Fig. 1.42 Galileo's problem of equal strength cantilevers for a load. His drawing of the thickness variation of a cantilever beam is close to the exact solution (Galileo, Discorsi e Dimostrazioni Matematiche [63])

law between stress and strain by Hooke that he published in 1678 in De Potentia Restitutiva [77], known as Hooke's law.

Without knowing Hooke's law, Mariotte [104] reformulated it in 1680 and remarked that the flexure of a beam arises from parts of its section which are in extension and in contraction (cf. Love [97]). He concluded that the position of the neutral curve, required in the solution of Galileo's problem, is at the center of the section.

• Equation of equilibrium of rods and beams: Considering Galileo s problem, Jacob Bernoulli in 1705 also concluded that the resulting elastic bent curve arises from the extension and contraction zones which are separated at the middle of the cross-sections by the neutral surface. This entails that the bending moment M is proportional to the curvature 1/R of the rod when bent, a result which was later assumed by Leonhard Euler (1707-1783). This local proportionality is expressed by where E is the Young modulus characterizing the material whose first notion was introduced by Young [177] in 1807 - who also introduced the concept of shear as an elastic strain -, and I the moment of inertia of the cross-section around the inertia axis which is perpendicular to the flexure plane and passes through the center of the section (this name is because of the analogy with an inertia mass around a straight line).

For a circular section of diameter 2a, its surface element dA is delimited by the contour and two parallel lines to the inertia axis. Denoting d the distance of dA to this axis, the moment of inertia is

For a beam of rectangular cross-section with sides 2a, 2b, the principal axes of inertia pass through its center and are parallel to the sides. The principal moments of inertia are

It was early known that the work done by a moment is the product M 89 of moment and the angle of rotation. Thus from (1.95), the strain energy per unit length of the rod may be denoted | EI/R2 where the varying quantity is the square of the curvature. In 1742, Daniel Bernoulli [13] suggested to Euler that the differential equation of the flexure could be found by making the integral of the square of the curvature along the rod a minimum. Euler, acting on this suggestion, considered thin rods of a constant circular section and straight in a unstressed state on which opposite forces F and moments M are applied to each end. Using curvilinear coordinates s, 9 along the rod where 9 is the angle between the tangent to the rod and the z-axis and where dy = dssin 9, dz = dscos 9, the general differential equation of the flexure is

where ci is a constant. Euler [51] derived this equation in 1744 and gave some solutions of elastica including zero, one or several loops (Fig. 1.43). He also classified them noticing that the curves may or may not include inflectional points.

Considering thin rods or thin beams, (1.97) leads to the curvilinear length d9

so the function 9(s) can be obtained in terms of elliptic functions (cf. for instance Landau and Lifshitz [92]). The parametric equations representing the flexure of the central line of the thin rod are y = 2EI(d -Fcos9)/F2 + C3, (1.98b)

Fig. 1.43 Plane flexures of a thin circular rod. These curves - also called elastica - were obtained by Euler [51] who classified the various shapes (after Love [97])

• Equal curvature bar of constant cross-section: If the force F is nulled and a moment M per unit length around the x-axis is applied to the free end, this moment is introduced in (1.97) by setting c1 = M. Hence l/R = \J2M/EI = constant and we obtain an equal curvature bar. The curvilinear length is s = \JEI/2M (e + constant). (1.99)

• Cantilever beam: Restricting to the case without loops i.e 0 < e < n/2 or -n/2 < e < 0, Eq. (1.98) allows solving the Galileo problem of a cantilever of constant-section clamped at one end along the y-axis, where s = 0, e = n/2, and deformed at the other free end by a force F parallel to the z-axis where M = 0 i.e. de/ds = 0 (Fig. 1.44-Lef). Denoting t the total length of the bar and e(t) = e0 its rotation at the free end, these conditions entail c1 = F cos e0, c2 = 0, c2 = 2EIc1/F2, c4 = 0, (1.100a)

where the negative sign is taken for the square root term of (1.98b) and the positive for the two other equations. The equation for e0 is r [ET rK/2 de t = ds = , e e . (1.100b)

J V 2F Je0 vcoseo - cose

We may consider now the case of a small deformation, i.e. where e0 is close to n/2. Using the complement angle ç = n/2 — e and approximating sin ç by ç, the integrations of t and z provide respectively at the free end of the cantilever

Ft2 Ft3

The flexure z(ç) may be represented by a polynomial form z(y) with coefficients defined by the boundaries. Hence, from (1.98b) and (1.98c), we obtain z = 6ET ( 3ty2 — y3 ), (1.100d)

which gives [d2z/dy2]y= = 0, accordingly to a null curvature at the free end.

If the force F is suppressed and a moment M around the x-axis is applied to the free end, the flexure of the central line is approximated by a parabola z = (M/2EI) y2; in fact this result formally corresponds to the previous case of equal curvature bars. For a cantilever where the force is applied at some distance from its end, or for cases differing from a cantilever such as bars with both free ends, similar representations of (p0, z(q>o) and z(y) are of classical use (cf. for instance Roark and Young [134]).

• Variable cross-section cantilevers - Equal constraint - Parabolic flexure: Beside determining the strength of a cantilever, Galileo's problem implicitly included the determination of the variable cross-section of a bar of equal strength, a case which we now call of equal constraint. He intuitively produced a drawing close to a correct solution (cf. Fig. 1.42).

Galileo's problem of a variable cross-section cantilever of equal constraint was first solved by Clebsch [34], in 1862, for a beam of constant width and a force concentrated at the unclamped end. If a beam is of width a=constant in the x-direction, the thickness solution in the z, y plane (i.e. the vertical plane if the force is a weight) is a parabola whose vertex is at the beam end.

Denoting My the bending moment along the y-axis, and Qy the shearing force acting in the bar, the statics equilibrium is represented by dMy

Considering both cases of equal constraint and parabolic flexure cantilevers, the maximum stress a and the curvature of the neutral line 1/R in the y, z plane of a bar are related by a = Et/2R, where t (y) is the thickness in this plane and E the Young modulus. A parabolic flexure is expressed by z = 1/2R y2. Hence the bending moment is expressed by d2z E a My = Elx-J = Ix- = 2IX — , (1.102)

7 dy2 R t where, from (1.96b), the inertia moments are Ix = nt4/64 for a rod, and Ix = at3/12 for a beam. The shearing force is determined at distance y as the sum of the external forces acting from the clamped origin of the bar to this distance. Denoting I the length of the cantilever, for a rod of total weight P, the shearing force is

4 Jo 4 Jy where ^ and g are the density and the gravity.

Considering bars with loading cases as a concentrated force F at the unclamped end, a line force f per unit length applied along the bar, and a flexure due to own weight, these shearing forces are

= , rod: F, (i - z)f, 4 gfdy, y 1 beam: F, (i — z) f, a^gL tdy.

where F, f, and g are negative since oriented towards negative, z.

After substitution of the moment and shearing force in (1.101), we obtain a set of 12 differential equations which can all be expressed in the general form dt fi tp--aG = 0, G = tqdy, (1.105)

dy y where p, q are positive integers given in Table 1.9 together with coefficient a and function G(t, y,i).

Table 1.9 Exponent p and terms a, G of the differential equation (1.105) of rod and beam cantilevers of equal constraint (g) or parabolic flexure (y2) flexed by various loads, with beam width a constant (after Lemaitre [95])

Bar Condition Exp. Point force F Line load f Own weight type p a G a G a G

Table 1.9 Exponent p and terms a, G of the differential equation (1.105) of rod and beam cantilevers of equal constraint (g) or parabolic flexure (y2) flexed by various loads, with beam width a constant (after Lemaitre [95])

Bar Condition Exp. Point force F Line load f Own weight type p a G a G a G


g =


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