eccomas2000_barcelona

8/8/2019 ECCOMAS2000_Barcelona

http://slidepdf.com/reader/full/eccomas2000barcelona 1/20

European Congress on Computational Methods in Applied Sciences and Engineering

ECCOMAS 2000

Barcelona, 11-14 September 2000

ECCOMAS

1

A COMPARATIVE STUDY OF SOME NON-LINEAR FORMULATIONS

FOR THE ANALYSIS OF PLANE FRAMES

Ricardo A. M. Silveira*, Alexandre S. Galvão

*, and Paulo B. Gonçalves

†

* Civil Engineering Department, School of Mines, UFOP

35400-000 Ouro Preto, MG, Brazil

E-Mail: [email protected];

† Civil Engineering Department, PUC-Rio

22453-900 Rio de Janeiro, RJ, Brazil

E-Mail: [email protected]

Key words: Non-Linear Finite Element Formulations, Incremental-Iterative Solution,

Stiffness Matrix, Internal Load Vector.

Abstract. This work presents a comparative study of geometrically non-linear formulations

for the analysis of two-dimensional frames. The non-linear formulations under consideration

employ different Lagrangian reference frames, different approximations of the Green-

Lagrange strains and potential energy, and various procedures to obtain the stiffness matrix

and the internal load vector. The importance of these different approaches is highlighted in

the finite element formulations proposed by Alves1 , Yang and Kuo

2 , and Pacoste and

Eriksson3 , which are tested in this work.

To solve the resulting non-linear equations and obtain non-linear equilibrium paths, the

Newton-Raphson’s method is employed together with path following techniques, such as the

arc-length schemes and the generalized displacement control technique. The performance of

the formulations is illustrated by means of three challenging numerical examples.



Ricardo A. M. Silveira, Alexandre S. Galvão, and Paulo B. Gonçalves.

2

1 INTRODUCTION

Recent developments in structural materials, more refined design methodologies and thelarge amount of research on stability of structures have led to increasingly slender structural

elements whose analysis necessitates a truly non-linear approach due to the presence of

geometric non-linearities. These systems may exhibit multiple solutions and may loose their

stability due to bifurcation or the existence of limit points along the non-linear equilibrium

path. The knowledge of the non-linear behaviour of slender structural elements, such as

columns, frames. rings and arches, is essential in the local or global stability analysis of

complex structural systems. The finite element method (FEM) has shown to be particularly

appropriate for the analysis of complex structural problems. The discretization process of non-

linear structures by the use of finite elements leads to a system of non-linear algebraic

equations that are often solved by Newton-type methodologies.In the analysis of steel frames, which is the topic of the present work, in order to evaluate

accurately numerically the instability behaviour, or critical structural aspects, some relevant

problems have to be studied:

• The development of reliable 2D and 3D non-linear finite element formulations1,2,3,4;

• Numerical procedures for determining the equilibrium path in both pre and post-buckling

ranges, which may include softening behaviour, the presence of load and displacement limit

points and bifurcations2,4,5;

• Bracketing procedures for the computation of singular points (limit points or bifurcations)

and techniques for branch switching4;

• Finite element formulations for 2D and 3D elastic-plastic and rigid-plastic frames6;

• Semi-rigid steel connections7; the moment-rotation characteristics of the connections can

strongly influence the response of individual members as well as the overall behavior of the

complete framing system;

• Graphical user interface during the model creation, analysis and post-analysis.

This work is mainly concerned with the first topic, but utilizes the knowledge of the second

and the last subject. Basically, the aim of the present work is to make a comparative study and

the computational implementation of the following geometrically non-linear formulations for

two dimensional frame elements: Alves’s updated Lagrangian formulation1, in which

complete expression of the Green-Lagrange strain increments and internal load vector based

on natural displacement were used; Yang and Kuo’s updated Lagrangian formulation2, where

a simplified planar frame element was adopted and two methodologies were used to obtain theinternal load vector: natural displacement and external stiffness approaches; and three of the

Pacoste and Eriksson’s formulations3, where a total reference frame (total Lagrangian),

following both total and co-rotational approaches, were tested.

These formulations were used together with the non-linear solution methodology

implemented initially by Silveira5, which solves the resulting non-linear equations and obtains

non-linear equilibrium paths through the Newton-Raphson’s method together with path

following techniques, such as the arc-length schemes of Crisfield4 and generalized

displacement control technique2. The computational performance and reliability of these




3

formulations are evaluated by means of several numerical examples.

2 NON-LINEAR METHOD OF SOLUTION

In this section, the incremental-iterative solution strategy for non-linear elastic problems

adopted in this work will be summarized. In the finite element context, the equilibrium of a

structural system can be expressed as

ri FuF λ =)( (1)

where Fi defines a set of generalized internal forces in terms of the corresponding generalized

displacement components u; λ is a scalar load multiplier, and Fr is a fixed load vector

(reference vector).

To obtain the equilibrium paths of a structure, an incremental solution technique should be

used to solve Equations (1). This is achieved by calculating a sequence of displacement

increments ∆u1, ∆u2, ∆u3,..... corresponding to a sequence of load parameter increments ∆λ 1,∆λ 2, ∆λ 3,..... In each increment, however, due to the non-linearities in Fi, the problem must be

solved iteratively.

To obtain the internal out-of-balance (or ‘residual’) forces acting on the structural system

in the iterative process and the new estimate for the displacements, the vector of residual

forces:

ri FuFg λ −= )( (2)

must be computed. This vector can be used to correct the displacements through the following

expression:

guK −=δ (3)

where, K is a representative stiffness matrix and δu is the residual displacement vector. This

work is particularly interested in the different definitions of K and Fi, which depend on the

adopted non-liner finite element formulation.

For an iterative algorithm to be able to deal with load and displacement limit points it must

include modifications of the load level during iterations. As a consequence, δu cannot be

obtained directly using Equation (3). Crisfield4 and other researchers2,9 advocate that the

residual displacement can be written as the linear combination of two vectors:

rg uuu δδλ +δ=δ (4)

where δλ is the load correction parameter that must be measured at each equilibrium iterative

cycle and δug and δur are obtained from: gK ug1−=δ and rr FK u 1−=δ . The definition of

δλ in (4) depends on the selection of an appropriate iterative strategy; in other words, an

additional constraint equation involving δλ with the aim of restoring equilibrium as rapidly as

possible is necessary. The numerical strategy used in this work to obtain δλ is based on the

concept of an ‘arc-length’ constraint equation4, which consist in considering, in addition to




4

Equation (1) the following relation:

2T2T l∆=λ ∆+∆∆ rr FFuu (5)

in which ∆l is the generalized ‘arc-length of the tangent at the converged state (i-1) in load-

displacement space. The iteration k gives the subincrements δλ k e δuk , which are used as

corrections for the load and displacement increments:

k )1k (k δλ +λ ∆=λ ∆ − and k )1k (k uuu δ+∆=∆ − (6)

If the convergence criteria are fulfilled after a number of iterations, a new equilibrium state of

the structural system is obtained.

In a computational context, a cycle of the proposed incremental-iterative strategy can be

summarized as follows:1. Considering the previous equilibrium configuration as a reference, the initial increment

of the load parameter ∆λ 0 is selected and used to calculate the initial increment of the

nodal displacements ∆u0:

T00 uu δλ ∆=∆ (7)

with rFK u 1T

−=δ . The initial increment of the load parameter ∆λ 0 can be selected

here using the constraint equation (5). The approximations ∆λ 0 and ∆u0 are termed

here “tangent incremental solution” or “predictor solution’. This solution rarely

satisfies the equilibrium equations, so additional iterative cycles are required. In this

work different load increment strategies will be used.2. The second step of the solution methodology deals with the geometric non-linearity of

the structural system. Here the Newton-Raphson’s method (full or modified) is used to

solve the discretized equilibrium equations making correction in the predictor solution

by using δλ k e δuk . Then, a new incremental solution is obtained and the correction

procedure is repeated until the convergence criteria are satisfied.

These procedures are better described in Figure 1.

3 NON-LINEAR FINITE ELEMENT FORMULATIONS

The stiffness matrix K and internal forces vector Fi depend on the adopted finite element

formulation. In this section three different formulations are presented, where special attentionis given to kinematics relations, the incremental potential energy and the interpolation

functions used to approximate the displacement fields.

3.1 Alves’s formulation1: AFI

Alves finite element formulation1 for non-linear frames is based on an updated Lagrangian

formulation and the complete form of the Green-Lagrange strain increments. In this case, the

axial deformation increments can be written as:




5

No

Initial configuration: tu and tλ

Stiffness matrix: K

Internal force vector:Fi

Yes

New increment

Update total and

incremental variablesCorrector: δλ k and δuk

Non-Linear

FormulationsPredictor: ∆λ 0 and ∆u 0

Res. vector: gg

∆λ Fr

≤ ζ

Iterative loop: k=1,2,...

Figure 1: Non-linear solution strategy.

]dx

vd

dx

ud[

2

1

dx

ud22

xx

∆

+

∆

+∆

=ε∆(8)

where u∆ is the axial displacement and ∆v the lateral displacement.

Using Bernoulli-Euler hypothesis: dx/vdyuu ∆−∆=∆ ; where, ∆u is due to the

extensional forces and the second part, y (d∆v/dx), is due to bending forces, one obtains fromequation (8):

∆εxx = ∆exx + ∆ηxx (9)

with ∆exx defining the linear component and ∆ηxx the non-linear component, which can be

expressed as:

2

2

xxdx

vdy

dx

ude

∆−

∆=∆ and ]dx

dx

vd

L

1

dx

vdy

dx

vd

dx

udy2

dx

ud[

2

1L

0

22

2

22

2

22

xx ∫

∆

+

∆+

∆∆−

∆

=η∆(10)

In order to remove or, at least ameliorate, the ‘membrane locking’ due to usually poor

shape functions adopted for u (see Section 3.1.2), the last term in (10) was uniformized

following Crisfield’s suggestion4.

In the updated Lagrangian formulation, the last calculated configuration t is selected as the

reference configuration. Therefore, it is important to know for the frame element considered

the stress field, or deformation configuration of the body, at configuration t. Following Alves1

proposal, the deformation at configuration t is obtained from:




6

( )

]xL

MM

M[EI

y

EA

P 21

1

t +

−+=ε

(11)

where P, M1, M2 are, respectively, the in-plane normal force and the bending moments acting

on the beam element at step t, as shown in Figure 2.

P

PM1

M2

Figure 2: Force and moments at equilibrium configuration t.

3.1.1 Potential energy

The incremental equilibrium equations can be derived by setting the first variation of the

increment of the total potential energy of the element equal to zero. Thus it is important to

define ∆Π, which can express by: VU ∆+∆=∆Π . Considering linear elastic behaviour

( ε∆=σ∆ E ) and an updated Lagrangian formulation, the strain energy increment, ∆U, and the

work of external forces, ∆V, can be written as:

dAdx)2

E E(U

vol

2t∫ ε∆+ε∆ε=∆ and ]dsuFdsuF[-dsuF-Vs s

iiiit

s

ii ∫ ∫ ∫ ∆∆+∆=∆=∆(12)

By substituting Equation (9) into (12) and by using the definitions in (10), it is possible towrite ∆Π as follow:

]dsuFdsuF[ -UUUUUs s

iiiit

21L0 ∫ ∫ ∆∆+∆++++=∆Π τ(13)

where:

dVoleE2

1U t

Vol

xxxxt

0t

∫ ∆ε= ; dVolEU txx

Vol

xxt

t

η∆ε= ∫ τ ; dVoleΔE2

1U t

Vol

2xxL

t∫ = ;

∫ η∆∆=

Vol

t

xxxx1 t

dVoleEU and dVolE2

1U t2

xxVol2 t

η∆=

∫ (14)

Here U0 is associated with the total force applied at configuration t; U τ corresponds to the

initial deformation influence and will generate the geometric stiffness matrix; UL is

responsible for the linear part of stiffness matrix; and, U1 and U2 will originate the stiffness

matrices which are linear and quadratic functions of the nodal displacement increments. All

these matrix components are presented in the next section.

However, the increment of the total potential energy can be rewritten, if the equality




7

dsuΔFdvoleΔεEs ii

tvol xxxx

t ∫ ∫ = is true, in the form:

∫ ∆∆+++=∆Π τ s ii21L dsuF-UUUU (15)

3.1.2 K and Fi expressions

Figure 3 shows the adopted planar frame element with the nodal displacements, which are

the degrees of freedom of the member. The displacements ∆u and ∆v along the centroidal axis

x can be related to those at the ends of the element by the following functions:

2211 uHuHu ∆+∆=∆ and ∆ ∆ ∆θ ∆ ∆θv H v H H v H= + + +3 1 4 1 5 2 6 2 (16)

where H1 and H2 are linear interpolation functions and H3, H4, H5 and H6 are the well-known

cubic Hermitian interpolation functions.

X

Y

∆v

∆u

∆θ

L x

y, ∆ v

∆u1

∆v1

∆θ1

1

x, ∆ u

∆θ2

∆u2

∆v2

2α

Figure 3: Frame element adopted.

Using, the previous relations (Alves1 and Silveira5), the energy functional (15) can be

written as a function of the nodal displacements and forces, that is:

uuuK uK K K u 21L ∆∆∆+∆++∆=∆Π τ )],(24

1)(

6

1

2

1

2

1[ eeeeT e

ri FuFu λ ∆−∆+ ∆+ ttTetT (17)

in which eeee e,, 21L K K K K ττττ can be obtained by differentiation of the energy terms in (14):

ji

L2

) j,i(L uu

U

k ∆∂∆∂

∂

= ; ji

2

) j,i( uu

U

k ∆∂∆∂

∂

=τ

τ ; k k ji

13

) j,i(1 uuuu

U

k ∆∆∂∆∂∆∂

∂

= ; lk lk ji

24

) j,i(2 uuuuuu

U

k ∆∆∆∂∆∂∆∂∆∂

∂

=

(18)

The incremental equilibrium equations obtained by setting the first variation of the

increment of the total potential energy of the element equal to zero ( 0 =∂∆Π∂ u∆∆∆∆ ), can be

expressed as: rii FFF λ =+ ∆+∆ ttetet , where increment of the internal forces vector is given by:




8

uuuK uK K K F 21L

e

i ∆∆∆+∆++= τ

∆

]),(6

1

)(2

1

[

eeeet (19)

Finally, the stiffness matrix K e in local coordinates is obtained from Equation (17) and is

given by:

]),(2

1)([ eeeee uuK uK K K K

21L∆∆+∆++=

τ

(20)

Here eLK is the conventional linear elastic-stiffness matrix and e

τK is commonly referred

to as the initial-stress matrix (geometric-stiffness matrix) and its terms depend on the initial

nodal forces. The elements of the matrix e1

K are linear functions of the incremental

displacements while the terms in e2K are quadratic functions of the incremental nodal

displacements. All these matrices are symmetric.

If a rigid body rotation is applied to the frame element, the forces in Equation (19) are not

equal to zero. Such result is inconsistent with the rigid body law. Trying to minimize this

inconsistency, the increment of the internal force vector may be rewritten using now the

natural displacement approach as:

nnn2n1Li uuuK uK K K F ∆∆∆+∆++=∆ )],(6

1)(

2

1[et (21)

where ∆∆∆∆un is the natural displacement increment vector , which can be expressed as:

[ ] 0 0 0 21T φδφ=∆ nu (22)

δ

φ1

φ2

y

x

∆θ1

∆θ2

A*

B*

L

BA

∆u1

∆u2L

t

t

t

t

∆v1

∆v2

Figure 4: Natural deformation of frame element.

Using the geometrical relations showed in Figure 4, is possibly to write:

L L ttt −=δ ∆+ ; Ψ−θ∆=φ 11 ; and Ψ−θ∆=φ 22(23)

where the rigid body rotation Ψ is given by: ]L)/vv([tan t12

-1 ∆−∆=Ψ .

Finally, the stiffness matrix and the internal forces vector of the structural system are




9

defined in global coordinates as:

R K R K t

m

eTt ∑= and ∑ ∆+∆+ =m

ettTtt iai FR F (24)

in which tR is the rotation matrix updated in configuration t and R a is the rotation matrix

updated in the last iteration.

3.2 Yang and Kuo’s formulations2: YGN and YGE

In this section two non-linear frame element formulations proposed by Yang and Kuo2 are

presented, which are different only in the way that the member forces at each incremental step

are obtained. One formulation uses the natural deformation approach, denoted here by YGN,

which has often been employed by researchers in the analysis of frame-type structures; the

other uses the external stiffness approach, denoted here by YGE, which has the advantage that

it can be generalized to deal with other types of structures.

These authors3 adopted for the frame element showed in Figure 3 two independent stress

components, the Cauchy stresses τxx and τxy, and two associated strain components, the

updated Green-Lagrange strain increments ∆εxx and ∆εxy. They are presented below

decomposed into their linear and non-linear components, i.e.,

xxxxxx e η∆+∆=ε∆ and xyxyxy e η∆+∆=ε∆ (25)

Based on Bernoulli-Euler hypothesis, these linear and non-linear components can be

expressed as follows:

2

2

xxdx

vdy

dx

ude

∆−

∆=∆ ; ]

dx

vd

dx

vdy

dx

vd

dx

udy2

dx

ud[

2

122

2

22

2

22

xx

∆

+

∆+

∆∆−

∆

=η∆

0exy =∆ ; and ]dx

vd

dx

vd y

dx

vd

dx

ud [

2

12

2

xy

∆∆+

∆∆−=η∆

(26)

The stress resultants (initial forces) are obtained through integration of the Cauchy stresses

τxx and τxy, that is,

∫ τ=A

xxtt dAP ; ∫ τ=A

xytt dAQ ; and ∫ τ=A

xxtt dAyM (27)

where tP is the axial force; tQ is the transverse shear; and tM is the bending moment. Based on

Figure 2, tM and tQ can be rewritten as follows:

( )x

L

MMMM 21

1t +

+−= and( )

L

MM - Q 21t +

=(28)




10


Considering the stress and the strain components defined in the last section, and assumingadditionally a linearized approach, it is possibly to express the increment of the strain energy,

∆U, as:

dAdx)e2

E(dAdx)2(U t

Vol

2xx

t

Vol

xyxyt

xxxxt

tt

∫ ∫ ∆+ε∆τ+ε∆τ=∆(29)

and the increment of ∆V, as:

]dsuFdsuF[-dsuF-Vs s

iiiit

s

ii ∫ ∫ ∫ ∆∆+∆=∆=∆ (30)

According to Alves1, the following condition is observed:

∫ ∫ ∆=∆τ+∆τs ii

t

Vol

txyxy

txxxx

t dsuF dVol)e2e(Et . Hence, ∆Π can be rewritten as:

sduF -UUs iiL ∫ ∆∆+=∆Π τ (31)

in which:

dVoleE2

1U t

Vol

2xxL

t

∫ = ; and

∫ ∫ ∆∆+

∆+

∆+

∆=τ

L

02

2

t

L

0

2

2

222

t dx]dx

vddx

udM[21 dx]

dxvd

AI

dxvd

dxud[P

21U

∫ ∆∆

−L

0

t dx]dx

vd

dx

udQ[

2

1

(32)

3.2.2 K and Fi expressions

Using the variables given in Figure 3 again, and substituting Equation (16), used here to

approximate the axial and lateral (∆u and ∆v) displacements, into Equation (32), it is possibly

to express ∆Π as a function of nodal displacements of the element. This leads to:

uK K u L ∆+∆=∆Π ]2

1

2

1[ eeT e

ri FuFu λ ∆−∆+ ∆+ ttTetT (33)

where eLK and e

τK are obtained using Equation (18). The increment of the internal forces

vector is obtained from (33), setting the first variation of the increment of the total potential

energy of the element equal to zero. Thus, this vector can be written as follows:




11

uK K F Li ∆+=∆ ][ eτ

eet (34)

The total internal forces vector in the equilibrium configuration is calculated

making etetett iii FFF ∆∆+ += . The global stiffness matrix of the planar frame element has a

linearized form, i.e., is independent of nodal displacement and can expressed by:

R K R K t

m

eTt ∑= , where:

][ eeeτ+= K K K L

(35)

Now, two procedures to evaluate the increment of the internal force vector etiF∆ will be

presented.The first approach, as mentioned before, is denoted by YGN, and allows the calculation of etiF∆ through Equation (34) using the incremental natural displacements vector ∆un (see

Equation (22)) in place of ∆u, as illustrated in the Section (3.1.2). The second approach,

formulation YGE, is based on the external stiffness matrix to account for the effect of rigid

body motions of the planar frame element. This matrix can be used to evaluate etiF∆ . Suppose

that the planar frame element is subjected to a rigid body rotation ψ , as shown in Figure 5,

which, when assumed to be small, can be represented by the following displacement vector:

[ ] T L 0 0 0 ΨΨΨ=∆ ru (36)

x

y

L

L

Ψ

Ψ

L tan (Ψ) ≈ L Ψ

Figure 5: Rigid body rotation.

Note that no forces will be generated by elastic stiffness matrix, that is, 0uK rL = .

However, the forces generated by the geometric stiffness matrix τK during the rigid body

rotation are not equal zero, i.e.,

T

r - -uK

ψ

+ψ ψ

+=τ 0P

L

MM 0P

L

MM 2121

(37)




12

If small rigid body rotations ψ are assumed, the following approximation can be adopted:

L/)vv( 12 −=Ψ , and it is possible to rewrite Equation (37) in the following form:

uK uK rer ∆=

θ∆

∆

∆

θ∆

∆

∆

=

++

++

++

++

τ

2

2v

2u

1

1v

1u

0 0 0 0 0 0

0

L

P

2L

2M1M 0

L

P-

2L

2M1M -

0 2

L

2M1M 0 0

2L

2M1M- 0

0 0 0 0 0 0

0 L

P-

2L

2M1M- 0

L

P

2L

2M1M

0 2

L

2M1M- 0 0

2L

2M1M 0

(38)

where K re is defined as the external stiffness matrix. As stated in Yang and Kuo2, by

subtracting the external stiffness matrix K re from tangent stiffness matrix (K L + K ττττ) a matrix

that accounts for the effect of member deformations can be obtained. Accordingly, the internal

force increments caused by deformation of the planar frame element can be calculated as:

uK K K F reLi ∆−+= τ∆ ][ eeeet (39)

3.3 Pacoste and Eriksson’s formulations3: PTT, PC1 and PC2

Pacoste and Eriksson3 proposed five different formulations for planar frames, following

both total and co-rotational approaches. In this work three of them are presented. These

authors explain that the total and co-rotational approaches only differ in the way they describe

the displacements of an element, the later eliminating or reducing rigid rotations. They are,

however, in both approaches, referred to as the initial shape of the element, and thus in a

‘Total Lagrangian’ reference system.

The first formulation presented here is denoted by PTT, and is based on Timoshenko type

strains; the second formulation PC1, is based on a co-rotational description and linear local

strains; and finally the third formulation PC2, is based on a co-rotational description and

‘shallow arch’ strain definitions.

For the PTT formulation, a highly non-linear frame theory, the deformations are written as:

)sen(dx

dv )cos(

dx

du 1 xx θ

+θ

+=ε ; )sen(

dx

du 1-)cos(

dx

dvθ

+θ

=γ ; and

dx

d k

θ=

(40)

The constitutive relations are taken as linear and can be written as:

P = EAεxx; Q = GAγ ; and M = EIk (41)

in which EA, GA and EI represent axial, shear and flexural rigidities, respectively.




13

In formulations PC1 and PC2, the central idea is to introduce a local coordinate system,

which continuously rotates and translates with the element. For PC1 a ‘linear’ definition

(a)

y, v

θ(x) = θr + θn(x)

r,x

a(x)

(S)

x, u

(S)

x

L

u(x)

v(x)

v1

P1

u1

u2

v2P1M1

Q1

P2

Q2

M2

(b)

y, v

x, u

x0

y0y

t+∆t

xt+∆t

u1

v1

M1P1

P2

v2

u2

L

θr

θ1

θn1θ2 = θr + θn2

Lt+∆t

0

M2

Figure 6: Element formulations: (a) total; (b) co-rotational.

for the local strain is assumed:

dx

du xx =ε ; 0=γ ; and

dx

d k

θ=

(42)

and for PC2, where an average axial strain to avoid membrane locking is introduced, the

following expressions are adopted:

∫

+=ε

L

0

2

xx dx]dx

dv

2

1

dx

du[

L

1; 0=γ ; and

dx

d k

θ=

(43)


In the total Lagrangian context, the total potential energy can be written as follows:

∫ ∫ −+γ +ε=Πs

ii

L

0

222xx dsuFdx]k EIGA[EA

2

1

(44)

where the first part of (44) is the strain energy ∫ ∫ ε ετ=vol 0 ijij

ij dVol)d(U and the last part is the

potential energy V due to applied forces. Note that in the PC1 and PC2 formulations γ = 0 in

(44).

3.3.2 K and Fi expressions – Total approach: PTT

For the finite element based on PTT formulations, linear interpolation functions are used to

approximate of the u, v and w (see Figure 6), that is: p = H1 q1 + H2 q2, where pT =[u v w], qi




14

and q j are the corresponding values of the displacement at the two nodes of the element, and

H1 = 1-x/L and H2 = x/L are the linear interpolation functions. If one-point Gaussianquadrature is used to perform the integral of the strain energy in (43), the following expression

is obtained for U:

]k EIGA[EA2

L U 222

xx +γ +ε=(45)

For a generic element, the components of the internal forces vector eifi and the components

of the tangent stiffness matrix eijk are obtained through successive differentiation of (45), that

is:

uUfi

i

ei ∂∂

= and ji

2

eij

uuU k ∂∂∂= (46)

The stiffness matrix K and the internal forces vector Fi are calculated in the local

coordinates system of the element and can be transformed into the global coordinate system

by the use of Equation (24), but adopting now for the calculation of R e the initial

(undeformed) configuration of the structural system.

3.3.3 K and Fi expressions – Co-rotational approach: PC1 and PC2

For PC1 and PC2 formulations, the basic idea is to write the strain energy U as a function

of the natural displacement components. As shown in Figure 6b, the natural displacement

vector has three components and can be written as follows:

T n2n1n2 ] u[ θθ=nu T

r 2r 10tt ]--LL -[ θθθθ= Λ+ (47)

where t+∆tL and 0L denote the initial and current length of the element; θr is the rigid body

rotation, which can be computed now from the total global nodal displacements3.

In both cases (PC1 and PC2) the following approximations for the axial and rotational

natural displacements are used:2n2n uHu = and

2n41n3n HH θ+θ=θ , where H2, H3 and H4

are given by:

L

x

H2 = ; )L

x3

L

x2

1(4

1

Hand);L

x3

L

x2

1(4

1

H 2

2

42

2

3 ++−=+−−=

(48)

The strain relations (42) and (43) can then be rewritten in terms of the natural

displacements. Under these assumptions, the strain energy expression for PC1 becomes:

)(L

EI2u

L2

EA U 2

n22n1n2n1

22n θ+θθ+θ+=

(49)

and for the ‘shallow arch’ model (PC2) the strain energy expression is given by:




15

)]2

1

(15

L

u[L2

EA

U

22

n22n1n

2

n12n θ+θθ−θ+= (L

EI

2 )

2

2n2n1n

2

1n θ+θθ+θ+

(50)

Expressions for the internal forces vector and tangent stiffness matrix in local (natural)

coordinates are:

T

2n1n2n

]U

U

u

U[

θ∂∂

θ∂∂

∂∂

=nF and3,2,1 j

3,2,1i;}

uu

U{

njni

2

==

∂∂∂

=K n(51)

where [ ]T21 MP M =nF .

According to Pacoste and Eriksson3, the internal forces vector and the tangent stiffness

matrix can be obtained in terms of the six nodal degrees of freedom in global coordinates

using the following expressions:

nci FAF T = and c3c2c1cnTc AAAAK AK MMP 21 +++= (52)

in which the transformations matrices Ac, Ac1, Ac2 and Ac3 are given by:

};u

u{

j

n k

∂∂

=cA };uu

u{

ji

2n2

∂∂∂

=c1A and6,5,4,3,2,1i,j

3,2,1k };

uu{

ji

r 2

==

∂∂θ∂

−== AA c3c2

(53)

4 NUMERICAL EXAMPLES

In this section three examples (two arches and one frame) were analyzed using the proposed non-linear finite element formulations. The main objective is to check their

computational performance when solving highly non-linear problems.

To solve the non-linear equilibrium equations (Section 2), the modified Newton-Raphson

approach together with one of the following iterative strategies was adopted: cylindrical arc-

length4 and generalized displacement control technique2. Also, the following ratio Id(desired

number of iterations)/I(actual number of iterations – previous load step) or the generalized

stiffness parameter (GSP ) were used to control the increment of the load parameter λ .

4.1 Circular arch with snap-through and snap-back

The shallow arch shown in Figure 7a has been used by several researches2,5,8

to investigatethe efficiency of incremental-iterative solution algorithms. This example, as illustrated in

Figure 7b, exhibits substantial geometric non-linear behavior: four load limit points and two

displacement limit points, and is used to test the non-linear finite element formulations here

implemented. In this work, the arch was modeled using meshes with 4, 6, 10 or 20 finite

elements and the results are compared with those given by Yang e Kuo2, obtained using 26

elements and formulation YGN.

The average error in the computation of the 6 limit points shown in Figure 7b is presented

in Table 1. In Table 1 it is also shown the CPU time necessary to obtain the equilibrium path




16

from A to B (see Figure 7b), for the arch modeled with 20 elements. For the calculations, a

Pentium II350/32MB was used. In general can be observed that all formulations performedwell, in particular YGN which uses a natural displacement approach and calculates the

internal force vector using an incremental procedure. Observe that PC1 and PC2 formulations

present higher CPU time, which can be explained by the calculation of transformation

matrices. A stiff behaviour was observed for the PTT formulation, mainly for 4 and 6 element

models. The 4 elements mesh error is not presented in the table because the load-deflection

curve obtained was different from the expected. Figure 8 shows the evolution of the results

obtained with this non-linear formulation, where the variation of p with the tangential

displacement u (see Figure7a) is plotted for increasing number of elements.

(a)

50 50w

P

5

E = 2000 G = E/2 A = 1 I = 1

M = 2 x P

u

(b)

0.0 1.5 3.0 4.5 6.0 7.5 9.0 10.5 12.0

w

-0.6

-0.3

0.0

0.3

0.6

0.9

1.2

1.5

P

A B

Present

YGN 20 elem.

Yang and Kuo

Load Limit Points

Snap-Back Points

2

Figure 7: Shallow arch: a. Geometric and material properties; b. Equilibrium path.

Form. Element Number time A-B (sec.)

4 6 10 20 20 elem.

AFI 20.37 9.71 3.76 1.30 86.0

YGN 20.79 10.13 4.20 1.72 35.9

YGE 19.82 9.44 3.64 1.89 36.6

PTT --- 14.64 3.55 1.49 54.9

PC1 22.37 6.66 2.57 1.79 125.2

PC2 17.43 7.72 3.26 1.56 169.2

Table 1: Average error and CPU time.

4.2 Lee frame

An L-frame with both ends hinged, as shown in Figure 9a, was analyzed with 4 different

meshes: 6, 8, 10 and 20 finite elements. The finite element models with 6 and 8 elements are

shown in Figure 9b. This structural system is usually refereed to as Lee frame3,9. The reference

results were obtained here using the AFI formulation and employing a very dense mesh of 100

finite elements; the resulting equilibrium path can be seen in Figure 9c.




17

Table 2 shows the CPU time and the average error in the computation of the 4 limit points

(2 load and 2 displacement limit points) obtained by AFI, YGN and PTT formulations. TheCPU time was evaluated from the beginning to the second load limit point, when P = 3.

Again, a Pentium II350/32MB was used for the calculations.

-0.10 -0.05 0.00 0.05 0.10u

-1.0

-0.5

0.0

0.5

1.0

1.5

P

(a)(b) (c)

(d)(b)(c)(d)

(a) 4 elem.

(b) 6 elem.

(c) 10 elem.

(d) 20 elem.

Present - PTT

Figure 8: Result evolution - PTT.

(a)

A

D

C

B 1 2 0

A = 6.0

I = 2.0

E = 720.0

ν = 0.3

P

24 96

(b)

6 elem.

24 48 48

8 elem.

2432 32 32

(c)0 20 40 60 80 100 120 140

w

-1.0

-0.5

0.0

0.5

1.0

1.5

P

A

B

C

D

LIMIT POINTS

AFI 100 elem.:A (48.791 , 1.856 )

B (61.006 , 1.192 )C (50.749 , -0.438 )D (58.188 , -0.942 )

Figure 9: Lee frame.

Form. Element Number

06 08 10 20

Error % t (sec.) Error % t (sec.) Error % t (sec.) Error % t (sec.)

AFI 14.63 26.47 7.32 30.60 6.92 40.87 1.87 81.9

YGN 14.49 11.86 7.54 13.40 6.87 16.09 1.68 32.5

PTT 36.22 11.59 16.90 15.38 17.80 18.23 4.26 37.9

Table 2: Error and CPU analysis - AFI, YGN and PTT formulations.

Once more, the average error obtained for AFI and YGN formulations were similar, but the

AFI results showed a higher CPU time; again, a stiff behaviour was observed for the PTT, and

only with 20 elements good results were obtained; convergence problems after the first limit

point were observed for the PC1 and PC2 formulations and the results are not shown in Table




18

2. However, as shown in Figure 10a, PC2 produced better results up to the first limit point

load. In Figure 10b the performance of the YGE formulation for different load increments isanalyzed.

(a)35 40 45 50 55 60 65

w

1.6

1.7

1.8

1.9

2.0

P

YGN

PC1

PC2AFI 100 elem.

6 elem.

(b) 35 40 45 50 55 60 65

w

1.6

1.7

1.8

1.9

2.0

P

∆λ = 1001

-2

YGN 20 elem.

AFI 100 elem.

∆λ = 5 1001

-3

∆λ = 1001

-3

YGE 20 elem.

∆λ = 1001

-4

∆λ = 1001

-2

Figure 10: First load limit point results: a. PC2; b. YGE.

4.3 Deep circular arch

The performances of the non-linear formulations are now analyzed studying the behavior

of the hinged circular arch shown in Figure 11a. It is assumed here that the load is displaced a

small distance from the apex node (π/50) producing a small load imperfection.

(a)

w

uθ

P

π/50

L =100

E = 2000

G = E/2

A = 10

I = 1

(b)0 15 30 45 60 75 90

w

-75

-50

-25

0

25

50

75

100

P

Imperfect Arch

Yang and Kuo

Complete archYGN 26 elem.

2

Present work Complete archYGN 26 elem.

(c)

3 L.P.o

1 L.P.o

2 L.P.o

Imperfect arch

.

Figure 11: Hinged deep circular arch – imperfect model.

This problem exhibits substantial geometric non-linearity with the equilibrium path

showing a complex looping behavior (see Figure 11b). The arch deformations associated with

first three load limit points are shown in Figure 11c. These figures were obtained employed




19

the YGN formulation and the mesh with 26 finite elements. The results obtained from Yang

and Kuo’s book

2

are used here as reference.Table 3 gives the values of the limit point loads calculated through the non-linear

formulations here studied. These results were obtained with a mesh of 26 finite elements.

Table 3 also shows the CPU times and the Yang and Kuo limits point load values. Again the

values obtained with the AFI and YGN formulations were close to each other, but CPU time

was rather different; PTT scheme was able to generate the complete load-deflection curve, but

after the first limit point load the results disagree with those reported by Yang and Kuo.

Again, PC2 formulation presents good results up to the first load limit point but due to

convergence problems, this non-linear approach wasn’t able to trace the complete equilibrium

path.

Form. 1o L.P. 2o L.P. 3o L.P. 4o L.P.. 5o L.P. 6o L.P. 7o L.P. 8o L.P. 9o L.P. t (sec.)

AFI 5.811 -8.484 16.116 -22.151 38.404 -49.783 64.414 -81.341 103.882 5798

YGN 5.811 -8.483 16.113 -22.152 38.391 -49.790 64.345 -82.189 103.679 1554

PTT 5.856 -8.681 16.869 -23.439 42.359 -56.512 78.582 -101.54 139.447 3535

PC2 5.795 --- --- --- --- --- --- --- --- ---

Ref 2. 5.813 -8.498 16.149 -22.162 38.566 -49.896 64.875 -82.42 104.611

Table 3: Error and CPU analysis.

5 CONCLUSIONS

In this paper geometrically non-linear finite element formulations proposed by Alves1,

Yang and Kuo2, and Pacoste and Eriksson3 were studied and implemented in the non-linear

finite element program developed by Silveira5, which includes several options of load

increment and iteration strategies. Special attention was given to the different procedures

available to obtain the stiffness matrix K and internal load vector Fi. Three examples

displaying complex geometrically non-linear behaviour including snap-through and snap-back

characteristics were used to verify the computational performance of these formulations.

Based on these examples, the performance of these non-linear formulations can be

summarized as follows: (i) AFI and YGN, which are based on a updated Lagrangian scheme

and obtain the internal load vector using natural displacements, exhibit virtually identical

results but different CPU time. The complete expression of the Green-Lagrange strain

increments utilized by Alves seems to be unnecessary and perhaps explain this difference; (ii)YGN showed very good performance in all the problems here analyzed. This formulation also

presents smaller CPU time; (iii) YGE, which obtain the internal load vector through an

external stiffness matrix, displayed a dependence on the average value of the load increment.

However, this technique seems to be more general and can be employed in the analysis of

other types of structures; (iv) PC1 and PC2, that are based on total Lagrangian and co-

rotational approaches, give excellent results but they presented convergence problems in two

examples. CPU time for these formulations was greater than for the others; the authors believe

that additional computational procedures should be used to make these approaches more




20

efficient; (v) PTT, different from PC1 and PC2, was able to trace the whole equilibrium paths.

However, a very dense mesh was necessary in all examples to obtain good results. The linear interpolations employed and the non-consideration of the rigid body effects may explain the

weak performance of the PTT formulation.

Acknowledgements

The authors are grateful for the financial support from USIMINAS, CAPES, and CNPq.

REFERENCES

[1] R.V. Alves, Non-linear elastic instability of space frames, D.Sc. Thesis, COPPE-Federal

University of Rio de Janeiro, (1995) (in Portuguese).

[2] Y.B. Yang and S.B Kuo, Theory & analysis of nonlinear framed structures, Prentice Hall,(1994).

[3] C. Pacoste and A. Eriksson, “Beam elements in instability problems”, Comput . Methods

Appl. Mech. Engrg ., 144, p. 163-197, (1997).

[4] M.A. Crisfield, Non-linear finite element analysis of solids and structures, John Wiley &Sons, Vol 1, (1991).

[5] R.A.M. Silveira, Analysis of slender structural elements under unilateral contact

constraints, D.Sc. Thesis, Catholic University, PUC-Rio, (1995) (in Portuguese).

[6] P.C. Olsen, “Rigid plastic analysis of plane frame structures”, Comput . Methods Appl.

Mech. Engrg ., 179, p. 19-30, (1999).

[7] W.S. King, “The limit loads of steel semi-rigid frames analyzed with different methods”,

Computers & Structures, 51, No 5, p. 475-487, (1994).

[8] M.J. Clarke and G.J. Hancock, “A study of incremental-iterative strategies for non-linear

analyses”, Int. J. Numer. Methods Eng ., 29, 1365-1391, (1990).

[9] K.H. Schweizerhof and P. Wriggers, “Consistent linearization for path following methods

in nonlinear FE analysis”, Comp. Methods Appl. Mech. Eng., 59, p. 269-279, (1986).

eccomas2000_barcelona

Documents