## P

Fig. 11-27. Stress and Strain. Stress is load, P, divided by area, A Strain is the change in length, AL, divided by the Initial length, L

We express it as the modulus of elasticity* or Young's modulus, E:

Values for E were shown in Table 11-52. Metals typically start by exhibiting a linear relationship between stress and strain. Strain in this region is termed elastic because it will return to zero after the load is removed.

Beyond a stress called the proportional limit (normally assumed to be the same value in tension and compression), a material's stress/strain curve is no longer linear. In other words, if we design our structure such that its material is stressed above the proportional limit, linear methods of analysis would no longer apply. This can be risky because so many of our methods of analysis are based on the assumption of linearity; any other assumption would make loads analysis, in particular, so cumbersome it would be impractical. Inelastic effects influence structural stability more than anything else. An effective guideline for preliminary design is to keep the design ultimate compressive stress below the material's proportional limit.

Above the elastic limit, which is often indistinguishable from the proportional limit but can be higher, the material will undergo residual strain {plastic strain), which remains after the load is removed. Convention has defined the yield stress to be the stress that would cause the material to have a residual strain of 0.2%. Although the material actually begins to yield at the elastic limit, such initial yielding is often not noticeable in a structural assembly. For design, we commonly use the traditionally defined yield stress, based on the assumption that 0.2% permanent strain would not be detrimental. To design to this value, we need to make sure our structure would still function properly if it sustained the corresponding permanent deformation.

A material that can yield substantially before rupturing is termed ductile. Ductile materials can survive local concentrations of strain without failing, resist crack formation, and allow parts to be shaped through hammering and bending. Conversely, brittle materials, such as ceramics, do not deform plastically before rupturing. In designing with brittle materials, we must make sure die local concentration of strain around a discontinuity, such as a drilled hole, does not cause an elastic stress that exceeds the

* Elasticity is die characteristic of a material to return to its original dimensions after an applied force is removed.

rupture stress (ultimate stress). Otherwise, a crack would form and possibly grow uncontrollably until the part ruptures. Brittle materials also are not as resistant to impact loads; the area under a material's stress/strain curve indicates how much energy it can absorb before it ruptures. Figure 11-28 shows representative stress/strain curves for ductile and brittle materials. Refer to Table 11-52 for statistically guaranteed design stresses for commonly used alloys.

Fig. 11-28. Representative Stress/Strain Curves for Ductile and Brittle Metals. We generate curves such as these from uni-axial tensile tests. The slope of the linear region is the modulus of elasticity. When the material is unloaded, even when stressed above the proportional limit, stress is again proportional to strain according to the elastic modulus. The yield stress is the stress that causes a permanent strain of 0.002.

Fig. 11-28. Representative Stress/Strain Curves for Ductile and Brittle Metals. We generate curves such as these from uni-axial tensile tests. The slope of the linear region is the modulus of elasticity. When the material is unloaded, even when stressed above the proportional limit, stress is again proportional to strain according to the elastic modulus. The yield stress is the stress that causes a permanent strain of 0.002.

Beams are very common structural members. We characterize a beam by how it is supported. Examples are described in Table 11-55 and can occur in various combinations.

 Name Constraints Examples Cantilevered One end constrained against translation and rotation; other end free Diving board Simply Supported Both ends constrained against translating, but free to rotate Plank placed across a stream for hikers to cross Rigidly Supported Both ends constrained against translation and rotation Floor joists Continuous Support Beam's entire span is supported Railroad track, sld

Loads on beams may be concentrated forces, distributed weights or pressures, or rotational loads, called bending moments. Figure 11-29 shows the symbols commonly used for beam characteristics. Figure 11-30 includes examples of sketches called free-body diagrams, showing beams in static equilibrium. Beams are said to be in static equilibrium when they fully react all applied forces and momentsâ€”a very important prerequisite for static structural analysis.

 Description Symbol Schematic Dimension Concentrated axial load Par F Wftf weight)
0 0