OS-V: 0080 Buckling of Shells and Composites with Offset

A test of influence of offset on buckling solution for shells, including composite with offset Z0 and element offset ZOFFS.

図 1. FE Model of the Beam with Boundary Conditions and Loadcases

Model Files

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Benchmark Model

Here, you solve several problems to calculate the critical load on different conditions. The model is a simply supported beam of height 1 mm, breadth 2 mm and length 100 mm with one end constrained in all DOFs and an axial load applied on the other end.

The material properties for the beam are:

MAT1
Young's Modulus: 1 x 10⁶ N/mm²
Poisson's Ratio: 0.0
Density: 2 kg/mm³
Thermal Expansion Coefficient: 1 x 10^-4 ºC^-1
Reference Temperature for Thermal Loading: 300ºC

The different case description of the problem are:

Buckling without offset.
Buckling with moment equivalent to offset.
Buckling with offset created by a frame.
Buckling with offset applied through ZOFFS.
Buckling of composite with non-symmetrical layup.
Buckling of composite with offset.

The theoretical critical buckling load is calculated using the Euler Buckling equation:

図 2.

f_{c r i t} = π \frac{E I}{{(K L)}^{2}}

Where,

$f_{c r i t}$: Maximum or critical force
$E$: Modulus of elasticity
$I$: Area moment of inertia (second moment of area)
$L$: Unsupported length of the beam
$K$: Column effective length factor (for one end fixed and the other end free, $K$ =2)

Results

図 3. First Four Buckling Eigenvalues for Non-offset (z₀ = -0.5)


Quantity	Theoretical	No-offset	Normalized
$λ$ _cr⁽¹⁾	4.1123	4.1208	0.997937
$λ$ _cr⁽²⁾	16.449	16.513	0.996124
$λ$ _cr⁽³⁾	37.011	37.701	0.981698
$λ$ _cr⁽⁴⁾	102.81	108.19	0.950273

図 4. First Four Buckling Eigenvalues for Non-offset + Moment . (the effect of offset is simulated by adding a moment at the end of the beam)


Quantity	Theoretical	No-offset + Moment	Normalized
$λ$ _cr⁽¹⁾	4.1123	4.1208	0.997937
$λ$ _cr⁽²⁾	16.449	16.513	0.996124
$λ$ _cr⁽³⁾	37.011	37.701	0.981698
$λ$ _cr⁽⁴⁾	102.81	108.19	0.950273

図 5. First Four Buckling Eigenvalues for C-Frame. (the effect of offset is simulated by creating a C-shaped frame)


Quantity	Theoretical	C-Frame	Normalized
$λ$ _cr⁽¹⁾	4.1123	4.1208	0.997937
$λ$ _cr⁽²⁾	16.449	16.513	0.996124
$λ$ _cr⁽³⁾	37.011	37.700	0.981724
$λ$ _cr⁽⁴⁾	102.81	108.19	0.950273

図 6. First Four Buckling Eigenvalues for z-offset (Z_offs = -0.5)


Quantity	Theoretical	ZOFFS	Normalized
$λ$ _cr⁽¹⁾	4.1123	4.1208	0.997937
$λ$ _cr⁽²⁾	16.449	16.513	0.996124
$λ$ _cr⁽³⁾	37.011	37.700	0.981724
$λ$ _cr⁽⁴⁾	102.81	108.19	0.950273

図 7. First Four Buckling Eigenvalues for Non-symmetric Layup . (since the top layer is very weak, the load is applied to the “strong” layer with an offset of 0.5)


Quantity	Theoretical	Non-symmetric Layup	Normalized
$λ$ _cr⁽¹⁾	4.1123	4.1203	0.998058
$λ$ _cr⁽²⁾	16.449	16.510	0.996305
$λ$ _cr⁽³⁾	37.011	37.663	0.982689
$λ$ _cr⁽⁴⁾	102.81	107.89	0.952915

図 8. First Four Buckling Eigenvalues for Composites with Offset (z₀ = -1)


Quantity	Theoretical	Offset Composite	Normalized
$λ$ _cr⁽¹⁾	4.1123	4.1203	0.998058
$λ$ _cr⁽²⁾	16.449	16.510	0.996305
$λ$ _cr⁽³⁾	37.011	37.663	0.982689
$λ$ _cr⁽⁴⁾	102.81	107.89	0.952915