non-linear finite element analysis for truss, frame

juuKHAL, u r a«„iK.NCE & TECHNOLOGY * No. 88 - 2012

NON-LINEAR FINITE ELEMENT ANALYSIS FOR TRUSS, FRAME STRUCTURES USING CO-ROTATIONAL APPROACH

WITH COMPATIBLE CONSTRAINT EQUATIONS PHAN TICH PHAN TU" H O U HAN PHI T U Y £ N CHO K^T CAU KHUNG,GIAN

SU' DUNG CACH TifeP CAN CO-ROTATION VOI CAC PHUONG TRINH RANG BUOC TU'ONG THICH

Dang Thien Ngon, Nguyen Hoai Son University Of Technical Education Ho Chi Minh City Received November 17,2011; accepted January 10.2012

ABSTRACT

TTie mosf commonly used formulations of non-linear analyses are the Total or Updated Lagrangian formulations, which have been incorporsted into many commercial finite element packages. However, for a special class of nonlinear problems, namely, large displacements and mtations but moderate strains, the Co-rotational formulation can be employed to further simplify the Lagrangian formulations A lot of iterative procedures for tracing the nonlinear equilibrium paths have been devised in nonlinear analysis. However in some cases, there are issues still existing so far, especially in geometric nonlinear problems possessing the snap-through and/or snap-back properties.

Thus, it has indeed improvement on the classical ones to achieve an effective solution. A novel constraint equation which is linear, simple and low cost of iteration compared with the classical am- length algorithm is suggested in this paper Several numerical examples are tested to verify the accuracy and pertormance of the proposed approach.

Keywords: Finite element analysis, arc-length method, co-rotational, nonlinear analysis, snap-through, snap-back, geometnc nonlinearity

TOM TAT

De phdn tich phi tuyen cac cong thirc Lagrange toan cue vd cap nhatthuang dugrc siJ" dgng.Cdc cong thirc nay cung da dugc kit hgp thanh cac goi (phin mim) phin tu- hdv han thuang m^i Tuy nhidn, ddi v&i l&p d$c bi0t cua cdc bai toan phi tuyin, eg thi nhu cac chuyen vi va goc xoay lan nhung biin dang trung binh. viic xdy dgng cong thuc xoaykit hgp (co-rotational) cd the dug^ su dgng di tiip tue dan giin hda cong thCrc Lagrange. Mqt lugng l&n cac phep tinh lap de can bdng phi tuyen da duac dua ra trong phan tich phi tuyin. Tuy nhien trong mqt sd tru&ng hgp mgt sd vin di vin con tdn tai cho den nay. d$c biet la trong cdc bdi toan hinh hgc phi tuyen co eac thudc tinh snap-through va snap-back. Vi the.cdc phep tinh tSp co thi cai thidn each giai eo dien de dat du%yc hidu qua: Mdt phucmg trinh rang bugc tuyin tinh, dan gian va chi phi thip cua phep tinh l$p Igi so v&i ede thudt todn arc-length cd dien difge de xuit trong bai bao nay. Mgt vai vi du sd duac trinh bay de xac minh tinh chinh xdc va hi$u qua cua phucmg phdp di xuat

1. INTRODUCTION introduced. However for stioictural systems . , _. . /• .. • .L exhibiting snap-back behavior, these lechniques

A lot of Iterative procedures for tracing the i j . -t- u. i ,. .... . '^ L L L lesd to error. To obtain a more general

nonlinear equilibrium paths have been . . , , . . . i • . • . j r , i 7 T w . , i T J . n J VI .. technique, the arc-length method, which is presented [1],[3],[11]. Load controlled Newton- ^ ' ^ Raphson method is one of Ihe oldest iterative

schemes, but it fails near the limit point (L), where after passing a limit point no static equilibrium existsbecause of load decrement,

originally developed by Riks (1972, 1979) and Wempner (1971), and later improved by Ramm (1981), Crisfield (1981, 1983) [II], is used.

Although all Ihe aforementioned methods have J? JT u J • 11 been demonstrated to be applicable for certain Therefore the structure dynamically snaps ,. , , ^ . • ., , . ^ , ^ ... , .,., . nonlinear problems, most of them in some cases through into the next post-critical equilibrium '^, , ^ -, , - t , ,

_ ..cr- ,. ., ,- •. are not stable, even tailed in tracing the load- position. To overcome difiiculties with limit , c • . . . r , . , . . . 1 . u • detlection curves containing limit points or high points, displacement control techniques were . , b t- e

computational cost.

JOURNAL OF SCIKNCK & TECHNOLOGY * No. 88 - 2012 2. KINEMATICS OF CO-ROTATIONAL

Consider co-rotational (CR) description as shown in Fig.l [3]: The reference configuration is "split". Strains and stresses are measured from the corotated configuration (C") whereas Ihe base configuration (C") is maintained as reference for measuring rigid body motions.

Corottted configuniion C- Defomied (ciurent)

configuntion C°

lotion splits into defomational and rigid Fig. I. The CR descriplion To analyze nonlinear problems Total Largrange (TL) or Update Largrange (UL) are the most commonly used formulations, which have been incorporated into many commercial finite element packages. The TL is applicable lo large displacement and rotation but small strain, otherwise tne UL is used for large strain situations [3]. However, for a special class of nonlinear problems, namely. large displacements and rotations but small strains, the CR formulation can be employed to further simplify the Lagrangian formulations. The attractiveness ofthe CR formulations is that it offers a nonlinear framework in which standard linear formulations can be used. The common linear theory is referred to a single element frame thai continuously rotates with the element and nonlinearity is introduced via Ihe rotation of this frame.

3. CLASICAL ARC-LENGTH METHOD In the standard arc-length method. Ihe equilibrium equations of nonlinear problems can be expressed as[I].

where; / is the internal force, u is the displacment, ^ is a load level parameter,9is a fixed total load vector

We reganj the load level as a variable, an extra governing cquaiion is required and is given by a constraint relationship in incremental form [1].

a = (Au'^Au + AAV^q''q)-A/'=0 (2) where Ihe vector Au and scalar A.^ an incremental and relate back to Ihc last converged equilibrium state, Ihe scalar if/ is Ihe scaling parameier. Fig. 2 shows a prescribed incremental arc length.

placemgt i

Fig. 2 Basic arc-length procedure and notatirm for Idofsystem (with i)/ ^\}

Following Riks and Wempner [7], we can solve for these variables by directly applying the Newton-Raphson method to Eq(l) and Eq(2).Assume «o is given and expand u around Uo. u„ =\x^-r (Vu so the truncated Taylor series of Eq( I) and Eq(2) with the subscript n meanii^

new and o meaning old leads to:

g ( u . A ) - f ( u ) - A q - 0 (i)

cw d2.

- K t5u -q(5/. = 0

(3)

JOURNAL OF SCIENCE & TECHNOLOGY • No. 88 - 2012

a„=ao-f2Au'5u + 2AA&l^^q''q=0 (4) In Ihe mabix form Eq(3) and Eq(4) can be rewritten as

[2A«' 2A^tc'q

-q

-Jfcl

⁽⁵⁾

where^ifand<$Ais the iterative change in displacement vector and load level parameter, respectively. Once the iterative change Su and S^

in Eq(5) have been computed, the displacement vector and load level parameter are updated.The above iterative procedure has not been put into practical use due to Ihe fact that Ihe quadratic constraint Eq{4) leads to poor iteration convergence and the augmented stifthess matrix in Eq(S) is not symmetric and banded as well. So instead of solving Eq(5), we may directly introduce the constraint of Eq(4) by following Batoz and Dhatt[IO] for displacement control at a single point. According to this technique, the iterative displacement,<5H, is split into two part.

The Newton change at the new unknown level load, /l, - ^o + ^^ become [ I ]:

6u = ~K:'[giu^J.^)-SAq\ = -K;% ^32,K~q = 5u-52.Su^ (6) wherefjff is the iterative change that comes from the standard load-controlled Newton Raphson method (at a fixed load level, A„) and^w, is the iterative displacement vector corresponding to the fixed load vector q .Once Su obtained Irom Eq(6), the new incremental displacements are A«„ = Aw„ +5u- A«„ + (?« + (5^ Su, (7) in which&l in Eq(7) now still is an unknown. On the other hand, Eq(2) can be rewritten as:

(AWJ^AH^-t-A/l;i^"q^q ) = ( A « J ^ A « „ + A ; i ; ^ ' q ' q )= A ( '

(8)

Substituting A«„ ft^om Eq(7) into Eq(8) leads to a scalar quadratic equation

a^SX-+a^Sl + a,=0 (9) where

u^ =Su[u, +>fr'q^q

c/j ={/iM^+mY(Au„+Su) -M- +AAjv/-q'"q

As a result both Crisfield and Ramm advocated settingv'=0. That kind of constraint is called

"cylindrical constraint" in [1]. The issue of finding an appropriate root to Eq(9) withv=0

• proposed by Crisfield [1]. If two real roots are obtained from Eq(8), Ihe correct solution is selected by computing and comparing both solutions, Au„, (that corresponds to SA^) and

Au„, that corresponds to Alj ).Then, the proper root would lay closest to the old incremental direction AUQ. This comparison can be made by finding the solution with Ihe minimum angle between A«„ and AH„ . If we denote $ as the angle between AH,, and Au,, Ihen the maximum cosine ofthe angle is

A«[Aa„

Afr

^.Ait'^Su.

<v.—^-T-=-q Ai'

(11)

4. CONSTRAINT EQUATION

In addition to many modified research on traditional arc-length method, here we present the novel constraint which is very simple, linear instead of quadratic, and easy and effective to implement. To this end, we assume that two converged equilibrium points reached at incremental load or displacement (n-I)"' and n"'as shown in Fig.3. Mathematically, we can introduce the constraint equation by the dot product of two vectors An and A«^ as following relationship.

A//_ AH = Af (12)

In (l2),Afrepresents a generalized step length for iteration from point n to (n+I) in R'"' space, which is prescribed in the computational implementation, A«^ is the displacement

JOURNAL OF SCIENCE & TECHNOLOGY * No. 88 • 2012 incremental for tracing a new equilibrium point

and Au represented the unit vector of last displacement incremental.

Fig. 3 Geometrical interpretation of new constraint

We can rewritten Eq (12) as:

( H - W J ' C „ + ( A - A , ) C „ . , =Af (13)

Eq(l3) is the new constraint equation using in computational procedure and obviously it possess Ihe linear form and it can be rewritten as

aitt.A)={u-u„Yc„

(14)

By introducing the truncated Taylor series with the subscript o meaning old values, (14) will be

a(u,A) = o(«„,AJ + r„d'u + C„.,(W=0 (15)

- - • • , : • 1 M 5 y 5 |-

' j B r i i o l d i - , r U ( e m » n T t i

The combination of Eq(3) and Eq(15) the Iterative change in displacement and load parameter is straightforward defined.

la-E ;:n::l

⁽¹⁶⁾

The iterative change in displacemait and load factor in Eq(t6) used for complete iteration procedure.

S. NUMERICAL EXAMPLES

5.*^ 2D Truss

The first of all we consider Ihe truss problem subjected two concentrated loads P , , Py as show in Fig.4. The truss members have the same uniform cross-section area A, and Young's modulus E, geometric parameters given by L = 13 (cm). EA = 1885 (N/cm), P. = 3000 (N), Py = 90 (N), v = 0.3. yg = 2 (cm). A load increment iteration of displacement convergence tolerance of 10^ was employed as Fig. 5.

Fig. 4. 2D Truss

1 Pieseni 1' . 1 A n s y i | : - • ,

\ , ^ \'--

' : :

13

5°^

3 0 4 02 0

1 PretM 1

: :

• " • " I " " " • • "

i/

*

Honzonlal di«plK«metil *

fig. 5 Load parameter, vertical and horizontal displacement of node 2

JOURNAL OF SCIENCE & TEC HNOLOGY • No. 88 - 2012 Fig. 5 shows the numerical responses

obtained from proposed solution procedure compared to those from ANSYS. A very good agreement is achieved for the case of y^ = 2cm as shown in Fig. 5 which demonstrates that our solution procedure is reliability. The proposed solution procedure can finish tracing on whole load-displacement path regardless of existing of special snap-back/snap-through on curves.

5.2 Beam

Consider a cantilever ofiai^e defiection subjected to a moment load M at its fi^e end with its span of L, bending^/ as shown in Fig.

6. wand vrepresent, respectively, Ihe horizontal displacement, vertical displacement at Ihe free end ofthe beam. The beam was computed by a mesh of 20 elements with /, = 20.0m, £/ = 10.0(l/m'), £4 = 10.0, M

=n-(N) respectively. A load increment iteration of displacement convergence tolerance of lO' vras employed.

Fig. 6. 4 cantilever beam: geometry and loading

Fig.7 shows the comparison of load- displacement curves of cantilever beam subjected to large rotation between the current solution with ANSYS. cleariy that a good agreement achieved. Fig. 8 shows the deformed configurations of cantilever beam corresponding to constant moments.

6. CONCLUSIONS

The main objective of this work was to study geometric nonlinear for truss, frame structures using Co-rotational approach, a novel constraint equation for arc-length method and give the numerical examples to demonstrate that our solution procedure is reliable, stability and effective in tracing equilibrium paths.

p:^;t

s 10 rt ai

Venicaf displacemeni u

Fig 7 Load parameter, vertical and horizonlal displacement at lis Fig 8. Deformed configuration free end for cantilever beam under

constant moment REFERENCES

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Meth.Engng. 19, 1269-1289(1983)

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JOURNAL OF SCIENCK & TECHNOLOGY * No. 88 - 2012

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Author's address: Dang Thien Ngon -Tel.: (+84)913 804 803. Email: ngondl'o hcmute.edu. vn University of Technical Education Ho Chi Minh City

No. I, Vo Van Ngan Sir., Thu Due District, Ho Chi Minh City