Teoria e applicazioni pratiche

Part III: RC PLATES

Theory of plates in bending. Rectangular and circular plates. Reinforcement design. Stability of plates. Limit analysis of plates. Strip method. Yield line theory.

Part II: MATERIALS

Behaviour of concrete and reinforcing steel. Concrete creep and shrinkage. Rheological models. Creep models in design codes. Age-adjusted effective modulus method.

Part V: DISCONTINUITY REGIONS

B- and D-regions. Truss analogy for RC beams. Discontinuous stress fields. Strut-and-tie modelling. Concrete effectiveness factor. Nodal zones. Reinforcement detailing. Load path method. Optimisation of strut-and-tie models. Generalised stress fields.

Theory of Plates under Bending

We can define thin plates as... They are constrained by shear Q1, torsional moment Mr, torque Mt, and they resist to in-plane deformations. In particular, torsional moment makes the plate stiffer because it produces apparent tension of the same geometric dimensions. (see before):

• Midplane: Plane equidistant from the bounding faces of the plate in the undeformed configuration
• Mid-surface: Transformation of midplane after loading

Kinematic Model

1. Straight normal (Kirchhoff-Love Hypothesis)
• A straight segment normal to the midplane will move normal to midplane in deformed shape, but not necessarily normal to the surface, other deformation occurs.
• Shear warping is neglected even if transverse shear deformation is allowed.
Un segmento retto normale al piano medio prima del primo retta di un'impennata istantanea, ma non necessariamente normale alla superficie media dopo che si è verificata una deformazione. Distorsioni di taglio sono trascurate, anche se deformazioni di taglio fuori piano sono consentite.

Displacements:

• displacements >>

...(u, v, w)αx, αy, wa

αx = β1f(x2) + β2f(x2) αy = g(x1) w(x1, x2) = g(x1) + g(x2)

Free Edge (2-D)

• No Reaction

V_q(a, b) = 0

M_q(a, b) = 0

• No Disruption

w(a, x) = 0

β₁(x, 0) = -2w_,x(x, 0)

P_x(ξ, 0) = 0

• Corner Reactions

R₁ & f₁

To calculate reactions at the corner of the plate R.

When R & f exist, reaction is zero.

R₁: if point support

R_i = 2 M_γz(x_j, y_j)

R = [P₁]

• R_i = [f_k]

R is on equilibrium: Bending moment in y-direction.

For simple support edge:

R_x = directed towards support of the corner.

Rectangular and Circular Plates

• Slide 05

Under transverse load:

Solution can be found by:

• Series expansion
• Disintegration

Formula: w = ∑w_nsin(nπx)

DUAL SERIES EXPANSION

w_n(y) = N_k1 sin(πny)

Can quickly choose mo = 0

To simplify: H_xy = D

M_φ(x, 0) = 0

IME is had defined as minimal distraction; A_IJ

Now given:

P(x, y) = P₊₁(x, y) + sin(nπx)

And here:

Explore the integration similar to displacement p_xy(y)

Therefore, final solution:

Again we can consider a generic face of a plate, inclined of β:

M_xydx

M_yydx

M_xdx

M_ydx

Since we had: M_x = M_xE + EXTENDED BENDING MOMENT

M_xE = M_xcos²θ + 2m₂cosθsinθ + m₂sin²θ = 0

M_xE = (M_x - M_y)sinθ + 2m₂cosθsinθ = 0

We can refer generic values of moments M_x and M_y to some critical values (leading moments). In particular leading moments are connected to top and bottom reinforcement amount.

TOP REINFORCEMENTM_max (θ) = max (m₁ (θ) ) = m₁(θ)BOTTOM REINFORCEMENTm_max(θ) = m_min(θ)

It’s clear the dependence on θ (“intrinsic surface”), face, that for a correct reinforcement (principal moment must be included in ultimate curve):

M_x, M_y and the TORSION MOMENT. We are studying RC plate element behaving, that can resist to torsion by developing some reinforcements added to the non-prestressed contributions:

• SHEAR STRESSES can be transferred through transversal arm according to Saint Venant.
• ORVIEL ACTIONS can be TENSING REINFORCEMENT after cracks that avoid the develop of mono crushing and the misreading of the distance between the reached rupture.
• AGGREGATE INTERLOCKING that works always as a unit: interlocking among them even if cracked, transfers extra tensile force proportionally.

For this reason we don’t consider the inequality about torsional moment. This is a note, and in course that always verified.

But what about design? We need only to respect bending reinforcement that left its satisfying, the inequalities m_max = m_max and that satisfy the elements yield criterion.Here we can introduce the REQUIRED METHOD THEOREM OF THE NORTH MOMENT.So reinforcement doesn’t undergo bending solicitation.

Heya 2FO

M_1min =

NORM. BENDING MOMENT time quantity

• m_max = M_xmax + hull
• M_xmin = M_x

M_max (θ) = m_max + 2m_y

As described in M_max and M_ummintensile moment and the simple Q referred to them is the principal directions ofdθ and θ

So, without modification to those on motion left to minimize the reinforcement, so to optimize it, and to make the plot of m_max (θ)_ref to M_max (θ)

θ = 0: m_max(0) = m_min(0)θ = 0⁺: derivative derivative point

(1) STATIONARITY

COUPLED OPTIMIZATION

2) plot m_max (θ)_ref for a certain field (x_n) and amplitude (m_min)

M_max = m_min + M_max(0) : m_max - M_minM_max(0): m’_max(0)(0)= m_max(0)

• This is the minimum sense (U)

# OPTIMAL DESIGN FOR OPERATION γ

M_max = m = m_H = σy·I_max

M_max + M_min = 2 [E] = 4mc +