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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