Similarly, the appropriate boundary conditions for an isolated stiffened plate element were obtained by subjecting a stiffened plate model to uniaxial compression, by monitoring the displacements at the boundaries of the elements. The obtained boundary conditions are used in the finite element analysis of the element model of an isolated stiffened plate, so that conclusions can be drawn on the failure mode, ultimate strength and stress distribution, together with a comparison between shell and cubic elements.
Fundamental Theory
Stiffened Plates
Stiffeners can be fixed on one side of the plate (single-sided) or on both sides (double-sided). The geometry of a longitudinally stiffened slab is fully described by: slab length (L), total slab width (B), span between two stiffeners (s), slab thickness (tp), web height (hw), web thickness (tw) and the width of the flange (bf) and the thickness of the flange for T and angle brace stiffened plate (tf).
Buckling of Stiffened Plates
- Collapse Mode of Stiffened Plates
- Overall Buckling vs Local Buckling
- Stiffened Plate’s Overall Buckling Stress and Stiffeners Beam Column Buckling
- Long Plate Buckling Stress
I=Moment of inertia of section consisting of bracing together with plate D=bending stiffness of plating,. Axial stress in the brace is greater than the external applied stress "σα" due to the reduced width of the slab.
Buckling Linear Finite Element Analysis
Nonlinear Finite Element Analysis
In nonlinear analysis, the stiffness matrix [K] and/or the load vector {f } in the structural equations, [K] {u} = {f }, become functions of the displacements, {u}. This property of the Riks method makes it possible to trace the behavior after a limit point is reached, even though the stiffness matrix is not positive definite.
FEM Analysis
Three-span Hull Model with Transverse Frame
- Model Geometry, Mesh and Material Properties
- Boundary Conditions
- Linear Analysis
- Nonlinear Analysis
The investigation will focus on the horizontal stiffened plate in the middle of the deck, as the hull is imposed on the overhang. The main dimensions of the stiffener vary and not all construction plates have the same thickness. There are five types of cuts in construction as shown in FIGURE 4.2 and their dimensions depend on the dimensions of the cross stiffener.
The first eigenmode located at the deck originates at the horizontal stiffened plate in the middle of the deck (FIGURE 4.6). During the analysis, the Reaction Moment-Rotation curve of the Hull reaches its peak on linear trend and decreases slowly. The stress-rotation curve of the horizontal stiffened plate in the middle of the deck reaches its peak at linear trend and drops rapidly.
It is noted that the maximum reaction moment of the torso and the maximum tension of the plate are not achieved at the same rotation. As shown in FIGURE 4.18, nodes on the longitudinal edge of the plate have a common displacement in the transverse direction.
One-span Hull Model
- Model Geometry, Mesh and Material Properties
- Boundary Conditions
- Analysis Considering Free Transverse Displacement for Transverse Section
The first eigenmode located to the deck occurs at the horizontal braced plate in the center of the deck. Critical buckling stress of the braced plate, which is part of the one-span model, is slightly lower compared to the corresponding result from the three-span model. The conditions at the plate boundaries are similar to those for the three-span model analysis.
As shown in FIGURE 4.32, when the plate is shortened, the node at the center of the transverse edge remains stationary. 48 It is observed (FIGURE 4.33) that nodes on the longitudinal edge of the plate have common displacement through the transverse direction. Collapse mode also differs from that predicted from three-span analysis of the model.
The critical buckling stress of the stiffened plate that is part of the single-span model is slightly higher compared to the equivalent derived from the three-span model. The plate boundary conditions are not similar to those of Hull's three-span model analysis.
One-span Deck’s Stiffened Plate Model
- Model Geometry, Mesh and Material Properties
- Boundary Conditions
- Transverse Edge Nodes are Free to Move in Transverse Direction, Longitudinal
- Transverse Edge and Longitudinal Edge Nodes have zero Transverse
The stress-strain curve of the stiffened plate is almost identical to the equivalent curve obtained from the analysis of the single-span hull model. Compared to the stress-strain curve of the stiffened plate obtained from the analysis of the three-span model, the maximum stress value is slightly lower and is reached for lower strain. The same collapse mode is observed for the stiffened plate in the equivalent analysis of the single-span hull model.
The boundary conditions used ensure that nodes on the longitudinal edge of the stiffened plate have a common displacement in the transverse direction. The stress-strain curve of the stiffened plate is almost identical to the equivalent curve obtained from the analysis of the single-span Hull model. Compared with the stress-strain curve of a stiffened plate obtained from the analysis of the three-span hull model, the maximum stress value is slightly higher and is achieved at a lower strain and with a completely different failure mode.
It should be noted that for the equivalent single-span hull model, the stiffened plate collapses in the same way. The displacement of the joints at the boundaries of the stiffened plate elements shall be monitored, to obtain appropriate boundary conditions that realistically represent the behavior of an isolated stiffened plate element.
Model mesh, Geometry and Material Properties
Boundary Conditions
Transverse Edge Nodes are Free to move in Transverse Direction, Longitudinal
- Linear Analysis
- Nonlinear Analysis
All the curves in FIGURE 5.7 reach the peak value for the same shear stress, the same deformation construct loses stability for it. The vertical displacement of the joints in Path 1 is greater than that of the joints in Path 2 in all three stiffeners. 77 tighten in place, but at the same time contribute to local twisting of the stiffener.
Again, by taking advantage of symmetry and investigating the vertical displacement of the nodes on the paths between the reinforcements, it is possible to infer the shape of the slab when the structure collapses. The key to approximating the conditions encountered by each stiffened panel element at its longitudinal edges is the relative transverse displacement between the stiffener and the longitudinal edges of the element. The transverse displacement of the reinforcement is measured through path 2 of FIGURE 5.8, and the element boundaries are the paths shown in FIGURE 5.10.
As shown in FIGURES 5.12 and FIGURE 5.13, each structural member has identical specimen conditions at its longitudinal boundaries. Therefore, it is observed that at the intersection nodes with the same longitudinal coordinate as the grid half-waves, the relative transverse displacement with the longitudinal edges increases or decreases rapidly.
Transverse Edge and Longitudinal Edge Nodes have Zero Transverse Displacement
- Nonlinear Analysis
The largest displacements when instability occurs show a combination of stumbling of the stiffeners with slight local buckling of the walls near the longitudinal ends of the stiffeners. Near the ends of the stiffeners, the transverse displacement of the nodes on path 3 is greater than that of the nodes on path 1 and path 2, there the web of the stiffener bends locally, which also contributes to local buckling of the stiffener. However, the boundary conditions force the nodes at the longitudinal edges of the stiffened plate not to move in the transverse direction.
Thus, a force acts on the nodes of the nodes in the longitudinal direction, preventing them from translating in the transverse direction (FIGURE 5.23). The ratio of the load acting on the transverse and longitudinal edges is not constant because the slab shortens in the longitudinal direction (FIGURE 5.24). As shown in FIGURE 5.25, a stiffened plate subjected to biaxial loading with σz/σx=6 is expected to fail under a failure mode in which the entire plate deflects between the stiffeners and stiffener tripping.
As shown in FIGURE, each of the construction's elements experiences equivalent conditions at its longitudinal boundaries. When structure collapses, stiffeners trip in the middle and web buckles locally near the ends of the stiffeners.
Model Geometry, Mesh, and Material Properties
This chapter aims to investigate the behavior (collapse mode, ultimate strength) of an isolated stiffened plate element under compressive loads. By observing how the isolated element reacts to compressive loads and comparing its behavior with that of a stiffened slab, it can be assessed whether a slab can be studied by analyzing only one of its elements (stiffener and attached cladding). The homogeneous solid model was discretized by elements of type C3D8R with length 20 mm, width 20 mm and height 4 mm.
At the junction between web and flange/plate elements 20 mm long, 4 mm wide and 4 mm high used. On the other hand, the size of the solid elements processed by the limitations due to the computational cost and the needs for an accurate analysis. It is an isotropic material having Young's modulus/modulus of elasticity (E) of 206 GPa, Poisson's ratio (ν) of 0.3 and yield strength (σy) of 315 MPa.
Boundary Conditions
Transverse Edge Nodes are Free to move in Transverse Direction, Longitudinal
- Linear Analysis
- Nonlinear Analysis
In the solid element model, the hourglass energy is higher than that of the shell element model, which is expected to result in a stiffer behavior for the solid element model. As shown in FIGURE 6.7, the theoretical buckling load is achieved neither for the model with shell elements nor for the model with solid elements. For both models, the braced plate element is predicted to collapse under nearly the same load and strain.
For the shell element analysis, the maximum compressive stress is equal to 258 MPa for a shear stress of 0.00144 and for the cube element analysis, 260 MPa and 0.00147, respectively. The maximum displacements when the member collapses match those obtained from the stiffened plate having equivalent boundary conditions. The results obtained from the single element analysis of a stiffened plate appear to be a good estimate of the flexural capacity of a stiffened plate.
The buckling capacity of an element is 258/260 MPa for shell/cubic elements and that of a stiffened plate with 6 stiffeners is 270 MPa. The shortening strain before failure of the element is for shell/cubic finite elements and 0.00144 for stiffened plate with shell elements.
Transverse Edge and Longitudinal Edge Nodes have Zero Transverse Displacement
Considering free transverse displacement of nodes on transverse cross-sections, the nodes on longitudinal edges of the stiffened plate have common displacement in the transverse direction. At the same time, transverse shear on the boundaries of the stiffened plate elements was investigated for both cases. By underestimating the frame's transverse stiffness, the ultimate strength of the tire's stiffened plate is also underestimated.
Similarly, an overestimated transverse stiffness of the transverse frames overestimates the ultimate strength of the stiffened plate on the deck. Transverse boundary conditions appear to have a large influence on the response of the longitudinal boundaries of a stiffened plate. For the cases studied in this work, the ultimate strength of the stiffened plate elements could vary up to 6.1% and 4.7% depending on the boundary conditions.
Boundary conditions simulating the behavior of an isolated stiffened plate element appear to be dependent on the boundary conditions of the stiffened plate. The collapse mode of the stiffened plate is accurately estimated by the analysis of an isolated element (stiffened with attached plate).