What every foundryman and designer should know.
Last month, part one of this two-part series focused on the four casting properties of foundry alloys, noting how widely they vary from alloy to alloy. These four properties [ILLUSTRATION FOR FIGURE 1 OMITTED] are metallurgical characteristics of molten metal, and they define alloy "castability." When the characteristics are of certain types and amounts, they combine to make alloys responsive to wide varieties of casting design geometry. Meanwhile, other combinations of these characteristics make successful geometry harder to find.
The essence of last month's article was that: 1) castability affects geometry but 2) well-chosen geometry affects castability. In other words, a geometry can be chosen that offsets the metallurgical nature of the more difficult-to-cast alloys. Knowing how to choose this "proactive" geometry is the key to consistently good casting designs - in any foundry alloy - that are economical to produce, machine and assemble into a final product.
Finally, last month's article was the foundry engineering spectrum of geometry for the benefit of design engineers; this article is the design engineering spectrum of geometry for the benefit of foundry engineers. Geometry found between these two spectrums offers boundless opportunity for castings.
Because castings can easily apply shape to structural requirements, most casting designs are used to statically or dynamically control forces. In fact, castings find their way into the most sophisticated applications because they can be so efficient in shape, properties and cost. Examples are turbine blades in jet engines, suspension components (in automobiles, trucks and railroad cars), engine blocks, airframe components, fluid power components, etc.
When designing a component structurally, a design engineer is generally interested in safely controlling forces through choice of allowable stress and deflection. Although choice of material affects allowable stress and defection, the most significant choice in the designer's structural arsenal is geometry. As we will see, geometry directly controls stiffness and stress in a structure.
The casting processes are limitless in their combined ability to allow variations in shape. Not many years ago, efficient structural geometry was limited by the designer's ability to visualize in 3-D. Now, computer generated solid models and rapid prototypes are greatly enhancing the designer's ability to visualize structural shapes. This technology often leads to casting designs.
Improved efficiency in solid modeling software has led to an interesting design dilemma. Solid models are readily applicable to Finite Element Analysis (FEA) of stress. FEA enables the engineer to quickly evaluate stress levels in the design, and solid models can be tweaked in shape via the software so geometry can be optimized for allowable, uniform stress. Figure 2 depicts a meshed solid model and a stress analysis via the mesh elements.
However, optimum geometry for allowable, uniform stress may not be acceptable geometry for castability. When a foundry engineer quotes a design that considered structural geometry only, requests for geometry changes are likely. At this point, the geometry adjustments for castability may be more substantial than the solid model software can "tweak." The result can be no-quotes, higher-than-expected casting prices, or starting over with a new solid model.
A practical solution to this problem is to concurrently engineer geometry considering structural, foundry and downstream manufacturing needs. The result can be optimal casting geometry. The most efficient technique is to make engineering sketches or marked sections and/or views on blueprints. The idea is to explore overall geometry before locking in to a solid model too quickly. Engineering sketches or markups are easy and quick to change - even dramatically - in the concurrent brainstorming process; solid models are not. A solid model should be the elegant result, not the knee-jerk start.
Our objective is to explore geometry possibilities, looking for an ideal shape that is both castable in the chosen foundry alloy and allowable in stress and deflection for that alloy. As learned in last month's installment, there is great variety in the four metallurgical characteristics that govern alloy castability. Similarly, great variety exists among metals in their allowable stress and deflection. Therefore, an ideal casting shape for all six of the casting design factors in Fig. 1 is not necessarily a trivial exercise. For alloys that have good castability, choosing geometry for allowable stress and deflection is the best place to start. For alloys with less than the best castability, it is better to first find geometry that assists castability, and then modify it for allowable stress and deflection.
Not all alloys are like ductile iron, which is both highly castable and relatively resistant to stress, and moderately resilient against deflection. For ductile iron, many geometries may be equally acceptable. Martensitic high-alloy steel has fair-to-poor castability, but can have amazing resistance to stress and can tolerate very large deflections without structural harm. Therefore, structural geometry is easy to develop, but a coincidental castable shape is more difficult to design. Premium A356 aluminum has good castability, but rather weak resistance to stress and low tolerance for deflection. Carefully chosen structural geometry, however, combined with solidification enhancements in the molding process, has resulted in extremely weight-effective A356 structural components for aircraft, cars and trucks.
5. Section Modulus
Playing with sketches before building a solid model means that we have to find another way to evaluate stress and deflection. This "other way" is the essence of efficient structural evaluation of geometry in casting design.
The equivalent of FEA for the design engineer's structural analysis is computerized "mold filling" and "solidification analysis" for the foundry engineer; the basis for both is a solid model.
The "other way" for the foundry engineer is the manual calculation of gating, solidification patterns and riser sizes; these are established, relatively simple mathematical techniques used long before the advent of solid models.
This "other way" for the design engineer is not so simple. To take full advantage of engineering sketching/print marking as a way to brainstorm geometry, we must be able to quickly evaluate stress and deflection at important cross-sections in the sketches. As the design engineer well knows, the classic formulas for bending stress, torsional stress and deflection are relatively simple. Each, however, contains the same parameter, Section Modulus, which is a function of shape and difficult to compute. Therefore, a quick, simple way to compute or estimate Section Modulus (more specifically, its foundational parameter, Area Moment of Inertia) is needed so that we can move from sketch to improved sketch in our casting geometry brainstorming.
Interestingly, the difficulty in computing Area Moment of Inertia for casting shapes is one of the hidden reasons for the design and use of fabrications. Fabrications are made from building blocks of wrought shapes, like I-beams, rectangular bars, angles, channels and tubes. These shapes, which are simple and constant over their length, have Area Moments of Inertia that are easy to calculate or are available in handbooks. Consequently, stress and deflection calculations are relatively easy. Fabricated designs, however, are heavy and nonuniform in stress compared to a casting well-designed for the same purpose.
Quick Method for Estimating Area Moment of Inertia from Sketches
Although there are five kinds of stress (tension, compression, shear, bending and torsion), the interesting ones for complex structures are bending and torsion, and their equations are shown in Fig. 3. (If more than one type of stress is involved in the same section, the Principle of Superposition allows the individual stress types to be analyzed separately and then added together; once again, the larger of the stresses to be combined are usually from bending or torsion.)
Equations for deflection are very complex-looking and different for each type of load geometry. An example of one of these formulas is shown in Fig. 4. Although it is not an equation, the simplified relationship that is proportional to deflection is also shown in Fig. 3.
In all three cases, the relationships apply to a cross-section of the geometry. It is easy to draw a scale cross-section, whether it be from an engineering isometric sketch or from a marked-up view on a blueprint.
If we can find a way to quickly estimate Area Moment of Inertia, we can readily estimate stresses in our brainstormed sketches as well as estimate whether deflection will increase or decrease. Note that Area Moment of Inertia is in the denominator in each relationship, meaning that increased Area Moment of Inertia reduces stress and deflection.
Maximum tensile stress in bending is often most critical in structural design. Section Modulus is defined as the Area Moment of Inertia divided by the maximum distance from the center of bending (centroid) to the outermost edge of the casting cross-section. Section Modulus is similar to a "stiffness index" because it considers not only magnitude of Area Moment of Inertia, but also maximum section depth. If maximum section depth increases faster than Area Moment of Inertia, a geometry change can actually increase maximum tensile stress, rather than reduce it. This "index" termed Section Modulus accounts for that potential problem.
The estimation method recommended is based on three principles. One is intuitive and the other two are from the mathematics of engineering mechanics. The principles are:
1. The design engineer's sense of load magnitudes and component size/shape - Engineers routinely use this sense to sketch sized shapes that are in the ballpark of the final design. Foundry engineers can learn this "sense," and when they do, they become effective concurrent engineering partners in their customers' casting designs.
2. The equation for Area Moment of Inertia [ILLUSTRATION FOR FIGURE 5 OMITTED] - Although the calculus for an interesting casting cross-section can be very difficult, the relationship expressed between "depth of section" (Y) and "change in cross-sectional area" ([Delta]A) is very simple.
The position and shape of the two rectangles in Fig. 5 (top) clearly demonstrates this simple yet powerful relationship. The change in shape of the inside of the tube (at bottom of the [ILLUSTRATION FOR FIGURE 5 OMITTED]) is an even more dramatic illustration. Calculations weren't made in either case, but the qualitative impact of [Y.sup.2][Delta]A on stiffness and stress is unmistakable.
3. Area Moment of Inertia - Once the engineering sense of structural size and [Y.sup.2][Delta]A have been applied qualitatively to a sketched cross-section, the Parallel Axis Theorem can be applied to simple building blocks in the cross-section to estimate Area Moment of Inertia quantitatively. A numerical value for Area Moment of Inertia is required to calculate the stress level in the sketched cross-section.
The Parallel Axis Theorem is illustrated in Fig. 6 (see Appendix for example equation).
6. Modulus of Elasticity
The measure of a material's stiffness (without regard to material geometry) is known as the Modulus of Elasticity. In the case of metals, it is a function of metallurgy, and it is a mechanical property of the metal alloy. Modulus of Elasticity varies widely among materials, and it varies significantly among metals; that is, some metals are considerably stiffer than others. Alloy groups tend to have the same modulus value; for example the entire family of steels (carbon, low alloy and high alloy) all have the same modulus value of 30 x [10.sup.6] lb/[in..sup.2].
Modulus of Elasticity is an important parameter in structural design, and it is directly involved in the relationship between casting geometry and deflection. A larger Modulus of Elasticity means less deflection. For example, a steel casting would deflect less than an aluminum casting of identical geometry simply because steel is stiffer than aluminum.
As an aside, foundrymen may know more about Modulus of Elasticity than they think they do; it is simply the elastic slope of the stress/strain diagram created when the foundry's metallurgical lab pulls a test bar. Figure 7 illustrates qualitatively the results of pulled test bars for common groups of foundry alloys. The steepness of the elastic slope of each graph indicates the alloy group's stiffness.
One subtlety about Modulus of Elasticity is that it is not affected by heat treatment. However, heat treatment can affect the height of the elastic slope. This is very important because the height at which the elastic slope begins to curve is called the metal's "yield stress." This is the stress level at which plastic deformation begins and the metal is permanently affected. Stresses should be designed below this level so that deflections in the casting under load do not damage it.
For example, consider the family of steels in Fig. 7; heat treatment can considerably raise the point at which an alloy steel yields. Although the steel is no stiffer at higher stress levels, it can withstand the additional stress without damage. The same is true for heat-treatable aluminum alloys, but the magnitude of heat treatment effect on yield stress is considerably less than that for steels.
Figure 8 and the Appendix illustrate a hypothetical casting design using the recommended six factors behind good geometry selection. The first four factors (metallurgical characteristics) detailed last month describe the alloy's "castability." The final two factors, the focus of this article, are from engineering mechanics and are Modulus of Elasticity and Section Modulus, an aspect of Area Moment of Inertia.
As a ductile iron casting design (see last month's article for the four characteristics of castability for ductile iron), this example is intended to illustrate structural geometry more than geometry for castability. As noted previously, for alloys that are highly castable like ductile iron, it is convenient to focus first on geometry for structure and let the alloy's friendly foundry characteristics adapt to the structural needs.
Briefly, as ductile iron, the casting could be made in a horizontally-parted sand mold with the center cylindrical section pointed down. One core would form the "tongue and groove" tabs, bolt holes and hollowed center of the cylinder. A second core would form the top side of the l-beam feature and the corresponding bottom side of the four-hole plate. Two risers would feed solidification shrinkage in the center section from the tab sides of the four hole plate. A third riser would follow the side of the second core and feed the cylindrical end of the I-beam section.
The appendix on page 51 illustrates the main point: This casting design is nothing more than an engineering sketch with a sense of size and proportion. Using the "quick method" of sketching cross-sectional areas, Area Moment of Inertia can be estimated with simple building blocks and minimal calculation. Once a value is known, stress can be easily calculated for the chosen cross-section. A relative measure for deflection can be easily calculated as well.
Final design would be a solid model, based on at least two or three sketched iterations of combined structural and castable geometry. Detailed structural evaluation could then be done via FEA. Any remaining stress problems could be easily solved by tweaking the solid model, which is already close to optimal geometry. Finally, the solid model could be modified to add risers and a gating system so that computer analysis of solidification and mold filling could verify the geometry chosen for castability.
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|Title Annotation:||Cost-effective Casting Design, part 2|
|Author:||Gwyn, Michael A.|
|Date:||Jun 1, 1998|
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