Galerkin method as a tool to investigate the planar and non-planar behavior of curved beams

Compurers & Srnrcrures Vol. 18, No. Printed in Great Britain.

I, pp. 16S174,

CO4>7949/84 Pergamon

1984

$3.00 + .MJ Press Ltd.

GALERKIN METHOD AS A TOOL TO INVESTIGATE THE PLANAR AND NON-PLANAR BEHAVIOR OF CURVED BEAMS A. ROSEN?and H. ABRAMOVICH$ Department of Aeronautical Engineering, Technion-Israel Institute of Technology, Haifa, Israel (Received

23 November 1981; received for publication 17 September 1982)

Abstract-The equations of equilibrium which describe the three-dimensional behavior of curved beams are derived. These equations include first order geometric nonlinear influences. The equations are solved by the Galerkin method. The method is very general and allows general variation of the geometric and structural properties along the beam. Any combination of boundary conditions is possible and the most general distribution of loads along the beam can be treated. Numerical results for a few examples are presented and compared to other theoretical and experimental results. The agreement between the results is generally good. It is shown that Galerkin method is an efficient method which enables one to solve the problem using small number of unknowns compared to other methods which are in use.

1. INTRODUCTION Curved beams, and in particular arches, are common structural elements which have gained a considerable amount of attention during the years. In past years most of the theoretical work in this field included the application of different numerical methods to solve various problems associated with curved beams. It seems that finite elements were the major tool in this research and since it is beyond the scope of this introduction to present any detailed literature survey, only typical examples of this kind of work are referred to [l-4]. Other numerical methods included finite differences (for example, [5,6]). The majority of the work in this field, like the references mentioned above, was restricted to the planar case where deformations only include bending in the plane of curvature of the beam. Very little has been done concerning the more general and more complicated problem on the non-planar behavior of curved beams. Two areas where this problem cannot be avoided are the well known lateral-torsional buckling of arches (for example [7,8] or coupled twist-bending vibrations of curved beams[S-101. A recent survey of the problem of vibrations of curved beams is presented in [l I]. There are cases where a curved beam is only a component of a more complicated structural system. If the behavior of this element is non-planar then a usual finite-element or finite-difference model of this component becomes very complicated with many degrees of freedom. In these cases the need for a model which, with a relatively few degrees of freedom, offers a good description of the behavior of the curved beam is evident. This need becomes even more apparent when the whole structural system is subjected to many different kinds of loading conditions or if the case of interest is a case of dynamic loading where integration with respect to time is performed. Such a case is, for example, the vertical axis Darrieus wind turbine. In this case the system is composed of a few curved blades (the element of interest here), tower, guying wires and generator. All the system is subjected to complex Wenior Lecturer. fInstructor.

gravity, inertia and aerodynamic loads which vary with time. It seems that the Galerkin method offers the possibility of obtaining a model of the non-planar behavior of the curved beam, which is relatively simple and has a small number of degrees of freedom, but is still accurate enough. In what follows, the derivation of the equilibrium equations of a curved beam, is presented. The present derivation is confined to the case where the elastic axis of the beam lies in a certain plane but besides that, the curvature, beam properties and loading are general. It should be pointed out that these equations contain first order nonlinear geometric influences. A general scheme for solving these equations, using the Galerkin method, is then presented. This scheme enables one to choose any combination of geometric or natural boundary conditions. Numerical results for a circular beam, loaded by a concentrated force at mid-spane, which varies in magnitude and direction, are presented. These results are compared with experimental and exact analytical solutions, and discussion of the results is given. It is shown that good results are obtained using a fairly small number of degrees of freedom. 2. THEORETICAL DERIVATION It is assumed that the position vector of each point of a slender curved beam, before the deformation, is given by F where:

r = T(x, y,

z)

(1)

x is a coordinate along the curved elastic axis of the beam, while the coordinate lines y and z are perpendicular to x at each point along the axis and also perpendicular one to the other. The present derivation will be restricted to cases where x and y lie always in a certain plane while z is perpendicular to this plane. This is the case in many practical applications and it results in a considerable simplification in the derivation. If needed, the derivation for the case where x is non-planar can be done while following the same procedure that is presented in what follows. 165

166

A. ROSEN and

The position vector of each point along the elastic axis is given by 0? where: 0’ = T;X,O.O).

(3)

One should note that the triad &, g,, and & is rotated about&along the beam. This rotation is defined by the initial curvature 1, where: L -. ^ *. * h = x,e,, ey,x= - xyex,qx = 0. (4) The deformation is described by the displacement of each point on the elastic axis. The components of this displacement are u, u, w, in the directions &,, t!,,and &, respectively. In addition each cross section is also rotated by an angle C$ about the elastic axis. Bernoulli-Euler assumptions are adopted meaning that each cross section which is plane and perpendicular to the elastic axis before the deformation remains plane and perpendicular to the curved elastic axis after the deformation. This assumption refers mainly to the bending of the beam. The twist is associated with warping of the cross section as it will be shown in what follows. Based on these assumptions a new triad of orthogonal unit vectors e^:,g; and t?:, is defined, which are tangent to the deformed coordinate lines x, y and z, respectively, along the deformed elastic axis. In addition to the Bernoulli-Euler assumptions, the derivation will be restricted to cases of small elastic rotations and small strains. According to this assumption squares of elastic rotations are of the order of magnitude of strains and both are neglected compared to unity. In [12] a detailed description of the ordering scheme, which results from these assumptions, is given. The results which will be presented in what follows incorporates the omission of small terms according to this ordering scheme. It is easily found that the triad of unit vectors after the deformation is given by: t?: = & + u&f

+

zC; + TI/@;

(8)

where Tt,b is the warping according to the theory of Saint-Venant torsion. The well-known warping function JI (see, for example[l3]) is a function of x, y and z. The derivation will be restricted to the usual cases of small warping where warping displacements are very small compared to cross-sectional dimensions. From this point regular derivation[l3] is performed in order to find expressions for the resultant forces and moments. Differentiation of the position vectors before and after the deformation yields the base vectors of undeformed and deformed beam. From these base vectors the metric tensors are calculated which then yield expressions for the strain components. Using the usual beam theories the stress components are finally obtained. All these steps are described in detail in [12]. The resultant force P which acts on each cross section of the beam is defined as:

P = Pt?: + v,e^; + v,;:.

(9)

The tensile force P is obtained by integration of the axial stress over the cross section, which implies: P = EA

u,, - vxy + $,X2 + w,,‘)

1 (10)

Where E is Young’s modulus and A the cross sectional area which is effective in carrying tensile stresses. The resultant moment about the elastic center of each cross section, d, is defined as: ti = M,e^: + Myt?; + M,e^;.

(11)

Again the components of I%?are obtained by integration over the cross section and their expressions will be given in what follows. The beam is acted upon by a distributed force ,G and distributed moment 4 per unit length, given by:

(5b)

4 = qg: + qye^;+ qg:.

(13)

(5c)

From equilibrium of forces and moments associated with a beam element, four equations of equilibrium in the four unknowns v, w, 4 and P are obtained. In order to obtain a non-dimensionalized form of the equations the following non-dimensional parameters are defined:

;; = - v,,& + Cy•l- +t?, 6; = -w,,e^, - c#$ + t$

The curvatures KY, K, and twist T of the deformed elastic axis are defined by the following relations:

-I e Y” = - Kj?; + Te^; e-’z.x =_ K,i; - T;;

R=,r+u~x+zli?y+w~z+yye^;

(12)

(5a)

^I

The position vector of each point of the beam after the deformation. becomes:

p = p,e^:+ p,e^;+ p,e^:

w,,e^z

eX,X= K,,C^;+ K,e^:

ABRAMOVICH

(2)

A triad of orthogonal unit vectors gX,5 and & will now be defined. gXis tangent to the elastic axis at each point along the beam, while C$ and & are tangent to the coordinate lines y and z, respectively. This implies that: e, = OT^,X; i, = TJx, 0, 0); f&= I& x K?~

H.

@a) (6b)

a

=;;

(EoA)=(EAlfii; (Eo,,,)+!

(k)

where according to the present assumptions:

(EOI,,) = 2,

(E”&) =E&;

(C’J) = g

(14)

(7a) (7b) (7c)

As one can see, nondimensional

variables are defined

Galerkin method as a tool to investigate the planar and non-planar behavior of curved beams by a small circle above them. b is a typical cross section dimension while L is the length of the beam. GJ is the well known Saint-Venant torsional stiffness while (EZ,,,,)(EI,,) and @I,,) are the bending stiffness components of the beam. For sake of completeness these stiffness elements are defined by the following equations:

tributions of the tensile force to the bending moments may be neglected. The equations of equilibrium are accompanied by the following boundary conditions at each edge of the beam,a=Oora=l: d = 0 or (E”Z,,,,)‘d”+ (E”Z,,,,)v”” + (E!Z,,z)‘(k,”- ,f#) + (EOZY,)(V - @I -id’)

GJ=

ssA

GKY + k)y

d’ = 0 or (E”Z,)d” + (E’Z,,)(ti” - &+) = 0

(18b)

- (E”Z,,);“’ - (E”Z,,)‘(k,”- &+) - (E’Z,,) Ez* dy dz

(154

Eyz dy dz.

(154

ssA EI,, =

(18a)

(15b) 3 = 0 or (G”J)~,,(&,3’ + 4’) - (E’Z,,,)‘d”

ssA EI, =

+ 4, = 0

ti&l dy dz UW

+ (z -

Ey* dy dz

EI, =

167

The components of the external distributed applied forces and moments are also nondimensionalized as follows:

x (6”’ -$4

-&&)+&=O

(18~)

3’ = 0 or (E”Z,,,)v”” + (E’Z,,)(~J”- &4) = 0

(184

f$ =0 or &+‘+t$‘=O.

(184

Another boundary condition

is:

+ &,i; -ii;‘* +G”)

da = 0 1

(16)

The equations of equilibrium

are then:

p’ + iY[(EoZ,$d” + (E”ZYY)ii”’ + (E0Z,)‘(3” - &&) + (E”ZY,)(k”’- i;d -

&4’>1=--A -i&

(174

[d - (E”ZYY)“],”- 2(E”Z,)‘6” - (E”Z,)d”” -(E”ZYz),,(,” - &4) - 2(E”Z,,,) x ($1” - Q#J - @‘) - (E”ZYz)(k)“” - iI’t$ - 2i;qY - &qY) + i,B = -&‘+ 4:

(17b)

or fi = 0 at one of the ends.

(18f)

The difference between the boundary condition (18f) and the other boundary conditions results from the following reason: This boundary condition, in its origin, is that at any boundary the axial displacement t is zero or that P equals some known value (which in the case of a free edge is zero). Since in the present derivation only 6 is an unknown, integration of eqn (10) is used in order to express J as a function of d. In addition, as a result of the integration only one boundary condition is obtained in place of the original two. In [12] the same equations of equilibrium and boundary conditions were also obtained by using the principle of virtual work. The components of the resultant moment at each cross section are given by:

- (E”Z,,)“d”- 2(E”ZY,)‘6”’- (E”Zyz)d”” + [p - (E’Z,,)“](ti” - iYb) - Z(E’Z,,) x (6”’ - i;f#J - ,f+,c$‘) - (PZ,,)(P - 2&4’ - @$“) + (G’J)i;(i$’

- &?$ + 4’)

+ (G”J)‘&(~y~’ + 4’) + (G’J)& x(Qv+&kv’+@‘)= (G”J)‘(@’

tl9b)

-fiz-4;

+ 4’) + (G”J)(i;3’

+ @”

U7c) + 4”)

+ iy(EoZ,,z)u”’+ iy(EoZ,,)(P - Q#J) = -6.

+

tE”~yzW’ - iy+>l.

(19c)

(17d)

In the last equations the notation ( )’ indicates differentiation with respect to LY. One should note that eqns (17b-c) contains nonlinear terms. Since in many cases of beam structures the axial force plays a major role in determining the stiffness of the structure these terms are retained while other smaller nonlinear terms have been neglected. It should also be pointed out that in deriving the equations it has been assumed that the offset between the elastic axis and the tension axis of the beam is small enough such that con-

3. METHOD OF SOLUTION

As indicated above, the equations will be solved using the Galerkin method. According to this method the solution is given by: (20a) ‘%

+= 1 w,.FW, k-l

t2Ob)

A.

168

ROSEN and

H.

5 pm FP,

B=

In=,

F”;, FW,, Fc$, and FP,,, are functions of a which are defined in the region 0 < a < 1 and satisfy the boundary conditions which d, 3, I#Jand P, respectively, must satisfy at the boundary points a = 0 and x = 1. In addition, it is assumed that these functions are continuous and have continuous derivatives as necessary. The unknowns now become the coefficients v , 4, and pm. The accuracy of the solution ;s I&endent very much on the number of terms in each series N,, N,, N@and N,. In general, if the number of terms is increased it also increases the accuracy of the solution. Equations (2Oa-d) are substituted into eqns (17a-d) and then the Galerkin method is applied. According to this method eqn (17a) is multiplied by FP,, and integration of both sides of the equation from a = 0 to a = 1 is carried out. By this procedure an algebraic equation of the unknowns is obtained which is not a function of s( anymore. Since there are N,, different functions FP,,,,, and multiplications by each one of them is possible, Np different algebraic equations are obtained. By using the same procedure, N, different equations are obtained by multiplying eqn (17b) by FVjn and carrying out the integration, NW equations are obtained by multiplying eqn (17~) by FW,, and N+ equations are obtained by multiplying eqn (17d) by Ftj~,~.The complete system of coupled algebraic equations may be described as: z HP lcmnJ, V/ +

,=I

2

k=l

wk+ 2 ,=I

HP2,,,,

x HP 3,,,,/, 4, + 3 HP 4,,,,,

pm = HP+,,,

(214

In=,

,!,[.;1, HV1,.W,‘P”! + Hv20”J,] +'I

ABRAMOVICH

In eqns (21ad) all the terms on the left side respresent structural contributions while the terms on the right side present the influence of the applied loads. These applied loads may contain gravity loads, inertia loads, aerodynamic loads, etc. Those terms on the right side are Galerkin integrals of the generalized forces P,, P,,, Pz and P, which are defined as:

k=l

PC)

p, = - 4.x

Wd)

VII HU lIJ,) + HU2,,

(21b)

= HV$,“,

Wb)

P:= -&4;

:I:, ,=I 1.v, +22wk ’wk, ’HU&,,

,=I

+ 2 HV5,,,9, WI=,

(224

P,= -p,,+g

It should be noted that according to eqns (12), (13) the components of the externally applied loads are in the directions of the deformed beam. Therefore in cases where the loads are given in other directions (for example the directions before the deformation) appropriate transformations based on eqns 5(ac) are required in order to obtain the right components of p and q. This introduces another kind of nonlinearity. It should be noted that in general one of the functions in the series describing d may also be a constant. In the cases where P is known at one of the ends, the value of this constant is determined (zero, for the case where at one of the ends fi is zero). But for the case where 3 = 0 at both ends, this constant must be found. Let us assume that FP, is a constant of unity magnitude. Formal treatment of this case is doomed to cause trouble. For example, if & = 0 then it is easy to note that the system of equations will become over determined since all the coefficients of p, will vanish (FP; = 0). Therefore in order to cope with this problem this constant will be determined by using the boundary condition (18f). By doing this the boundary condition of 3 = 0 at both ends is satisfied, as it should. Mathematically eqn (21a) for mn = 1 is replaced by the following equation which is obtained by substitution of eqns (20a-d) into (18f) and carrying out the integration:

wk + 2 HV4,,0 .+I

+ 2 HV3,,,

P, = - d-r- 54:

k=l

kl=l

NP +

1

In=,

4, = HB’%,

wk

k=l

+

2

I=,

W

30.0 . 4, = H4 4~

All the coefficients of eqns (21a-d) Appendix A by eqns (alHa8).

(21d)

are given in

(23)

All the coefficients which appear in eqn (23) are given by eqns (a-8,9) of Appendix A. In order to solve the problem, the system of eqns (2lad), together with eqn (23) if necessary, is solved in finding all the vj, w,, 4, and pm. Since this system is nonlinear, where nonlinearity is represented by the double summation in eqns (21 b, c) and (23), it will be solved iteratively. Another kind of nonlinearity appears in the expressions for the generalized forces as has been described above. A computer code based on the above-described method has been prepared. The operator of the program supplies the functions in the series of eqns (2Oad), and their derivatives. The number of the

1 (214 VI +1 H42wc, ’ ,E, WJ 1~1. Nw

pm + HW5,,,,

HU4,,, ‘p,,, = 0.

Galerkin method as a tool to investigate the planar and non-planar behavior of curved beams

terms in each series is determined by the operator. Then an input which includes all the details on the distribution of the structural properties is fed in. The code calculates all the Galerkin integrals which appear in the Appendix using Gauss method of integration. The number of Gauss points can be changed according to the nature of the functions which are integrated. It should be pointed out that because of the modular structure of the code it is possible, if desired, to replace the Gauss method by any other method with only minor changes. Since in many cases one is interested in the behavior of a certain structure under different kinds of load conditions, the integrals and the coefficients of eqns (21) are stored and called whenever needed. This procedure contributes to the efficiency of the code. As mentioned previously, because of the nonlinear nature of the problem iterative process is used. During each iteration the generalized forces are calculated and the Galerkin integration of these forces is performed. In the calculations of the generalized forces the values of the displacements according to previous iterations are used whenever needed. After the terms on the right side of eqns (21a-d) are known the system is solved as a linear system of equations. Again P,,, in the square brackets of eqns (21bc) are taken from previous iterations. In the present form of the code Gauss-elimination method with complete pivoting is used in order to solve the linear system. The iterative process is terminated when a pre-determined convergence criterion has been achieved. This convergence criterion is flexible and is determined by the operator of the code. 4. NUMERICAL AND EXPERIMENTALEXAMPLE

The curved beam of the present example is a circular beam. The elastic axis of the undeformed beam is a section of a circle, as shown in Fig. 1. The spatial location of each point of the beam is described by the Cartesian fixed in space system of coordinates X, Y, Z, also shown in Fig. 1. The center of the circle is positioned at (X = X,, Y = Y,) and its radius is R. Therefore xv =

I I

Xl

------

-’

R'

a-0

ELASTIC AXIS

X Fig. 1. The geometry of the circular beam.

169

The properties of the circular beam, which is uniform and has rectangular cross sections, are as follows: L = 440 mm, R = 233.6 mm, /I = 36”, cross sectional width = 9.6 mm, cross sectional thickness = 3.2 mm, E = 2. 10’ N/mm*, G = 0.76. lo5 N/mm*. The beam is loaded by a concentrated force at its midspane (x = 0.5). The loading directions will include X, Y and Z directions. At the boundary points, displacements and rotations are restrained except in cases where, at the edges, the beam is free to rotate in its plane. In what follows, cases where all rotations are restrained, will be denoted Bl while cases where planar rotations (about cz direction) are allowed, will be denoted B2. The terms of the series describing z?,3,4 and fi were chosen as follows (see eqns (20a-d)): FV, are the natural modes of vibration of a uniform straight slender beam. For the case Bl these are the modes of a clamped-clamped beam, while for the case B2 the modes of a beam simply supported at both ends are taken. These modes can be found in [14, 151. FW, are identical to FV, for case B 1. F&J,are the natural modes of torsional vibrations of uniform straight slender beams given by: F4, = sin da

(25)

FP, are given by: FP, = 1.0 FP, = cos xa FP, = sin na FP, = cos 2na FP, = sin 2na.

(26)

As previously indicated, the theoretical work was accompanied by experiments with the same model of circular beam. A concentrated force was applied at the midspane of the beam and its direction was varied according to the cases which are described in what follows. The experimental boundary conditions were also varied between complete clamping (B 1) and free planar rotations (B2). During the experiment the displacements at different points along the beam were measured. In addition pairs of strain gages along the beam were used to measure the bending strains at different points. These bending strains can be translated into resultant bending moment at the cross section. 4.1 Load in the X direction This is a planar case where the behavior of the beam is asymmetric with respect to the middle point. Therefore, in the series describing 8 and d only asymmetric terms were included. Figure 2 presents the displacement i; at the point a = 0.75 as a function of the load PC for the two kinds of boundary conditions. Theoretical results were obtained for analyses, containing two, three, four and five terms in the series. The approximations are so good that the difference between two and five terms cannot be seen in the figure. The agreement between theory and experiment is good and the deviations are probably as a result of the fact that

A. ROSENand H.

170

15

I

the results of Ref. 16 and the experimental results are also good. In doing the comparison, it should be noted that the case of a concentrated force presents an example which has some difficulties. There is a basic discontinuity in the shearing force and therefore discontinuity in the high derivatives of the displacement. A description, even approximate, of such a displacement, by using a series of continuous functions with continuous derivatives, is very difficult. Therefore the good results in this case are very encouraging.

Theory Experiment

/

5

0

i

IO

20

30

40

ABRAMOVICH

50

Fig. 2. Transverse displacement-force

60

70 N

PC

in the X direction.

the boundary conditions in the experiment are slightly different from the ideal conditions that are assumed in the theory. Figure 3 presents the spanwise distribution of the nondimensional bending moment IU, at a load PC= 10 N. Theoretical results for series containing three, four and five terms are compared with experimental results and accurate theoretical results for this case which are presented in [ 161.From the results it is clear that convergence with five terms is very good. The agreement between the theoretical present results,

4.2 Load in the Y direction This is also a planar case but the difference from the previous case is that the behavior of the beam is symmetric with respect to the middle point. Therefore in the series describing P and 6 only symmetric terms were included. In Fig. 4 the displacements at different points along the beam at the two boundary conditions are plotted as a function of the applied load. The good convergence is evident by comparing the results for two and five terms in the series. The difference between the results for three and five terms are less than 3%. The theoretical results are compared to experimental results. For the Bl boundary conditions the agreement is very good. In the case of the B2 boundary conditions, there exists a small deviation between theory and experiment which probably results from the fact that because of friction, the edges are not fully free to rotate. The results of Fig. 4 cannot be fully understood without knowing the general picture of the transverse deflection. This picture, for the two boundary conditions, is given in Fig. 5. The more complicated behavior of a curved beam, compared to a straight beam, is evident. 4.3 Load in the Z direction This is an interesting case since it presents a non-planar behavior of the beam. In this case the

100

---- -

I

Three terms Four ond five terms REF. 16 Experiment 75

0

b x

5c

O>

25

C

Fig. 3. Bending moment distribution-force direction.

in the X Fig. 4. Transverse displacement-force

in the Y direction.

171

Galerkin method as a tool to investigate the planar and non-planar behavior of curved beams 24

r

20

--- --.-

P,:ION A

t

,L.

Two terms Three terms Four terms -.--- Five terms I.REF 17

’ I

Prlsent theory

2’

Fig. 5. Transverse displacement distribution-force Y direction.

in the

boundary conditions are B 1. This is a symmetric case relative to the midpoint of the beam and therefore only symmetric functions were included in the s@es.

In Fig. 6 the nondimensional moment MY is presented. The calcultions were made with series containing two, three, and five terms in each series. The present calculations were compared to accurate results according to [17]. It is shown that except at the middle of the beam (0.42 < tl c 0.58) using even only three terms yields very good results. At the middle point, convergence is slow and difficult as may be expected, remembering the basic discontinuity which exists there. Even so, for series of five terms the error is less than ten per cent. In Fig. 7 the torsional moment I%?*along the beam, at a load of 10 N, is presented. The results of the present theory, using two, three, four and five terms are compared to accurate results from [17]. It is shown that convergence with five terms is good. Five terms give results which agree very well with the

IO

P,=ION A

-24

L

Fig. 7. Torsional moment distribution-force direction.

results of [17] except at the neighbourhood boundary points.

in the Z

of the

5. CONCLUSIONS Based on the results of this paper it may be concluded that the set of equations which were derived in this work describes appropriately the planar and non-planar behavior of curved beams. The Galerkin method, which was used to solve the equations, presents an efficient method which requires a relatively small number of unknowns in order to obtain satisfactory results. Still, the method, as presented, is very general and allows any variation of the geometry and structural properties along the beam’s span. Also the loading may present any combination of loads along the beam while the boundary conditions may include any combination. It seems that Galerkin’s method as presented here is specially appropriate to describe the non-planar nonlinear behavior of curved beams in cases where one needs a technique which involves a minimal number of unknowns but yields accurate enough results. Such demands appear, for example, in cases where the curved beam is one of many other elements in a large structural system or, on the other hand, in cases where a curved beam is subjected to dynamic loads and integration with time is required. Acknowledgements-The authors would like to thank Mr. 0. Rand for his very helpful participation in the programming work. They would also like to acknowledge Mr. S. Nachmani, Mr. A. Grunwald and Mr. G. Rubin for their assistance in doing the experimental work, and Mr. A. Aronson for typing the manuscript.

-10

-I 2

---.-

Two terms Three terms pEu;,o;; five terms

-I 4

Fig.

6.

Bending moment distribution-force direction.

in the 2

REFERENCES 1. A. B. Sabir and A. C. Lock, Large deflexion, geometrically non-linear finite element analysis of circular arches. Znt. J. Mech. Sci. 15, 3747 (1973).

A. ROSENand H.

172

ABRAMOVICH

2. D. G. Ashwell and A. B. Sabir, Gn the finite element calculation of stress-distributions in arches. Inr. J. Mech. Sci. 16, 21-29 (1974). 3. Y. Yamada and Y. Ezawa, On curved finite elements for the analysis of circular arches. Inr. J. Numer. Merh. Eng. 11, 1635-1651 (1977). 4. A. K. Noor, W. H. Greene and S. Hartley, Nonlinear finite element analysis of curved beams. Compur. Merh.

iy2. (E’l,,)’

(Al.])

HP2,,,,, = GP3,,,k, + GP4+,k,

(A1.2)

FP,,,, FPb

GP8 (mnm)=

where

GP4,~ =

GP 3,,,, =

I’

&. (EOZ,) FP,,

(A2.7) (A2.8)

da

P,

GP9,m, =

FP,,

da

(A2.9)

I’0 The

coefficients of eqn (21b) are given by: (A3.1)

HVI (FrnJ) = GVLwI HVZcbJ, = - GV2,,,

- 2 GV3,,,

- GV4,,“, (A3.2)

HV3,,,,

= - GV5cp.w - 2

G&n,,, - GV7,,,,,, (A3.3)

HV4,,n,0 = GV8,,,

+ 2. GV9,,,0 + 2. GVIO,,n,O

+ GVI1,,,~+2.GV12,,“,~+GV13,,“, (A3.4) HV5,,,, HV6,,,

= GV’4,,,,

(A3.5)

= GV15,,“,

(A3.6)

where GVI (FrnJi = GV2, jnn =

FVF. FP,,, FV;’

da

(A4.1)

I’0

’ (E’&,)”

FV,” FV;’

da

(A4.2)

FVfl. FV”‘, da

(A4.3)

s0 (E’I,)’

G%w1=

I'0 I

GV%w~

=

GV$p,k,

=

(E”I,,y) FVF. FVY.

da

(A4.4)

s 0

(E’I,)”

FY,

FW;

da

(A4.5)

GV$,,,,

=

(EOI,)’

FV,,, FW;” da

(A4.6)

(E’I,)

FVF. F&”

da

(A4.7)

S'0 I = f 0

G%dJ

=

GV%,O

=

,&.

(EOI,)”

FVF

Fq$. da

(A4.8)

i;.

(EOI,)

FVF

FI$, . da

(A4.9)

S'0

j'0

(A2.1)

da

,$. (EOZ,)

GV1O,,,O =

FVF. F4;.

da (A4.10)

I’0 FV;

. da

(A2.2)

0

& (ED&)’ . FP,,

da

I’0

(A1.5)

HP 5,,,, = GP9,,,,

FV;

F4,. da(A2.6)

I’0

(A1.3)

(A1.4)

FP,,

FP,,

t’. (E'I,,) . FP,,,, F4;

7,,,,, =

GP

'=7(,,k,

- GP6,,,,

= GP8,,,,

& (EOI,)

(A2.5)

I'0

HP1 c,,w~= GP l,mnn+ GP2w,

GPI (mwI=

(E’I,,)

,fi. 1.

=

APPENDIX A

HP4,,,,

da

s 0

Vib. 66(21. 219-225 (19791.

HP3 ,,,,qr,= - GP5,,,,, - GP7,,,,

F+,

I

Gqm",,

6. I. Sheinman, Forced vibration of a curved beam with viscous damping. Compur. Srrucrures 10, 499-503 (1979). 7. F. J. Tokarz and R. S. Sandhu, Lateral-torsional buckling of parabolic arches. Proc. ASCE, J. Srr. Diu. 98(STS), 1161-1179 (1972). 8. Z. Celep, On the lateral stability of a bar with a circular axis subjected to a non-conservative load. J. Sound &

Additions lo Secrion 3 The coefficients of eqn (21a) are given by:

FP,,

0

Appl. Mech. Engng 12, 289-307 (1977). 5. Y. Tene, M. Epstein and I. Sheinman, Dynamics of curved beams involving shear deformation. Inr. J. Solids Structures 11, 827-840 (1975).

9. I. U. Ojalvo, Coupled twist-bending vibrations of incomolete elastic rines. Inr. J. Mech. Sci. 4. 53-72 (1962). 10. K. Suzuki, H. Aid: and S. Takahashi,‘Vibratibns of curved bars perpendicular to their planes. Bull. JSME 21(162), 1685-1695 (1978). 11. S. Markus and T. Nanasi, Vibrations of curved beam. The Shock and Vib. Digest 13(4), 3-14 (1981). 12. A. Rosen and H. Abramovich, Structural behavior of curved beams, Department of Aeronautical Engineering, Technion, Israel Institute of Technology, TAE No. 452, June 1981. 13. G. Wempner, Mechanics of So&&, With Applications to Thin Bodies. McGraw-Hill, New York (1973). 14. D. Young and R. P. Felgar, Jr., Tables of characteristic functions representing normal modes of vibration of a beam. The University of Texas Publication, No. 4913, July 1949. 15. E. Volterra and E. C. Zachmanoglou, Dynamics of Vibrations, DP. 319-320. Charles E. Merrill, New York (1965). __ 16. J. J. Tuma and R. K. Munshi, Theory and Problems of Advanced Structural Analysis, pp. 208-209. Schaum’s Outline Series. McGraw-Hill, New York (1971). 17. R. J. Roark and W. C. Young, Formulas for Stress and Strain, International Student Edition, 5th Edn, p. 256. McGraw-Hill, Kogakusha, India.

5’

GP5,m,,=

I

i$

GV1lcj.,n=

. (E”Iyz) FV,+, Fq%,. da (A4.1 I)

s 0

FW;.

da (A2.3)

GV12,,,0

i; . (EOZ,,) FVF

=

Ftj ; da (A4.12)

S’0

I’0 I

GP4,,,,

=

I 0

G . (E”‘ys) . FP,, . FW;" . da (A2.4)

i, (E'I,)

GV13,,,0 = I’0

FL’,,, F+;‘. da (A4.13)

Galerkin method as a tool to investigate the planar and non-planar behavior of curved beams

s I

GV14,,,

I

,&. FV,=*FF, . da

=

173

(A4.14)

GWb,, =

II

f; (E”Z,, ’ FW,, . F#; . da i

(A&16)

0

1

I

GV15’,,n,= o P,,.FV,n.dct. I

(A4.15)

&.

GW17,kx,.q =

(E"Z,,) . FW,,,

. Fc$;’*da

(A6.17)

Fc$; ‘da

(A6.18)

s0 L

The coefftcients of eqn (21~) are given by:

GW%kn,,

J_;.(G’J). FW,.

= s0

HWI (knJ> = - GWl,.n

(A5.1)

- 2. GW2,,,,,, - GW3o.n

1

GWt9*,, N w2,k,.m,,

~Y.(GoJ)‘.FWk,+F$;.da

=

(A5.2)

= GW4,&n.mn.k~

(A6.19)

I0 I

H w3
-GWso,k,

- 2

GW$k,,kj

GW20,,,o =

- G@'7,,k,

&. (G”J) . FW, s

+ 2 ' GWb,

+

GW9(,,

+ GWlO,,,

(A5.3)

NWS,,,,

= GW12,,,0 + 2. CW13,,,0 + 2. GW14C,,,,

(A6.20)

I GW21,=

(A5.4)

HW4+,,0 = - GWl l,~n.m.,,

Fr#; f da

0 P; FW,,.dn

(A6.21)

s0 The

coefficients of eqn (21d) are given by:

+ GW)5C,,n + 2. GW16tkn,o+ GW17,,n,o + GW18,,,, + GW19,,,,, + GW20,,0 (A.5.5) (A7.3)

(A5.6) where:

(A7.4) GWI ckMn =

’ (E”l,$ . FW,, . FV;’ 4da J-0

(A6 I)

(E’I,) . FW, ‘ FH$” . da

GW7(,,,, =

where:

(A6.2)

(A8.1)

(A6.3)

(A8.2)

(A6.4)

(A&3)

(A6.5)

(A8.4)

(A6.6)

(A8.5)

(A6.7)

G#6t,_.n=

(6.8)

GrP7~n.0=

’ (G’J)’ I F&, . FqS;. da

I-'

GW%,,k, =

!0 i,;. lly (G”J) *FW,, . FW; . da

GW~M.N =

i: . (GOJ)’ I FWk,, . FW; 1da

(AM)

’ (G’J) . F& 1F+; da s0

1 iy2 (E”I,) . F& I F#, ’ da

G#+,a =

=

(A8.7)

(AM)

s0 i

I

GwI",k,k,

(AS.6)

s0

5’0

1

i;.(G’J).

FW,; FW,” .da

(A&10)

0

G4

9,,, = 5

0 P+.F4,.daS

(A8.9)

I

GWlt (&f&0=

iY ’ FW,, . Fp,. Frj~,. da

(A6.11)

The coefficients of eqn (23) are given by:

s0 1

f3qk,,~

. FW, . F& ’ da

(A& 12)

ii. (E”fll)’ . FW,, . FQl,. da

(.46.13)

&

=

. (E”f,)”

s0

G w

13gm.o

= 1’0

HUl,,,,

= - 1 I GUl,,,,,

HU2<,, = GU2,,,

(A9.1)

(A9.2)

(A9.3) 1 GW~%n.c,

=

i_z

(E'I,) . FW,, I Fr$,*da

(A6. f 5) NU4,, = GU4,,

(A9.4)

174

A. ROSENand H.

ABFCAMOVICH

where: GU~W,,, = GCJl(MO=

FV;.

FV;,

I’0 x,. FV, da I’0

(A10.1) GCJ4,,, =

*

GlJ2,,, =

da

(A10.2)

FW;.

FW;,

da

S’0

’

~

I

s o (E’A)

FP,,, . da

(A10.3) (A10.4)