## 63 Elastic deflection of beams and panels

(a) Isotropic solids

When a beam is loaded by a force, F, or moments, M, the initially straight axis is deformed into a curve. If the beam is uniform in section and properties, long in relation to its depth and nowhere stressed beyond the elastic limit, the deflection, S, and the angle of rotation, B, can be calculated from elastic beam theory. The differential equation describing the curvature of the beam at a point x along its length for small strains is d2y dx2

where y is the lateral deflection, and M(x) is the bending moment at the point x on the beam. E is Young's modulus and I is the second moment of area (Section 6.2). When M is constant, this becomes

/ \R Ro where R0 is the radius of curvature before applying the moment and R the radius after it is applied. Deflections, S, and rotations, B, are found by integrating these equations along the beam. Equations for the deflection, S, and end slope, B, of beams, for various common modes of loading are shown below.

The stiffness of the beam is defined by F B1EI

s e3

It depends on Young's modulus, E, for the material of the beam, on its length, l, and on the second moment of its section, I. Values of Bi are listed below.

(b) Metal foams

The moduli of open-cell metal foams scales as (p/ps)2, that of closed-cell foams has an additional linear term (Table 4.2). When seeking bending stiffness at low weight, the material index characterizing performance (see Appendix) is E1/2/p (beams) or E1/3/p (panels (see Figure 6.2)). Used as beams, foams have approximately the same index value as the material of which they are made; as panels, they have a higher one, meaning that the foam panel is potentially lighter for the same bending stiffness. Their performance, however, is best exploited as cores for sandwich structures (Chapter 10). Clamping metal foams requires special attention: (see Section 6.7).

B1 B2

8 =

Fi3

Mi2

B1EI = Fi2

B1EI Mi

b2ei Figure 6.2 Elastic bending of beams and panels

 E = Youngs modulus (N/m2) 48 16 S = Deflection (m) F = Force (N) 384 5 M = Moment (Nm) i = Length (m) 192 - b = Width (m) t = Depth (m) 384 - e = End slope (-) I = See Figure 6.1 (m4) 6- y = Distance from N.A.(m) R = : Radius of curvature (m)

Figure 6.2 Elastic bending of beams and panels

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