In Chap. 11 Betti addressed the problem which Clebsch in [45] had attributed to Saint Venant. Quite strangely, considering the contact of Betti with Riemann and with the German scientific community and mainly for the popularity of Clebsch’s treatise, Betti made no reference to Saint Venant. The case studied by Betti was however the same as that found in [46,77,78]: the linear elasto-static problem for the cylinder of Fig.3.2loaded at the basis ω1 andω2 by regular surface forces(L1,M1,N1) and(L2,M2,N2). The cylinder was described with respect to a system of Cartesian coordinates having its origin in the barycenter ofω1, axiszorthogonal toω1and coinciding with the axis of the cylinder, according to that shown in Fig.3.2.
Betti started by considering an arbitrary field of displacements(u,v,w), which satisfies the field equations of the elasto-static problem for the cylinder assumed free of volume forces. The reciprocal theorem gives [20]37:
ω1
(L1u1+M1v1+N1w1)dω1 +
ω2
(L2u2+M2v2+N2w2)dω2
=
ω1
(L1u1+M1v1+N1w1)dω1 +
ω2
(L2u2+M2v2+N2w2)dω2
+
σ(L0u0+M0v0+N0w0)dσ
(3.27)
where σ is the lateral surface of the cylinder;(u,v,w)is the displacement field providing stresses that equilibrates the contact forcesL,M,N;L,M,Nis the field of forces due to the displacements (u,v,w). The subscripts indicate the value assumed by the two fields on the two basis (1 and 2) and on the lateral surface of the cylinder (0), respectively.
To use the relation (3.27) Betti had to characterize the field(u,v,w), which assumes the role of the ‘virtual’ displacement field; he wrote the equations of local and boundary equilibrium for the forcesL,M,N, as a function of(u,v,w).38Of all the fields (u,v,w)which satisfy local and boundary equilibrium Betti chose the one for which the associated stress state have no components in the plane of the cylinderz=const.
36p. 83.
37Equation 59, p. 84.
38This is done in all groups of the not numbered equations enclosed among the (59) and (60) of the Chap. 10. The first group represents the linear elastic homogeneous and isotropic relationship between the components of stress and the partial derivatives of the components of the displacement,
Fig. 3.2 Solid of Saint Venant
base 1
base 2
lateral surface
y,v x,u
ω1
ω2
z,w G
This condition with the consequences that it implies in the local equilibrium equations was expressed as [20]39:
dG
dz =0, dF dz =0 dG
dx +dF dy +dC
dz =0
(3.28)
and40:
A=0, H=0, B=0 (3.29)
whereC,F,Gindicate the stresses parallel to thezaxis whileA,H,Bthe stresses orthogonal to it, that is parallel to the section of the cylinder. Betti used for the stress a notation close to that of Cauchy and also adopted by Piola in [74] forty years before;
though the letters are slightly different. Betti’s notation of stresses recalls the letters he had used for strains; more precisely the stressesA,B,C,F,G,Hcorrespond to the straina,b,c,f,g,h. Notice that at Betti’s time there were already in use notations with two subscripts, present in later work of Cauchy, Clebsch, Saint Venant, William Thomson and Tait, and Kirchhoff that were related to the theory of determinants [89]. Betti however was not directly interested in the concept of stresses so he did not care ‘details’ about them.
(Footnote 38 continued)
the second group expresses the local equations of static equilibrium, the third group characterizes the components of the normal (oriented toward the interior according to the convention in the 19th century) to the outer surface of the cylinder, and the fourth group expresses the boundary conditions on the specialized components of the normal just characterized.
39Equations 60–62, p. 85.
40Betti did not use a semi-inverse method based on the so-called hypothesis of Clebsch-Saint- Venant, Eq. (62). The vanishing of the stresses on the plane of the section (today indicated with the symbolsσx,σy,τxy) is a condition for the auxiliary field of displacement useful to be introduced in the reciprocal theorem.
Betti thus chose(u,v,w)so that only the components of the stress in the direction of the axis of the cylinder are different from zero. By imposing local equilibrium he obtained that the tangent stresses were independent ofz. The only equation remaining to satisfy the equilibrium was that obtained by projecting the local equilibrium equa- tion along thezaxis. Betti required that the state of stress satisfied the condition of the lateral surface free of surface forces [20]41:
G1=G2, F1 =F2. (3.30)
He characterized the field of displacements(u,v,w)in such a way that it looked like the solution of the problem, leaving free only the surface forces on the basis associated with it. On the other hand, Betti introduced the hypothesis (3.28) as a free choice of the field of ‘virtual’ displacements-stresses which appears in the formula of reciprocity. Consequently, Betti’s results might be more general than that of Saint Venant, since it does not postulate that the ‘true’ is characterized by the canceling of the components of the stress parallel to the plane of the section.
Later in chapter, Betti followed Clebsch and Saint Venant [46].42Moreover, want- ing only to determine the ‘virtual ingredients’ to be introduced in relation (3.27), Betti used technicalities of integration for systems of linear differential equations in partial derivatives similar to others in the literature of his time. With the aid of elements of complex analysis, Betti found the components of the field(u,v,w):
u =h+kz+az2 2 +bz3
6 +(c+ez)x−(c+ez)y +τ
(a+bz)x2−y2
2 +(a+bz)xy
v =h+kz+az2 2 +bz3
6 +(c+ez)y+(c+ez)x +τ
(a+bz)y2−x2
2 +(a+bz)xy
w= −cz+ez22
τ −
az+bz2 2
x+
az+bz2 2
y +e
τ(x2+y2)+bxy2+byx2+U,
(3.31)
whereτis Poisson’s coefficient,U(x,y)is the solution of a harmonic problem with Neumann’s boundary conditions on the bases of the cylinder. The relations (3.31), considering the due correspondences, coincide with the solutions already found by Clebsch and Saint Venant, even though Betti did not point out the fact.
The fields of ‘virtual’ displacementsu,v,w depend on the constants of inte- gration a,a,b,b,c, c,e,e,h,h,k,k. About the meaning of the constants of
41Second not numbered equation after the equation 62, p. 86.
42A discussion in absolute form in the subject is for example found in [76].
integration Betti did not put forward any interpretation. Only six of them, the same number of the components of the resultant actions on the basis, are used.
Betti obtained first the mean values, simple or weighted with the positionx,y,z, of the difference of the axial displacement between the two bases of the cylinder as a function of the surface forces on the bases [20].43This dependence is not trivial to make explicit; Betti so examined the simple case, coinciding with one of the Saint Venant’s case, in which the contact forces on the basis have opposite directions [20]44 so that for any point ofω1≡ω2it is:
L1+L2=0, M1+M2=0, N1+N2=0. (3.32) He then inferred that the mean values are directly proportional to the resultant actions on the ends of the cylinder [20].45In particular, the simple mean gives the elongation of the cylinder, the variation of the area of the section and the coefficient of lateral contraction. He also commented on the results of the weighted mean, but there is no mechanical meaning for them.
Betti broke down the fieldUof the last of the relations (3.31) in the three addends, harmonic solutions of problems with boundary conditions of Neumann type on the sections of the cylinder. Each addend is proportional to one of the constantse,b,b, identifiable with the torsional and non-uniform bending curvatures for the axis of the cylinder. Betti considered only sections with two axes of symmetry, for which the area integrals weighed with the odd powers of the coordinates vanish; he thus obtained the equations for the components of the torque46 and bending moment resulting on the bases.47He did not comment on their results in general, but only the particular solutions for cylinders with elliptical sections, for which a solution closed form exists for the addendsU. For the non-uniform bending he found the expression of the ‘deflection of bending’, that he particularized to the circular sections.48
Betti did not seem to care about the fact of not presenting a complete solution (for instance, the analysis of uniform bending is lacking). He had clearly applied the formula of reciprocity, which led to some solutions of technical interest, and this was probably enough for him.