<p> Gaining insight into the pressworking properties of laminated sheet metal is the aim of this work. One of the deformation processes in which the difference in behaviour between single and laminated sheet metal is most distinct and possibly the easiest to analyze is that of pure plastic bending. A bending theory, initially proposed by Crafoord, is further developed to analyze the pure bending of laminated metals. The bending behaviour of single and laminated nonstrain hardening and strain hardening sheets, with and without Bauschinger effect, is treated extensively from a theoretical point of view. </p> <p> Stretch forming, bending and deep drawing tests on laminated sheets are also performed experimentally. It is found that the orientation of the laminated sheet during the deformation process has a significant influence on the bending behaviour and the deep drawability of laminated sheet. The change in deep drawability can be qualitatively predicted from the bending behaviour. </p> / Thesis / Master of Engineering (MEngr)

The Layerwise Shell Theory is used to model discretely stiffened laminated composite cylindrical shells for stress, vibration, pre-buckling and post-buckling analysis. The layerwise theory reduces a three-dimensional problem to a two-dimensional problem by expanding the three-dimensional displacement field as a function of a surface-wise two-dimensional displacement field and a one-dimensional interpolation through the shell thickness. Any required degree of accuracy can be obtained by an appropriate, independent selection of the one-dimensional interpolation functions through the thickness and the two-dimensional interpolation of the variables on the surface. Using a layerwise format, discrete axial and circumferential stiffeners are modeled as two-dimensional beam elements. Similar displacement fields are prescribed for both the stiffener and shell elements. The contribution of the stiffeners to the membrane stretching, bending and twisting stiffnesses of the laminated shell is accounted for by forcing compatibility of strains and equilibrium of forces between the stiffeners and the shell skin. The layerwise theory is also used to model initial imperfections of the unstressed configuration. A finite element scheme of the layerwise model is developed and applied here to investigate the effect of imperfections on the response of laminated cylindrical shells. Using a finite element model of the layerwise theory for shells and shell stiffener elements, the accuracy and reliability of the elements is investigated through a wide variety of examples. The examples include laminated stiffened and unstiffened plates and shells and stand-alone beams under different types of external destabilizing loads. Finally, the study identifies the particular types of problems where the layerwise elements possess a clear advantage and superiority over the conventional equivalent single-layer models. / Ph. D.

The effect of laminate thickness on the fracture behavior of laminated graphite/epoxy (T300/5208) composites has been studied. The predominantly experimental research program included the study of the [0/±45/90]_ns and [0/90]_ns laminates with thicknesses of 8, 32, 64, 96 and 120 plies and the [0/±45]_ns laminate with thicknesses of 6, 30, 60, 90 and 120 plies. The research concentrated on the measurement of fracture toughness utilizing the center-cracked tension, compact tension and three point bend specimen configurations. Fracture toughness was computed using the stress intensity factor results of a finite element stress analysis of each specimen geometry which treated the composite as homogeneous but anisotropic. The development of subcritical damage at the crack tip was studied nondestructively using enhanced x-ray radiography and destructively using the laminate deply technique. The test results showed fracture toughness to be a function of laminate thickness. The fracture toughness of the [0/±45/90]_ns and [0/90]_ns laminates decreased with increasing thickness and asymptotically approached lower bound values of 30 ksi√in (1043 MP a√mm) and 25 ksi√in (869 √mm) respectively. The fracture toughness of the [0/±45/90]_ns laminate was independent of crack length at 8 and 120 plies. The fracture of the thin and thick [0/±45/90]_ns laminates were self-similar, macroscopically. However, the [0/90]_2s laminate (8 plies) exhibited fracture toughness values that increased sharply as a function of increasing crack size. This was attributed to large axial splits which formed perpendicular to the crack tip in the 0° plies and extended in the direction of applied load. The fracture toughness of the [0/90]_ns laminate was independent of crack length at 90 plies. The axial splits in the 0° plies of the thicker specimens were confined to the surface and the final fracture was self-similar. For both the [0/±45/90]_ns and [0/90]_ns laminates, the center-cracked tension, three-point bend and compact tension specimens gave comparable results. In contrast to the other two laminates, the fracture toughness of the [0/±45]_ns laminate increased sharply with increasing thickness but reached an upper plateau value of 40 ksi√in (1390 MP a√mm) at 30 plies. Fracture toughness was independent of crack size at 6 and 90 plies. The 6 ply specimens failed by an apparent uncoupling mechanism where the two interior -45° plies delaminated from the adjacent +45° plies and failed by matrix splitting parallel to the fibers. The surface 0° plies failed by broken fibers along a +45° line in association with matrix splitting parallel to the fibers in the +45° plies. The thick [0/±45]_ns laminates exhibited a surface boundary layer in which 45° fiber breaks and splits were evident along with delaminations. However, the interior of the specimens failed in a self-similar manner with fibers in the 0° plies breaking along a line collinear with the starter notch. The compact tension and three-point bend specimens defined a constant fracture toughness at about 15% below the plateau exhibited by the center-cracked tension specimens. The general toughness parameter model, a strain criterion developed by C. C. Poe, Jr. of NASA Langley, was the only candidate thin laminate failure criterion that was successful in using thin laminate parameters to predict the fracture of thick laminates. The "universal” general toughness parameter value of 1.5 √mm quite closely predicted the fracture of the thick laminates. / Ph. D.

The area of penetration of phenol-formaldehyde resin into aspen (<i>Populus tremuloides</i>) and Douglas-fir (<i>Pseudotsuga menziesii</i>) flakes, and the layer properties of yellow poplar (<i>Liriodendron tulipifera</i>) flakeboard were investigated to determine how they were influenced by various pressing parameters. The evenness of penetration was found to be a function of the natural variability of the wood and was not influenced by the pressing parameters of temperature, moisture content, pressure, or time. These four parameters were found to influence the area of penetration by controlling the viscosity and flow of the resin. The temperature, gas pressure, and platen pressure history at any particular plane through a flakeboard panel thickness were found to be directly and interactively determined by the pressing parameters of platen temperature, initial mat moisture content, and press closing time. The specific gravity profile was observed to be a function of the press closing time while platen temperature and the mat moisture content influenced the amount of springback which the panel exhibited. Layer-shear and the corresponding specific gravity at a particular plane were found to be similarly dependent on environmental history. The layer-shear strength increased consistently toward the surface of the panel and exhibited a lower coefficient of variation nearer the surface. / M.S.

Two straight and one curved Southern Pine glued-laminated beams are subjected to bending, bending and compression, and biaxial bending. The beams are tested at service loads and only one beam is loaded to failure. The analysis is performed by using the finite element program ABAQUS. The 3-D beam finite element model used is formulated on engineering theory and includes shear deformations. The engineering elastic constants needed to specify the constitutive matrix of the beam element are obtained experimentally. Strain measurements in wood with small bonded strain gages are not accurate. A simple clip-type transducer is fabricated and used to measure the strains in the test glulam beams. The distribution of normal stresses is investigated for symmetrically applied concentrated loads. Experimental and analytical values agree very well. / M.S.

The strength reduction of a graphite-epoxy laminate due to dropping plies is investigated experimentally and analytically. Laminates were tested under uniaxial tension and compression. All the laminates were flat on one side with the plies dropped from the middle of a [( ± 45/0/90)_s (N_d) ( ± 45/0/90)_s]_T laminate, where N_d denotes the number of dropped-plies and their orientations. A total of 54 specimens were tested consisting of eight dropped-ply configurations, or values of N_d. This geometry creates an eccentric load path which causes local bending moments in the region of the ply drop-off. The strength of a laminate with dropped-plies is less than the strength of its thin section, and the compression specimens exhibited a lower strength than a tension specimen of the same configuration and width. For the laminates in this study, the reduction in strength is directly related to the axial stiffness change between the thick and thin sections of the laminate. The three-dimensional state of stress in the laminate was evaluated by the finite element method. The magnitude of the interlaminar stresses at the ply drop-off for N_d = [0₄]_T are greater than for N_d = [90₄]_T . The initial failure event for N_d = [0₄]_T was a delamination between the dropped plies and upper sublaminate at the drop-off. The tensile interlaminar failure criterion predicts this as the critical location in the finite element model as well. However, the tensile interlaminar criterion underestimates the failure initiation load. The cause for this may be due to the inaccuracies in the modeling of the ply drop-off geometry in the finite element analysis. The magnitude of the stresses at the ply drop-off are sensitive to changes in the finite element mesh geometry. The N_d = [90₄]_T laminates were stronger than the N_d = [0₄]_T laminates and the N_d = [90₄]_T laminates failed in the thin section away from the drop-off. / M.S.

Three dimensional contact problems of square orthotropic laminates indented by a rigid spherical indenter are solved. Simplified problems of indentations of beam and isotropic square plate are studied first to develop an efficient numerical technique and to gather the knowledge of the shape of the contact area in order to solve for the three dimensional orthotropic cases. The approach combines an exact solution method in conjunction with a simple discretization numerical scheme. Numerical sensitivity due to the ill-posed nature of the problem was experienced but was cured by enhancing the numerical approach with a least square spirit. Well agreement is obtained by comparing the results of these simplified studies with available published solutions. For isotropic plate, contact area is found to be either a circle or a hypotrochoid of four lobes featured with a shorter length of contact along the through-the- corner directions of the plate. Hertz's theory fails earlier than assuming the contact area to be a circle. In-plane dependence of the contact stress is presented to illustrate the difference of contact behavior between a square plate and a circular plate. Load-indentation relation reveals indenting a square plate is harder than indenting a circular plate of a diameter equal to the side length of the square plate. Solutions of multi-layered orthotropic cases are achieved by employing a modified analytical approach with the same numerical method. Three different configurations of plate are implemented for the orthotropic case, namely, a single layered magnesium (Mg) plate, which is slightly orthotropic, and a single and double layered plates of graphite-epoxy (G-E), which are highly orthotropic. Results for the (Mg) plate agrees with the previous isotropic case. Concept of modifying the previous hypotrochoids is introduced to seek for the contact stresses for comparatively large indentation conditions. Single-layered (G-E) plate was implemented for small indentations. The result supports the validity of Hertz's theory for small indentation and shows a relatively longer contact length in the direction of less stiffness. Two layered (G-E) plate illustrates similar distributions for the contact stresses along both of the in-plane directions with a smaller range of validity of Hertzian type behavior than the previous cases. The boundary effect prevails at the initial stage of indentation but is overcome by the effect of material orthotropy as the indentation proceeds. Thus, the contact area for small indentation appears to be the same kind of hypotrochoids as located in the isotropic case but changes to be the other type of hypotrochoids as the indentation advances. / Ph. D.

Free-edge effects in composite laminates were studied experimentally. Strains were determined and compared on a ply-by-ply basis for the curved edges of a hole in thick composite panels and along the straight free-edge of the panels. The experimental technique of moire interferometry was extended to take measurements of in-plane deformations on singly-curved surfaces. A replication scheme was developed to produce high-frequency diffraction gratings on singly-curved surfaces. Two different techniques were developed to interrogate specimen gratings on 25.4 mm (1 in.) diameter holes. Eight thick composite laminates from three material systems were tested in uniaxial compression on a screw-driven testing machine. Interlaminar deformations were measured at the straight free-edge on four of the specimens. Strain distributions on the straight free-edge were compared with FEM results for two specimens. Good agreement was obtained for one specimen while poor agreement was obtained for the other. / Ph. D.

The effects of layer waviness on the compression response of T3001P1700 carbon/polysulfone composite laminates were investigated both experimentally and analytically. A three-step procedure was used to fabricate isolated layer waves into the central 0° layer of [90₂,/0₂/90₂/0₂/90₂/0_2w]_S laminates. The influence of various layer wave geometries on the static compression strength and compression fatigue life were determined experimentally. Moire interferometry was used to investigate the disturbance in the displacement fields and the modes of deformation associated with layer waviness under compression loading. The state of stress in the vicinity of the layer waves and the influence of the layer waves on static compression strength were predicted using a planestrain finite element analysis which included material nonlinearity. / Ph. D.

The free vibration and nonlinear transient analysis of imperfect laminated structures with emphasis on computational methods for accurate and efficient analysis are studied. The evaluation of interlaminar stresses is also studied by approximating global displacements of laminated plates. Free vibration response of imperfect laminated structures is studied in the presence of geometric and stress imperfections. The stress imperfections are the initial stresses as caused by preloads. Using a 48 degrees of freedom thin shell element, the effect of complex, arbitrary in-plane and out-of-plane loads on the transverse vibrations of thin arbitrarily laminated plates, cylindrical panels, and hyperbolic shells without and with geometric imperfections is analyzed. The, effects of geometric parameters (aspect ratio and panel curvature) and material properties (varying the number of layers but keeping the same laminate thickness) of imperfect plates are examined. The nonlinear transient response of imperfect structures is next obtained using the direct time integration schemes as applied to the full set of equations and also using reduction methods. Two time integration schemes, the Newmark method and the Wilson () method, are first tested on a series of linear and nonlinear examples without and with geometric imperfections. Reduction methods using the normal modes and Ritz vectors as the base vectors are employed to reduce the size of the nonlinear problem and thus save computational resources. The resulting reduced (but still coupled) set of equations is integrated in a step-by-step fashion using the aforementioned time integration schemes along with an iterative scheme for dynamic equilibrium. Also, the nonlinear dynamic response of imperfect plates subjected to impact loads is studied. The evaluation of the loads (due to a projectile) depends on a contact law which relates contact forces with indentation. The well-known Hertzian law and its previously proposed modification are incorporated. The transient response of an example problem is obtained using both full and reduced equations of motion. Finally, for accurate determination of interlaminar shear and normal stresses of laminated structures, a postprocessor for displacement-based finite element solutions of laminated plates under transverse loads is developed. The postprocessor can be used for the finite element solutions that have been obtained using either the classical laminated plate theory or the first order shear deformation theory. The equilibrium equations of elasticity are integrated directly. These equations include the influence of the products of in-plane stresses for geometrically nonlinear problems. To obtain accurately the derivatives of in-plane stresses the finite element nodal displacement data is first interpolated using polynomials with global support (Le., the interpolating polynomials are defined over the whole domain). Two types of polynomials, Chebyshev and a class of orthogonal polynomials that can be generated for a given location of known data points are used. / Ph. D.