A Corotational Procedure That Handles1987

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A corotational procedure that handles

large rotations of spatial beam structures

Article in Computers & Structures December 1987

Impact Factor: 2.13 DOI: 10.1016/0045-7949(87)90290-2

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Kuo Mo Hsiao

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Cm$~ers 6 S~~~rures Vol. 27. No. 6. pp. 769-781. 1987 aa-7949187 13.00 + oxa

Printed in Great Br itain.

0 1987 Pcrgamon Journals Lid.

A COROTATIONAL PROCEDURE THAT HANDLES

LARGE ROTATIONS OF SPATIAL BEAM STRUCTURES

Kuo-MO HSIAO HORNG-JANNHORNGand YEH-REN CHEN

~~rtment of Mechanic& Engineering, National Chiao Tung University, Hsinchu.

Taiwan, Republic of China

(Recei ved 5 Januar y 1987)

Abstract-A practical motion process of the

t hree

dimensional beam element is presented to remove the

restriction of small rotations between two successive increments for large displacement and large rotation

analysis of space frames using incremental-iterative methods. In order to improve convergence properties

of the equilibrium iteration, an n-cycle iteration scheme is introduced.

The nonlinear formulation is based on the corotational formulation. The transformation of the element

coordinate system is assumed to be accomplished by a translation and two sueesssive rigid body rotations:

a transverse rotation

followed by an axial rotation. The element formulation is derived based on the small

deflection beam theory with the inclusion of the effect of axial force in the element coordinate system.

The membrane strain along the deformed beam axis obtained from the elongation of the arc length of

the beam element is assumed to be constant. The element internal nodal forces are cakulated using the

total defo~atioM1 nodal rotations. Two methods, referred to as direct method and incremental method,

are proposed in this paper to calculate the total defo~~ona~ rotations.

An incremental-iterative method based on the Newto~Rap~n method combined with arc length

control is adopted. Numerical studies are presented to demonstrate the accuracy and efficiency of the

present method.

1. INTRODUCTION

The development of new and efficient formulations

for the nonlinear analysis of beam structures has

attracted the study of many researchers in recent

years, and different alternative formulation strategies

and procedures to accomodate large rotation capa-

bility during the large defo~ation process have been

presented [I-lo]. These formulations can be divided

into three categories: Total Lagrangian (TL) formu-

lation, Updated Lagrangian (UL) formulation and

Corotational (CR) formulation. It should be noted

that within the corotating system either a

TL or a

UL formulation, or even a formulation based on a

small deflection theory may be employed. The large

number of publications on the nonlinear analysis of

beam structures is, at least partially, due to the fact

that various kinematic nonlinear formulations can

be employed. It seems that large rotations in plane

frames pose no major problem. Hsiao and Hou [IO]

introduced a simple and effective corotational formu-

lation of beam element and numerical procedure,

which can remove the restriction of small rotations

between two successive increments for the large

displacement analysis of plane frames using incre-

mental-iterative methods. Unfortunately, the method

presented in [lo] cannot be applied to three dimen-

sional frames. The difficulty of obtaining effective

solutions is particularly pronounced in the analysis of

spatial beam structures; a general three dimensional

nonlinear formulation is not a simple extension of a

two dimensional fo~ulation, because large rotations

in three dimensional analysis are not true vector

quantities; that is, they do not comply with the rules

of vector operations and the result will in general

depend on the order in whch the rotations are

taken. This point has been throughly discussed by

Argyris [ 1 I] and Wempner [121.

The problem of large rotations on space structures

has received wide attention in the ~terature; many

different strategies based on the TL, the UL, or the

CR formulations have been reported, those of

[Z-S, 6, 1 -241 being only a small fraction of the total.

In 1191Hughes and Liu developed a specialized shell

element which can handle arbitrarily large rotations.

Argyris has covered the subject of corotational coor-

dinates extensively including a lengthy discourse on

the subject of large rotations [20,21]. Belytschko er

al. [3,4, 14 have applied corotational formulation

to the dynamic analysis of space frames where arbi-

trarily large rotations can be expected. Horrigmoe

and Bergan (IS] have su~~fully applied a co-

rotational approach to their shell elements Rankin

and Brogan [23] have introduced a corotational

procedure which may enable existing shell element

formulations to be used in problems that contain

arbitrarily large rotations. Recently Hsiao 1241 has

proposed a motion process for triangular shell ele-

ments to remove the restriction of small rotations

between two successive increments for nonlinear

shell analysis using incremental-iterative methods.

The wide range of numerical examples studied

in [3 4,7,9, 11 , 13, 15, 16,18,20,21,23-261 indicates

that the corotation approach, first described by

Argyris et al. [25], may be very useful in the analysis

of spatial structures containing arbitrarily large


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770 Kuo MO HSIAOet al.

rotations. However, most strategies based on the

corotational formulation suffer from one inherent

drawback: they are restricted to small rotations be-

tween two successive load increments during the

deformation process. This limitation arises because

the incremental nodal rotations are considered to be

vector quantities. Although the method introduced in

[24] may remove this restriction for triangular shell

elements, unfortunately, this method cannot be

applied to the space beam elements, because, unlike

the shell elements, the element coordinate of the

space beam elements cannot be determined using

only nodal coordinates.

The objective in this paper is to present a practical

motion process of the three dimensional beam ele-

ment which can remove the restriction of small

rotations between two successive increments for large

displacement and large rotation analysis of space

frames using incremental-iterative methods. Here

the motion process presented in [24] is modified to

accomodate the characteristics of the motion of

beam elements. In this paper the transformation

between the element coordinate systems is assumed to

be accomplished by a translation and two successive

finite rotations: a lateral rotation about an axis

perpendicular to the current beam axis followed by

an axial rotation about the current beam axis. Two

methods, termed direct method and incremental

method, are introduced to describe the motion pro-

cess of the beam element and to determine the total

deformational nodal rotations.

The dominant factors in the geometrical non-

linearities of space structures are attributable to finite

rotations, the strains remaining small. For space

frames discretized by finite elements, this implies that

the motion of the individual elements with proper size

will, to a large extent, consist of rigid body motion.

If the rigid body motion part is eliminated from the

total displacements, the deformational part of the

motion is always a small quantity relative to the local

element axes; thus, incorporated with the corotational

formulation, the small deflection beam theory with

the inclusion of the effect of axial force is adopted

here to deal with the large rotations but small strains

problems.

The numerical algorithm used here is an

incremental-iterative method based on the Newton-

Raphson method combined with constant arc length

of the incremental displacement vector [27,28]. In

order to improve the convergence properties of the

equilibrium iterations, an n-cycle iteration scheme,

which is an extension of the two-cycle iteration

scheme proposed in [lo], is introduced. Numerical

examples are presented and compared with the results

reported in the literature to demonstrate the accuracy

and efficiency of the proposed method.

2.

COORDINATESYSTEMS

One of the basic considerations in formulating

nonlinear structural problems is the selection of a

description of motion. In the present study, the

corotational (CR) formulation is adopted. In this

formulation, each element is associated with a local

Cartesian element coordinate system 2, (i = 1,2, 3) as

shown in Fig. 1, that rotates and translates with the

element but does not deform with the element. The

element coordinate system used herein is a right

handed one and is defined as follows. The origin of

the element coordinate system is located at the local

node 1, the 2, axis passes through the centroids of

member end sections, and the Z2 and 5, axes are

parallel to the principal directions of the undeformed

end cross section.

Fig. 1. Coordinate systems, member deformations and associated forces.


4/14

Corotational procedure for rotations of beam structures

771

Also shown in Fig. 1 is a fixed global coordinate

system xi (i = 1,2,3) used to define the location of the

nodal points. We note that the equilibrium equations

of a structure are written in the fixed global coordi-

nate system, and the incremental nodal parameters of

the system of equations are also calculated in this

coordinate system. The element equations are first

formulated in the element coordinate system, and

then transformed to the global coordinate system

using standard procedure [29] prior to the element

assemblage process.

If we consider a vector B (or B if measured in the

_Zi coordinate system) with global components Bi

(i = 1,2,3), and element coordinate components Si

(i = 1,2,3), we have the following transformation:

where clti= cos(xi, Xj).

The beam element employed here has two nodes

with six degrees of freedom per node (Fig. 1): these

are the translations 0, in the fi (i = 1,2,3) directions

at nodes j (j = 1,2), and the rotations flj about the

Zi axes at nodes j (j = 1,2). The global nodal par-

ameters for the system of equations associated with

the individual elements are chosen to be the trans-

lations Vii in the xi ( i = 1,2,3) directions at nodes j

(j = 1,2), and rotations 0, about the xi

( i =

1,2,3)

axes at nodes j (j = 1,2).

In this study an incremental-iterative method is

used to solve the nonlinear equilibrium equations.

Both the incremental nodal translations and the

incremental nodal rotations are regarded as vector

quantities. Thus, using eqn (l), the nodal vectors,

AUj= {AU,j, AU,, AUjj) and AtIj = {A8ij, A&, Ae,},

referred to the global coordinate system can be

transformed to AUj = {AO,j, AUq, AOJj} and A4 =

{A&, A&, Ae,}, referred to the element coordinate

system, respectively.

It should be noted that Aej cannot be interpreted

as component rotations about Cartesian axes &.

In this study, Ae, = {Ae,,

0,0},

he components of

Ah along the 2, axis, and Afj, = (0, A&, AB,), the

components of A4 perpendicular to the R, axis, are

considered to be rotation vectors to define rotations,

details of which will be discussed later.

For convenience of the later discussion, the term

rotation vector is used to represent a finite rotation.

Figure 2 shows a vector R which as a result of the

application of a rotation vector 4n is transported to

a new position R. The relation between R and R

may be expressed as [30]

R=cos+R+(l -cos4)(n.R)n

+ sin 4(n x R), (2)

where

*

and x denote the dot and the cross product

9

P

n

Fig. 2. Finite rotation of vector.

respectively; 4 is the angle of counterclockwise rota-

tion, and n is the

unit vector along

the axis of

rotation.

3. CR-FORMULATION OF BEAM ELEMENT

The formulation of the beam element developed

here is applicable to arbitrarily large rotations but

restricted to small rotations relative to the element

axis. The beam element is formulated in the element

coordinate system based on the small deflection beam

theory with the inclusion of the effect of axial force.

The element, as shown in Fig. 1, has two nodes with

six degrees of freedom per node, and can transmit an

axial force, two shear forces, two bending moments

and a torque. Herein the beam element is assumed to

be straight and of constant cross section. The cross

section is doubly symmetric, thus excluding coupling

of the torsional stiffness to that of bending and axial

stiffness. Shearing deformations and warping effects

are neglected. The material is assumed to be linearly

elastic.

The element stiffness matrix is obtained by super-

imposing its bending, geometric, torsional and axial

stiffness matrices. The element internal nodal forces

are evaluated using the total deformations.

3.1. Ki nemat i cs of beam el ement

It

is assumed that the lateral deflection curves of

the beam member are the cubic Hermitian poly-

nomials in the & and .?s directions of the element

coordinates (the principal directions of the un-

deformed end cross section), and that the axial

rotation varies linearly along the member. The mem-

brane strain along the deformed element axis is

assumed to be constant. Thus, the membrane strain

can be evaluated from the elongation of the arc

length of the member.

The lateral deflection curves of the beam member

may be given by

where a bar over a quantity denotes that it is defined

in the element coordinate system. P and Ware lateral
https://www.researchgate.net/publication/246780688_Matrix_and_Finite_Element_Displacement_Analysis_of_Structures?el=1_x_8&enrichId=rgreq-026eaa3de7790f1cbb90a5aa906a8739-XXX&enrichSource=Y292ZXJQYWdlOzIyMzY0MDkwODtBUzoyNzk2OTU5MDA4NTYzMzNAMTQ0MzY5NjEwNzU4OA==https://www.researchgate.net/publication/246780688_Matrix_and_Finite_Element_Displacement_Analysis_of_Structures?el=1_x_8&enrichId=rgreq-026eaa3de7790f1cbb90a5aa906a8739-XXX&enrichSource=Y292ZXJQYWdlOzIyMzY0MDkwODtBUzoyNzk2OTU5MDA4NTYzMzNAMTQ0MzY5NjEwNzU4OA==


5/14

IWO-MOHstao et al

2

/

undeformed end section

Fig. 3. Deformational nodal rotations.

deflections in the & and i, directions, respectively. iii,

and ej (i = 2,3; j = f , 2) are the nodal displacements

and rotations shown in Fig. 1. N, i = i ,4) are shape

functions and are given by

N, = l/4(1 - (2 + t)

N2 = c/8(1 - r)(l - r)

N,= l/4(1 +S)(Z--r)

Nl = c/8(1 + c*)(l + r),

(4)

where c = Zi, - %ir is the current chord length of

the beam member, and f, are the 2, coordinates of

nodes j (j = I, 2) in the element coordinate system;

5 = - I + 23,/c is a nondimensional coordinate.

Note that in this study, the relative displacements

of the elements are referred to the element coodinate

system, Due to the definition of the element co-

ordinate system, the lateral nodal displa~ments 041

i = 2,3) at the nodal points j (j = 1,2) are identical

to zero. The nonzero deformational nodal displace-

ments of an element can be divided into the axial

relative displacement, the axial relative rotation and

lateral deformational rotations. The element defor-

mation can be decomposed into the membrane defor-

mation, the torsional deformation and the flexural

deformation. Herein the flexural deformation is de-

termined by the lateral deformational nodal rotations

using elementary beam theory and the torsional

deformation is determined from the axial relative

nodal rotation, while the membrane deformation is

obtained from the change of arc length of the beam

axis which can be calculated from the lateral

deflection.

If the arc length of the beam axis is expressed by

i

1

S=c/2

(1 + p2 -i- @2)2 d&,

(5)

-1

where c is the current chord length of the beam

member, ( ) denotes x,-derivatives and P and rii are

given in eqn (3), then from the assumption of con-

stant membrane strain along the deformed beam axis,

the membrane strain of the beam axis can be written

as

Cl = (S - SJ)/&, (6)

where S, = L is the initial arc length of the beam axis.

Figure 3 shows that the normals of the undeformed

element end sections at nodes j (j = 1,2), 6, are

rotated to g,,,, the current deformed normals of the

element end sections by the rotation vectors e)l,,

which is ~~ndicular to the f,axis of the element

coordinate system. The representations of the lateral

deformational nodal rotations are based on the

assumptions that these rotations are small. On the

basis of this assumption e,,, the 5, i = 2,3) compon-

ents of the rotation vectors 6,,j are chosen to be the

Iateral defo~ational nodal rotations about Zi axes at

nodes j. Note that the direction of the undeformed

normal of the element end sections coincides with the

positive direction of the 5, axis. Thus, the second

subscript j of gq is omitted throughout this paper.

3.2. Determination of element coordinate system and

element de~ormat~o~al otations

Assume that the incremental-iterative method is

used for the solution of nonlinear equilibrium equa-

tions and the equilibrium configuration of the Ith

increment is known. Let AUj and A@,(j = 1,2) be

the incremental nodal displacement and rotation

vectors of an element at nodes j extracted from the

incremental nodal parameters of the system of equa-

tions. At this point, an interesting and relevant

question arises. Given the incremental nodal dis-

placements and rotations, how are the current

element coordinate system, the axial and lateral

defo~ationa1 nodal rotations dete~ined?

Let xj (j = 1,2) denote the node coordinate

vectors of an element in its Ith equilibrium con-

figuration; the current node coordinate vectors xj

are obtained by adding the incremental nodal dis-

placement vectors AUj, so that

xj = x, + AU,.

(7)

The 3, axis of the current element coordinate

system can then be constructed using x, given in

eqn (7) and the definition of the element coordinate

system. But, unlike the cases of triangular shell

elements fl5, 18,241, the .?r and 5 axes cannot be

determined using only the node coordinates. The

determination of the current element coordinate

system will be discussed in the process of element

motion.

For determining the element deformational nodal

rotation and element rigid body rotations, we pro-

pose two methods to describe the process of the

element motion in this paper. In the first method,

referred to as the direct method, the lateral defor-

mational nodal rotations are determined from the


6/14


773

(d)

(b)

orientations of the undeformed and deformed nor-

mals of the element end sections. In the second

method, referred to as incremental method, the total

lateral deformational nodal rotations are calculated

by incrementation. For both methods, the total twist

nodal rotations are determined by incrementation.

The processes of element motions and the methods

corresponding to these motion processes to determine

the deformational nodal rotations and element

coordinate system are described as follows.

(a)

Direct mefhod.

The process of element motion

is divided into the following six steps.

Fig. 4. Process of element motion.

1. A rigid body translation by AU,. The whole

element is translated by AU,, where AU, is the

incremental nodal displacement vector of node 1.The

origin of the 3, axes is translated to the origin of the

Zi axes as shown in Fig. 4(a).

2. A lateral rigid body rotation by the rotation

vector 8. The rotation vector 6 (referred to *ii

coordinate system) is given by

=cos-(e,*e,)

42, e,

II

el x e, I

(8)

where e, and e, are unit vectors associated with the


7/14

774

Kuo MO

HSIAO f al.

.f, and x, axes rcspcclivcly. The rotation vector oi,

passing through node 1, is applied to the whole

element except tQ, the deformed normals of the

element end sections at nodes j (j = 1,2) at the Ith

equilibrium configuration. Here it is assumed that e,,,

are not rotated by the rotation vector ci, but are

translated with the motion of nodes j (Figs 4(b) and

(d)). The & axes are rotated by an angle about the

axis perpendicular to the 2, and f, axes (Fig. 4(a)),

and the resultant coordinate system is labeled 2; axes

(Fig. 4(c)). As can be seen, the 2; axis coincides with

the 2, axis.

3. Finite rotations of C,+by the rotation vectors

Afig. The deformed normals c?~are rotated to &,

by the application of the rotation vectors A6:, as

shown in Figs 4(c) and (d), where AtiLj are the

components of Afij (j = 1,2) (the given incremental

nodal rotation vectors referred to 2: coordinate

system) perpendicular to the Xi axis.

4. Twist rotations by A4j. The rotation vectors

A4j are applied to nodes j (j = 1,2), where AtJj are

given by

A& = A&, - /I

(9)

/? = l/2(6&, + A&,),

(10)

in which tI,, are the components of A& along the 5;

axis as shown in Fig. 4(c).

5. A stretch by (c - c)e,. Node 2 is translated

along the 2, axis (Fig. 4(d)), where c and c are chord

lengths of the element corresponding to the current

configuration and the equilibrium configuration of

the Ith increment.

6. An axial rigid body rotation by g. The rotation

vector (eqn (10)) passing through node 1 is applied to

the whole element. The intermediate axes, ai, are

rotated about the 2; axis by an angle 11i 1) o produce

the & axes of the current element coordinate system

as shown in Fig. 4(f). Vectors El and 5; shown in

Fig. 4(d) are rotated by this rotation vector to reach

their final positions Z, and &, (not shown in Fig. 4(f)).

Iheorientations of the current element coordinate

axes fi may be obtained from the rigid body rotations

caused by the rotation vectors given in steps 2 and 6

of the above process. The orientation of the deformed

normals gdi and the undeformed normal 5, of the

element are also determined by the above process

of motion. Thus, the lateral deformational nodal

rotation vectors, e,, referred to the current element

coordinate system may be expressed as

0

e;I,= IT4cos-(E:~~)

11

ii,*dj

(11)

O3j

II& x ~, II

The current axial deformational nodal rotations

may be obtained by

(12)

whcrc 4, is the axial dcformational nodal rotation

vector of the Zth equilibrium configuration at nodes

j (j = 1,2); A$j is given in eqn (9).

(b) Incremental method In this method, the motion

process is also divided into six steps. Only steps 2 and

3 of this process are different from those in the direct

method, and are described below.

3. Deformational rotations by (68;, - a). The

intermediate deformed normals *EL are rotated to

other intermediate positions 5; as shown in Fig. 4(e)

by the application of the rotation vectors (A&, - o?),

2. A rigid body rotation by o?. The rotation vector

6i (eqn (8)) passing through node 1 is applied to the

in which A Aj are the components of AJj (j = 1,2)

whole element. As can be seen in Figs 4(b) and (e),

the deformed normals of the Ith increment gdl are

perpendicular to 2; axis, and OSs the rotation vector

rotated to an intermediate position I&;.

given in eqn (8) (referred to 2: coordinate system).

Vectors AgAj, oi and 6; (Fig. 4(e)) are rotated by

the rotation vector [ (eqn (10)) to their final positions

Afimj,6 and 4 (referred to the current element co-

ordinate system &), which are not shown in Fig. 4(f).

The element coordinate obtained from this process

is the same as that constructed by the direct method.

The orientations of the deformed and undeformed

normals of the element end sections can be deter-

mined from this process of motion as well. Thus,

the deformational nodal rotations can be calculated

using eqn (11). Due to the assumption of small

deformational nodal rotations, the vector operations

might be valid for the deformational nodal rotation

vectors. Thus, an alternative, referred to as incre-

mental method, is introduced here. The concept of

this method is similar to that in [l&24]. The total

deformational rotations of the current configuration

referred to the current element coordinate are ob-

tained by adding the incremental nodal rotations

(A&j-oS) to the deformational nodal rotations of the

equilibrium configuration of the Ith increment, and

are expressed as

0

e;.=

11

f$, = 4 + (A8, - a).

(13)

03,

For simplicity of computation, only eqn (13) is

used to calculate the total lateral deformational nodal

rotations for the numerical examples studied in this

paper. It is believed that identical results will be

obtained when eqn (11) is used.

3.3. Element st1~ne s smatrix

The

total element stiffness matrix is formulated

by superimposing the bending stiffness matrix i&

and geometric stiffness matrix $ of the basic beam

element, and the axial stiffness matrix g,,, and the

torsional stiffness matrix g, of the linear bar element.

The derivation of these matrices is well documented


8/14


115

in the textbooks and thus will not be repeated here.

However, these matrices are given as follows.

(a) Bending stiffness matrix K,:

(14)

where

r 12 -6L -12 -6L 1

-6L 4L= 6L 2L

-12 6L

12 6L (1%

and

L-6L 2~~ 6L 4~2J

f 12 6L -12

6L 1

6L 4L -6L

2L2

-12 -6L

12 -6L 9 (16)

6L 2L2 -6L

4L2

where L is the initial length of the beam axis, and

EI, and E13are the flexural rigidities about the 4 axis

and 2, axis respectively. The degrees of freedom

corresponding to $ are

ob= {~,d2,, &2.42* ~2,r&,, 022,42}r

(17)

where uU are nodal translations and gU are nodal

rotations as shown in Fig. 1.

(b) Geometric stiffness matrix &:

rt,=

2 0

[

1

r t , ,

(18)

36 -3L

-36 -3L

-3L 4L2 3L -L2

-36 3L

36 3L

-3L -L2 3L

4L2

1

19)

and

K*&

[ -36 6

3L 4L2 -3L

-L2

-3L L

-36 6 -3L L

9 (20)

3L

-L2

-3L

4L2

I

where L is the initial arc length of the beam axis and

F is the axial nodal force at node 2. The degrees of

freedom corresponding to KE are the same as that

corresponding to izb.

(c) Axial stiffness matrix KM:

it_=:

1 -1

[ 1

1 1

(21)

where AE is the axial rigidity and L is the initial arc

length of the beam axis.

(d) Torsional stiffness matrix K,:

z( GJ

1 -1

=L -1 1

1

(22)

where GJ is the torsional rigidity and L is the initial

arc length of the beam axis.

3.4. Element nodal force vectors

The nodal force vectors of the elements corre-

sponding to the global coordinate system are evalu-

ated first in the current element coordinate system,

and then transformed to the global coordinate system

using standard procedure. Since small deformations

are assumed, the element nodal forces can-in the

element coordinate system-be evaluated in much the

same way as in linear analysis. For linearly elastic

material properties the element nodal force vectors

can be calculated as follows.

(a) Bending nodal force vector t,:

&i=@,+K$,, (23)

where ~b={F31rn221,F,2,11-3,,F2,,~,,,F22r1U32} is

shown in Fig. 2; Kb is the bending stiffness matrix

given in eqn (14); ab is the total bending deformation

vector given in eqn (17). Note that due to the

definition of the element coordinate system, the only

nonzero elements in ab are f$, the deformational

nodal rotations at nodes j (j = 1,2) about the fi

(i = 2,3) axes, which may be obtained by using eqns

(11) or (13).

(b) Axial nodal force vector Fm;,:

The element internal nodal forces are calculated by

the total nodal deformation rotations. The axial

nodal force vector F,,, = {I? ] F,2} (see Fig. 1) can be

evaluated by introducing nodal virtual displacements

80, = 6 {D,,

, u12}

at nodes 1 and 2 in the f, direc-

tion, and equating the work done by the axial nodal

force F,,, going through the virtual displacement

JO,,, to the work done by the internal stress resultant

T

going through the virtual strain St,,, (that corre-

sponds to the imposed virtual displacement) along

the deformed beam axis as

I

cm,Fm =

Tsr

dS, (24)

0

where S is the current arc length of the beam axis.

The stress resultant T may be given by

T = AEF-,,

25)

where A is the cross section area, E is Youngs

modulus and r,,, is the membrane strain of the current

deformation. From eqn (6), the virtual strain a


9/14

776

Kuo MO HSIAO l

in which 6c is the variation of the chord length of the

beam axis with respect to SD,,,.

Substituting eqns (S), (25), and (26) into eqn (24)

gives

Since the virtual nodal displacements aa, are

arbitrary, the axial nodal forces are obtained from

eqn (27) as

Because the assumption of the small strain and

small deformation, S2/S,c in eqn (28) is approxi-

mated by unity in this paper, and eqn (28) is thus

reduced to

for numerical computation.

(c) Torsional nodal force vector F,:

F,={;;;}=T{_;}, (30)

where 4 = ($,, - $,,) is the total relative rotation of

the member about the 2, axis, r&j are the total twist

rotations at nodes j (j = 1,2) about the fi axis and

are obtained in eqn (12), and &?,j (j = 1,2) are twist

moments shown in Fig. 1.

4 EQUILIBRIUM EQUATIONS AND

CONVERGENCE CRITERION

The nonlinear equilibrium equation may be ex-

pressed by

=F-IP=O,

(31)

where $ is the unbalanced force between the internal

nodal force vector

F

nd the external nodal force

1P;

1

s a loading parameter and

P

is a normalized

loading vector. The internal nodal force vector is

obtained by summing up the element nodal force

vectors in the global coordinate system.

In this paper, a weighted Euclidean norm of the

unbalanced force [31] is employed as the error mea-

sure of the equilibrium state during the equilibrium

iterations, and the convergence criterion is given by

(32)

where N is the number of degrees of freedom for the

disceretized structure and p,& is a prescribed value of

error tolerance. Unless it is stated otherwise, the error

tolerance is set to lo- in this paper.

5 SOLUTION ALGORITHM

An incremental-iterative method based on the

Newton-Raphson method is adopted here. In order

to deal with the limit points and snap through, the

arc length of the incremental displacement vector

is kept constant during the equilibrium iteration

using Crisfields method [27,28]. An n-cycle iteration

scheme is introduced here to improve the con-

vergence characteristics of the equilibrium iteration.

If the equiiibrium configuration of the Ith in-

crement is assumed to be known, the system tangent

stiffness matrix KT hen can be calculated at this

configuration and an initial displacement increment

Aq for the next increment may be obtained by using

Euler predictor as

Aq = A%, (33)

where Al is the initial incremental loading parameter

and

qT= Kf'P

s the tangential displacement of unit

loading P. or all increments other than the first, Ai.

is obtained in much the same way as that mentioned

in [27] and is given by

A1 = fAa(q;q,)2,

(34)

where the sign is chosen following an approach due

to Bergan and Ssreide [32] in which the sign follows

that of the previous increment unless the determinant

of the tangent stiffness matrix has changed the sign,

in which case a sign reversal is applied. Aa is the

incremental arc length used for the next increment,

and is determined by

and

Au = C, (JD/J,)2Au,

(35)

(36)

where: Au, is the arc length used for the Ith in-

crement; J, is the number of iterations required to

achieve equilibrium for the Ith increment; JD is the

desired number of iterations; the safety factor, C,,

lies between 0.7 and 1.0, and the cut parameters C,

and C, are chosen to be 0.2 and 1.5, respectively, to

prevent yielding of an incremental displacement

which is too large or too small.

Using the displacement increment obtained in eqn

(33) and the method described in the previous section,

the internal nodal force vector F in eqn (31) associ-

ated with the current configuration can be calculated.

The loading parameter corresponding to the current

configuration is given by I = A+ AL, where I is the

convergent loading parameter at the Ith increment

and Al the loading parameter increment. Then the

unbalanced force $ can be obtained from eqn (31).

If the convergence criterion (eqn (32)) is not satisfied,

a displacement correction r and loading parameter


10/14

Corotationai procedure for rotations of beam structures

777

correction 61 [27,28] are added to the previous Aq

and AL respectively to obtain a new incremental

displacement and incremental loading parameter for

the next iteration. The values of r and 612 may be

determined by

r=K;(-$ +61P)

(37)

and

Au* = (Aq + r)(Aq + r),

(38)

where Kr may be the tangent stiffness matrix at some

known configuration. This procedure is repeated

until the convergence criterion is satisfied.

It should be mentioned that, during the first few

equilibrium iterations, the values of the element axial

nodal forces obtained from the current deformation

using eqns (6) and (29) may be several orders larger

than their convergent values for certain problems.

This may cause di~culty in convergence or even

divergence for a large increment. In [IO] a two-cycle

iteration scheme is introduced to overcome this

difficulty. This scheme was proven to be very effective

by numerous examples studied in [IO]; however, in

the present study, it is found that the accuracy of

convergent solutions obtained by this scheme is not

sufficient for some problems. This difficulty probably

arises due to the fact that the difference between the

values of convergent element axial forces at the first

and second cycles is not small for some problems. In

order to overcome this difficulty an n-cycle iteration

scheme, an extension of the two-cycle iteration

scheme, is proposed in this study and can be

described as follows.

Let fdenote the element axial force corresponding

to the current deformation, h denote the convergent

element axial force of thejth cycle, and& denote the

convergent element axial force of the fth increment.

For the iterations of thejth cycle the element axial

force F,, (eqn (29)) is replaced by

F;,==(l-CJJ;_,+CJ

j21,

(39)

where Ca f [0, 1, s a prescribed parameter; the FL2

required for the evaluation of $ in eqn (18) and Fb

in eqn (23) are replaced by

J$= fit*

I

j=l

(I--C&,+C&,, ja2,

(40)

where Cbe [0, I] is a prescribed parameter.

The equilibrium iterations of the jth cycle are

performed until the Euclidean norm of the un-

balanced force vector in eqn (31) is smaller than a

prescribed value, which may be chosen to be larger

than the error tolerance given in eqn (32). Then the

following inequality is checked:

where f, and 4_ , are m x 1 column matrices contain-

ing the convergent values of element axial forces 4

andA_, , m is the number of the elements used for the

discretization of the structure, and p, is a prescribed

parameter. If eqn (41) is not satisfied, the iterations

for next cycle are performed. Otherwise, the final

cycle of iterations is carried out until eqn (32) is

satisfied, At the final cycle, the element axial force f

calculated from the current defo~ation is used in

eqns (29) and (23) to obtain the element axial and

bending forces. The convergent solution of the final

cycle is used as the solution of the corresponding

increment.

6. NUMERICAL STUDIES

Example 1. Canti lever beam with an end moment

This example considered is a cantilever beam sub.

jetted to a concentrated moment at the free end as

shown in Fig. 5. The beam was d&ret&d by 10

elements. The results shown in Fig. 5 are obtained by

using only three increments. The number of iterations

is about six per increment. As can be seen, the

agreement with analytical solution is quite good. It

should be mentioned that the two-cycle iteration

scheme is used for this example because the element

axial forces are zero for this problem.

This example is extensively studied in the literature

to demonstrate the efficiency of numerical methods

and the large rotation capability of the beam, plate

and shell elements. To the authors knowledge, only

the present authors have achieved bending of the

cantilever beam into a full circle by using only three

increments.

Example 2. Cantilever 4Megree bend with an end wd

The bend as illustrated in Fig. 6 is curved in the

horizontal pfane and subjected to a vertical load. The

+=====&=I

W

I

I

E = l.2X10skN/m2

V=O

L = 10.0 m

b = 1.0 m

h = 0.1 m

-Analytical solution

Present analysis

0 2 4 6 8 10

12

Displacement (ml

Fig. 5. Cantilever beam with an end moment.


11/14

Kuo-MO HSIAO t (11.

R = 100. in

u = 0.

I2 t

E = 1Opsi

l-l-1

BEAM CROSS SECllON

5

t

.6

- Ref [ 6 ]

*

Present analysis

0. 1. 2. 3. 4. 5. 6. 7.

LOAD PARAMETER k = Pi/El

Fig. 6. Forty-five-degree circular bend with an end force.

bend has an average radius of 100 in. and cross

section area 1 it?.

The bend is idealized using eight equal beam

elements. At each increment, the stiffness matrix

updating is only performed at the first five iterations

of each cycle. Only four increments are used in this

analysis. Three cycles of iteration are used per in-

crement. The average number of iterations per in-

crement is about 12. The results for the end displace-

ments versus applied load are shown in Fig. 6

together with the solutions given by Bathe and

Bolourchi [6] using eight beam elements and 60

equal load increments. Very good agreement be-

tween these two solutions is observed. The deformed

configurations of the bend at various load levels are

shown in Fig. 7.

Example 3. Space arch fr ame

Figure 8 shows the structure load system and the

load displacement curve. In addition to four vertical

loads

P,

the structure is subjected to two lateral loads

equal to 0.001P. For all members the major principal

axis of inertia x is normal to the plane of either arch

rib. The symbols GJ, 1, and 1, denote, respectively,

the torsional rigidity, the major and the minor prin-

cipal moments of inertia, and the subscripts 1 and 2

denote the member groups as noted in the figure.

Each member of the structure is idealized by four

equal elements. The results of the present study

shown in Fig. 8 are obtained by using two increments

with the error tolerance p,,,, = lo-. Three cycles of

iteration are used for both increments and the total

number of iterations used for each increment are four

and five, respectively. The present results are in

excellent agreement with the solutions given in [8]

which are obtained using 17 equal load increments

(transcribed by the authors).

Case

A

:

6 boundary nodes free in

trondatlonal movsrnent

Case B :

6 boundary nodes restrained

against tronslotlonol movsment

(16.4 , 46.3 , 53.6)

A (24.7 , 60.6 ,

35.6)

-439600 lb/in2

=159000 lb/in2

10.494 iI+

PO.02 inz

-0.02 it?

=0.0331 in

Fig. 7. Deformed shapes of a 45-degree circular bend.

Fig. 9. Geometry of 12-member hexagonal frame.

c5jgft$;*

69.26 I 61.44 I 69.26

+ L=2OO.Oh

E=4.32XlO A,=.500 A,=.100

(GJ), =4.15x105 (I,), =.400 (Ix), =.05

(GJ),=1.66X10s (ly ), =.133 (Iy)2=.05

PL2mJ,

2*o7----

1.5

t/

- Ref [ 6 ]

1 ;-

p w,Lx,oJ

0.0 1.0 2.0 3.0 4.0

5.0

Fig. 8. Space arch frame.
https://www.researchgate.net/publication/275188347_Nonlinear_Elastic_Frame_Analysis_by_Finite_Element?el=1_x_8&enrichId=rgreq-026eaa3de7790f1cbb90a5aa906a8739-XXX&enrichSource=Y292ZXJQYWdlOzIyMzY0MDkwODtBUzoyNzk2OTU5MDA4NTYzMzNAMTQ0MzY5NjEwNzU4OA==https://www.researchgate.net/publication/275188347_Nonlinear_Elastic_Frame_Analysis_by_Finite_Element?el=1_x_8&enrichId=rgreq-026eaa3de7790f1cbb90a5aa906a8739-XXX&enrichSource=Y292ZXJQYWdlOzIyMzY0MDkwODtBUzoyNzk2OTU5MDA4NTYzMzNAMTQ0MzY5NjEwNzU4OA==


12/14


179

J

5

DEfTLECllON V (in)

Fig. 10. Load-deflection curves for hexagonal frame.

Exampl e 4. Tw el ve member hexagonal fr ame

The hexagonal frame depicted in Fig. 9 is subjected

to a concentrated load at the crown; two boundary

conditions are considered: (a) six boundary nodes are

free in translational movement and (b) six boundary

nodes are restrained against translational movement.

The frame was idealized using 12 (one element per

member) and 36 (three equal elements per member)

beam elements. For both boundary conditions, the

results (not shown) using 12 elements are in close

agreement with the solutions of Meek and Tan [9],

who used a similar element but did not mention the

number of elements used for discretixation.

The results of case (a) using 36 elements are shown

in Fig. 10, together with those reported by Meek and

Tan [9] and Papadrakakis [7]. It is observed that

the present results are in agreement with that of

Papadrakakis, which is in exact agreement with ex-

perimental results given by Griggs [33]. The present

results are obtained using five increments; the number

of cycles used is about four per increment, and the

Case B

250- o

Preset7 t

- Ref [S]

200-

Fig. 11. Load-deflection curves for hexagonal frame.

&

2-

x,,u

V

I

E-3.03X1 0 N/cm*

G=l.O96Xl@ N/cd

x ,,w

Fig. 12. Geometry of 24-member shallow dome.

total number of iterations used per increment is about

11. The stiffness matrix is updated only at the first

two iterations of each cycle.

The results of case (b) using 36 elements are shown

in Fig. 11. The present results are obtained using six

increments; the number of cycles used per increment

is about four, and the total number of iterations used

per increment is about 13. The dashed curve shown

in Fig. 11 is also obtained by the present study using

16 increments. The stiffness matrix is updated only at

the first two iterations of each cycle. Also shown in

Fig. 11 are the solutions reported by Meek and

Tan [9]. The discrepancies between these two solu-

tions may be explained by suggesting that the number

of elements used in [9]is insufficient.

Exampl e 5. Tw ent y- four -member hexagonal st ar-

shaped shall ow dome

Figure 12 shows the geometry of a 24member

hexagonal star-shaped shallow dome. The supports

of the dome are assumed to be pinned and restrained

against translational motion. The dome is idealized

using 24 (one element per member) and 72 (three

equal elements per member) beam elements. For all

loading conditions, the results (not shown) using 12

elements are in close agreement with the solutions of

Meek and Tan[9] who used a similar element but

did not mention the number of elements

used

for

discretixation.

The first loading condition considered is that of a

concentrated vertical load at the apex of the dome.

The present results using 72 elements, shown in

Fig. 13, are obtained by using three increments. The

number of iterations for each cycle is also given in

parenthesis beside each point on the graph. The

average number of iterations required for one in-

crement is about eight. The stiffness matrix is only

updated at the Crst two iterations at each cycle. By

keeping the same member cross sectional area but

decreasing the tlexural stiffness in the vertical plane,

the structure is reanalyzed. The graph of the load-
https://www.researchgate.net/publication/222145714_Geometricall'_nonlinear_analysis_of_space_frames_by_an_incremental_iterative_technique?el=1_x_8&enrichId=rgreq-026eaa3de7790f1cbb90a5aa906a8739-XXX&enrichSource=Y292ZXJQYWdlOzIyMzY0MDkwODtBUzoyNzk2OTU5MDA4NTYzMzNAMTQ0MzY5NjEwNzU4OA==https://www.researchgate.net/publication/222145714_Geometricall'_nonlinear_analysis_of_space_frames_by_an_incremental_iterative_technique?el=1_x_8&enrichId=rgreq-026eaa3de7790f1cbb90a5aa906a8739-XXX&enrichSource=Y292ZXJQYWdlOzIyMzY0MDkwODtBUzoyNzk2OTU5MDA4NTYzMzNAMTQ0MzY5NjEwNzU4OA==https://www.researchgate.net/publication/223111389_Post-buckling_analysis_of_spatial_structures_by_vector_iteration_method?el=1_x_8&enrichId=rgreq-026eaa3de7790f1cbb90a5aa906a8739-XXX&enrichSource=Y292ZXJQYWdlOzIyMzY0MDkwODtBUzoyNzk2OTU5MDA4NTYzMzNAMTQ0MzY5NjEwNzU4OA==https://www.researchgate.net/publication/222145714_Geometricall'_nonlinear_analysis_of_space_frames_by_an_incremental_iterative_technique?el=1_x_8&enrichId=rgreq-026eaa3de7790f1cbb90a5aa906a8739-XXX&enrichSource=Y292ZXJQYWdlOzIyMzY0MDkwODtBUzoyNzk2OTU5MDA4NTYzMzNAMTQ0MzY5NjEwNzU4OA==https://www.researchgate.net/publication/222145714_Geometricall'_nonlinear_analysis_of_space_frames_by_an_incremental_iterative_technique?el=1_x_8&enrichId=rgreq-026eaa3de7790f1cbb90a5aa906a8739-XXX&enrichSource=Y292ZXJQYWdlOzIyMzY0MDkwODtBUzoyNzk2OTU5MDA4NTYzMzNAMTQ0MzY5NjEwNzU4OA==https://www.researchgate.net/publication/222145714_Geometricall'_nonlinear_analysis_of_space_frames_by_an_incremental_iterative_technique?el=1_x_8&enrichId=rgreq-026eaa3de7790f1cbb90a5aa906a8739-XXX&enrichSource=Y292ZXJQYWdlOzIyMzY0MDkwODtBUzoyNzk2OTU5MDA4NTYzMzNAMTQ0MzY5NjEwNzU4OA==https://www.researchgate.net/publication/222145714_Geometricall'_nonlinear_analysis_of_space_frames_by_an_incremental_iterative_technique?el=1_x_8&enrichId=rgreq-026eaa3de7790f1cbb90a5aa906a8739-XXX&enrichSource=Y292ZXJQYWdlOzIyMzY0MDkwODtBUzoyNzk2OTU5MDA4NTYzMzNAMTQ0MzY5NjEwNzU4OA==https://www.researchgate.net/publication/222145714_Geometricall'_nonlinear_analysis_of_space_frames_by_an_incremental_iterative_technique?el=1_x_8&enrichId=rgreq-026eaa3de7790f1cbb90a5aa906a8739-XXX&enrichSource=Y292ZXJQYWdlOzIyMzY0MDkwODtBUzoyNzk2OTU5MDA4NTYzMzNAMTQ0MzY5NjEwNzU4OA==https://www.researchgate.net/publication/222145714_Geometricall'_nonlinear_analysis_of_space_frames_by_an_incremental_iterative_technique?el=1_x_8&enrichId=rgreq-026eaa3de7790f1cbb90a5aa906a8739-XXX&enrichSource=Y292ZXJQYWdlOzIyMzY0MDkwODtBUzoyNzk2OTU5MDA4NTYzMzNAMTQ0MzY5NjEwNzU4OA==https://www.researchgate.net/publication/222145714_Geometricall'_nonlinear_analysis_of_space_frames_by_an_incremental_iterative_technique?el=1_x_8&enrichId=rgreq-026eaa3de7790f1cbb90a5aa906a8739-XXX&enrichSource=Y292ZXJQYWdlOzIyMzY0MDkwODtBUzoyNzk2OTU5MDA4NTYzMzNAMTQ0MzY5NjEwNzU4OA==https://www.researchgate.net/publication/222145714_Geometricall'_nonlinear_analysis_of_space_frames_by_an_incremental_iterative_technique?el=1_x_8&enrichId=rgreq-026eaa3de7790f1cbb90a5aa906a8739-XXX&enrichSource=Y292ZXJQYWdlOzIyMzY0MDkwODtBUzoyNzk2OTU5MDA4NTYzMzNAMTQ0MzY5NjEwNzU4OA==https://www.researchgate.net/publication/223111389_Post-buckling_analysis_of_spatial_structures_by_vector_iteration_method?el=1_x_8&enrichId=rgreq-026eaa3de7790f1cbb90a5aa906a8739-XXX&enrichSource=Y292ZXJQYWdlOzIyMzY0MDkwODtBUzoyNzk2OTU5MDA4NTYzMzNAMTQ0MzY5NjEwNzU4OA==


13/14

780

Kuo MO

HSMO er al.

2.5?

(I 2.1, I )

. Present

0

2.0

- Rsf [ 9 ]

P*V

A

. Loaded node

0

1.0

2.0 3.0 co 5.0

DEFIJXTION V (cm)

Fig. 13. Load~efl~tion curves for con~ntrat~ central

Ioad.

deflection curves using 72 elements is shown in

Fig. 14. As can be seen, only three increments are

used. The average number of iterations per increment

is nine.

The second load condition is al1 nodes loaded

symmetrically. The results using 72 elements are

shown in Fig. 15. Four increments are used and the

2.0

0 Prsssnt

-Rsf [S]

1.5 *

F

a

0.

12=2.377cm'

13=0.295cm4

9 1.0,

J PO.91 Bcm

s

/

Loaded

node

/

l/---j

f

1.0 2.0 3.0

4.0 5.0

DEFLECTION

Fig.

14. ~ad~efl~tion curves

load.

2.0

0 Present

- Ref [ 9 ]

V

(cm)

for ~n~trat~ central

0

1.0

2.0 3.0 4.0 5.0

DENCTION V (cm)

0

6.0

Fig. 15. Load-deflection curves for symmetrical loading.

2.0

I

0 Present

-Ref [9]

wc&ed

12=2.377cm4

13 =0.295cm4

J =0.918cm4

OK

1.0 2.0

3.0 4.0 3.0

DEFLECTION V (cm)

Fig. 16. Load-deflection curves for unsymmetrical loading.

average number of iterations used per increment

is eight. For the unsymmetrical loading condition

shown in Fig. 16, five increments are used and

the average number of iterations used is 10 per

increment.

7. CONCLUSIONS

A practical motion process of the three dimen-

sional beam element is presented to remove the

restriction of small rotations between two successive

increments for large displacement and large rotation

analysis of space frames using incremental-iterative

methods.

The nonlinear fo~uiation is based on the co-

rotational formulation by which the major geometric

nonlinearities were shown to be embodied in the

coordinate transformation when forming the element

assemblage. The transformation of the element co-

ordinate system is assumed to be accomplished by a

~anslation and two successive rigid body rotations:

a transverse rotation followed by an axial rotation.

The element formulation is derived based on the

small deflection beam theory with the inclusion of

the effect of axial force in the element coordinate

system. The element internal nodal forces are calcu-

lated using the total defo~ational nodal rotations.

Two methods, referred to as direct method and

incremental method, are proposed in this paper to

calculate the total deformational rotations.

Despite the fact that the formulation of the beam

element is very simple, highly accurate solutions are

obtained. It is believed that the use of a simple

element combined with the corotational formulation

and the process of element motion proposed in this

paper may represent a valuable engineering tool for

the solution of nonlinear spatial beam problems.

REFERENCES

t. K. H. Chu and R. H. Rampetsreiter, Large deflection

buckling of space frames. J. Srrucr. Do., AXE St%,

2701-2722 (1972).

2. C. Oran, Tangent stillness in space frames. J. Srrucf.

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A Corotational Procedure That Handles1987