Experimental investigation and FEA of AlMg3-stiffened rectangular plate subjected to concentrated load

In this paper, AlMg3-plates are studied through experimental and numerical using finite element representation under concentrated load at the center point. The plates of (300- (cid:215) 200 mm) are clamped at the shorter ends and strengthened longitudinally by one rib at the centerline and two at different spans. the stiffened plates were modeled using a 3-D 10-node tetrahedral element with a non dimensional analysis. The models were validated using the results of tests on full-size stiffened plate specimens and were subsequently used to perform the study of the parameters presented in this paper. The parameters investigated are: the maximum stress, deflection of the plate and the position of ribs. Effect of the investigated parameters on the concentrated load strength were studied within elastic range. FEA give closer results with those of experimental and these results show that the use of two parallel ribs with a 40-mm span improves the strength of the plate. Due to these results, further investigation is presented to show the optimum thickness of the ribs at the best span.


Introduction
For a given loading, the stiffness of a plate or shell structure can be increased significantly by the addition of ribs or stiffeners. Previously, the optimization techniques are mainly on the sizing of the ribs. The more important issue of identifying the optimum location of the ribs has received little attention. Thin aluminum plates that are stabilized in one direction by stiffeners are used extensively in the structures in which a high strength-to-weight ratio is important. More than 7000 series wrought aluminum alloys are used for aerospace applications because of their combination of high strength, stress-corrosion-cracking resistance, and toughness [1]. Being economical and strong enough, stiffened plates are widely used in all kinds of circumstances, such as bridges, ship hulls or decks, and aircraft structures. Numerous studies have analyzed stiffened plates. Earlier researchers simulated stiffened plates with orthotropic models [2] or grillage models [3]. However, these models did not achieve satisfying results in solving generalized stiffened plate problems. Hence, subsequent researchers tended to regard the plate and the stiffener separately and then combine them by imposing the displacement compatible conditions between the plate and the stiffener. Several methods have been developed, such as the Rayleigh-Ritz method [4][5][6][7], the finite element method (FEM) [8][9][10], and the constraint method based on finite elements [11]. Of all these methods, finite element methods are the most convenient and can be used in large, complex structures due to the advances in computing that have made the methods extensively used in industry. As an alternative to FEMs, the element-free method [12], mesh-free or meshless methods [13][14][15][16] have recently been made popular in engineering analysis. Unlike the meshing process in the FEM, the mesh-free methods only discretize the domain of a problem to a set of scattered points. The mesh-free methods are flexible in many instances, such as moving boundary problem, crack growth with arbitrary and complex paths, and phase transformation problems. In such cases, FEMs can encounter difficulties in dealing with discontinuities that do not coincide with the original mesh lines. It is obvious that re-meshing will be needed in each step of the solution procedures, which will be quite involved. Mesh-free methods construct the approximation solution of a problem entirely in terms of a set of nodes that are distributed in the problem domain, and no elements or any other interrelationship among the nodes are needed. Without the need for meshes, the meshless methods avoid the disadvantages of FEMs. By Lam and Santhikumar [17], a methodology has been developed for the automatic determination of the optimum locations of the ribs for a given set of design constraints.
Graciano and Johansson [18] present a design procedure for the determination of the ultimate resistance of longitudinally stiffened girder webs to concentrated loads. The influence from the longitudinal stiffener is considered in the slenderness parameter, through the buckling coefficient. This procedure is harmonized with other design procedures currently used for describing buckling problems in steel structures. An expression is developed for the buckling coefficient based on finite element analysis. The interaction between the web plate with flanges and a longitudinal stiffener was considered in the analysis. The ultimate strength according to the design procedure presented and the results are compared with available experimental results. Also they investigated the interaction with bending.
Peng et. al. [19] present an element-free Galerkin (EFG) method for the static analysis of concentrically and eccentrically stiffened plates based on first-order shear deformable theory (FSDT). The stiffened plates were regarded as composite structures of plates and beams. Their results show good agreement with the existing analytical and finite element solutions.
Virág [20] investigated the minimum cost design of longitudinally stiffened plates using the strength calculation methods. Deflections due to lateral pressure, compression stress and shrinkage of longitudinal welds are taken into account in the stress constraint. The self-weight is added to the lateral pressure. The local buckling constraint of the base plate strips is formulated as well.
Three-dimensional finite element analyses are used by Moreira et. al. [22] to calibrate the stress intensity factor in a cracked stiffened plate subjected to remote uniform traction. An accurate numerical determination of the stress field and stress intensity variation through the thickness of a central cracked plate was first carried out in order to evaluate three-dimensional effects. A stiffened cracked plate was then analyzed, taking into account the results and the conclusions obtained in the previous study. Such a structure was chosen due to the growing interest for large integral metallic structures for aircraft applications, following the continuous need for low cost and the emergence of new technologies. The J-Integral technique was used to calculate the values of the stress intensity factor along the plate thickness. The plane strain behavior near the crack front and the variation of the opening stress are discussed.
Based on the higher-order global-local theories, a finite element model is proposed by Li Li and Ren Xiaohui [23] to study the bending behavior of stiffened laminated plates. The proposed model treats the embedded stiffeners as the part of laminated plate, so that the compatibility of displacements and stresses between the plate and the stiffeners can be automatically satisfied. Distributions of the displacements and stresses through the thickness of laminates were also given for the first time, which can serve as references for future investigations as such information is lacking in the published literature. In addition, the impact of the stiffeners on the bending response of the stiffened laminated plates is also studied in terms of the quantity, the collocation and the geometry of stiffeners. Numerical results showed that the higher-order global-local theories are more suitable for predicting the bending response of thick and moderately thick stiffened laminated plates compared to the first order theory commonly used in engineering. By varying the quantity, the collocation and the geometry of stiffeners, the stiffness and the strength of stiffened laminated plates can be remarkably improved.

Experimental Work:
The test rig and the test specimen's plates are shown in figure (1) shows. The plates are (300-× 200-mm) and are made of AlMg3 (3.1 Mg, 0.03 Mn, 0.31 Fe, 0.09 Si). Thickness of the flat one (unstiffened) are 2-mm. The other plates (stiffened) are machined -at the Shaheed Companyfrom 4mm-thick plates to get equal thickness, 2-mm, for plates and stiffeners. First, one longitudinal stiffener is used at the centerline of the plate. Other models are stiffened by two parallel ribs with span, w, equal to 40, 80, and 18 mm as shown in figure (2). After machining, twostage annealing was applied -at the Labs of Mechanical Engineering Dept, University of Anbarin order to develop as much as possible homogenous microstructure of the tested plats. During the first step of annealing simulating homogenization, the dissolution of the intermetalic particles at 565°C takes place as it was found in previous papers [24]. During subsequent annealing at 450°C precipitation and particle coarsening should take place, leading to increased mobility of grain boundaries and the development of a rather homogenous grain structure [25].
Load is concentrated at the mid point of each tested plate and increased gradually from zero to the maximum (within elastic limit). Finally, the Baty dial-gauge (see Fig.1&2) is used to measure the deformation of the models.

Numerical Analysis
The numerical analysis was done using Finite Element technique. The models used in the experimental part are simulated using the finite element software, ANSYS11. The plates are clamped at the shorter ends and meshed with a 3-D 10-node tetrahedral element. At the center of the model, load is concentrated then nonlinear with large displacement analysis is used to solve each case. Figure (3) shows contour plots of the total displacement (left column, d1-d5) and of the Mises stress distribution (right column, S1-S5) for all models under maximum load.

Result And Discussion
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Further Investigation :
Based upon the results, advanced analysis may be presented to study the strengthened plate with two neighbored ribs, which is the best case. Thickness of the rib (t) is studied numerically (see figure 5).

Conclusions
There were several ways to increase the strength of plates. One of these ways is the use of stiffeners. Stiffeners are used not only to strengthen the plates, hulls, structures, and other parts, but they also used to minimize the weight and cost. For these reasons and others, strengthening plates by using of longitudinal ribs is one of the important and practical ways and need to deep investigation. Position, number and dimensions of the stiffeners play an important role to get optimum results. As it is noted from the results of this scientific research, there is an optimum use of these factors or in the other side; it may be no intention to use ribs. In this work, strengthening of

Thickness of the stiffener, mm
Non-dimensional Unity Stress Deflection the plate with two parallel ribs and a certain thickness at a small span give best results, whereas other choices give poor results. No doubt the modification in boundary conditions may change these results, so more studies are recommended for other cases.