final project1 3/19/2010 isogrid buckling with varying boundary conditions jeffrey lavin rpi masters...
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Final Project 13/19/2010
Isogrid Buckling With Varying Boundary Conditions
Jeffrey Lavin
RPI Masters Project
Final Project 23/19/2010
Topic Overview
• Geometry Definitions• Simplification• Buckling Analytical Solution• Model Creation And Meshing• Mode Shape Verification• Modeling Approach• Modeling Results• Summary
Final Project 33/19/2010
Isogrid Geometry
• Created by individual equilateral triangular panels– Triangle geometry defined by height (h) and side (s)– Geometry is reducible to single panel for all h and s
• Non-dimensional parameters identified– Based on NASA report CR-124075– No cap shown =0, =0
h
a
tb
d
1.000”
Section View
δ =d
t =
c
tα =
bd
th =
wc
th
h =a32
Final Project 43/19/2010
Isogrid Simplification
• Reduce isogrid to single sheet – Maintain bending (D) and tensile (K) stiffness– Transformed area with parallel axis theorem
• Solve K and D simultaneously for t* and E*
– t* is equivalent single sheet thickness– E* is equivalent single sheet modulus
• E* and t* provide equivalent stiffness– Do not provide equivalent stress
D =1
1−ν 2 E0 I
K =1
1−ν 2 E0A t* =12IA
=tβ
1+α +
β 2 = (1 +α + μ ) 3(1 + δ )2 + 3μ (δ + λ )2 +1+αδ 2 + μλ 2⎡⎣ ⎤⎦− 3 (1+ δ ) − μ (δ + λ )[ ]2
E* =E0
At*D =
E*t*3
12(1−v2 )
K =E*t*
1−ν 2
Final Project 53/19/2010
xy
a
b
N x =ba2π 2D
m2
m2
a2 +n2
b2
⎛
⎝⎜⎞
⎠⎟
2
D =Et3
12(1−ν 2 )
Analytical Solution
• Calculate critical load required for elastic instability• Multiple load cases established• Simple supported plate• Half wave buckling modes vary
– M controls load direction half wave– N controls load perpendicular half wave
Final Project 63/19/2010
Model Creation And Meshing
• Model Created Using Something
• Mesh Creation– Determine mesh density
Final Project 73/19/2010
Mode Shape Visualization
M=2
N=1
F
F
M
M
F
F
M
MN
N
N
N
M=1
N=1
Final Project 83/19/2010
+ +
=
Final Modeling Approach
• Model single isogrid panel (4.000” x 4.618”)• Represents specific geometry based on h, s, d, b
– Similar configuration to existing hardware
• Multiple mode shapes produced
Final Project 93/19/2010
Final Modeling Results
• Compare analytical solution to numerical solutions– Plate model completed using E*t*
– Isogrid model completed based on geometry
• Analytical solution comparison– Results show good correlation between models and
predicted solution
Calculated Percent Error Comsol Plate Isogrid Percent Error m n1882 -1.28 1858 1946 3.39 1 12458 -1.11 2431 2543 3.44 2 17529 -1.27 7433 7620 1.21 2 24102 -0.94 4063 4223 2.96 3 16492 -0.87 6435 6568 1.18 4 1
Final Project 103/19/2010
Rib Buckling
• Study completed for 4 different rib heights– .050”, .100”, .250”, .500”
• Each model run with load applied to both edges• Parameter α used for comparison of E*t* applicability
–
α < .23 shows good correlation to E*t* method
α vs. 1st Critical Buckling Load
0
50000
100000
150000
200000
250000
0.00 0.20 0.40 0.60 0.80
α
1st Critical Buckling Load
(lb)Plate mode
Isogrid Mode
α =bd
th
Final Project 113/19/2010
Conclusions
• Analytical solution matches numerical approximation– Plate model critical loads off 1.3%– Isogrid model critical loads off 3.5%
• Model accuracy does not decrease as deflection increases– Mesh density appropriate to calculate complex deflections
• Highly dependant on boundary conditions– Small change creates large difference in results
• Rib buckling can dominate final results– Increased rib stiffness dominates buckling solution
Final Project 123/19/2010
Questions
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