Theory of Structures - Teoria

Revision of St. Venant's Problem

Object of the elastic problem: the object of the problem is a cylindrical solid in static conditions...

1. Section is constant along length l; 2. Material of body is linear and elastic; 3. Material is also homogenous and isotropic; 4. There are no body forces (not even inertia); 5. There are no thermal effects and no inelastic effects in general; 6. No constraints: it is free to move where it wants;

No traction on the lateral surface Γ; - Forces are only applied on the top section; - In x = 0 and x = l, so they are generally equilibrated; - Principio di equilibrio elastico afferma che una certa distanza dalle sezioni esterne la soluzione dipende solo dalle risultanti (per cui sei il cilindro è MOLTO affusolato NON vi interessa prorpio comme delle forze

Section n defined as a plane S which divides...

S_x dA = 0

Definitions

• Stat...
• Centroidal...

S_x = y dA

S_y = x dA

I_y = x² dA (I^C_xx)

I_xy = xy dA (0 origin)

Area = A

I_x = x² dA

I_xy = xy dA (origin)

Now, let's assume that on the cross section...

dy dy dy S_x x A = N

0 0

Here omitted an important point...

In a rigid body without curvature, the external action must coincide with the neo equation.

Overall, a globally

N(z) = t_ext t_nb

Tx(z) = Tx cos t

Ty(z) = Ty cos t

H_z (z) = Tz + H_I (z)

Now imagine a numerical method, in fact

h_i = f₀ 1 to S

The linear-elastic problem, now there is generalization in it.

Small displacements, small deformations (strains)

E.P. Indeterminate Equilibrium (3)

• d²u₀
• d²v₀
• d²w₀

Compatibility Condition (C)

• d²ε₀
• d²γ₀

E.P. Resolution (3)

• τ_ij, + f_S = 0
• ε_ij = f (S_i + S_ij)

Here is general configuration, but twist-torsion problem is a peculiar case as one part represented as simplified in a specific pattern of manner.

And made intuitively analogous with two formulations of consideration, the monotonic line can be found in the unique solution of definition according to the von Neumann theorem.

It must be simply connected and it mustn't have any sort of hole inside, to guarantee that condition.

Explanation of elements.

Now E_i, A_k + ER_ik, is symmetrically defined for constrained edge.

Boundary conditions are to be noted, let's adopt specific selectorate formulation.

Elastic Relationship in V (6)

• ε_ij = f(γ₀ - νγ₀ - νγ₀)
• γ₀ = f(λ+ν)
• 1/2 E (λ+ν)

In F₀, boundary

Shear stress

Is a contour test.

Stress Approach

Ma la figura: ℌ_c da idea: … invece??

Contro H^c:

Di convergenza fra cui due famme importanti:

Ma … tale bene… M_c introdotto come costante in modo lungo tutta la luce dell'aria?

Introduce anche come stabilizzare permanentemente?

Disegno pure per definire un mio indifferentemente con un percorso modo lungo e:

M_c = EF° - GSB°

Ma:

M_c = costante apparato in un punto -> IV ordine

mt -> distribuito con corrente assegnata IV ordine - mℓ = ^dHt/_dz

Soluzione di Levy

w= - (x - ωo)

V₀ = 0; T₀ = 0; M₀ = -D

w = sin (2 π x / a)

V_N = A cos (m π x / a) + B sin (m π x / a)

y = ω₀; θ₀ = 0

H_x = 0: V(x=0) = 0 H_x0

H_x = z, cogliendo i coefficienti non nulli

T_x = H_x = (A - A_x x) cos (m π x / a)

soluzione del tipo V₀ = 0; T₀ = 0; M₀ = 0

Soluzione: P/Fu}N/F/A/(x=0)