the Galerkin method as was done previously in eqn (21). 4.1. Element stiffness matrix, [K] The strain and the curvature changes can be expressed in terms of shape functions. Thus, the element internal energy, U*e, may be calculated easily in the form
The matrix [KA 1, in eqn (26), depends on the shape function and the variation of the rod cross-section with respect to s. For the shape function employed herein, we have the strain energy as ;
+ so’(in*L}7al{&*L] {k*L}~z]{k*L})ds
s
o’Cie)Vl{e)
+w
++Mlw~
++NdJ
=i{cYt~~l{cJ = f{v*“}T[R]~KA][R]{u*~}
U” =
‘~{c*L)?A]{s*L} + {k*~}~f]{k*‘})d.r s0
=f{~*“}r[Dl’{~*~}.
(28)
Spatial buckling of arches
513
Hence, the matrix [K4] can be written in the following form:
(29) where
IN =
[~~Cl[~l,
where the matrix [Z’JaX6 is the transformation matrix from the xy axes to the u,az axes. Also, the internal forces at each section (needed for calculations of the matrices [D,], [DJ and [Q]) must also be first transformed into the natural coordinate system. If
EIJ
0
0
0
E~~12/2
0
0
EIJ
0
0
0
EI, 12/2
0
0
cr,r
0
0
0
0
0
0
EAI
0
0
E&l212
0
0
0
EIJ3/3
0
0
E&l’/2
0
0
0
EZ,,i3/3
WI =
Recalling eqn (25) the element stiffness matrix is [kr]e= [DY + [D1’.
(30)
In calculating the elements of the matrix [KS]c the shape function is expressed in the local principal coordinate system. Thus, if the spring stiffness matrix is expressed in a constant coordinate system it must be transformed into the local coordinate system. 4.2. The perturbation load vector (P*>
relation (33) is inserted into eqn (21), we obtain I vG1’ = FIT
-
1s 0
([~l~([~,l[yl+
P'l*@UEYl
P2lIn
- P,IrlxlN
ds WI,
1
(34)
in which VI = P-WI
Let (PEf’ represent the vector of element nodal forces due to the applied distributed forces across the element, then the element and internal forces vector (Ps} can be calculated according to fP*L)e = [K]e(v*t}6 - (P&Lil”,
(33)
(31)
in which the vector {Pe:“) is determined as
(F*L) and {E*L) are the ~~urbations in the distributed forces and moments along the length of the element and are expressed in the local coordinate system. If they are expressed in the global system, they must also be transformed.
WI = VI [Cl’ + I17’w3
In the special case of a straight beam the matrix [T’j is an identity matrix; therefore, for such cases relations (34) would become WI = [Cl,
WI = w.
(35)
The elastic support is assumed to be of a constant directional type. Otherwise, its influence must be incorporated into [&]e. Such is also the case for the type of defo~ation-dependent loadings such as the follower-type load [ 13, 151. 5. ILLUSTRATIVE
EXAMPLES
4.3. The geometrical stiffness matrix, [IC,p
5.1. In-plane buckling of a simply supported semi-circular arch
Once the internal forces at the ends of the element are evaluated, the matrix ~~~b~urn equations can be used to find the internal forces at any section along the length of the element. The forces arising from elastic supports can be calculated using the spring stiffness multiplied by the appropriate deformation field. To develop the element geometric stiffness matrix, the set of eqns (21) are employed. Again, since the shape function matrix [N] is expressed in the local coordinate system, it must first be transformed to the natural coordinate system, i.e.
A semi-circular arch under a constant direction pressure load is considered, as shown in Fig. 3. The cross-sectional properties are taken to be E = 207 GPa, G = 95 GPa, R = 5 m, A = 0.070686 m2, Z, = 10e6, Z, = 0.3976 x 10-4, Z, = 0.7952 x 10m4m4. In the present example out-of plane buckling is restrained. A curved element as well the available straight beam elements are use to discretize the arch. To show the rate of convergence by increasing the number of elements, as well as to check and compare the accuracy of the two kinds of elements used, the
G. KARAMI
574
et al.
P d
T
Sm
Q
1 I I I I I xc ::.: :.:.:.:.:.
Fig. 3. A simply-supported semi-circular arch.
6m
Y
results for the critical buckling loads are plotted in Fig. 4. As we observe, the results for the curved elements converge very rapidly, while the convergence of straight prismatic elements is not satisfactory. The accuracy in comparison with analytical solutions [21,22] is very satisfactory. The buckling modes are shown in Fig. 5. 5.2. An archedframe
t
under its own weight
In this example, the stability of an arched frame under its own weight is considered, as shown in
.I +-?-?=in=mmq
L
I-
Fig.
6.
lOm-
1 J
i
A simply-supported arched frame under its own
weight.
Fig. 6. The assumed cross-sectional properties are E = 207 GPa, G = 95 GPa, A = 0.7068 m2, Z, = Z,,= 10e3 m4, Z, = 21,. Altogether, six curved elements are used to model the problem. The critical
Mode II
n 0 Prismatic elements + * Curved element
0
r
I
Mode I %
Number of elements Fig. 4. The in-plane buckling critical loads of modes 1 and 2 for a simply-supported semi-circular arch.
Fig. 5. The in-plane buckling modes of the semi-circular arch.
575
Spatial buckling of arches
Fig. 7. The first two buckling modes of the arched frame.
Fig. 11. The first mode of buckling for the spatial arched structure.
T
Sm
6m
1 i--'OmI Fig. 8. An arched frame with partial elastic foundation.
buckling distributed loads corresponding to the first and the second modes are found to be qCrl= 0.47826 x IO6N/m and qer2= 0.281936 x 10’ N/m, respectively. The first and the second modes of buckling are also plotted in Fig. 7. 5.3. An arched frame with partial elastic foundations The previous example is considered again. Now, the sides and bottom are assumed to rest on partial elastic foundation (Fig. 8). The geometrical properties are the same as in example 2, except for Z, and I,,, which are assumed to be equal to 0.5 m4. Also, the axial stiffness of the elastic support is assumed to be 0.25 x lo9 N/m2. Using six curved elements to model the problem, the first critical buckling load is found to’ be q_, = 0.23635 x lo9 N/m. The first mode of buckling is shown in Fig. 9; the deformed buckled state is also shown. This example may be considered as a satisfactory model for a semi-buried tunnel and/or framework. 5.4. A spatial arched structure
Mode I
Deformed state
Fig. 9. The first mode and the deformed state of the arched frame on partial elastic foundation.
As the final example, a spatial arched structure as shown in Fig. 10 under a concentrated loading P is considered. The cross-sectional properties are the same as in example 2. Twelve curved elements are used to model the problem. The first and the second mode critical buckling loads are to be
Fig. 10. A spatial arched structure.
G. KARAMI et al.
516
Perl = 0.83759 x log and Pw2= 0.10742 x 1O’N. The first mode of buckling is shown in Fig. 11. This example may he considered as a model of a dome of revolution.
6. CONCLUSION
A master curved element to be used with FE modeling for the linear stability analysis of spatial curved elements and arches has been presented. The illustrating examples show that in the modeling of structural problems, even with a minimum number of such elements, the results are quite satisfactory.
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