Numerical simulations and comparisons
4.2 Fully-restrained warping
Chapter 4 – Numerical simulations and comparisons
60 Figure 4.19 shows how warping vanishes at the ends, because of the stiffeners, and at mid-length, because of the symmetry.
Figure 4.19 – Displacement DZ.
61
1 43.34 kN 0.28E-04
2 120.72 kN 0.14E-06
3 120.72 kN 0.37E-07
Table 4.7 – Buckling loads of the first 3 mode.
The undeformed and deformed models of the first mode are shown below:
Figure 4.21 - From the left: undeformed, deformed, upper extremity, lower extremity and top view of the deformed.
It is possible to notice that the instability of mode 1 is of the torsional type, as already observed in the experimental campaign, while the ones of mode 2 and of mode 3 is of a flexural type.
Focusing on the ends of the beam, the effect of the type of constraint is visible. As this is a case of restrained warping, the ends are not deformed.
Contour plot and values in figure 4.22 show that warping is null at the ends and around the stiffened zone. A detail near the end sections is shown in figure 4.23.
Figure 4.22 – Displacement DZ.
Chapter 4 – Numerical simulations and comparisons
62
Figure 4.23 – Displacements DZ near the end sections (warping is null).
Model M8
Model M8 has the stiffener positioned in the interval 380-475 mm, measured from the base and it is analysed in the fully-restrained warping condition.
Table 4.8 shows the results obtained from the buckling analysis of the structure for the first 3 modes.
Mode Ncr Error
1 45.41 kN 0.22E-06
2 118.42 kN 0.37E-06
3 118.44 kN 0.28E-07
Table 4.8 – Buckling loads of the first 3 mode.
The undeformed and deformed models of the first mode are shown below:
Figure 4.24 - From the left: undeformed, deformed, upper extremity, lower extremity and top view of the deformed.
It is possible to notice that the instability of mode 1 is of the torsional type, as already observed in the experimental campaign. Even that of mode 2 is of the torsional type while the instability of mode 3 is of the flexural type.
Figure 4.25 shows that the behaviour is similar to that of Model M8, exception made for the symmetry.
63
Figure 4.25 – Displacement DZ.
Model M9
Model M9 has the stiffener positioned in the interval 95-190 mm, measured from the base and it is analysed in the fully-restrained warping condition.
Table 4.9 shows the results obtained from the buckling analysis of the structure for the first 3 modes.
Mode Ncr Error
1 53.21 kN 0.95E-04
2 98 kN 0.59E-04
3 114.84 kN 0.32E-07
Table 4.9 – Buckling loads of the first 3 mode.
The undeformed and deformed models of the first mode are shown below:
Figure 4.26 - From the left: undeformed, deformed, upper extremity, lower extremity and top view of the deformed.
It is possible to notice that the instability of mode 1 is of the torsional type, as already observed in the experimental campaign. Even that of mode 2 is of the torsional type while the instability of mode 3 is of the flexural type.
Chapter 4 – Numerical simulations and comparisons
64 Figure 4.27 shows that warping vanishes at the ends, near the stiffened portion, and between the top and the stiffeners. The behaviour is, thus, similar to that of Model M2, but here with no warping at the top, and therefore with a stiffer response.
Figure 4.27 – Displacement DZ.
Model M10
Model M10 has the two stiffeners positioned in the interval 95-190 mm from each end. This model is analysed in the fully-restrained warping condition.
Table 4.10 shows the results obtained from the buckling analysis of the structure for the first 3 modes.
Mode Ncr Error
1 84.26 kN 0.54E-04
2 116.40 kN 0.30E-07
3 116.40 kN 0.65E-06
Table 4.10 – Buckling loads of the first 3 mode.
The undeformed and deformed models of the first mode are shown below:
Figure 4.28 - From the left: undeformed, deformed, upper extremity, lower extremity and top view of the deformed.
65 It is possible to notice that the instability of mode 1 is of the torsional type, as already observed in the experimental campaign, while the ones of mode 2 and of mode 3 is of a flexural type.
Figure 4.29 shows that warping occurs only in the part comprised between the two stiffened zoned, and vanishes at the mid-section because of the symmetry.
Figure 4.29 – Displacement DZ.
Model M11
Model M11 has the stiffener positioned in the interval 0-95 mm, measured from the base and it is analysed in the fully-restrained warping condition.
Table 4.11 shows the results obtained from the buckling analysis of the structure for the first 3 modes.
Mode Ncr Error
1 44.10 kN 0.11E-03
2 80.08 kN 0.32E-04
3 119.50 kN 0.32E-07
Table 4.11 – Buckling loads of the first 3 mode.
The undeformed and deformed models of the first mode are shown below:
Chapter 4 – Numerical simulations and comparisons
66
Figure 4.30 - From the left: undeformed, deformed, upper extremity, lower extremity and top view of the deformed.
It is possible to notice that the instability of mode 1 is of the torsional type, as already observed in the experimental campaign. Even that of mode 2 is of the torsional type while the instability of mode 3 is of the flexural type.
Figure 4.31 shows that warping vanishes at the top, around the stiffened zone, and in an intermediate region.
Figure 4.31 – Displacement DZ.
Model M12
Model M12 has the two stiffeners positioned in the interval 0-95 mm from each end. This model is analysed in the fully-restrained warping condition.
Table 4.12 shows the results obtained from the buckling analysis of the structure for the first 3 modes.
Mode Ncr Error
1 52.35 kN 0.15E-04
2 96.49 kN 0.59E-05
3 123.26 kN 0.26E-08
Table 4.12 – Buckling loads of the first 3 mode.
The undeformed and deformed models of the first mode are shown below:
67
Figure 4.32 - From the left: undeformed, deformed, upper extremity, lower extremity and top view of the deformed.
It is possible to notice that the instability of mode 1 is of the torsional type, as already observed in the experimental campaign. Even that of mode 2 is of the torsional type while the instability of mode 3 is of the flexurall type.
Figure 4.33 shows that warping vanishes around the stiffened zones and at mid-length, because of the symmetry.
Figure 4.33 – Displacement DZ.