Theory of Structures - Teoria

Revision of St. Venant's problem

In one of the static problems, the object of the problem is a cylindrical solid in twist contention (which eliminates war immediately). The body has a uniform cross section. In the cross section, we define a point G, where the torsors are applied with the direction of torsion that lets us define a natural reference. Because of this, we make three main considerations:

• Section is constant along length;
• Material of body is linear and elastic;
• Material is also homogeneous and isotropic;
• There are no body forces (for admission). Real means no heavy and that body doesn't have weight;
• There are no thermal effects and no inelastic effects in general;
• No constraints - it is free to move where it wants and the equilibrium is ensured for any static coupling in the space;
• No action on the lateral surface F;
• Forces are only applied on the top sections x = 0 and x = l, and they are generally equivalent;

If we consider the principal stress affecting a certain distance in the sections, we reach a solution that the body is fully restraint for any section. It is initially attractive that we do not need additional calculations for these sections.

Now, I know that on the cross section:

• To exploit the momentum of forces \(\int_S \sigma_{xz} da = T_x\) \(\int_S \sigma_{yz} da = T_y\) \(\int_S \sigma_{xz} y da = T_y\)
• I_x = \(\int_S x^2 da\)
• I_xy = \(\int_S xy da\)
• A_S =\(I_x\) = \(\int y^2 da\)
• I_yx = \(\int xy da\)
• \(\bar{x}G\) = \(\frac{1}{A} \int x da\)
• \(\bar{y}G\) = \(\frac{\bar{y} da}{A}\)

Equilibrium on the fixed point in V₀. Torque equations. Those included at different forces appear to produce new results.

Revision of St. Venant's problem

It's one of the elastic problems. The object of the problem is a cylindrical solid in those conditions (...elide eliminative more unrealistic...) the body forms the dimensions. Before these, the 2 cross (...). The texted section we define a point C, where in general there should be a set of internal forces and a result of stresses that act in the origin of residues (...).

Because of this, I can make these some considerations:

• Section is constant along "x";
• Material of body is linear and elastic;
• Material is also homogeneous and isotropic;
• There are no body forces (for the division) that means for example that body doesn't add any (...)
• There are no thermal effects and no investigate effect in general;
• No constraints: it is free to move where it wants and the equilibrium is ensured for any situation implying in the space;
• No traction on the internal surface Γ;
• Forces are only applied on the top sections (...);
• Forces are applied on the top sections, and they are generally equilibrated.

Il principio dell'invarianza elastica, afferma che a una certa distanza dalle sezioni estreme la soluzione di opionde ed olo dell'restraint, per cui se il cilindro é alleviato relativamente non vita interessare alcune delle 2;

Now, I know that in the cross section: ... to express the resultant of force:

• _Sn dA = 0
• _{Sn x dA = Tx}
• _{Sn yn dA = Ty (because of the right-hand rule according to x j convention ...)}

T_x, y_n are the moments of the force respect to the and (...).Then instinct at are distinct forces: (...)

Overview of geometry

In a rigid body without curvature, the torsion axes must coincide with the axis equations of motion. Mass defines a momentum in m × x and _bi = ¹/_N

3D motion:

• M > 0 → flat tension
• T > 0 → if the element rotates.

Centralization in more modes: the physical definition of work: ^{M2 = F∙S}

The linear static position has this formulation (15 unknowns in 15 equations) so it is overall:

Elastic Relationship in VΞ_J,ik = Eκ_ik,ij = Eκ_ik,ij+Ξκ_ik,ij small perturbations (stations)

Indefinite Equilibrium (3):

• ∂_xΞ_xx=0
• ∂_yΞ_yx=0
• ∂_zΞ_zx=0

Compatibility Condition (6):

• ∂_yxΞ_xy=0
• ∂_zyΞ_yz=0
• ∂_zxΞ_xz=0

And more applications summarize in the formulation of... Stress A