Plastics Engineered Product Design 1 Episode 7 pot

220 Plastics Engineered Product Design In the following formula P = axial load; L = length of column; I = least moment of inertia; 12 = least radius of gyration; E = modulus of elasticity; y = lateral deflection, at any point along a larger column, that is caused by load P. If a column has round ends, so that the bending is not restrained, the equation of its elastic curve is: d2Y El - = -pv dx2 (4-12) When the origin of the coordinate axes is at the top of the column, the positive direction of x being taken downward and the positive direction of y in the direction of the deflection. Integrating the above expression twice and determining the constants of integration give: El P = Q7L2 - 12 (4-13) which is Euler’s formula for long columns. The factor 0 is a constant depending on the condition of the ends. For round ends i2 = 1; for fixed ends i2 = 4; for one end round and the other fixed B = 2.05. Pis the load at which, if a slight deflection is produced, the column will not return to its original position. If P is decreased, the column will approach its original position, but if P is increased, the deflection will increase until the column fails by bending. For columns with value of L/k less than about 150, Euler’s formula gives results distinctly higher than those observed in tests. Euler’s formula is used for long members and as a basis for the analysis of the stresses in some types of structural parts. It always gives an ultimate and never an allowable load. Torsi o n A bar is under torsional stress when it is held fast at one end, and a force acts at the other end to twist the bar. In a round bar (Fig. 4.9) with a constant force acting, the straight-line ab becomes the helix ad, and a radial line in the cross-section, ob, moves to the position ad. The angle bad remains constant while the angle bod increases with the length of the bar. Each cross section of the bar tends to shear off the one adjacent to it, and in any cross section the shearing stress at any point is normal to a radial line drawn through the point. Within the shearing proportional limit, a radial line of the cross section remains straight after the twisting force has been applied, and the unit shearing stress at any point is proportional to its distance from the axis. 4 . Product design 221 Round bar subject to torsion stress ya dA I The twisting moment, T, is equal to the product of the resultant, P, of the twisting forces, multiplied by its distance from the axis, p. Resisting moment, T,, in torsion, is equal to the sum of the moments of the unit shearing stresses acting along a cross section with respect to the axis of the bar. If dA is an elementary area of the section at a distance of z units from the axis of a circular shaft [Fig. 4.9 (b)], and c is the distance from the axis to the outside of the cross section where the unit shearing stress is Z, then the unit shearing stress acting on dA is (ZZ/C) dA, its moment with respect to the axis is (zz2/c) dA, and the sum of all the moments of the unit shearing stresses on the cross section is f (rz2/c) dA. In this expression the factor fz2 dA is the polar moment of inertia of the section with respect to the axis. Denoting this by J the resisting moment may be written zJ/c. The polar moment of inertia of a surface about an axis through its center of gravity and perpendicular to the surface is the sum of the products obtained by multiplying each elementary area by the square of its distance from the center of gravity of its surface; it is equal to the sum of the moments of inertia taken with respect to two axes in the plane of the surface at right angles to each other passing through the center of gravity section of a round shaft. The analysis of torsional shearing stress distribution along noncircular cross sections of bars under torsion is complex. By drawing two lines at right angles through the center of gravity of a section before twisting, and observing the angular distortion after twisting, it has been found from many experiments that in noncircular sections the shearing unit stresses are not proportional to their distances from the axis. Thus in a rectangular bar there is no shearing stress at the comers of the sections, and the stress at the middle of the wide side is greater than at the middle of the narrow side. In an elliptical bar the shearing stress is greater along the flat side than at the round side. 222 Plastics Engineered Product Design It has been found by tests as well as by mathematical analysis that the torsional resistance of a section, made up of a number of rectangular parts, is approximately equal to the sum of the resistances of the separate parts. It is on this basis that nearly all the formulas for noncircular sections have been developed. For example, the torsional resistance of an I-beam is approximately equal to the sum of the torsional resistances of the web and the outstanding flanges. In an I- beam in torsion the maximum shearing stress will occur at the middle of the side of the web, except where the flanges are thicker than the web, and then the maximum stress will be at the midpoint of the width of the flange. Reentrant angles, as those in l-beams and channels, are always a source of weakness in members subjected to torsion. The ultimate/failure strength in torsion, the outer fibers of a section are the first to shear, and the rupture extends toward the axis as the twisting is continued. The torsion ula for round shafts has no theoretical basis after the shearing stresses on the outer fibers exceed the proportional limit, as the stresses along the section then are no longer proportional to their distances from the axis. It is convenient, however, to compare the torsional strength of various materials by using the formula to compute values of z at which rupture takes place. Sandwich The same or different materials are combined in the form of sandwich structures (Fig. 4.10). They can be used in products with an irregular distribution of the different materials, and in the form of large structures or sub-structures. A sandwich material composed of two shns and a different core material is similar to RP laminates. Overall load-carrying capabilities depend on average local sandwich properties, but materials failure criteria depend on local detailed stress and strain distributions. Design analysis procedures for sandwich materials composed of linear elastic constituents are well developed. In principle, sandwich materials can be analyzed as composite structures, but incorporation of viscoelastic properties will be subject to the limitations discussed throughout this book. Structures and sub-structures composed of a number of different components and/or materials, including traditional matcrials, obey the same principles of design analysis. Stresses, strains, and displacements within individual components must be related through the character- istics (anisotropy, viscoelasticity, and so on) relevant to the particular material, and loads and displacements must be compatible at component 4 . Product design 223 Honeycomb core sandwich structure (Courtesy of Plastics FALLO) interfaces. Thus, each individual component or sub-component must be treated. Load and support conditions for individual components depend on the complete structure (or system) analysis, and are unknown to be deter- mined in that analysis. As an example, if a plastic panel is mounted into a much more rigid structure, then its support conditions can be specified with acceptable accuracy. However, if the surrounding structure has comparable flexibility to the panel, then the interface conditions will depend on the flexural analysis of the complete structure. In a more localized context, structural stiffness may be achieved by ribbing and relevant analyses may be carried out using available design formulae (usually for elastic behavior) or finite element analysis, but necessary anisotropy or viscoelasticity complicate the analysis, often beyond the ability of the design analyst. Design A structural sandwich is a specially shaped product in which two thin facings of relatively stiff, hard, dense, strong material is bonded to a relatively thick core material. With this geometry and relationship of mechanical properties, facings are subjected to almost all the stresses in transverse bending or axial loading. The geometry of the arrangement provides for high stiffness combined with lightness, because the stiff facings are at a maximum distance from the neutral axis, similar to the flanges of an I-beam. Overall load-carrying capabilities depend on average local sandwich properties, but material failure criteria depend on local detailed stress and strain distributions. Design analysis procedures 224 Plastics Engineered Product Design and fabricating procedures for sandwich materials composed of linear elastic constituents are well developed and reported in the literature. In principle, sandwich materials can be analyzed as RP composites. The usual objective of a sandwich design is to save weight, increase stiffness, use less expensive materials, or a combination of these factors, in a product. Sometimes, other objectives are also involved such as reducing tooling and other costs, achieving smooth or aerodynamic smoothness, reducing reflected noise, or increasing durability under exposure to acoustic energy. The designers consider factors such as getting the loads in, getting the loads out, and attaching small or large load-carrying members under constraints of deflection, contour, weight, and cost. To design properly, it is important to understand the fabrication sequence and methods, use of the correct materials of construction, the important influcncc of bond between facing materials and core, and to allow a safety factor that will be required on original, new developments. Use of sandwich panels are extensively used in building and, construction, aircraft, containers, etc. The primary function of the face sheets is to provide the required bending and in-plane shear stiffness, and to carry the axial, bending, and in-plane shear loading. In high-performance structures, facings most commonly chosen are RPs (usually prepreg), solid plastic, aluminum, titanium, or stainless steel. The primary function of a core in structural sandwich parts is that of stabilizing the facings and carrying most of the shear loads through the thickness. In order to perform this task efficiently, the core must be as rigid and as light as possible, and must deliver uniformly predictable properties in the environment and meet performance requirements. Several different materials are used such as plastic foam, honeycomb [using RE’, film (plastic, steel, aluminum, paper, etc.), balsa wood, etc.]. Different fabricating processes are used. These include bag molding, compression molding, reinforced reaction rejection molding (RRIM), filament winding, corotational molding, etc. There is also the so-called structural foam (SF) that is also called integral skin foaming or reaction injection molding. It can overlap in lower performance use with the significantly larger market of the more conventional sandwich. Up until the 1980s in the U.S., the RIM and SF processes were kept separate. Combining them in the marketplace was to aid in market penetration. During the 1930s to 1960s, liquid injection molding (LIM) was the popular name for what later became RIM and SF (Chapter 1). These structures are characterized as a plastic structure with nearly uniform density foam core and integral near-solid skins (facings). When 4 - Product design 225 these structures are used in load-bearing applications, the foam bulk density is typically 50 to 90% of the plastic’s unfoamed bulk density. Most SF products (90wt%) are made from different TPs, principally PS, PE, PVC, and ABS. Polyurethane is the primary TS plastic. Unfilled and reinforced SFs represents about 70% of the products. The principal method of processing (75%) is modified low-pressure injection molding. Extrusion and RIM account for about 10% each. In a sandwich design, overall proportions of structures can be established to produce an optimization of face thickness and core depth which provides the necessary overall strength and stiffness requirements for minimum cost of materials, weight of components, or other desired objectives. Competing materials should be evaluated on the basis of optimized sandwich section properties that take into account both the structural properties and the relative costs of the core and facing materials in each combination under consideration. For each combination of materials being investigated, thickness of both facings and core should be determined to result in a minimum cost of a sandwich design that provides structural and other hnctional requirements. Sandwich configurations are used in small to large shapes. They generally are more efficient for large components that require significant bending strength and/or stiffness. Examples of these include roofs, wall and floor panels, large shell components that are subject to compressive buckling, boat hulls, truck and car bodies, and cargo containers. They also provide an efficient solution for multiple fimctional requirements such as structural strength and stiffness combined with good thermal insulation, or good buoyancy for flotation. Sandwich materials can be analyzed as composite structures. Structures and sub-structures composed of a number of different components and/or materials, including traditional materials, obey the same principles of design analysis. Stresses, strains, and displacements within individual components must be related through the characteristics (anisotropy, viscoelasticity, etc.) relevant to the particular material; also loads and displacements must be compatible at component interfaces. Thus, each individual component or sub-component must be treated using the relevant methods. Load and support conditions for individual components depend on the complete structure (or system) analysis. For example, if a panel is mounted into a much more rigid structure, then its support conditions can be specified with acceptable accuracy. However, if the surrounding structure has comparable flexibility to the panel, then the interface conditions will depend on the flexural analysis of the complete structure. 226 Plastics Engineered Product Design In a more localized context, structural stiffness may be achieved by ribbing, and relevant analyses may be carried out using available design formulae (usually for elastic behavior) or finite element analysis. But necessary anisotropy or viscoelasticity complicate the analysis, often beyond the ability of the design analyst. Primary structural role of the face/core interface in sandwich con- struction is to transfer transverse shear stresses between faces and core. This condition stabilizes the faces against rupture or buckling away from the core. It also carries loads normally applied to the panel surface. They resist transverse shear and normal compressive and tensile stress resultants. For the most part, the faces and core that contain all plastics can be connected during a wet lay-up molding or, thereafter, by adhesive bonding. In some special cases, such as in a truss-core pipe, faces and core are formed together during the extrusion process, resulting in an integral homogeneous bond/connection between the components. Fasteners are seldom used to connect faces and core because they may allow erratic shear slippage between faces and core or buckling of the faces between fasteners. Also, they may compromise other advantages such as waterproofing integrity and appearance. For RP-faced sandwich structures the design approaches includes both the unique characteristics introduced by sandwich construction and the special behavior introduced by RP materials. The overall stiffness provided by the interaction of the faces, the core, and their interfaces must be sufficient to meet deflection and deformation limits set for the structures. Overall stiffness of the sandwich component is also a key consideration in design for general instability of elements in compression. In a typical sandwich constructions, the faces provide primary stiffness under in-plane shear stress resultants (N.), direct stress resultants (N, Ny), and bending stress resultants (M, My). Also as important, the adhesive and the core provide primary stiffness under normal direct stress resultants (Nz), and transverse shear stress resultants (& Q). Resistance to twisting moments (Tx. TYZ), which is important in certain plate configurations, is improved by the faces. Capacity of faces is designed not to be limited by either material strength or resistance to local buckling. The stiffness of the face and core elements of a sandwich composite must be sufficient to preclude local buckling of the faces. Local crippling occurs when the two faces buckle in the same mode (anti- symmetric). Local wrinkling occurs when either or both faces buckle locally and independently of cach other. Local buckling can occur 4 - Product design 227 ~~~ under either axial compression or bending compression. Resistance to local buckling is developed by an interaction between face and core that depends upon the stiffness of each. With the structural foam (SF) construction, large and complicated parts usually require more critical structural evaluation to allow better prediction of their load- bearing capabilities under both static and dynamic conditions. Thus, predictions require carehl analysis of the structural foam's cross-section. The composite cross-section of an SF part contains an ideal distribution of material, with a solid skin and a foamed core. The manufacturing process distributes a thick, almost impervious solid skin that is in the range of 25% of overall wall thickness at the extreme locations from the neutral axis where maximum compressive and tensile stresses occur during bending. When load is applied flatwise the upper skin is in compression while the lower one is in tension, and a uniform bending curve will develop. However, this happens only if the shear rigidity or shear modulus of the cellular core is sufficiently high. If this is not the case, both skins will deflect as independent members, thus eliminating the load-bearing capability of the composite structure. In this manner of applying a load the core provides resistance against shear and buckling stresses as well as impact (Fig. 4.11). There is an optimum thickness that is critical in designing this structure. When the SF cross-section is analyzed, its composite nature still results in a twofold increase in rigidity, compared to an equivalent amount of solid plastic, since rigidity is a cubic hnction of wall thickness. This TP 4 Core thickness vs density impact strength Thickness : 'I." Impact Strength 1 % Oenwty Aeducllon 228 Plastics Engineered Product Design Sandwich and solid material construction x- X Sandwich cross-section with and without a core T T increased rigidity allows large structural parts to be designed with only minimal distortion and deflection when stressed within the recommended values for a particular core material. When analyzing rigidity and the moment of inertia (I) can be evaluated three ways. In the first approach, the cross-section is considered to be solid material (Fig. 4.12). The moment of inertia (Ix) is then equal to: ix = ~/12 (4- 14) where b = width and h = height. This commonly used approach provides acceptable accuracy when the load-bearing requirements are minimal. An example is the case of simple stresses or when time and cost constraints prevent more exact analysis. The second approach ignores the strength contribution of the core and assumes that the two outer skins provide all the rigidity (Fig. 4.13). The equivalent moment of inertia is then equal to: ix = qh3 - h3112) (4- 1 5) This formula results in conservative accuracy, since the core does not 4 - Product design 229 re Sandwich and I-beam Cross-section contribute to the stress-absorbing function. It also adds a built-in safety factor to a loaded beam or plate element when safety is a concern. A third method is to convert the structural foam cross-section to an equivalent I-beam section of solid resin material (Fig. 4.14). The moment of inertia is then formulated as: /x = [bh3 - (b - bl)(h - 2 tx)3] /I 2 (4-16) where bl = b(E,)/(€J, E,= modulus of core, E,= modulus of skin, t, = skin thickness, and h, = core height This approach may be necessary where operating conditions require stringent load-bearing capabilities without resorting to overdesign and thus unnecessary costs. Such an analysis produces maximum accuracy and would, therefore, be suitable for finite element analysis (FEA) on complex parts. However, the one difficulty with this method is that the core modulus and the as-molded variations in skin thicknesses cannot be accurately measured. The following review relates to the performance of sandwich constructions such as those with RP skins and honeycomb core. For an isotropic material with a modulus of elasticity (E), the bending stiffness factor (EI) of a rectangular beam b wide and h deep is: A rectangular structural sandwich with the same dimensions whose facings and core have moduli of elasticity Efand E,, respectively, and a core thickness c, the bending stiffness factor EI becomes: E/ = ~(bh3/12) (4-1 7) E/=(f&/12)(h3 -6) +(fcb/12) 6 (4-18) This equation is OK if the facings are of equal thickness, and approximate or approximately equal, but the approximation is close if the facings are thin relative to the core. If, as is usually the case, E, is much smaller than Efi the last term in the equation is deleted. 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Number of Teeth 12 13 14 15 16 17 18 19 20 21 22 24 26 28 30 50 10 0 15 0 300 Rack - 14 lIz-deg Involute or Cycloidal 20-deg Full Depth Involute 20-deg Stub Tooth lnvolu te 20-deg Internal Full Depth Pinion Gear 0. 210 0.220 0.226 0.236 0.242 0.2 51 0.2 61 0. 273 0.283 0.289 0.292 0.298 0.3 07 0. 314 0.320 0.352 0.3 71 0. 377 0.383 0.390 0.245 0.2 61 0. 276 0.289 0.259 0.302 0.308 0. 314 0.320 0.3 27 0.330 0.336 0.346... sills Running boa rdslsafety a ppln Brake system Misc hardware Trucks Material Total weight 6800 4 410 640 2 910 460 510 10 70 a60 18 00 9640 4420 16 50 15 70 10 60 212 00 59000 11 .5 7. 5 1. 1 4.9 0.8 0.9 1. a 1. 5 3.0 16 .3 7. 5 2.8 2 .7 1. 8 35.9 10 0.0 x 10 0 z RP component percentages = 30 c Steel component percentages = 70 F W process was used to fabricate both Glasshopper car bodies It was determined that this process... 650 880 1, 070 1, 325 1, 575 1, 800 2,500 3,200 Outside diameter, inches Nominal weight, pounds Maximum weight, pounds 1. 50 2.50 4.44 6.25 8.06 9. 87 12 .56 16 .00 20.06 24.50 28 .75 32 .75 44. 81 56.50 Overall length, inches 1. 60 2.63 4.62 6.56 8.48 10 .35 13 .18 16 .80 21. 07 25 .73 30 .1 5 34.42 47. 06 59.32 For a sphere with the stresses uniform in all directions, it follows that the fibers require equal orientation... cylinder shell support against 248 Plastics Engineered Product Design ~- ~ ~ Figure 4 .1 7 Example of a design for 10 ,000 gallon gasoline RP storage tank t ,r K 3Q ! aA &i ! lank Wall: 2iY Spray-up 12 S Mat Woven Roving Rib Top: 12 ' Filament Winding 12 S Mat Woven Roving Sides: 25' Mat Woven Roving Feet: 12 S Mat Woven Roving O W Filament Winding 12 S Mat Woven Roving Bottom: 12 5' Mat Woven Rovina added over... negotiability, uncoupled Cubic capacity Tare weight Gross rail load AAR clearance diagram 50 ft 3% in 51 ft 55/8in 52 ft 11 in 55 ft 6l/2 in 53 ft 7/ a in 42 ft 3 in 10 ft 8 in 15 ft 1 27/ 32 in 12 in 15 ft 6 in 4 20 in x 44 ft 73 /4 in 15 0 ft 500 ft3 59,000 Ib 263,000 Ib Plate 'C' The second prototype car Glasshopper I1 that was later put into service had three compartments The tare weight of the second car was... and I= moment of inertia For a cylinder: I = R (04 - d4)/64 in.4 (4- 27) 242 Plastics Engineered Product Design Figure 4 .1 6 Cylinder comparison of thickness for a flat end and a hemispherical end 1. 8 I I - - Flat 5 5 5 - P 0.6 0.2 - I I I end Hemispherical end 1. 6 1. 4 n u 1. 2 c 1. 0 0 c x U 0.8 0.4 0, 0 2 4 6 8 Radius, inches 10 12 This stress must then be considered in addition to the longitudinal... tanks has widened the size range from 2,0 81 to 18 1,632 liter (550 to at least 48,000 gal) and the range ? in typical diameters from about 1. 22 to 3.35 m (4f? to a t least 11 f) The tank configuration is basically cylindrical, in order to provide the required design volumes within the established envelope of heights and widths Length ranges are from 5.5 to 11 m (18 to 36 ft); they are well within practical... force of 22 ,70 0 kg (50,000 Ib) applied at the coupler pulling face - 4 Product design 253 3 Lift the car free of the truck bolster by jacking at the coupler shank, 11 ,0001b) was required a vertical force of 50,400 kg (1 4 Lift the car free of the truck bolster by jacking at the lifting lug/jacking pad assembly to verify torsional rigidity and stability, a vertical force of 31, 78 0 kg (70 ,000 Ib) was... equipment can handle including thicknesses and product complexity and capability to package and ship to the customer 238 Plastics Engineered Product Design (Chapters 1 and 2) The ability to achieve specific shapes and design details is dependent on the way the process operates and plastics to be processed Generally the lower the process pressure, the larger the product that can be produced With most labor-intensive . 1. 25 0.80 0.50 Light shock 1. 25 1. 5 1 .oo 0.80 Medium shock 1. 5 1. 75 1. 25 1 .oo Heavy shock 1. 75 2.00 1. 5 1. 25 234 Plastics Engineered Product Design filled or reinforced laminated. 30 50 10 0 15 0 300 Rack 0. 210 0.220 0.226 0.236 0.242 0.2 51 0.2 61 0. 273 0.283 0.289 0.292 0.298 0.3 07 0. 314 0.320 0.352 0.3 71 0. 377 0.383 0.390 0.245 0.2 61 0. 276 0.289. 0.308 0. 314 0.320 0.3 27 0.330 0.336 0.346 0.352 0.358 0.480 0.446 0.459 0.4 71 0.484 0. 311 0.324 0.339 0.348 0.3 61 0.3 67 0. 377 0.386 0.393 0.399 0.405 0. 41 5 0.424 0.430