Tube : beam model : analytical

Tube : beam model : analytical Tube = simply supported homogeneous & isotropic beam Load = point load (man) & distributed load (wind) z a L I x = 1 [ BH 3 bh 3] ; I z = 1 [ HB 3 hb 3] 12 12 F y x z b, B q h, H 0 < y a : u z (y) = F [ (L a)y 3 a(2l a)(l a)y ] 6EI x L a y < L : u z (y) = F [ (L a)y 3 a(2l a)(l a)y L(y a) 3] 6EI x L 0 y < L : u x (y) = q [ 1 12EI 2 y4 + Ly 3 1 2 L3 y ] z () 2 / 59

Tube : beam model : deformation and stress L = 10 [m] ; B = 1 [m] ; H = 2 [m] ; D = 0.01 [m] E = 20 GPa ; a = 0.75L ; F = 5000 [N] ; q = 500 H [N/m] 0 x 10 4 1 2 disp [m] 3 4 5 6 0 2 4 6 8 10 y [m] F q σ x,max = 0.29 [MPa] ; σ z,max = 0.56 [MPa] () 3 / 59

Tube : beam model in MARC/Mentat element 52 point load & global load 0 x 10 4 1 2 disp [m] 3 4 5 6 0 2 4 6 8 10 y [m] () 4 / 59

Tube : shell model Inc: 0 Time: 0.000e+00 Y job1 Z X 1 () 5 / 59

Tube : shell model Tube = simply supported homogeneous & isotropic shell Load = point loads on top & face loads (wind) Inc: 0 Time: 0.000e+00 4.848e-03 3.223e-03 1.598e-03-2.724e-05-1.652e-03-3.277e-03-4.902e-03-6.527e-03-8.152e-03-9.777e-03-1.140e-02 Y job1 Displacement Y Z X 1 () 6 / 59

Tube : shell model vs beam model disp [m] 0.01 0 0.01 0.02 0.03 0.04 0.05 disp [m] 0 x 10 4 1 2 3 4 0.06 0.07 0.08 0 2 4 6 8 10 y [m] 5 6 0 2 4 6 8 10 y [m] σ x,max 15.5 [MPa] ; σ z,max 2.5 [MPa] () 7 / 59

Tube : buckling Inc: 0:1 Time: 0.000e+00 Fac: 2.353e+00 Y lcase1 Z X 1 () 8 / 59

LINEAR PLATE BENDING AND LAMINATES

Linear plate bending () 10 / 59

Geometry the mid-plane is planar in the undeformed state, the mid-plane coincides with the global coordinate plane z = 0, the thickness h is uniform. z y x s θ h n r () 11 / 59

External loads z p x θ r y s f z (s) m b (s) s f s (s) m w (s) n f n (s) s x (x, y), s y (x, y) or s r (r, θ), s t (r, θ) p(x, y) or p(r, θ) f n (s), f s (s), f z (s) m b (s), m w (s) () 12 / 59

Displacements z θ y θ θ x T P P z S R z y x u Q v w Q u x = u z sin(θ x )cos(θ y ) u y = v z cos(θ x )sin(θ y ) u z = w + z cos(θ) z () 13 / 59

Simplifications No out-of-plane shear Kirchhoff hypotheses straight lines, perp. to mid-plane remain straight straight lines, perp. to mid-plane remain perp. to mid-plane Small rotation Constant thickness : z = z z φ x P Q z z P Q u x (x, y, z) = u(x, y) zw,x φ x u y (x, y, z) = v(x, y) zw,y u z (x, y, z) = w(x, y) + η(x, y)z 2 x () 14 / 59

Strains and curvatures ε xx = u x,x = u,x zw,xx = ε xx0 zκ xx ε yy = u y,y = v,y zw,yy = ε yy0 zκ yy γ xy = u x,y + u y,x = u,y + v,x 2zw,xy = γ xy0 zκ xy ε = ε 0 zκ z y ε xx0 ε xy0 ε xy0 ε yy0 x κ xx κ yy κ xy κ xy () 15 / 59

Stresses z plane stress state σ zz = σ xz = σ yx = 0 x y σ xx σ yx σ xy σ yy Linear elastic material behaviour ε xx ε yy = 1 1 ν 0 ν 1 0 E γ xy 0 0 2(1 + ν) σ xx σ yy σ xy = E 1 ν 2 σ xx σ yy 1 ν 0 ν 1 0 1 0 0 2 (1 ν) σ xy ε xx ε yy γ xy ; ε zz = ν E (σ xx + σ yy ) short notation : ε = Sσ σ = S 1 ε = Cε = C(ε 0 zκ ) () 16 / 59

Cross-sectional forces and moments Ñ = M = D x = N xx N yy N xy M xx M yy M xy h/2 h/2 = h/2 = h/2 h/2 h/2 σ dz = h/2 σ zx dz ; D y = {C(ε 0 h/2 zκ )} dz = hcε 0 h/2 1 σ z dz = {C(ε 0 zκ )}z dz = h/2 12 h3 Cκ h/2 h/2 σ zy dz z y N xx N xy N xy D y N yy x D x M xx M yy M xy M xy () 17 / 59

Stiffness- and compliance matrix [ Ñ M ] = [ Ch 0 0 Ch 3 /12 ][ ε 0 κ ] [ ε 0 κ ] [ ][ S/h 0 Ñ = 0 12S/h 3 M ] () 18 / 59

Orthotropic plate 2 y α 1 x ε 11 ε 22 γ 12 σ 11 σ 22 σ 12 = E1 1 ν 21 E2 1 0 ν 12 E1 1 E2 1 0 0 0 G 1 1 = 1 ν 21 ν 12 12 σ 11 σ 22 σ 12 E 1 ν 21 E 1 0 ν 12 E 2 E 2 0 0 0 (1 ν 21 ν 12 )G 12 ε 11 ε 22 γ 12 () 19 / 59

Transformation c = cos(α) ; s = sin(α) ε xx ε yy = c2 s 2 cs s 2 c 2 cs 2cs 2cs c 2 s 2 γ xy σ xx σ yy σ xy = c2 s 2 2cs s 2 c 2 2cs cs cs c 2 s 2 ε 11 ε 22 σ 11 σ 22 σ 12 γ 12 = T ε 1 ε ε = T σ 1 σ σ ε = T 1 ε ε = T 1 ε S σ = T 1 ε S T σ = S σ σ = T σ 1 σ σ = T 1 σ C ε = T 1 σ C T ε = C ε ε () 20 / 59

Laminates () 21 / 59

Laminates z k 1 z k y x () 21 / 59

Ply strains z k 1 z k y 2 y x x α 1 strains in ply k with ε = ε 0 k(z) zκ T ε = k ε (z) = T ε,k ε k(z) s 2 c 2 cs c2 s 2 cs 2cs 2cs c 2 s 2 () 22 / 59

Ply stresses Constitutive relation in material coordinate system for ply k k σ = C k ε k σ k = T 1 σ,k C k T ε,k ε k = C k ε k = C k (ε 0 zκ ) transformation matrices T σ = c2 s 2 2cs s 2 c 2 2cs ; T 1 cs cs c 2 s 2 σ = c2 s 2 2cs s 2 c 2 2cs cs cs c 2 s 2 z k 1 z k y x () 23 / 59

ABD-matrix z k 1 z k y x zk Ñ k = σ k dz = (z k z k 1 )C k ε 0 1 2 (z2 k z2 k 1 )C k κ = A k ε 0 + B k κ z k 1 zk k M = σ k z dz = 1 2 (z2 k z2 k 1 )C k ε 0 + 1 3 (z3 k z3 k 1 )C k κ = B k ε 0 + D k κ z k 1 summation over all plies Ñ = [ Ñ M n Ñ k = Aε 0 + B κ ] [ ] [ A B = ε 0 B D κ k=1 ] ; M = [ ε 0 κ n k = Bε 0 + M Dκ k=1 ] [ ] [ ] a b Ñ = b d M () 24 / 59

ABD-matrix A 11 A 12 A 13 B 11 B 12 B 13 A 22 A 23 B 12 B 22 B 23 A 33 B 13 B 23 B 33 D 12 D 13 D 11 D 22 D 23 D 33 () 25 / 59

Stacking cross-ply orthotropic plies material directions = global directions. A 13 = A 23 = 0 angle-ply orthotropic plies each ply material direction 1 is rotated over α o w.r.t. global direction x. regular angle-ply orthotropic plies subsequent plies have material direction 1 rotated alternatingly over α o and α o w.r.t. the global x-axis. even number of plies A 13 = A 23 = 0 symmetric symmetric stacking w.r.t. mid-plane B = 0 anti-symm. anti-symmetric stacking w.r.t. mid-plane D 13 = D 23 = 0 quasi-isotropic α k = k π n with k = 1,.., n (n = number of plies) () 26 / 59

Damage fibre rupture fibre buckling matrix cracking fibre-matrix de-adhesion interlaminar delamination : Inter Laminar Shear Stress (ILSS) ils xx = σ xxb σ xxt ; ils yy = σ yyb σ yyt ; ils xy = σ xyb σ xyt b = bottom of top layer ; t = top of bottom layer () 27 / 59

Random 4-ply laminate : Matlab LAM : IL4r.m ============================================================ Laminate build-up (lam) z- z+ ang El Et nutl Gl 2.000 3.000 90.000 150.000 30.000 0.300 10.000 1.000 2.000 45.000 100.000 25.000 0.200 20.000 0.000 1.000 0.000 110.000 21.000 0.300 15.000-1.000 0.000 30.000 90.000 17.000 0.200 10.000 ------------------------------------------------------------ Mechanical load (ld) [Nxx Nyy Nxy Mxx Myy Mxy] = [ 100.00 0.00 0.00 100.00 0.00 0.00 ] ------------------------------------------------------------ Stiffness matrix 2.57e+08 4.45e+07 4.15e+07-1.83e+05-3.98e+04-1.71e+04 4.45e+07 2.52e+08 2.83e+07-3.98e+04-4.62e+05-2.37e+04 4.15e+07 2.83e+07 7.55e+07-1.71e+04-2.37e+04-6.53e+04-1.83e+05-3.98e+04-1.71e+04 3.77e+02 9.80e+01 5.17e+01-3.98e+04-4.62e+05-2.37e+04 9.80e+01 1.11e+03 4.73e+01-1.71e+04-2.37e+04-6.53e+04 5.17e+01 4.73e+01 1.43e+02 ------------------------------------------------------------ Strains in the mid-plane (e0) [ exx eyy exy ] = [ 3.810e-04-1.079e-04-3.102e-04 ] Curvatures of the mid-plane (kr) [ kxx kyy kxy ] = [ 4.784e-01-6.892e-02-2.637e-01 ] ============================================================ () 28 / 59

w Random 4-ply laminate : results 3 2 xx yy xy 3 2 11 22 33 z [mm] 1 z [mm] 1 0 0 1 4 2 0 2 4 6 σ [Pa] x 10 7 1 4 2 0 2 4 6 σ [Pa] x 10 7 0.5 0.4 3 2 xx yy xy 3 2 11 22 33 0.3 0.2 0.1 z [mm] 1 z [mm] 1 0 0.1 0 0 0.2 1 0.5 0 y 0.5 1 1 0.5 x 0 0.5 1 1 1.5 1 0.5 0 0.5 1 ε [ ] x 10 3 1 1.5 1 0.5 0 0.5 1 ε [ ] x 10 3 () 29 / 59

MSC MARC/MENTAT SHELL ELEMENTS

Cross-sectional forces and moments LAM x z y N xx N xy D x M xx N xy M yy D y N yy M xy M xy MSC.Marc/Mentat x z y N x N y N x M x N y M y M y M x () 31 / 59

Bending of a strip H W z y L F z x F x M y length L 1 m width W 0.1 m heigth H 0.05 m Young s modulus E 150 GPa Poisson s ratio ν 0.3 - axial end force F x 100 N lateral end force F z 100 N bending end moment M y 100 Nm () 32 / 59

Bending of a strip Inc: 0 Time: 0.000e+00 Inc: 0 Time: 0.000e+00 Inc: 0 Time: 0.000e+00 2.000e+02 1.900e+02 1.800e+02 1.700e+02 1.600e+02 1.500e+02 1.400e+02 1.300e+02 1.200e+02 1.100e+02-2.443e+06-2.693e+06-2.943e+06-3.192e+06-3.442e+06-3.692e+06-3.942e+06-4.192e+06-4.442e+06-4.692e+06 4.362e+06 3.494e+06 2.626e+06 1.759e+06 8.911e+05 2.343e+04-8.442e+05-1.712e+06-2.580e+06-3.447e+06 1.000e+02 Z -4.941e+06 Z -4.315e+06 Z job1 X Y job1 X Y job1 X Y Beam Bending Moment Local Y 4 Comp 11 of Stress Layer 1 4 Comp 11 of Stress 4 exact beam shell 3D axial end displacement u x 0.133 0.133 0.133 0.133 µm lateral end displacement u z 0.533 0.533 0.531 0.392 mm end rotation φ y 0.00096 0.00096 0.00097 - deg maximum axial stress σ 4.82-4.94 4.35 MPa stripbeam.proc ; stripplate.proc ; strip3dsol.proc () 33 / 59

Local/global stress components Inc: 0 Time: 0.000e+00-2.443e+06-2.693e+06-2.943e+06-3.192e+06-3.442e+06-3.692e+06-3.942e+06-4.192e+06-4.442e+06-4.692e+06-4.941e+06 Z job1 X Y Comp 11 of Stress Layer 1 4 Inc: 0 Time: 0.000e+00 Inc: 0 Time: 0.000e+00 1.067e+03-2.443e+06-4.932e+05-2.693e+06-9.874e+05-2.943e+06-1.482e+06-3.192e+06-1.976e+06-3.442e+06-2.470e+06-3.692e+06-2.964e+06-3.942e+06-3.459e+06-4.192e+06-3.953e+06-4.442e+06-4.447e+06-4.692e+06-4.941e+06 Z -4.941e+06 Z job1 X Y job1 X Y Comp 11 of Stress Layer 1 4 Comp 11 of Global Stress Layer 1 4 () 34 / 59

Orthotropic plate 2 = t y z = 3 = t α 1 = l x () 35 / 59

Orthotropic plate 2 = t y α z = 3 = t 1 = l x MSC E 1 E 11 E l E 2 E 22 E t E 3 E 33 E t G 12 G 12 G lt E t G 23 G 23 2(1 + ν tt ) G 31 G 31 G lt ν 12 ν 12 ν lt ν 23 ν 23 ν tt E 3 ν 31 ν 31 ν 13 = E t ν lt E 1 E l ν 21 ν 21 ν 32 ν 32 ν 13 ν 13 () 35 / 59

Orthotropic plate 2 = t y MSC α z = 3 = t y 1 = l x E 1 E 11 E l E 2 E 22 E t E 3 E 33 E t G 12 G 12 G lt E t G 23 G 23 2(1 + ν tt ) G 31 G 31 G lt ν 12 ν 12 ν lt ν 23 ν 23 ν tt F x M y z F x x E 3 ν 31 ν 31 ν 13 = E t ν lt E 1 E l ν 21 ν 21 ν 32 ν 32 M y ν 13 ν 13 () 36 / 59

Orthotropic plate : deformation and stress-11 in layer 1 Inc: 0 Time: 0.000e+00-3.710e+07-3.714e+07-3.718e+07-3.721e+07-3.725e+07-3.729e+07-3.733e+07-3.736e+07-3.740e+07-3.744e+07-3.748e+07 Z job1 Comp 11 of Stress Layer 1 X Y 4 plate20x20ortm1p4.proc ; ortm1p4.m () 37 / 59

Laminated plate : ply properties For each ply : (MAIN MENU) (PREPROCESSING) MATERIAL PROPERTIES ORIENTATION NEW EDGE12 ANGLE 0 MATERIAL PROPERTIES NEW (type) STANDARD STRUCTURAL ELASTO-PLASTIC ORTHOTROPIC E1 E2 E3 N12 N23 N31 G12 G23 G31 OK () 38 / 59

Laminated plate : stacking NEW (type) COMPOSITE (DATA CATAGORIES) GENERAL ABSOLUTE THICKNESS Toggle! APPEND material 1 THICKNESS 0.001 ANGLE 90 APPEND material 2 THICKNESS 0.001 ANGLE 45 etc OK (ELEMENTS) ADD (ALL) EXIST. RETURN () 39 / 59

Laminated plate : loading and results y M y F x z F x M y x Inc: 0 Time: 0.000e+00-2.404e+07-2.407e+07-2.409e+07-2.412e+07-2.414e+07-2.417e+07-2.419e+07-2.421e+07-2.424e+07-2.426e+07-2.429e+07 Z job1 Comp 11 of Stress Layer 1 X Y 4 plate20x20ortm4p4.proc ; ortm4p4.m () 40 / 59

RECTANGULAR TUBE

Tube : buckling : FEM K ũ = f e ũ = K 1 f e σ K g (σ) K : linear stiffness matrix ũ : nodal displacements σ : stresses : proportional with f e K g : geometric or stress stiffness matrix : proportional with σ K + K g : total stiffness matrix () 42 / 59

Tube : buckling : FEM K ũ = f e ũ = K 1 f e σ K g (σ) K : linear stiffness matrix ũ : nodal displacements σ : stresses : proportional with f e K g : geometric or stress stiffness matrix : proportional with σ K + K g : total stiffness matrix [ K + λ Kg (σ) ] α = 0 λ i α i λ i : buckling factors λ i f e : buckling forces α i : buckling modes minλ i : collaps load factor () 42 / 59

Tube : buckling : MARC/Mentat LOADCASE NEW BUCKLE RETURN JOB PROPERTIES select loadcase STRUCTURAL buckle OK RUN SUBMIT RESULTS OPEN DEFAULT NEXT () 43 / 59

Tube : deformation Inc: 0 Time: 0.000e+00 4.848e-03 3.223e-03 1.598e-03-2.724e-05-1.652e-03-3.277e-03-4.902e-03-6.527e-03-8.152e-03-9.777e-03-1.140e-02 Y job1 Displacement Y Z X 1 Buckling modes () 44 / 59

Inc: 0:1 Time: 0.000e+00 Fac: 2.353e+00 Y lcase1 Z X 1 () 45 / 59

Inc: 0:3 Time: 0.000e+00 Fac: -4.675e+00 Y lcase1 Z X 1 () 47 / 59

Tube : inside loads and wind load Inc: 0 Time: 0.000e+00 4.877e-03 3.467e-03 2.057e-03 6.473e-04-7.627e-04-2.173e-03-3.583e-03-4.993e-03-6.402e-03-7.812e-03-9.222e-03 Y job1 Displacement Y Z X 1 () 49 / 59

Tube : deformation disp [m] 0.01 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0 2 4 6 8 10 y [m] Buckling modes () 50 / 59

Inc: 0:2 Time: 0.000e+00 Fac: -3.769e+00 Y lcase1 Z X 1 () 52 / 59

OPTIMIZATION

Optimization theory and procedures See slides Pascal Etman for background information. Use examples from download page as starting point Optimization with MSC.Marc/Mentat is also possible. This will not be discussed here. () 54 / 59

Cyclic analysis 4-ply laminate : laminate0 lam.m matrix lam with laminate stacking 3 z [mm] 2 1 0 xx yy xy z [mm] 2 1 0 11 22 12 2 1 1 w 1 0 1 2 3 1 0.5 0 y 0.5 1 1 0.5 x 0 0.5 1 z [mm] 2 5 0 5 7 σ [Pa] x 10 2 1 0 1 xx yy xy 2 0.01 0.005 0 0.005 0.01 ε [ ] z [mm] 2 2 0 2 4 6 7 σ [Pa] x 10 2 1 0 1 11 22 12 2 0.02 0.01 0 0.01 0.02 ε [ ] Two design variables : ply angles Cyclic variation of design variables in 2 nested loops : laminate0 cyc.m Plot results in 3-d plots and contour plot () 55 / 59

Surface plots x 10 7 6 5 Curvature 4 2 0 2 4 Stress xx 4 3 2 1 6 100 80 60 x2 40 20 0 100 80 60 x1 40 20 0 0 100 80 60 x2 40 20 0 100 80 60 x1 40 20 0 0.94 1.5 0.96 1 Stress 11/Tl 1 0.98 1 1.02 ilsxx1/ilss 1 0.5 0 0.5 1.04 100 80 60 x2 40 20 0 100 80 60 x1 40 20 0 1 100 80 60 x2 40 20 0 100 80 60 x1 40 20 0 () 56 / 59

Contour plot 90 80 70 60 50 x2 40 30 20 10 0 90 80 70 60 50 40 30 20 10 0 x1 () 57 / 59

Searching algorithms laminate0 opt.m ; laminate0 obj.m ; laminate0 con.m fmincon : output and results Max Line search Directional First-order Iter F-count f(x) constraint steplength derivative optimality Procedure 0 3 3.09808e+08 2.701 Infeasible start point 1 5 3.09808e+08 2.701 2-5.01e+09 5.51e+09 2 7 3.09808e+08 2.701 2-5.01e+09 5.51e+09 Hessian not updated 3 9 3.09808e+08 2.701 2-5.01e+09 5.51e+09 Hessian not updated 4 11 3.09808e+08 2.701 2-5.01e+09 5.51e+09 Hessian not updated 5 14 411620 0.4799 1-2.8e+08 1.85e+08 Hessian not updated 15 44 1.65446e+08 0 1-5.48e+08 3.52e+08 16 47 1.6372e-05 0 1 152 1.43e+07 Hessian modified twice 17 69 5.11047e-08-2.376e-06 1.91e-06 1.12 1.13 18 72 5.58125e-10-2.232e-06 1-7.75e-10 0.0455 Optimization terminated: magnitude of directional derivative in search direction less than 2*options.TolFun and maximum constraint violation is less than options.tolcon. No active inequalities. ========================================================== Initial ply angles : -60.0000 50.0000 Initial curvature : 1.76 Initial constraints : -0.8302-0.4656-0.6674-1.0000-1.0000 : 0.2993-0.8302-0.4656-0.6674 2.7009 Optimized ply angles : -0.0000 0.0000 Optimized curvature : 2.362e-09 Optimized constraints : -1.0000-1.0000-1.0000-1.0000-1.0000 : -1.0000-1.0000-1.0000-1.0000-0.9995 ========================================================== () 58 / 59

Searching algorithms laminate0 opt.m ; laminate0 obj.m ; laminate0 con.m patternsearch : output and results max Iter f-count f(x) constraint MeshSize Method 0 1 3.09808e+08 2.701 1 1 12 4.11222e+07 0 0.001 Increase penalty 2 112 18.7301 0 1e-05 Increase penalty 3 200 0.943907 0 1e-07 Increase penalty Maximum number of function evaluations exceeded: increase options.maxfunevals. ========================================================== Initial ply angles : -60.0000 50.0000 Initial curvature : 1.76 Initial constraints : -0.8302-0.4656-0.6674-1.0000-1.0000 : 0.2993-0.8302-0.4656-0.6674 2.7009 Optimized ply angles : -4.7072 23.5991 Optimized curvature : 9.715e-05 Optimized constraints : -0.5595-0.8952-0.6647-1.0000-1.0000 : -0.4970-0.5595-0.8952-0.6647-0.9989 ========================================================== () 59 / 59