Calculus of the Elastic Properties of a Beam Cross-Section

Presented at the COMSOL Conference 2009 Milan Calculus of the Elastic Properties of a Beam Cross-Section Dipartimento di Modellistica per l Ingegneria Università degli Studi della Calabria (Ital) COMSOL Conference Milan 2009

Subjects Introduction The classical linear elastic solutions, as the Saint-Venànt one for slender bodies, can be exploited to generate frame invariant nonlinear modelings due to the recent proposal of the Implicit Corotational Method.

Subjects Introduction The classical linear elastic solutions, as the Saint-Venànt one for slender bodies, can be exploited to generate frame invariant nonlinear modelings due to the recent proposal of the Implicit Corotational Method. Implicit Corotational Method (ICR) ICR splits the motion of the neighbor of each continuum point (or of the cross-section in the case of beam structures) in a rigid part filtered through a change of observer a strain part considered small and described using linear solutions

Subjects Introduction The classical linear elastic solutions, as the Saint-Venànt one for slender bodies, can be exploited to generate frame invariant nonlinear modelings due to the recent proposal of the Implicit Corotational Method. Implicit Corotational Method (ICR) ICR splits the motion of the neighbor of each continuum point (or of the cross-section in the case of beam structures) in a rigid part filtered through a change of observer a strain part considered small and described using linear solutions The aims of this work are the calculus of the elastic properties of a beam cross-section through the numerical solution of Saint-Venànt problem as proposed in [2]; the use of the coefficients so evaluated in geometric nonlinear analsis of 3D beam structures as proposed in [1].

Stress distribution Cross-section flexibilit matrixes Shear sstem The stress distribution Saint-Venànt clinder M z x N G M x 0 T x M 0 z M l l T T T x M x l N M z

Stress distribution Cross-section flexibilit matrixes Shear sstem The stress distribution Saint-Venànt clinder M z M x 0 l N T x G z M 0 T x M l T T x N M z M x l the clindrical isotropic bod is subjected to surface loading on its end sections the local barcentric Cartesian sstem is oriented according to the principal directions of the cross-section Saint-Venànt assumptions impl σ xx = σ = τ x = 0.

Stress distribution Cross-section flexibilit matrixes Shear sstem The stress distribution Saint-Venànt clinder M z M x 0 l N T x G z M 0 T x M l T T x N M z M x l the clindrical isotropic bod is subjected to surface loading on its end sections the local barcentric Cartesian sstem is oriented according to the principal directions of the cross-section Saint-Venànt assumptions impl σ xx = σ = τ x = 0 The not zero stress components σ zz = D σ t σ, τ = { τxz τ z }. = D τ t τ

Stress distribution Cross-section flexibilit matrixes Shear sstem The stress distribution where N T x t σ = M x[z] tτ = T M [z] M z D σ = [ ] 1/A, /J x, x/j D τ = [ ] d x, d, d z d x = ψ x b x r xd z d = ψ b r d z d z = ( ψ z b z)/r z b x = 1 2J { } x 2 ν 2 0 { } b = 1 0 2J x 2 νx 2 { } b z = 1 2 x r j = 2 {b j ψ j} T b zda, A j = x,, z.

Stress distribution Cross-section flexibilit matrixes Shear sstem The stress distribution where N T x t σ = M x[z] tτ = T M [z] M z D σ = [ ] 1/A, /J x, x/j D τ = [ ] d x, d, d z d x = ψ x b x r xd z d = ψ b r d z d z = ( ψ z b z)/r z N, T x, T : normal and shear forces, M x[z], M [z], M z: bending couples and torque, A, J x and J : cross- section area and inertia moments, : gradient operator in the {x,} plane. b x = 1 2J { } x 2 ν 2 0 { } b = 1 0 2J x 2 νx 2 { } b z = 1 2 x r j = 2 {b j ψ j} T b zda, A j = x,, z. ν = ν/(1 + ν) depends on the Poisson coefficient ν of the material.

Stress distribution Cross-section flexibilit matrixes Shear sstem The stress distribution The Neumann-Dini problems Functions ψ j have to satisf the Laplace equation for three different Neumann boundar conditions: 2 ψ j x + 2 ψ j = 0, on A 2 2 ψ j ψj nx + x n = bt j n, on Γ n = {n x, n } T is the external normal to the cross-section contour Γ.

Stress distribution Cross-section flexibilit matrixes Shear sstem The stress distribution The Neumann-Dini problems Functions ψ j have to satisf the Laplace equation for three different Neumann boundar conditions: 2 ψ j x + 2 ψ j = 0, on A 2 2 ψ j ψj nx + x n = bt j n, on Γ n = {n x, n } T is the external normal to the cross-section contour Γ. Remark the axial and bending factors can be easil obtained the description of the torsional and shear behavior requires the solution of a sstem of partial differential equations

Stress distribution Cross-section flexibilit matrixes Shear sstem Cross-section flexibilit matrixes The complementar strain unitar energ The elastic properties of the cross-section are defined through the energ balance φ z = 1 σzz 2 da + 1 τ T τ da = 1 2E A 2G A 2E tt σh σt σ + 1 2G tt τ H τt τ, E where E and G = are the Young and the tangential elasticit 2(1 + ν) coefficients of the material, H σ = D T σd σ da, H τ = D T τ D τ da or, for components H σ = diag A [ 1 A, 1 J x, J A ] 1, H τij = d T i d j da. A

Stress distribution Cross-section flexibilit matrixes Shear sstem The shear and torsional factors A x, A, J t used in technical applications The shear sstem In the case of smmetric cross-sections, H τ is a diagonal matrix: [ ] 1 1 1 H τ = diag,,. A x In the hpothesis of general cross-section the shear unitar energ is composed of three independent contributions onl in the shear principal sstem defined b A J t x c = Hτ23 H τ33, c = Hτ13 H τ33, tanα t = c2 c1 + c 2 2 2c1c2 + c2 1 + 4c2 3 2c 3, c 1 = H τ11 2H τ13 c + H τ33 2 c, c 2 = H τ22 + 2H τ23 x c + H τ33 x 2 c, c 3 = H τ12 + H τ13 x c H τ23 c H τ33 x c c.

Stress distribution Cross-section flexibilit matrixes Shear sstem The shear and torsional factors A x, A, J t used in technical applications The shear sstem Let s denote with T x = T x c + T s, T = T x s + T c, Mz = M z + T x c T x c the shear strengths and the torque in shear sstem, c and s being the cosine and the sine of the angle α t. The energ is then where A x = φ T z = 1 { T x 2 2G A x + T 2 A + M 2 z J t 1 c 1 c 2 + c 2 s 2 + 2c 3 c s, 1 A = c 1 s 2 + c 2 c 2 2c 3 c s, Jt = 1 H 33 are the necessar shear and torsional factors. },

Calculus of the shear flexibilit matrix through COMSOL Multiphsics The modulus COMSOL Multiphsics-PDE The following quantities are defined: A 2D geometr in which x and are the independent variables; ν and ν as constants; The three terms b T j n as Boundar Expressions; Quantities useful to obtain the area, the inertia moments and the expressions of r j between the Integration Coupling Variables on the domain; The components of vectors d j and the dot products d T i d j on the domain; The three differential Laplace equations for the domain and the Neumann conditions for the contour; A mesh for the domain using Lagrange-Quadratic elements. Having found the ψ j functions, the Postprocessing menu is ver useful to evaluate the integrals H τij.

Validation tests Comparison tests Validation tests Rectangular and Trapezoidal cross-sections 2 3 d2 G x 1 2 3 x x C G ν = 0 ν = 0.3 ν = 0.5 k x 1.2000 1.2748 1.3561 k 1.2000 1.2006 1.2012 k t 1.8220 The numerical values of k x = A, k A = A, k x A t = Jp J t are in perfect agreement with the available analtical values: k x = k = 6/5 for ν = 0 k t = 1.8220 for an ν. J p = A (x2 + 2 )da is the polar inertia. ν = 0 ν = 0.3 ν = 0.5 k x 1.3468 1.3771 1.4100 k 1.1841 1.1856 1.1871 k t 1.6481 x c 0.0091 c 0.2429 α t (rad) -0.0996-0.0847-0.0729 d1 The flexural sstem indicated in red is defined b d 1 = 1.8571, d 2 = 1.2857, α = 1.1695rad. It does not coincide with the shear sstem indicated in blue. Numerical results agree with those proposed in [2].

Validation tests Comparison tests Validation tests C-shaped thin walled section and Bridge section 10 90 G x C x ν = 0 ν = 0.3 ν = 0.5 k x 2.8614 2.8716 2.8826 k 2.2693 2.2693 2.2693 k t 59.0380 x c 30.3727 10 d 45 10 The barcentric sstem is defined b d = 37.6250. 0.30 0.60 7.60 0.75 0.45 G x C 3.30 x d 2.50 0.30 3.65 4.50 3.75 3.50 2.00 3.45 ν = 0.3 k x 1.6686 k 4.3291 k t 4.4489 c -0.5863 The barcentric sstem is defined b d = 2.1552.

Validation tests Comparison tests Comparison tests C-shaped beam under axial force COMSOL was used in performing the calculus of the compliance operators H σ and H τ for the C-shaped section. The analsis was performed using the codes proposed in [1] and through ABAQUS where the cantilever was modeled as a plate-assemblage. 200 200 160 120 Beam Abaqus 160 120 80 40 x z 80 40 h = 7 G t = 1 b = 3 x 0-40 -30-20 -10 0 u z Geometr and equilibrium path. Axial displacement u z at the edge of the beam 0 0 20 40 60 80 u x Geometr and equilibrium path. Lateral displacement u x at the edge of the beam

Validation tests Comparison tests Comparison tests C-shaped beam under shear force In this case too the equilibrium paths obtained through the beam model agree with those furnished b ABAQUS. 120 120 80 Beam Abaqus 80 40 x z 40 h = 7 G t = 1 x b = 3 0-6 -4 u z -2 Geometr and equilibrium path. Axial displacement u z at the edge of the beam 0 0 0 4 8 12 16 20 u x Geometr and equilibrium path. Lateral displacement u x at the edge of the beam

Conclusion Conclusion ICR allows to generate a nonlinear 3D beam model that contains information gained from Saint-Venànt solution; The possibilit to evaluate the beam section elastic compliance matrixes in a simple wa and reuse these quantities gives a series of advantages in nonlinear analsis of 3D beam structures with respect to the classical beam models; In [1] a mixed 3D beam element is proposed. The numerical results agree with those furnished b more complex structural modelings such as the shell one.

Conclusion Conclusion ICR allows to generate a nonlinear 3D beam model that contains information gained from Saint-Venànt solution; The possibilit to evaluate the beam section elastic compliance matrixes in a simple wa and reuse these quantities gives a series of advantages in nonlinear analsis of 3D beam structures with respect to the classical beam models; In [1] a mixed 3D beam element is proposed. The numerical results agree with those furnished b more complex structural modelings such as the shell one. References G. Garcea, A. Madeo, R. Casciaro The implicit corotational method, Part I/II, submitted to Computer Methods in Applied Mechanics and Engineering A. S. Petrolo R. Casciaro 3d beam element based on Saint-Venànt rod theor, Computers & Structures, 82, pp. 2471-2481, (2004)