Computational Methods in Engineering

---

In this study, Buckling Limit of Strain (B.L.S.) is introduced as one of the most important limiting factors in cold roll forming process. B.L.S. is calculated by the finite element procedure. Then for two particular processes with existing analysis and experimental results, B.L.S. has been determined and evaluated. LUSAS 12.3 is used for finite element analysis. The results show that when buckling of the sheet metal is the limiting factor, B.L.S. is in good agreement with practical limits. It has also been shown that flower pattern can be well predicted when B.L.S. is obtained and this idea is another new outcome from this study. Using this criterion to define and determine B.L.S. and to design the flower pattern is a new concept accomplished for the first time. Keywords: Cold Roll-Forming, Nonlinear Finite Element Analysis, Local Buckling

---

It is empirically established that, due to a number of factors involved, a classical (linear) analysis of buckling pressure is impossible. Nonlinear theories of buckling are, therefore, required that involve effective factors such as imperfections and welding effects. In this study, models are developed which are as close to allowable standard deviations as possible. In the next stage, their buckling behavior is investigated both experimentally and numerically using finite element packages ADINA, ANSYS, COSMOS, and MARC based on specific capabilities of each. Results show that reasonable estimates of real buckling pressure will become possible when material and geometrical nonlinearities and initial imperfections are introduced into the analytical system. Finally, in the light of the results obtained, a submarine pressure hull is analyzed.

---

This paper is concerned with elastic local buckling of rectangular plates subjected to intermediate and end inplane loads. Since closed form solution for buckling analysis of plates with different end conditions and subjected to intermediate loads is complicated, numerical methods are more useful. Because of restrictions on the two finite strip methods, the longitudinal B3 spline expressions combined with conventional transverse shape functions are used as displacement functions. This method is computationally more efficient than the finite element method, more flexible in boundary treatment, and more accurate in dealing with point forces and axial loads than the conventional finite strip method. Local buckling coefficients are presented for plates under intermediate and end inplane loads which are useful for design of steel walls or plates that support intermediate floors/loads.

---

Considering the vulnerability of double-layer grid space structures to progressive collapse phenomenon, it is necessary to pay special attention to this phenomenon in the design process. Alternate path method is one of the most appropriate and accepted methods for progressive collapse resistant design of structures. Alternate Path Method permits local failure to occur but provides alternate paths around the damaged area so that the structure is able to absorb the applied loads without overall collapse. Following the sudden initial local failure event, severe dynamic effects may arise which should be taken into account in determining the realistic collapse behavior of the structure. In this paper, a new methodology based on alternate path method is presented to apply dynamic effects of initial local failure. The method is called nonlinear dynamic alternate path method. Due to its capability to take account of dynamic nature of the failure, this method can be used to evaluate realistic collapse behavior of the structure and to investigate the vulnerability of the structure to progressive collapse phenomenon.

---

Buckling under compression is a common phenomenon for thin-walled structures. Under certain conditions, they can also buckle locally under tensile loads. In this paper, the buckling of cracked panels is investigated. Effects of some parameters such as the length and orientation of crack, length, width and thickness of the panel and boundary conditions on the compressive and tensile buckling loads are determined. The results show that the dimensions of panels have the least and characteristics of crack have the most effect on the buckling.

---

In this paper, buckling behavior of functionally graded carbon nanotube-reinforced composite (FG-CNTRC) plates is studied in line with the plates thikness. All  governing equations are presented incrementally, based on a First-order Shear Deformation Theory (FSDT) of plates and von Karman strain field. In order to find the critical buckling load, the axial load is applied to the plate incrementally and the equilibrium equations are solved by Dynamic Relaxation (DR) method. Parametric study of the effects of volume fraction of Carbon Nanotubes (CNTs), CNTs distribution, plate width-to-thickness ratio and aspect ratio of nano composite plates is done in detail. The results show that functionally graded distribution of CNTs causes a significant increase of critical buckling load.

---

---

A semi-analytical finite strip method was developed for the buckling analysis of laminated composite plates based on zigzag and third order shear deformation theories. The displacement functions of the plates were evaluated using a continuous harmonic function series in the longitudinal direction that satisfied the simply supported boundary conditions and a piecewise interpolation polynomial in the transverse direction. By considering the displacement-strain relations and strain-stress relations, the standard and geometric matrices were evaluated using the virtual work principle. The numerical results related to the buckling of single-layer and multi-layer plates were presented based on two different plate theories. The effects of different  boundary conditions, length to thickness ratio, fiber orientation and modulus of elasticity were also investigated through numerical examples.

---

In this paper, three-dimensional displacement response of a cylindrical sandwich panel with compressible core under the action of dynamic pulse loading is addressed using the extended high order sandwich panel theory. Also, local dynamic pulse buckling of facesheets is studied by considering the Budiansky-Roth buckling criterion. It is assumed that the sandwich panels consist of orthotropic face sheets and an isotropic viscoelastic foam core layer. The effects of various parameters including the panel span, core and facing thickness, pulse duration and maximum pressure on the non-linear dynamic response and buckling strength of the sandwich cylindrical panel are studied. The results obtained from the present method are compared with finite element solutions using the commercial software ANSYS and those reported in the literature, showing a good agreement. It is revealed that applied core non-linear theory could be satisfactory for the dynamic pulse response of sandwich viscoelastic panels. It is also shown that the pulse buckling strength of panel increases with a decrease of the panel radius or an increase of the panel thickness.

---

Plates made of laminated composite materials with variable stiffness can have wide applications in various branches of engineering due to such advantages as high strength /stiffness to weight ratio. In these composites, curved fibers are used to reinforce each lamina instead of the straight fibers. In this paper, the application of finite strip method for the buckling analysis of moderately thick composite plates with variable stiffness is investigated. For buckling analysis, a semi-analytical finite strip method based on the first-order shear deformation theory is employed. In this method, all displacements are presumed by the appropriate harmonic shape functions in the longitudinal direction and polynomial interpolation functions in the transverse direction. The minimum potential energy method has been used to develop the stability formulations. This analysis examines the effect of using curved fibers instead of straight fibers on the laminate composites. The critical loads obtained from this analysis are compared with those of other researchers and the efficiency and accuracy of the developed finite strip method are confirmed. Comparison of the analysis results of these plates shows that changing the slope of the fibers can lead to a significant change in the buckling response. Also, increasing the number of the terms of shape functions in the longitudinal direction has a significant effect on the convergence to the desired results.

---

Continuously varying cross-section members have found wide applications in engineering for cost and resistance optimization. Since steel structures generally have more slender members compared to concrete structures, buckling analysis of steel members is of more importance. Determining the critical load of functionally varying cross-section columns using the analytical solution is a time-consuming process. In this paper, buckling analysis of non-prismatic steel columns is conducted using the meshless local Petrov-Galerkin (MLPG) method. In meshless methods, the scattered nodes are used rather than the elements to model the problem domain and its boundaries. The change of the inertia moment within the length of a column is characterized by introducing a power function with variable taper ratio and exponent. The radial basis function is used to discretize the differential equation governing the buckling. The penalty method is used for the imposition of the boundary conditions. Numerical examples of the critical buckling load for prismatic and non-prismatic columns using the proposed method are compared with the analytical solution, and the effectiveness of the MLPG method for buckling analysis of non-prismatic columns is validated. Also, buckling analysis of muscle column members subjected to non-uniform axial load is carried out to show the efficiency of the proposed method. The effect of several parameters such as non-uniformity of the load and variation of the cross-section on the buckling load of the column is discussed in details.

---

In this paper, static buckling of homogeneous beams coated by a functionally graded porous layer with different boundary conditions is investigated based on the Timoshenko beam theory. The principle of virtual work has been used to obtain the governing equations. Two different methods, namely analyticalsolution and numerical solution are used to solve the governing equations and extract the buckling force. The governing equations are coupled as a series of ordinary differential equations. In the analytical solution, these equations are first uncoupled using a series of mathematical operations, and are then solved. The obtained solution has a series of parameters and unknown constants. Using the boundary conditions at the boundaries of the beam, a homogeneous system of equations is extracted, from which the axial buckling force is obtained. In the numerical solution, the generalized differential quadrature method is used to solve the static equations. Finally, the numerical results are presented and the effects of various parameters such as thickness to beam length ratio, porous layer thickness, porosity parameter, etc. on the buckling of the beam are investigated. Comparison of the results obtained from the two analytical and numerical solution methods confirms the accuracy and validity of both methods.
 

---

In this article, mechanical buckling analysis of tapered beams having constant width and variable thickness, made of two-dimensional functionally graded materials is studied. The beam is assumed to be made of metal and ceramic, where their volume fractions vary in both longitudinal and thickness directions based on the power law. The beam is generally subjected to combined concentrated and distributed axial loads. The set of governing equations are derived using the Principle of Minimum total Potential Energy (PMPE), and are solved numerically using Differential Quadrature Method (DQM) for clamped-free boundary conditions. Convergence and accuracy of the presented solution are confirmed for both cases of concentrated and distributed axial loads. The effects of different parameters on the critical buckling load of the beam for both load cases are studied including geometrical parameters, gradation indices in longitudinal and thickness directions, and variation of thickness. Also buckling analysis of the beam under a combination of concentrated load and distributed axial loads of linear, quadratic and exponential types are investigated. Numerical results show that the highest values of the critical buckling load belong to the linear distributed load, and the lowest value is owned by exponential load.