PRACTICAL IDEAS FOR MATERIAL SAVING IN DESIGN

Very rarely can any product be developed completely on an analytical basis. Judgment and the designer’s personal choice add a touch of “originality”. In this section we present several practical concepts concerning strength and weight. Each concept should be viewed as a special interpretation or application of the general theories.

4.8.1 Slimming Down

Slimming down or carve-out means “shave material from where it is not needed”. A good designer can see through the part and visualize the flow of forces. He can cut holes or thin down sections where there is little flow of force. Experimentally, if many samples of an existing product are available, holes can be made and material can be removed in suspected low-stress areas. The

original and modified samples will be tested for failure, and their strength-to-weight ratios will be compared. The one with the highest strength-to-weight ratio is the most efficient. This structure does not have to be the strongest. If the strength of the most efficient design is lower than the required strength, its shape should be retained.

4.8.2 Achieving Balance through Reinforcement

Reinforcement means “patching up” the weak spots. Through such an operation, a balanced design in which there are no weak spots or every section has the same safety factor, can be obtained. Even though reinforcement can be achieved by adding webs of the same material, more often composite materials are used to take advantage of the strengths of different materials. A material which is weak in tension (e.g. concrete or plastic) may be strengthened by steel, glass or carbon fibres. The stiffer material will carry most of the force. The reinforcing material should be stiffer than the reinforced material (the matrix). A typical relationship for fibre and plastic is shown in Fig. 4.12. The uniaxial tensile strength and modulus are in between the values for the fibre and the plastic. The fracture strain is about the same as that for the fibre. The fibre-reinforced plastic is stronger and more rigid than the matrix, but may not increase resistance to impact. Since concrete is weak in tension, steel bars are embedded at places of high, tension to strengthen concrete.

Fig. 4.12 Stress–strain diagrams for fibre, plastic and composite. 4.8.3 Cardboard Model of Plate Structures

Cardboard models can be used to show the relative deflections and stress distribution in steel plate structure. Failure modes, weak spots and stress concentrations can be estimated by simulation. Since paper models are inexpensive, such an experimental method is very useful. Deflections are especially easy to observe because of the flexibility of paper material.

It has long been observed by designers that a box frame with diagonal braces (see Fig. 4.13(a)) has exceptional torsional rigidity. Stiffness in lathe beds provides a typical example of diagonal bracing. Figure 4.13(b) shows that torsion creates shear stresses along the horizontal and vertical lines on the surface of the cylinder. Maximum compressive stress occurs along right-hand helices and maximum tensile stresses occur along left hand helices on the cylinder. When the braces in Fig. 4.13(a) make an angle of 45° with the sides, the braces are subjected to in-plane bending. The behaviour of various members of the frames can be evaluated simply by cutting them or replacing them with more flexible members.

Fig. 4.13 Torsion of a box frame and a tube.

4.8.4 Soap Film Analogy of Torsional Shear Stresses

The following discussion is based on research done by Agrawal [2].

The stresses in a cylindrical member under a torsional load are easy to visualize (Fig. 4.13(b)). Fortunately, most torsional members, such as shafts and pulleys are indeed, circular in cross-section. If the section is non-circular, the stress distribution is so complex that most designers lose conceptual feel. During a practical study of the behaviour of a split tube (roll pin) under torsion, it is seen that a transverse plane which is initially flat becomes warped (Fig. 4.14).

Fig. 4.14 Torsion of a split tube.

The membrane analogy first stated by Ludwig Prandtl (1903), in Germany is based on the similarity of the stress function (f) and the height function Z as shown in Fig. 4.15(a). The crosssection of the member is replaced by a wire frame, which is an exact duplicate of the circumferential contour of the section. The wire frame is covered by membrane. Pressure is applied to one side of the membrane so that a dome is formed. The maximum slope of the membrane at a point represents the shear stress at that point. In twisting of shells, the protruding convex corners carry very little shear stresses, whereas sharp internal and external comers should be avoided because of stress concentration. The centre of a circle or the middle of a narrow web are locations of zero slope and zero stress. The directions of the shear stresses in the cross-section are contour lines (constant z-lines) on the membrane. The volume under the membrane is proportional to the torsional strength of the member. The analogy is extended to torque and direction of shear stress in Figs. 4.15(b) and 4.15(c), respectively.

Fig. 4.15 Prandtls’ analogy for non-circular prisms in torsion.

Figure 4.16 shows that sections which are open (with a split on the circumference) are much weaker than closed sections, the volume under the dome increases greatly if the section is generally round or closed. Agrawal [2] states: “To increase strength, increase the volume of the analog bubble”. If the section is an open shape, such as a thin blade, an I-section, T-section or U-section or a Z-section, then the torsional strength is proportional to the product of the sectional area, A, and the web thickness t. This implies that the shape of the section is rather unimportant.

Fig. 4.16 Analogy for torsion of hollow and solid non-circular sections.

If the section is hollow instead of solid, the loss in strength is very small. The internal hole can be represented by the thin weightless plate which is cemented to the membrane. Because of this plate, the hollowed area has a zero slope and no stress. The web adds nothing to the torsional strength because it bisects an internal hollow space. The membrane model of the web is flat, and the volume under the dome remains constant regardless of the presence of the web.