Theory & Results Louie L. Yaw, Sashi Kunnath and N. Sukumar University of California, Davis Department of Civil and Environmental Engineering Minisymposium 47 Recent Advances in Modeling of Engineering Materials/Systems USNCCM9 San Francisco July 24, 2007
Acknowledgements Funding provided by Walla Walla College N. Sukumar acknowledges research support of NSF (Grant CMMI-0626481) Helpful discussions with Dr. Michael Puso, Lawrence Livermore National Laboratory Helpful discussions with Professor Boris Jeremic, UC Davis
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Motivation Goal to advance collapse simulation technology Current FE technology unsatisfactory for large deformations at collapse limit states To explore feasibility of meshfree approach
Outline Motivation 1 Motivation 2 3 4 5
MLS Shape Functions Derivation Start with displacement approximation u h (x) = n φ a (x)d a φ T d a=1 Shape function φ a is of the form (Belytschko et al 1996) φ a (x) = P T (x a )α(x)w(x a ), where P(x) = {1 x y} T is a linear basis in two dimensions, α(x) is a vector of unknowns to be determined and w(x) 0 is a weighting function
MLS Shape Functions Derivation The φ s must satisfy reproducing conditions P(x) = n P(x a )φ a (x) a=1 Substitution of φ a into P(x) and solving for α yields α(x) = A 1 (x)p(x) Finally, substituting α into φ a gives φ a (x) = P T (x a )A 1 (x)p(x)w(x a ), where A = n a=1 P(x a)p T (x a )w(x a )
Enforcing boundary conditions MLS (meshfree) shape functions do not have the Kronecker-delta property Hence MLS shape functions are blended with quadrilateral FE shape functions at essential B.C. s Then essential boundary conditions are enforced on the finite element nodes in the standard way The blending technique proposed by Huerta and Fernández-Méndez (2004) is adopted
1 Weigth function for node 5 1 1D MLS shape functions 0.8 w 5 0.8 Shape Function Node 5 Weight function 0.6 0.4 0.2 Shape functions 0.6 0.4 0.2 0 Radius of support ρ = 0.3125 0-0.2 0 0.2 0.4 0.6 0.8 1 (a) -0.2 0 0.2 0.4 0.6 0.8 1 (b) 1.2 1D Blending example 1 Shape Function 0.8 0.6 0.4 0.2 0 MLS Shape Functions Blended FE Shape Region Functions -0.2 0 0.2 0.4 0.6 0.8 1 x
Nodal Integration Smoothed strain tensor for node a (per Chen et al (2001)) ε ij (x a ) = 1 (u i,j + u j,i ) dv = 1 (u i n j + u j n i ) ds 2A a V a 2A a S a Strain-displacement relation ε(x a ) = 6 B b (x a )d b Bd b=1
Nodal Integration Strain-displacement definitions ε = [ε 11 ε 22 2ε 12 ] T and d a = [d a1 d a2 ] T b b1 (x a ) 0 B b (x a ) = 0 b b2 (x a ) b b2 (x a ) b b1 (x a ) 3 x 2 2 V a a = 1 4 n x 1 S a 6 5 b bi (x a ) = 1 A a S a φ b (x)n i (x) ds Nodally integrated stiffness matrix K bc = n B T b (x a)cb c (x a )A a t a=1
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Stabilization of Nodal Integration Nodal integration w/o stabilization leads to a. hourglass modes b. spurious low energy modes c. and locking Following Puso and Solberg (2006) stabilization is provided to the stiffness matrix as follows: K s = (1 α s )K MLS + α s K FE, where K s is the stabilized matrix and α s = 0.05 is called the stabilization factor
Nonlinear analysis Loads applied incrementally At global level a Newton-Raphson scheme is used to iterate the linearized system of equations until equilibrium is achieved K t(ν) (ν) n+1 d n = f ext n+1 f int(ν) n+1 At the constitutive level for J2 plasticity a radial return scheme is used (Simo and Hughes (1998))
Meshfree analysis of wide-flange steel sections Based on the meshfree nodal discretization of a beam a Voronoi diagram is generated. A thickness is specified for each Voronoi cell. To get wide-flange behavior a web thickness and a flange thickness is specified.
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Results for normalized tip displacement and maximum bending stress, where δ theor. = 0.0308 in and σ theor. = 25.0 ksi. Grid δ/δ theor. (in) σ xx /σ theor. (ksi) 11 3 1.045 0.77 21 5 1.025 0.89 31 7 1.016 0.94 41 9 1.012 0.95 51 11 1.010 0.96 61 13 1.009 0.97 P
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Elasto-plastic cantilever I-beam J2 plasticity with linear hardening 12 10 σ P (kips) 8 6 4 σ m σ y E ε y H ε m ε P 2 Meshfree Analytical 0 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 Tip Displacement δ (in)
Frame corner connection: Load deflection response 150 125 P 11 ft 3 in Load P (kips) 100 75 50 25 Deflection Gage 11 ft 3 in P Experimental (Beedle 1964) Meshfree - Exponential Hardening Meshfree - Linear Hardening 0 0 2 4 6 8 Displacement δ (in)
Frame corner connection: Displacement and stress
Inelastic frame analysis 12 10 8 8 ft P δ P P (kips) 6 8 ft 4 16 ft 2 Experimental (Baker 1952) MLS Shape Functions Maxent Shape Functions 0 0 2 4 6 8 δ (in)
Demonstrated feasiblity of wide-flange beam analysis under plane stress A coupled FE and meshfree method shows promise for inelastic frame analysis Demonstrated success of Maxent shape functions Further research is ongoing to extend this for large deformations to enable collapse simulations.