Constructal Design

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- BUCKLING AND POSTBUCKLING OF PLATES

Buckling is an instability phenomenon that can occur if a slender and thin-walled plate (plane or curved) is subjected to axial compression. At a certain critical load, the plate will suddenly buckle in the out-of-plane transverse direction. However, plate buckling has a post-critical load-carrying capacity that enables for additional loading after elastic buckling has occurred. A plate is in that sense inner statically indeterminate, which makes the collapse of the plate not coming when elastic buckling occurs, but instead later, at a higher loading level reached in the elasto-plastic buckling. This is taken into consideration in the ultimate limit state design of plates because the elastic buckling does not restrict the load carrying capacity to the critical buckling stress, instead the maximum capacity consists of the two parts: the buckling load added to the additional post-critical load (Åkesson, 2007). In other words, when a load level called critical (Pcr) is achieved the plate is subjected to the elastic buckling, however the ultimate loading capacity (Pu) of plates is not restricted to the occurrence of elastic buckling.

This capacity to carry additional load after elastic buckling is due to the formation of a membrane that stabilizes the buckle through a transverse tension band. When the central part of the plate buckles, it loses the major part of its stiffness, and then the load is forced to be “linked’’ around this weakened zone into the stiffer parts on either side. Additionally, due to this redistribution a transverse membrane in tension is formed and anchored (Åkesson, 2007).

The relative magnitude of the post-buckling strength to the buckling load depends on various parameters such as dimensional properties, boundary conditions, types of loading, and the ratio of buckling stress to yield stress (Yoo and Lee, 2011). There is an analytical solution for the problem of the elastic buckling of a simply supported plate (without perforations) of length L, width H, thickness t, and subjected to a distributed uniaxial compressive load, which is given by (Vinson, 2005):

[pic 1] (1)

where Pcr is the critical load per unit length, π is the mathematical constant, k is the function of aspect radio H/L, m is the wavelength parameter, defined as:

[pic 2] (2)

and D is the plate bending stiffness, determined as:

[pic 3] (3)

being E the Young’s Modulus and v the Poisson’s ratio of the plate material. The optimum value of m that gives the lowest σcr depends on the aspect ratio H/L. For a plate with a large aspect ratio, k = 4.0 serves as a good approximation (Yamaguchi, 1999).

The stress at which elastic buckling occurs, σcr, is defined by the average stress that is equal to the uniformly applied compressive load, Pcr, divided by the thickness of the plate, t. This stress is called elastic buckling stress, defined by:

[pic 4] (4)

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- COMPUTATIONAL MODELS

There are many practical engineering problems for which exact solutions cannot be obtained. This inability to obtain an exact solution may be attributed to either the complex nature of governing differential equations or the difficulties that arise from dealing with the boundary and initial conditions. To deal with such problems, one may resort to numerical methods (Moaveni, 1999). In this context, the ANSYS® software, which is based on the Finite Element Method (FEM), was used to solve elastic and elasto-plastic plate buckling problems.

In the present work the 8-Node Structural Shell finite element called SHELL93 was used. This element is particularly well suited to model curved shells. The element has six degrees of freedom at each node: translations in the nodal x, y and z directions and rotations about the nodal x, y and z axes. It was employed in its quadrilateral form and the deformation shapes are quadratic in both in-plane directions. The element has plasticity, stress stiffening, large deflection, and large strain capabilities (Ansys®, 2005).

- Elastic Buckling

Eigenvalue linear buckling analysis is generally used to estimate the critical buckling load of ideal structures (Vinson, 2005). This numerical procedure is used for calculating the theoretical critical buckling load of a linear elastic structure. Since it assumes the structure exhibits linearly elastic behavior, the predicted critical buckling loads are overestimated. So, if the component is expected to exhibit structural instability, the search for the load that causes structural bifurcation is referred to as a critical buckling load analysis. Because the critical buckling load is not known a priori, the finite element equilibrium equations for this type of analysis involve the solution of homogeneous algebraic equations whose lowest eigenvalue corresponds to the critical buckling load, and the associated eigenvector represents the primary buckling mode (Madenci and Guven, 2006).

The strain formulation used in the analysis includes both the linear and nonlinear terms. Thus, the total stiffness matrix, [K], is obtained by summing the conventional stiffness matrix for small deformation, [KE], with another matrix, [KG], which is the so-called geometrical stiffness matrix (Przemieniecki, 1985). The matrix [KG] depends not only on the geometry but also on the initial internal forces (stresses) existing at the start of the loading step, {P0}. Therefore the total stiffness matrix of the plate with load level {P0} can be written as:

[pic 5]. (5)

When the load reaches the level of {P} = λ{P0}, where λ is a scalar, the stiffness matrix can be defined as:

[pic 6]. (6)

Now, the governing equilibrium equations for the plate behavior can be written as:

[pic 7]. (7)

where is the total displacement vector that may therefore be determined from:[pic 8]

[pic 9]. (8)

At buckling, the plate exhibits a large increase in its displacements with no increase in the load. From the mathematical definition of the inverse matrix as the adjoint matrix divided by the determinant of the coefficients