Riassunti Teoria Advanced Structural Design

1 - THEORY OF PLATES UNDER BENDING

• _{Solid bodies bounded by symmetrical surfaces, the distane of which (thickness t), is small with respect to the minimum in-plane dimension (L) RC Slab: L/t ≥ 5}

• THICK PLATES: L/t ≤ 8-10
• THIN PLATES: 10 ≤ L/t ≤ 80-100
• MEMBRANES: L/t ≥ 80-100

In general they are subjected to transverse loads which cause bending and torsion To resist them, shear forces, bending and torsional moment are developed.

• Plate as generator of beam

  ^{PLATE AS A CONTINUOUS STRUCTURE GENERATION OF BEAM}

  _{It is here assumed to apply}

  Twisting stiffness → Beam from plate of zero thickness.

• To resist in-plane loads, membrane forces are developed. ^{They cause additional bending in case of plates.}

• Def: MIDPLANE → The plane equidistant from the bounding faces of the plate under the un-deformed configuration.

  Under transverse loading conditions it becomes a convenient mid-surface.

BEHAVIOUR OF THIN PLATES

_ASSUMPTIONS: ^{TRANSVERSE LOAD & SMALL DEFLECTION}

KINEMATIC MODEL

Displacement functions:

• w(x, y, z) = z β_x(x, y)
• v(x, y, z) = z β_y(x, y)
• w(x, y, z) = w(x, y)

• Generalized displacements: ^{β_x = β_x(x, y) β_y = β_y(x, y)}

#1 STRAIGHT NORMAL A straight normal → the midplane was normal, straight and inextensible. But not recovered to the mid... When after deformation we use linearized final displacements:...

We found 3 displa. cannot functions that are unknown... each depends on 2 coordinates.

1. N(x, z) = 0
2. N(x = 0) = 0

COMPATIBILITY EQUATIONS

6 components of strains:

• 3 component with respect to LENGTH VARIATION
• 3 component with respect to ANGLE VARIATION

1. ε_xx = du/dx = ∂u/∂x
2. ε_yy = dv/dy = ∂v/∂y
3. ε_zz = dw/dz = ∂w/∂z
4. γ_xy = ∂u/∂y + ∂v/∂x
5. γ_yz = ∂v/∂z + ∂w/∂y
6. γ_zx = ∂w/∂x + ∂u/∂z

represent the shear strains in the plane represent the shear strains in the vertical plane ≈ 0 (associated to bending mov)

1 - THEORY OF PLATES UNDER BENDING

Solid bodies bounded by symmetrical surfaces, the extent of which, thickness (t), is small with respect to the minimum in-plane dimension (h).

• THICK PLATES : L/h ≥ 2 → 10
• THIN PLATES : 10 ≤ L/h ≤ 80-100
• MEMBRANES : L/h ≥ 80-100 ->∞

RC Slab : L/h ≥ 5

BEHAVIOUR OF THIN PLATES

ASSUMPTIONS: TRANSVERSE LOAD & SMALL DEFLECTION

FEATURES

• In general they are subjected to transverse loads which cause bending and torsion.
  • To resist them, shear forces, bending and torsional moments are developed.
  • Plate → evolution of beam
  • PLATE | |- ↔ BENDING STRENGTH

    at the corner → crust phenomenon (3D effects)

    To resist in-plane loads, membrane forces are developed.

    They cause additional bending in case of force perturbation.

    Def: MIDPLANE is the plane equidistant from the bounding faces of the plate under the un-deformed configuration.

    Under transverse loading conditions it become an equilibrium mid surface.

KINEMATIC MODEL

Displacement functio