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Procedia Engineering
ProcediaProcedia Engineering 00 (2011) Engineering 29 000–000 (2012) 3703 – 3707 www.elsevier.com/locate/procedia
2012 International Workshop on Information and Electronics Engineering (IWIEE)
Three-Node Euler-Bernoulli Beam Element Based on Positional FEM Liu Jiana*,b, Zhou Shenjiea, Dong Meiling c, Yan Yuqin c a
School of Mechanical Engineering, Shandong University, Jinan 250061, PR China b School of Science, Shandong Jianzhu University, Jinan 250101, PR China c School of Mechanical & Electronic Engineering, Shandong Jianzhu University, Jinan 250101,PR China
Abstract Based on positional FEM, three-node Euler-Bernoulli beam element for large deflection 2D frame analysis is researched. Solution strategy of geometric nonlinear static analysis with three-node Euler-Bernoulli beam element is introduced to the frame structures, and the program flow chart is given, then Matlab language is used to compile program. Simple example is shown at the end of the paper, comparing the numerical results achieved with the analytical and other numerical solutions found in the literature.
© 2011 Published by Elsevier Ltd. Selection and/or peer-review under responsibility of Harbin University of Science and Technology Open access under CC BY-NC-ND license. Key words: positional FEM; geometric nonlinearity; three-node Euler-Bernoulli beam element
1. Introduction Whether static or dynamic non-linear analysis for frame structures by FEM (finite element method), the calculation precision depends, to a great extent, on the element types (Belytschko et al. 2000). In the past geometric non-linear analysis, traditional two-node beam element is often used (Crisfield 1991; Clough 2004). Ref. [4] (Coda and Greco 2004) presents a simple formulation to treat large deflections by FEM based on position description. The simplicity of the resulting formulation may be considered as its main attribute. But get a more satisfactory accuracy, it must need more element number, and the different calculation result of different element number indicates that it has an important influence on the result accordant to the practical situation to choose the element number properly. So it is necessary to search a
* Corresponding author. Tel.: +86-13791027226 E-mail address:
[email protected].
1877-7058 © 2011 Published by Elsevier Ltd. Open access under CC BY-NC-ND license. doi:10.1016/j.proeng.2012.01.556
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Liuname Jian et al. / Procedia Engineering 29 (2012) 3703 – 3707 Author / Procedia Engineering 00 (2011) 000–000
new beam element type that has the following characteristics: higher precision and less divided element number. 2. Three-node Euler-Bernoulli beam element and strain calculation The central line general geometry of a curve over a plane (the accepted geometric approximation) is mapped in Figure 1 (a). The curve of Figure 1 (a) can be parameterized as a function of a nondimensional variable ξ (varying from 0 to 1). In this study a square approximation for position in x direction and a quintic approximation for y direction are imposed. Center node is node 3, now ξ = 0.5 , one writes x = X 1 + c1ξ + c2ξ
2
(1)
y =Y1 + e1ξ + e2ξ + e3ξ + e4ξ + e5ξ 2
3
where = lx X 2 − X 1 ，
4
5
(2)
l= Y2 − Y1 . y
In this study the physical reference configuration is a straight frame bar, as follows: ly lx X 3 − X1 = ， Y3 − Y1 = 2 2 with c1 = lx , c2 = 0 ,
(3)
e1 = l x tan θ1 , e2 =-6l x tan θ1 - l x tan θ 2 - 8l x tan θ 3 + 15l y ,
e3 = 13l x tan θ1 + 5l x tan θ 2 + 32l x tan θ 3 - 50l y , e4 =-12l x tanθ1 -8l x tanθ 2 -40l x tanθ 3 +60l y e5 =4l x tanθ1 +4l x tanθ 2 +16l x tanθ 3 -24l y
Applying Eq. (1), for central line one calculates = x X 1 + l xξ
(4)
As Euler-Bernoulli hypothesis is adopted, only the longitudinal strain is considered. Imagine a fiber parallel to the central line with an initial length defined by ds0 . After deformation, its length becomes ds and the following non-linear engineering strain referred to the non-dimensional space represented here by variable ξ is defined (Ogden 1984):
εe =
ds − ds 0 ds 0
=
ds / dξ − ds 0 / dξ
(5)
ds 0 / dξ
In the initial configuration for the central line passing trough the mass center of the bar, one has ds
0
( dx
dξ =
dξ ) + ( dy0 dξ ) = 2
0
2
(l ) + (l ) 0
2
0
x
y
2
= l0
(6)
where l0 is the initial length of the finite element. A general configuration, for any instant, is described by the approximation defined in Figure 1 (a). For this case the central line auxiliary stretch in computed as ds dξ
central
=
( dx
dξ ) + ( dy dξ ) = 2
2
(
l x + e1 + 2e2ξ + 3e3ξ + 4e4ξ + 5e5ξ 2
2
3
4
)
2
(7)
Substituting Eqs. (6) and (7) in Eq. (5) results
ε central=
1
⎡l x 2 + ( e1 + 2e2ξ + 3e3ξ 2 + 4e4ξ 3 + 5e5ξ 4 ) ⎤ − 1 ⎦ l0 ⎣ 2
1/ 2
(8)
Liu Jian et al. // Procedia – 3707 Author name ProcediaEngineering Engineering29 00(2012) (2011)3703 000–000
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Following usual engineering procedures, for Euler-Bernoulli (originally straight) frame element, the strain at a fiber which is z distant from the central line can be written as = ε e ε central + z r (9) where 1 / r is the exact curvature of the central line depicted in Figure 1(a). This curvature is given as a function of ξ only, as one can see in (Wylie and Barrett 1995): dx d y ⎛ ⎛ dx
2 2 ⎞ ⎛ dy ⎞ ⎞ ⎟ +⎜ ⎟ ⎟ 2 ⎜⎜ dξ dξ ⎝ ⎝ dξ ⎠ ⎝ dξ ⎠ ⎠ 2
1 r
−3/ 2
(10)
Replacing the known approximations, i.e. Eqs. (2)-(4), Eq. (10) becomes
(
)
3
1 2 3 2 2 3 4 2 = l x 2e2 + 6e3ξ + 12e4ξ + 20e5ξ l x + e1 + 2e2ξ + 3e3ξ + 4e4ξ + 5e5ξ (11) r Placing Eqs. (8), (11) and (9) together, and remembering that the initial configuration is a straight element, one has nonlinear engineering strain ε e , corresponding to the engineering stress σ e .
(
)
(
)
3. The simple positional formulation
Adopting linear constitutive relation for hyper-elastic materials, the strain energy can be written for the reference volume V as 0
= Ue
1
dV ∫ = u σ ε dV ∫ ∫= 2
1
ε e dV0 2
(12) 2 In order to calculate the strain energy it is necessary to integrate the specific strain energy ue( espec ) over the initial volume of the analysed body, as the proposed strain measure is, by nature, a Lagrangian variable. Substituting Eq. (9) into Eq. (12) results in e (espec)
V0
0
V0
e
e
0
V0
2 2 E⎛ 1 1 ⎞ ⎞ 2 ⎞ ⎛ (13) ue( espec= ⎜ ε central + z ⎟= ⎜ ( ε central ) + 2ε central z + ⎜ z ⎟ ⎟ ) 2⎝ r ⎠ 2⎝ r ⎝r ⎠ ⎠ It is necessary to integrate the specific strain energy ue( espec ) , Eq. (13), over the bar volume. Integrating
E⎛
1
it over the cross-section area results in 2
EI ⎛ 1 ⎞ (14) ⎜ ⎟ central 2 2 ⎝r⎠ Now one integrates the strain energy per unit of length, Eq. (14), along the original length of the bar, i.e.
ue =
EA
(ε
)
2
+
2 1 ⎡ EA EI ⎛ 1 ⎞ ⎤ 2 Ue = ∫ ⎢ ( ε central ) + ⎜ ⎟ ⎥ l0 dξ = l0 ∫0 ue dξ 0 2 ⎝r⎠ ⎦ ⎣ 2 1
The potential energy of applied conservative concentrated forces is written as W= Fx1 X 1 + Fy1 + M 1θ1 + Fx 2 X 2 + Fy 2Y2 + M 2θ 2 + Fx 3 X 3 + Fy 3Y3 + M 3θ 3
(15)
(16)
or W = Fi X i
(17)
where Fi represents forces (or moments) applied in i direction and X i is the i th coordinate parameter of the point where the load is applied.
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Liuname Jian et al. / Procedia Engineering 29 (2012) 3703 – 3707 Author / Procedia Engineering 00 (2011) 000–000
The total potential energy is written as = ∏ l0
∫ u dξ − F 1
x1
e
0
X 1 − Fy 1 − M 1θ1 − Fx 2 X 2 − Fy 2Y2 − M 2θ 2 − Fx 3 X 3 − Fy 3Y3 − M 3θ 3
(18)
Figure1: (a) Curve in a 2D space; (b) Flow chart of the nonlinear analysis for beam structures; (c) Relation between the applied bending moment and the rotation at the loaded end. where ( X 1 , Y1 , θ1 , X 2 , Y2 , θ 2 , X 3 , Y3 , θ 3 ) are nodal positions and their conjugate forces are
(F
X1
, FY 1 , M 1 , FX 2 , FY 2 , M 2 , FX 3 , FY 3 , M 3 ) . As there is no singularity in the strain energy integral one
can derive Eq. (18) regarding nodal positions. The next equation shows this procedure for position X 1 . ∂∏ = l0 ∂X 1
∫
1
0
∂u e ∂X 1
dξ= − Fx 1 0
or in a compact notation: ∂∏ = g i ( p= fi ( p j ) −= Fi 0 j ) ∂pi
(19)
(20)
where pi is a generalized parameter and indices are related to nodal positions by
(X
1
, Y1 , θ1 , X 2 , Y2 , θ 2 , X 3 , Y3 , θ 3 ) = (1, 2, 3, 4, 5, 6, 7, 8, 9 ) 。
In a vector representation one has g ( p, F ) = 0 f ( p) − F = 0
(21) (22)
Liu Jian et al. // Procedia – 3707 Author name ProcediaEngineering Engineering29 00(2012) (2011)3703 000–000
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The vector function g ( p ) is non-linear regarding nodal parameters ( p and F ). To solve Eq. (21) one can use the Newton-Raphson procedure, flow chart of the nonlinear analysis for beam structures is as figure 1 (b). A, E, I and L represent cross-sectional area, Elastic Modulus, inertial moment, cantilever beam length respectively. ElementNum, eps, IterMax respectively represent total structure element number, convergence standard and maximum iterating times (when iteration number more than this, the end). n and i are iteration times and element number. X and F are ( i + 2 ) × 3 matrixes, respectively represent position and force vector. Norm (.) is the calculation vector norm. 3. Numerical example
The numerical example is an Euler beam initially horizontal, clamped at one end and subjected to an applied moment at the other. It has been presented in Ref. [4] (Coda and Greco 2004). The adopted 2 6 2 4 physical properties are: L = 500cm , A = 20cm , E= 2 × 10 N / cm and I = 2000cm . Six load steps and two finite elements were used to run this problem until an entire lap of the geometry. Some important values are compared with Ref. [4] (Coda and Greco 2004). In Figure 1 (c) results for this study and Ref. [4] (Coda and Greco 2004) values are compared with the analytical solution for bending moment/rotation relation. One can see that the global structure usually needs to be divided into more divided element number by using traditional two-node beam element than three-node beam element to achieve the same precision. More divided element number means order number of system balance equations increased twofold; it must bring a lot of difficult to solve. Therefore, the precision of three-node beam element is much higher than one of two conventional two-node beam elements in nonlinear analysis. 4. Conclusions
In geometry nonlinear static and dynamic analysis of Structures that exhibit large displacement, only simply increasing divided element number must bring a lot of solving difficult, and cannot solve all problems, especially in nonlinear dynamics. Although three-node increased computational complexity of element level, the computers’ treatment polynomial function efficiency is very high, and in the overall level, not increased amount of calculation. Therefore, three-node beam element has a great advantage over two-node one, and this article provides a new solving method to the large displacement geometry nonlinear static and dynamic analysis. References [1] Belytschko T, Liu W K, Moran M. Nonlinear finite elements for continua and structures. New York: John Wiley & Sons Ltd, 2000 [2] Crisfield M A. Non-linear Finite Element Analysis of Solids and Structures - Volume 1: Essentials. Chichester, UK: John Wiley & Sons, 1991 [3] Clough R W. Speech by professor Clough R W early history of the finite element method from the view point of a pioneer. International Journal for Numerical Methods in Engineering. 2004, 60(1): 283-287. [4] Coda H B, Greco M. A simple FEM formulation for large deflection 2D frame analysis based on position description. Computer Methods in Applied Mechanics and Enginee ring. 2004, 193(1): 3541-3557. [5] Ogden R W. Non-linear Elastic Deformation, Ellis Horwood, England, 1984. [6] Wylie C R , Barrett L C. Advanced Engineering Mathematics, sixth ed., McGraw-Hill, New York, 1995.
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