Appunti di Modeling of space structures notes (Space structures)

[SPACE STRUCTURES]

MODELING ON SPACE STRUCTURES

NOTATION

VECTORS

A vector is associated with a no of components and the set of a base.

V = V1 E1 + V2 E2 + V3 E3 in 3D Space

Vec Ci : i = 1,2,3 ➔ Vectors

They build up an orthonormal base

ORTHOGONAL ➔ C1 ⋅ C2 = 0

C1 ⋅ C3 = 0

C2 ⋅ C3 = 0

NORMAL ➔ C1 ⋅ C1 = 1

Since they’re properly normalized

V = ∑_i=1³ V_i C_i = V_i C_i

∑ EINSTEIN NOTATION

RMK Whenever an index is repeated then there is a summation implicit

V_i C_i ➔ no ∑

V_m C_n ➔ ∑

RMK The letter used as index is not important, it's the relative position that matters → it's possible to rename indices but never change the positions.

RMK The bounds of the Σ are expressed only if there is ambiguity, and they are not important in the context.

RMK v_e1 → component

V = V·c₁

→ V = V·e₁

The components are the projection of V onto the basis, if we change the orthonormal basis also the components will be → BUT the vector doesn't change since it's independent from the choice of the reference system.

V ⇒ standard vector

doesn't depend on reference system

{V} = V =

Mistake vectr

depends on the reference system

V₁ V₂ V₃

When we formulate a physical pb we don't want to be dependent on the reference system.

SCALAR PRODUCT

u·v = w where

vector

scalar

w = ||u||·||v|| cosΘ

VOIGT NOTATION

In this representation we represent 2^nd order tensors rearranging them into vectors (6x1) & 4^th order tensors of matrixes (6x6), (even if they are not).

• ̅ = [_{11 22 33 23 13 12}]
• ̅ = [_{11 22 33 23 13 12}]

w̅here ₁₃ = 2_{13 23} = 2₂₃ so we can write = ¹/₂ (̅ : ̅ = ¹/₂ (^T : = ¹/₂ '( : )

⧉⧉⧉

= Voigt =>

⧇ ⧇ ⧇ = [C] [B̅] secuript - matrix

GRADIENT AND DIVERGENCE

GRADIENT

grad(•) = (•),_i ⊗ e_i

where i indicates the derivative respect to i. Eg given a vector ⦈ ➝ grad ℕ = grad (⦈,_i e_i) = ℕ,_i e_i ⊗ e_j RRM The gradient RISES UP the order of the object

DIVERGENCE

div(•) = (•),_i • e_i

Given a vector M ➝ div m = div (m_i e_i) = m_i,_i e_i • e_i = = M,₁₁ + M,₂₂ + M,₃₃ RRM Divergence LOWERS the order of the object.

PROBLEM FORMULATION

SF

We have

S = SF ∪ SO

SF ∩ SO = ∅

GF → region where we prescribe the forces

GO → region where we prescribe the displacements

SE = SF ∪ SO → means that where we don’t prescribe displacements we prescribe forces. Indeed we can see the portions of the surface with no implementation of displacements or forces as a region where we prescribe zero force so Γ = SF.

SE ∩ SO = ∅ means that it is impossible to prescribe both a force and a displacement because they are not unrelated and so if we fix the force there will be only one value of displacement obtained and vice versa.

If we want a certain displacement we will have to apply a specific force.

We have to solve:

1. GEOMETRATION
2. COMPARTORIZING
3. CONSTITUTIVE RELATION
4. BOUNDARY CONDITIONS

RMK Over the boundary u = u^u and S_u.

Introducing a virtual variation -> u + Su = u^u into Su.

unperturbed perturbation

But since u = u^u must be valid in Su -> δu = 0 in Su.

So -> * = ∫_SF Su . t ds_F - ∫_V ε_ik σ_ik dV = *.

Let’s now introduce

∫_V ε_ik σ_ik = 1/2 (Su_ik + ∑u_ik) σ_ik = 1/2 ∑ M_ik σ_ik + 1/2 (Su_ik σ_ik.t).

= 1/2 ∑ u_ik σ_ik - 1/2 δu_ik σ_ik = 1/8 Mu_ik - δm.

can have the law of

this vector after

<= both always in

otherwise

...the relative rotation

Instead he respects this

= σ_ik. 1/2 (δu_ik + ∑ u_ik).

= C_ik δ Σ_ik.

λ 1

δΣ_ik from the definition of

strain tensor

=> * = ∫_SF Su . t dl_SF - ∫_V d ε . σ dl_V.

Let’s see now it the initial expression of the pb:

∫_V (δu - β ε . σ) dV - ∫_SF ε M_i dl_SF - ∫_SF ε . 7 dl_SF -> ∫_SF ε . 7 ds = 0.

=>

∫_V ε . σ dV = ∫_V Su₁ . f dV + ∫_SF Su_M . 7 dl_SF, ∀ ε

PRINCIPLES OF VIRTUAL WORKS

If we consider the external work we have:

∂W_e = -∂V with V = potential of the external loads

So substituting in the PCW:

∂W_i = ∂W_e => ∂U = -∂V => ∂(U + V) = 0

Def (total potential energy) => Ṽ = U + V

=> ∂Ṽ = 0

Minimum potential energy theorem

We have equilibrium when ∂Ṽ = 0 so the observed extreme values (max or min) we can prove it happens when it's a minimum.

Now we can summate these two contributions back to the total PLS and so we get Timoshenko's kinematic model:

• Timoshenko's kinematic model
• U(x,z) = U₀(x) + zφ_x
• W(x,z) = W₀(x)

M₀, V₀, φ_x → Generalized displacement components (e.g. the 2 displacements and a rotation)

Remark (linearization of the rotation)

Let's consider the portion of the beam dx

• Before
• After

dl cos θ = dx

dl sin θ ≈ dw

→ dw = θ dx = tan^-1 f' = this explains why the sign f_{0 1 v0}

Remark

We can see the expansion U(x,z)= U₀(x) + zφ_x is a 1st order Taylor expansion

We could also have a higher order expansion, for example:

U(x,z,t) = U₀(x) + zφ_x(x) + z²φ_{0 x} + z³φ_x(x) + ...

The quantities φ_x and V_x have no physical meaning, but they are still generalized displacement components

We now substitute in this expression the identity

= ∫_e (δN_o u̇_x + δQ_x u̇^′_x + δV_o u̇_z)dx

So δW_e = ∫_e (δN_o u̇_x + δQ_x u̇^′_x + δV_o u̇_z) dx

Where:

û_x = ∫_A F_x dA -> axial force per unit length

ṁ^′ = ∫_A zF_x dA -> bending moment per unit length

V_z = ∫_A F_z dA -> shear force per unit length

EQUILIBRIUM CONDITIONS

Now we impose the satisfaction of equilibrium by noting use of the PTW

(dev ∂ + δ_w e) -> stays for f_n, e_n d_n is so big

Since it's difficult working with the stays for use with PTW (weak form)