/
static-joined-wing.jl
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/
static-joined-wing.jl
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# # [Static Analysis of a Joined-Wing](@id static-joined-wing)
#
# In this example we consider the joined-wing model proposed by Blair in "An Equivalent
# Beam Formulation for Joined-Wings in a Post-Buckled State" and optimized by Green
# et al. in "Structural Optimization of Joined-Wing Beam Model with Bend-Twist
# Coupling using Equivalent Static Loads".
#
# 
#
#-
#md # !!! tip
#md # This example is also available as a Jupyter notebook:
#md # [`static-joined-wing.ipynb`](@__NBVIEWER_ROOT_URL__/examples/static-joined-wing.ipynb).
#-
#
using GXBeam, LinearAlgebra
## Set endpoints of each beam
p1 = [-7.1726, -12, -3.21539]
p2 = [-5.37945, -9, -2.41154]
p3 = [-3.5863, -6, -1.6077]
p4 = [-1.79315, -3, -0.803848]
p5 = [0, 0, 0]
p6 = [7.1726, -12, 3.21539]
## get transformation matrix for left beams
## transformation from intermediate frame to global frame
tmp1 = sqrt(p1[1]^2 + p1[2]^2)
c1, s1 = -p1[1]/tmp1, -p1[2]/tmp1
rot1 = [c1 -s1 0; s1 c1 0; 0 0 1]
## transformation from local beam frame to intermediate frame
tmp2 = sqrt(p1[1]^2 + p1[2]^2 + p1[3]^2)
c2, s2 = tmp1/tmp2, -p1[3]/tmp2
rot2 = [c2 0 -s2; 0 1 0; s2 0 c2]
Cab_1 = rot1*rot2
## get transformation matrix for right beam
## transformation from intermediate frame to global frame
tmp1 = sqrt(p6[1]^2 + p6[2]^2)
c1, s1 = p6[1]/tmp1, p6[2]/tmp1
rot1 = [c1 -s1 0; s1 c1 0; 0 0 1]
## transformation from local beam frame to intermediate frame
tmp2 = sqrt(p6[1]^2 + p6[2]^2 + p6[3]^2)
c2, s2 = tmp1/tmp2, p6[3]/tmp2
rot2 = [c2 0 -s2; 0 1 0; s2 0 c2]
Cab_2 = rot1*rot2
## beam 1
L_b1 = norm(p2-p1)
r_b1 = p1
nelem_b1 = 5
lengths_b1, xp_b1, xm_b1, Cab_b1 = discretize_beam(L_b1, r_b1, nelem_b1;
frame = Cab_1)
compliance_b1 = fill(Diagonal([1.05204e-9, 3.19659e-9, 2.13106e-8, 1.15475e-7,
1.52885e-7, 7.1672e-9]), nelem_b1)
## beam 2
L_b2 = norm(p3-p2)
r_b2 = p2
nelem_b2 = 5
lengths_b2, xp_b2, xm_b2, Cab_b2 = discretize_beam(L_b2, r_b2, nelem_b2;
frame = Cab_1)
compliance_b2 = fill(Diagonal([1.24467e-9, 3.77682e-9, 2.51788e-8, 1.90461e-7,
2.55034e-7, 1.18646e-8]), nelem_b2)
## beam 3
L_b3 = norm(p4-p3)
r_b3 = p3
nelem_b3 = 5
lengths_b3, xp_b3, xm_b3, Cab_b3 = discretize_beam(L_b3, r_b3, nelem_b3;
frame = Cab_1)
compliance_b3 = fill(Diagonal([1.60806e-9, 4.86724e-9, 3.24482e-8, 4.07637e-7,
5.57611e-7, 2.55684e-8]), nelem_b3)
## beam 4
L_b4 = norm(p5-p4)
r_b4 = p4
nelem_b4 = 5
lengths_b4, xp_b4, xm_b4, Cab_b4 = discretize_beam(L_b4, r_b4, nelem_b4;
frame = Cab_1)
compliance_b4 = fill(Diagonal([2.56482e-9, 7.60456e-9, 5.67609e-8, 1.92171e-6,
2.8757e-6, 1.02718e-7]), nelem_b4)
## beam 5
L_b5 = norm(p6-p5)
r_b5 = p5
nelem_b5 = 20
lengths_b5, xp_b5, xm_b5, Cab_b5 = discretize_beam(L_b5, r_b5, nelem_b5;
frame = Cab_2)
compliance_b5 = fill(Diagonal([2.77393e-9, 7.60456e-9, 1.52091e-7, 1.27757e-5,
2.7835e-5, 1.26026e-7]), nelem_b5)
## combine elements and points into one array
nelem = nelem_b1 + nelem_b2 + nelem_b3 + nelem_b4 + nelem_b5
points = vcat(xp_b1, xp_b2[2:end], xp_b3[2:end], xp_b4[2:end], xp_b5[2:end])
start = 1:nelem
stop = 2:nelem + 1
lengths = vcat(lengths_b1, lengths_b2, lengths_b3, lengths_b4, lengths_b5)
midpoints = vcat(xm_b1, xm_b2, xm_b3, xm_b4, xm_b5)
Cab = vcat(Cab_b1, Cab_b2, Cab_b3, Cab_b4, Cab_b5)
compliance = vcat(compliance_b1, compliance_b2, compliance_b3, compliance_b4,
compliance_b5)
## create assembly
assembly = Assembly(points, start, stop;
compliance = compliance,
frames = Cab,
lengths = lengths,
midpoints = midpoints)
Fz = range(0, 70e3, length=141)
## pre-allocate memory to reduce run-time
ijoint = nelem_b1 + nelem_b2 + nelem_b3 + nelem_b4 + 1
prescribed_points = [1, ijoint, nelem+1]
static = true
system = StaticSystem(assembly)
linear_states = Vector{AssemblyState{Float64}}(undef, length(Fz))
for i = 1:length(Fz)
## create dictionary of prescribed conditions
prescribed_conditions = Dict(
## fixed endpoint on beam 1
1 => PrescribedConditions(ux=0, uy=0, uz=0, theta_x=0, theta_y=0,
theta_z=0),
## force applied on point 4
nelem_b1 + nelem_b2 + nelem_b3 + nelem_b4 + 1 => PrescribedConditions(
Fz = Fz[i]),
## fixed endpoint on last beam
nelem+1 => PrescribedConditions(ux=0, uy=0, uz=0, theta_x=0, theta_y=0,
theta_z=0),
)
_, linear_states[i], converged = static_analysis!(system, assembly;
prescribed_conditions = prescribed_conditions,
linear = true)
end
reset_state!(system)
nonlinear_states = Vector{AssemblyState{Float64}}(undef, length(Fz))
for i = 1:length(Fz)
## create dictionary of prescribed conditions
prescribed_conditions = Dict(
## fixed endpoint on beam 1
1 => PrescribedConditions(ux=0, uy=0, uz=0, theta_x=0, theta_y=0,
theta_z=0),
## force applied on point 4
nelem_b1 + nelem_b2 + nelem_b3 + nelem_b4 + 1 => PrescribedConditions(
Fz = Fz[i]),
## fixed endpoint on last beam
nelem+1 => PrescribedConditions(ux=0, uy=0, uz=0, theta_x=0, theta_y=0,
theta_z=0),
)
_, nonlinear_states[i], converged = static_analysis!(system, assembly;
prescribed_conditions=prescribed_conditions, reset_state=false)
end
reset_state!(system)
nonlinear_follower_states = Vector{AssemblyState{Float64}}(undef, length(Fz))
for i = 1:length(Fz)
## create dictionary of prescribed conditions
prescribed_conditions = Dict(
## fixed endpoint on beam 1
1 => PrescribedConditions(ux=0, uy=0, uz=0, theta_x=0, theta_y=0,
theta_z=0),
## force applied on point 4
nelem_b1 + nelem_b2 + nelem_b3 + nelem_b4 + 1 => PrescribedConditions(
Fz_follower = Fz[i]),
## fixed endpoint on last beam
nelem+1 => PrescribedConditions(ux=0, uy=0, uz=0, theta_x=0, theta_y=0,
theta_z=0),
)
_, nonlinear_follower_states[i], converged = static_analysis!(system, assembly;
prescribed_conditions=prescribed_conditions, reset_state=false)
end
#!jl nothing #hide
# Note that we incrementally increased the load from 0 to 70 kN in order to ensure that we
# obtained converged solutions.
#
#-
#
# To visualize the differences between the different types of analyses we can plot the
# load deflection curve.
#md using Suppressor #hide
#md @suppress_err begin #hide
using Plots
pyplot()
plot(
xlim = (0, 7),
xticks = 0:1:7,
xlabel = "Vertical Displacement at the Joint (m)",
yticks = 0:10:70,
ylim = (0, 70),
ylabel = "Load (kN)",
grid = false,
overwrite_figure=false
)
uz_l = [linear_states[i].points[ijoint].u[3] for i = 1:length(Fz)]
uz_nl = [nonlinear_states[i].points[ijoint].u[3] for i = 1:length(Fz)]
uz_fnl = [nonlinear_follower_states[i].points[ijoint].u[3] for i = 1:length(Fz)]
plot!(uz_l, Fz./1e3, label="Linear")
plot!(uz_nl, Fz./1e3, label="Nonlinear with Dead Force")
plot!(uz_fnl, Fz./1e3, label="Nonlinear with Follower Force")
plot!(show=true) #!nb
#md savefig("../assets/static-joined-wing-deflection.svg") #hide
#md closeall() #hide
#md end #hide
#md nothing #hide
#md # 
#-
# This plot matches the plot provided by Wenbin Yu in "GEBT: A general-purpose nonlinear
# analysis tool for composite beams".
#-
# We can also visualize the deformed geometry and inspect the associated point and element
# data for any of these operating conditions conditions using ParaView. To demonstrate
# we will visualize the 70kN follower force condition and set the color gradient to
# match the magnitude of the deflections.
airfoil = [ #FX 60-100 airfoil
0.0000000 0.0000000;
0.0010700 0.0057400;
0.0042800 0.0114400;
0.0096100 0.0177500;
0.0170400 0.0236800;
0.0265300 0.0294800;
0.0380600 0.0352300;
0.0515600 0.0405600;
0.0669900 0.0460900;
0.0842700 0.0508600;
0.1033200 0.0556900;
0.1240800 0.0598900;
0.1464500 0.0640400;
0.1703300 0.0675400;
0.1956200 0.0708100;
0.2222100 0.0733900;
0.2500000 0.0756500;
0.2788600 0.0772000;
0.3086600 0.0783800;
0.3392800 0.0788800;
0.3705900 0.0789800;
0.4024500 0.0784500;
0.4347400 0.0775000;
0.4673000 0.0759600;
0.5000000 0.0740900;
0.5327000 0.0717400;
0.5652600 0.0691100;
0.5975500 0.0660800;
0.6294100 0.0627500;
0.6607200 0.0590500;
0.6913400 0.0551100;
0.7211400 0.0508900;
0.7500000 0.0465200;
0.7777900 0.0420000;
0.8043801 0.0374700;
0.8296700 0.0329800;
0.8535500 0.0286400;
0.8759201 0.0244700;
0.8966800 0.0205300;
0.9157300 0.0168100;
0.9330100 0.0134200;
0.9484400 0.0103500;
0.9619400 0.0076600;
0.9734700 0.0053400;
0.9829600 0.0034100;
0.9903900 0.0019300;
0.9957200 0.0008600;
0.9989300 0.0002300;
1.0000000 0.0000000;
0.9989300 0.0001500;
0.9957200 0.0007000;
0.9903900 0.0015100;
0.9829600 0.00251;
0.9734700 0.00377;
0.9619400 0.00515;
0.9484400 0.00659;
0.9330100 0.00802;
0.9157300 0.00941;
0.8966800 0.01072;
0.8759201 0.01186;
0.8535500 0.0128;
0.8296700 0.01347;
0.8043801 0.01381;
0.7777900 0.01373;
0.7500000 0.01329;
0.7211400 0.01241;
0.6913400 0.01118;
0.6607200 0.00951;
0.6294100 0.00748;
0.5975500 0.00496;
0.5652600 0.00217;
0.532700 -0.00092;
0.500000 -0.00405;
0.467300 -0.00731;
0.434740 -0.01045;
0.402450 -0.01357;
0.370590 -0.01637;
0.339280 -0.01895;
0.308660 -0.021;
0.278860 -0.02275;
0.250000 -0.02389;
0.222210 -0.02475;
0.195620 -0.025;
0.170330 -0.02503;
0.146450 -0.02447;
0.124080 -0.02377;
0.103320 -0.02246;
0.084270 -0.0211;
0.066990 -0.01913;
0.051560 -0.0173;
0.038060 -0.01481;
0.026530 -0.01247;
0.017040 -0.0097;
0.009610 -0.00691;
0.004280 -0.00436;
0.001070 -0.002;
0.0 0.0;
]
section = zeros(3, size(airfoil, 1))
for ic = 1:size(airfoil, 1)
section[1,ic] = airfoil[ic,1] - 0.5
section[2,ic] = 0
section[3,ic] = airfoil[ic,2]
end
mkpath("static-joined-wing-visualization")
write_vtk("static-joined-wing-visualization/static-joined-wing-visualization", assembly, nonlinear_follower_states[end];
sections = section)
#md rm("static-joined-wing-visualization"; recursive=true) #hide
# 