Appunti di Theory of structures

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Appunti presi durante il corso.Grade 30/30Syllabus:1. Saint Venant’s problem2. Saint Venant’s case of torsion3. Non-uniform torsion and theory of thin-walled beams4. Theory of curved …

Esame Theory of Structures

Facoltà Ingegneria civile ambientale e territoriale

Dal corso del Prof. Ardito Raffaele

Università Politecnico di Milano

A.A. 2021-2022

115 pagine

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Revision of St. Venant problem

The St. Venant's problem regards the definition of the stress and displacement field in an elastic beam with a rectilinear axis and subjected to actions applied only to the extreme sections

Hypothesis

Small strains & displacements
Linear elastic homogeneous material
Long beam with a constant cross section
Free body in the space (no kinematic constraints)
No body forces acting on the body, F=0
No surface forces on Γ: f=0
Loads act only on the extreme sections

Reference system

It is located in the centroid of one of the extreme section and it is characterized by having:

S_x = ∫_A y dA = 0
S_y = ∫_A x dA = 0
I_x = ∫_A y² dA
I_y = ∫_A x² dA

It is a Principal Reference System

Revision of St. Venant problem

The St. Venant's problem regards the definition of the stress and displacement field in an elastic beam with a rectilinear axis and subjected to actions applied only to the extreme sections.

Hypothesis

Small strains and displacements
Linear elastic homogeneous material
Long beam with a constant cross section
Free body in the space (no kinematic constraints)
No body forces acting on the body, F = 0
No surface forces on Γ : f = 0
Loads act only on the extreme sections

Reference system

It is located in the centroid of one of the extreme section and it is characterized by having:

S_x = ∫_n y dA = 0
S_y = ∫_n x dA = 0
I_x = ∫_n y² dA
I_y = ∫_n x² dA

It is a Principal Reference System

Surface force and action resultants

Since we have F.o on V and f.o on M the only acting force is the Surface force P acting on the External bases and it must be Self-equilibrated (since there are no constraints)
Note: the distribution of P is not specified BTW the St. Venant prb ensures that at a certain distance from the sections the solution depends only on the action resultants (this is why the rod must be sufficiently long)
The internal actions which defines the Stress resultants are defined as:

N: ∫_A σz dA
T_x: ∫_A τ_xz dA ; T_y = ∫_A τ_yz dA
H_x: ∫_A σz y dA ; H_y ∫_A σz x dA
M_t = T_x x g + T_y x = ∫_A (τ_yz x - τ_xz y) dA

Surface force P and internal actions

Since we have F.o su V and f.o su \[M\] the only acting force is the Surface force \[P\] acting on the External bases and it must be Self-equilibrated (since there are no constraints)
Note: the distribution of \[P\] is not specified BTW the St. Venant prb assures that at a certain distance from the sections the solution depends only on the Actions resultants (this is why the rod must be sufficiently long)
The Internal actions which defines the stress resultants are defined as:

\[N = \int_{A} \sigma z \, dA\]

\[T_x = \int_{A} T_{xz} \, dA\] \[T_y = \int_{A} T_{yz} \, dA\]

\[M_x = \int_{A} \sigma z \cdot y \, dA\] \[H_y = \int_{A} \sigma z \cdot x \, dA\]

\[M_t = \int_{A} T_{xz} \cdot y + T_{yz} x - \int_{A} (\sigma z \cdot x - T_{xz} y) \, dA\]

Static equilibrium equations

x since there are No constraints, the internal actions must satisfy the static equilibrium equations: ΣF_i = 0; ΣM_i = 0

T_x - T_x0 = 0 → T_x = T_x0
T_y - T_y0 = 0 → T_y = T_y0
N - N₀ = 0 → N = N₀
M_y + T_x - H_y0 = 0 → M_y = H_y0 - T_x · z
M_x - T_y - H_x0 = 0 → M_x = V_x0 + T_y · z
H_e - H_t0 = 0 → H_e = H_t0

Governing equations

The governing equations of the Saint Venant prob are:

a) Equilibrium:

σ_x,x + τ_xy,y - τ_xz,z + F_k = 0
τ_yx,x - σ_y,y + τ_yz,z + F_y = 0 on V

Anteprima

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