Riassunti Teoria Advanced Structural Design

1 - THEORY OF PLATES UNDER BENDING

Solid bodies bounded by geometrical surfaces, whose at least thickness (t), is small with respect to the maximum in-plane dimension (a).

• THICK PLATES: L/B ≤ 1 by Alem-Mohammadi & Mullen.
• THIN PLATES: 1,0 < L/B ≤ 80-100 ... opt.usually: t < ...
• MEMBRANES: L/B > 80-100 ... membrane structures

RC SLAB: L/B ≥ 5

BEHAVIOUR OF THIN PLATES

• in general, they are subjected to transverse loads
• which cause BENDING and TORSION
• V
• to resist them, SHEAR FORCES, BENDING and TORSIONAL MOMENTS are developed

plate ≡ generalized form of beam

Def: MIDPLANE = the plane equidistant from the bounding faces of the plate under the undeformed configuration.

KINEMATIC MODEL

ASSUMPTIONS: TRANSVERSE LOAD & SMALL DEFLECTION

Displacement functions:

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

Generalized displacements:

• β_x = B_x(x, y)
• β_y = B_y(x, y)

^#1 STRAIGHT NORMAL

As straight normal at the midplane was thrown straight and unmodified (angle), but was recorrected to the mid-config. after deformation occurs.

N(x = 0) = x = 0

N(z = 0) = x = 0

we found 3 displac. component functions that are unknown, each depends on 2 coordinates

COMPATIBILITY EQUATIONS

6 components of strain

• 3 components with respect to LINEAR VARIATION
• 3 components with respect to ANGLE VARIATION

represent the SHEAR STRAINS IN-THE-PLANE

represent the SHEAR STRAINS in the VERTICAL PLANE (ex: xy associated to bending mov.)

The previous compatibility equations become:

_εx = z _∂βx / _∂x = z _∂²w / _∂x² = χ_x bending curvature in plane xz

_εyy = z _∂βy / _∂y = z _∂²w / _∂y² = χ_y bending curvature in plane yz

_γxy = z (_∂βx / _∂y + _∂βy / _∂x) = 2z _∂²w / _∂x∂y = χ_xy flexural curvature (twisting curvature)

_εxz = _γyz = 0

It is important because it describes the kinematic model and due to that it describes, its correspondent distribution of bending and torsional curvatures.

STATIC FIELD

#2 _Gz = 0

the intensity of normal stresses _σz is generally small and it is NEGLIGIBLE

Stress components: _σx, _σy, _σxy

Shear components: _τxy, _τyz, _τzx, _τxz, _τyx, _τzy

→ 5 Stress components

we want ε to be consistent with the kinematic field... stress equilibrium local quantities (in plane) ↔ local quantities (in plane)

#1 STRAIGHT NORMAL (Kirchhoff-Love)

a straight segment to the midplane will remain straight and normal after deformation occurs...

TRANSVERSE SHEAR DEFORMATIONS ARE NEGLECTED

_χxz = _γyz = 0

Plates rigid in shear with small deflections

β = VARIATION OF THE DEFLECTION ω

{_βx = _∂ω / _∂x

_βy = _∂ω / _∂y}

Shear deformability: _γ

⊥ _β ⊃ counterclockwise

_δθ clockwise

ω ⊃ counterclockwise

_β, ω concave

_βx = -2ω / _∂x

_βy = -2ω / _∂y

u = - _∂ω / _∂x

v = - _∂ω / _∂y

w = ω (x,y)

OSS: units of these are different from beams

Bending moments:

• _Mx = ∫_h/2^n/2 _σz dz
• _My = ∫_h/2^n/2 _σz dz

Torsional moments (or twisting moments, or torques):

• _Mxy = ∫_n/2^h/2 _τyz dz = ∫_n/2^h/2 _τzx dz = 0

Shear forces:

• _Qx = ∫_n/2^h/2 _τzx dz
• _Qy = ∫_n/2^h/2 _τyz dz

Kinematic field ⇒ 3 FUNCTIONS (bending and torsional curvatures)

Static field ⇒ 5 FUNCTIONS (stresses, forces, moments)

Outcomes compatibility ⇔ stress strain

Equilibrium ⇔ shear forces