2. The Problem of Saint Venant

In the Theory of integration we prove that, decomposing an area B in k subareas B ₁ , …, B _k such that \(\cup_{i = 1}^{\,k} B_i\ \textrm{and}\ meas_2\ (B_i \cap B_j) = 0\ \forall i,j\epsilon \{ 1, \cdots ,k\} \), it results \(\int_B {fdB = } \int_{B_1 } {fdB + \cdots + } \int_{B_k } {fdB.}\)

From the Integration theory it follows that, since meas ₂ (B) > 0, it results \(\int_B {y^2 dA > 0.}\)

We have from the Fig. 2.1.18 that x' = y sin α + x cos α and from the Fig. 2.1.19 that y cos α = y' + x sin α.

In Analytical geometry we prove that the tangent to ellipse in \(T^\prime\) is parallel to t.

Adhémar Jean Claude Barré, count of Saint Venant, Filliers en Brie 1797 – Saint Ouen 1886.

The axis of a couple is the line normal to the plane on which the couple acts.

We assume positive the anticlockwise moments. About M _t , we refer to Fig. 2.2.13.

Otherwise the deviate bending is the already examined right bending.

We easily have \(k=(a^{2}+b^{2})^{1/2}\).

We are imposing the equilibrium of the part [z, l] of the beam to the translation according z.

We are imposing the equilibrium of the part [z, l] of the beam to the rotation around the axis n.

The (2.4.8) is also called second monomial formula of the non centroidal axial load.

We are imposing the equilibrium of the part [z, l] of the beam to the rotation around the axis s.

It deals with a linearization consequent to the hypothesis of small deformations. In the (nonlinear) field of the great deformations, in the problem of the torsion the lines parallel to z but distinguished by the axis z turn into cylindrical helixes of the space.

We consider the cross section z = k, with 0 ≤ k ≥ l, a frame of reference 0, x, y and a generic point P ₀ = (x ₀ , y ₀) of the cross section (Fig. 2.5.6). When we project on the plane z = k the point of S in which P ₀ is moved to happened deformation, evidently we obtain the point P ₂ = (x ₀ + u, y ₀ + v). We consider the displacement s ₁ = P ₀ P ₂ = \( (u,v) = ( - \vartheta ky_0,\vartheta kx_0 )\) Since s ₁ × OP ₀ = u x ₀ + v y ₀ = 0, P ₀ is moved orthogonally to the line passing through O and P ₀. Since \(|{\textit{\textbf{s}}}_{{\textbf{1}}} | = (u^2 + v^2 )\frac{1}{2} = \vartheta k\,dist(O,P_0 )\), the length of the displacement s ₁ is equal to the length of the arc P ₀ P ₁ of the circumference of center O and radius \(dist(o,p_0 )\), obtained rotating the cross section of \( \alpha= \vartheta k\) radians around O. Since the hypothesis of small displacements, \((u^2 + v^2 )^{\frac{1}{2}} \) is small, so that \(\alpha = \vartheta k\) is small, so that we can approximate the segment P ₀ P ₂ with the arc P ₀ P ₁, so that we can approximate P ₂ with P ₁. In other words, in the hypothesis of small displacements the arc P ₀ P ₁ can be linearized with the segment P ₀ P ₂.

Obviously \(I_p = \int_A {(x^2 + y^2 )dA = \int_A {x^2 dA + } \int_A {y^2 dA} = 2\int_A {y^2 dA = 2} I_x = \pi\frac{{r_0^4 }}{2}.} \)

Let \(\textit{\textbf{v}} = \left( {v_x ,v_y ,v_z } \right):\Omega \subseteq \) ℜ³ → ℜ³ . We call divergence of v and denote with the symbol div v the real function, defined in Ω, \(div\;{\textit{\textbf{v}}} = \frac{{\partial v_x }}{{\partial x}} + \frac{{\partial v_y }}{{\partial y}} + \frac{{\partial v_z }}{{\partial z}}\). We call curl (or rotation) of v and we denote with the symbol curl v (or rot v) the vectorial function \({\textit{\textbf{curl\; v}}} = \left( {\frac{{\partial v_z }}{{\partial y}} - \frac{{\partial v_y }}{{\partial z}}} \right){\textit{\textbf{x}}} + \left( {\frac{{\partial v_x }}{{\partial z}} - \frac{{\partial v_z }}{{\partial x}}} \right){\textit{\textbf{y}}} + \left( {\frac{{\partial v_y }}{{\partial x}} - \frac{{\partial v_z }}{{\partial y}}} \right){\textit{\textbf{z}}}\), defined in Ω and having values in ℜ³. In the bidimensional case we adopt the analogous definitions \({\textit{\textbf{v}}} = (v_x ,v_y ),div\;{\textit{\textbf{v}}} = \frac{{\partial v_x }}{{\partial x}} + \frac{{\partial v_y }}{{\partial y}},{\textit{\textbf{curl\;v}}} = \left( {\frac{{\partial v_y }}{{\partial x}} - \frac{{\partial v_x }}{{\partial y}}} \right){\textit{\textbf{z}}},\) and we also write, more concisely, \({\textit{\textbf{curl\;v}}} = \frac{{\partial v_y }}{{\partial x}} - \frac{{\partial v_z }}{{\partial y}}\).

We call streamline every curve γ such that, whatever a point P of γ is chosen, the velocity v (P, t) of P at the instant t is tangent to γ (Fig. 2.5.16). In general γ doesn’t coincide with the path β of the fluid particle, that is situated in the point A at the instant t _A (Fig. 2.5.17). Clearly, if the flow is steady, the path β of the fluid particle is also a streamline (Fig. 2.5.18).

We consider a vectorial function \({\textit{\textbf{v}}}\,:\,\Omega \subseteq \) ℜ² → ℜ² and a regular plane curve δ having first end Q ₁ and second end Q ₂, such that \(\delta \subseteq \Omega \) (Fig. 2.5.19). We denote in every point Q of δ with t [resp. n] the tangent [resp. normal] line to δ. We call flow of v through δ the real number

If v is the velocity of a fluid, clearly such curvilinear integral is the area of fluid that goes through δ throughout an unitary time interval.

We denote with n the normal to the tangent t to γ in P (Figs. 2.5.31 and 2.5.32).

In the Integration theory we prove that such curvilinear integral measures the length of the arc of γ individualized by the points W, P.

We denote with A, B [resp. C, D] the end points of the chord c ₁ [resp. c ₂]; with ξ ₁ [resp. ξ ₂] the line passing through A and B [resp. C and D]; with η ₁ [resp. η ₂] a line normal to ξ ₁ [resp. ξ ₂] and z.

Sir George Gabriel Stokes, Skreen 1819 – Cambridge 1903. The theorem of Stokes decrees that: Let D be a portion of the three-dimensional space, S be a regular surface contained in D and bounded by a closed curve γ, \({\boldsymbol\nu}\) be a vectorial function defined in D. Then the circulation of \({\boldsymbol\nu}\) along γ is equal to the flow of curl \({\boldsymbol\nu}\) through S (Fig. 2.5.46).

This is the motive for which such problem is called non uniform torsion.

To perform such construction it is necessary to know two points of the parabola and the tangents to the parabola in such points.

It is well known that the tangents to a parabola in two every points of abscissa x ₁ and x ₂ intersect in a point of abscissa (x _l + x ₂)/2.

Insofar in the closed thin walled cross section a chord doesn't generally divide the cross section into two parts.