Web Analysis - Buckling

Web Analysis - Buckling

The analysis has been made using standard hand-made calculation methods from Bruhn manuals using F.E.M. model freebodies - interface loads - as applied loads. Each panel, except the ones with the filled circle inside no buckling zones -, has been analyzed extracting on each side the freebody interface loads for all LCs considering all the web shell elements and the interface grids.

Fig. 2.1 Rear Bulkhead Representation & Panel Identification

Then, averaged tangential forces and normal plane moments on the panel uprights have been considered for compression and bending action, while the averaged upright radial forces and tangential forces on the flanges have been considered for shear action. Although bending, compression and shear effects have been taken into account to calculate the combined action, the web thinness cannot support all those loads, so this will be analyzed for the shear action while the compressive and bending effects will be recalculated to act on the bulkhead flanges, overloading the items compared to the flange FEM internal loads.

.The picture 2.2 shows the generic bulkhead panel, the applied loads, the constraints and used coordinated systems (general cylindrical Xo (tangential direction), Yo (radial direction); local rectangular X, Y).

Fig. 2.2 Panel Loads & Coordinate System Representation

The F.E.M idealization has been placed on top of the previous CAD bulkhead representation giving the following picture result.

Fig. 2.3 Fem Idealization vs Cad Representation

The following table shows the considered panels CAD geometry:

Table. 2.4 CAD Geometry

while the following one gives the FEM geometry:

Table. 2.5 FEM Geometry (see fig. 7.2.7.1.2.3)

Since the maximum active temperature on the fwd bulkhead is T = 80C, all panel critical stresses have been calculated for this ambient condition. The critical stresses and the allowable combined formulation outcome, as already explained in par. 7.1.2, from ref. . and ref. .

Table. 2.6 Buckling Coefficients & Allowable Stresses

The following tables group the applied loads & stresses for the most critical shear RF:

see fig. 2.2

Table 2.7 Applied loads and RFs

where:

F_sx = (F_x¹/L_{1 FEM} + F_x³/L_{3 FEM})/2;

F_sy = (F_x²/L_{2 FEM} + F_x⁴/L_{4 FEM})/2;

f_xc = (F_x¹ + F_x³) /2/A_x;

f_xb = (((M_z² - M_z⁴) /2)(b/2))(1/J_zx

f_x = f_xc+ f_xb;

f_s = (F_sx + F_sy)/2/t;

Table 2.8 contd - Applied loads and RFs

where:

RF_s = combined formulation for RF with R_s only;

f_xc¹ = inner flange stress due to compression = f_xc/A/2;

f_xb¹ = inner flange stress due to bending = (M_z² - M_z⁴) /2/b;

f_x¹ = total inner flange stress = f_xc¹ + f_xb¹

f_xc² = outer flange stress due to compression = f_xc/A/2;

f_xb² = outer flange stress due to bending = - f_xb¹

f_x² = total outer flange stress = f_xc² + f_xb²

The f_x¹ and f_x² columns will be considered for each load condition - the previous table returns the loads for the most critical shear actions only - along all the flanges and the maximum compressive envelope load will be added to the internal flanges-BAR f.e.m. forces and moments to evaluate the effective post-buckling actions on the items.

Here, the most critical full detailed panel (nr. 8) calculation RF_s = 1.10 from Table 2.8 - has been reported:

Fig. 2.9 Rear Bulkhead Panel 8 - Zoom

Where, for buckling analysis,:

t = 0.8 mm; a = 235 mm; b = 227 mm;

E₈₀ = 108550 N/mm²; clamped uprights considered;

K_c = 3.73 - ref. . fig. C5.2;

K_b = 21.50 - ref. . fig. C5.14;

K_s = 8.22 - ref. . fig. C5.15;

F_c = K_c E₂₈₁ (t/b)² = 3.73 x 108550 x (0.8/227)² = 5.03 N/mm²;

F_b = K_b E₂₈₁ (t/b)² = 21.5 x 108550 x (0.8/227)² = 28.99 N/mm²;

F_s = K_s E₂₈₁ (t/b)² = 8.22 x 108550 x (0.8/227)² = 11.08 N/mm²;

F_sx = (F_x¹/L_{1 FEM} + F_x³/L_{3 FEM})/2 = - 2221/186.5)/2 = -8.54 N/mm

F_sy = (F_x²/L_{2 FEM} + F_x⁴/L_{4 FEM})/2= (-2204/275 -2216/313.2)/2 = -7.58 N/mm;

f_xc = (F_y² + F_y⁴) /2/A_x = (659 + 562)/2/181.6 = 3.36 N/mm²;

f_xb = (((M_z² - M_z⁴) /2)(b/2))(1/J_zx) = (((19007+70132)/2)(227/2))(1/779805) = -3.7 N/mm² ;

f_x = f_xc+ f_xb = 3.36 3.7 = 0.88 N/mm²;

f_s = (F_sx + F_sy)/2/t = (-8.54 7.58)/2/0.8 = -10.07 N/mm²;

R_c = f_xc/F_c = 0/5.03 = 0.000;

R_b = f_xb/F_b = 3.7/28.99 = 0.128;

R_s = f_s/F_s = 10.07/11.08 = 0.909;

RF = 2/(R_c + (R_c² + 4(R_s² + R_b²))^.5) = 1.10 ref. . Par. 6.2.12 eq. 2;

RF Summary

Table. 2.10 Minimum RF