Teoria - Advanced Structural Design

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.

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 when: a σ = t/10 L/R < 60 ÷ 100 ⇒ NEGLIGIBLE SHEAR DEFORMATIONS.

They are characterized by their form G₁ (infinite number), type M₁ and they need nomenclature identifying them – in particular, transversal dimension makes the plate either a "prismatic or apparent because of the same geometric dimensions.

New define:

• MIDPLANE: PLANE EQUIDISTANT FROM THE BENDING FACES OF THE PLATE IN THE UNDEFORMED CONFIGURATION
• MOISATURE: TRANSFORMATION OF MIDPLANE AFTER LOADING (Terza dimensione trascurabile)

Reference system x,y,z unique in a corner of the midplane.

These plates are loaded by TRANSVERSE LOAD (out of plane) ⟂ NORMALLY TO MIDPLANE which cause bending:

We assume:

• 1 – ONLY TRANSVERSE LOAD
• 2 - DEFLECTIONS ARE SMALL ⇒ no 2nd order effects.

KINEMATIC MODEL

• 1 STRAIGHT NORMAL (KIRCHHOFF – TIMOSHENKO)
• A straight segment normal to the midplane, which remains straight and of undistorted length; it is not necessarily normal to midplane anymore, after deformation occurs. These mappings: in neglected terms it becomes shear deformation is allowed.

"UN SEGMENTO RETTO NORMALE AL PIANO MEDIO RIMANE RETTO ED IN UNDAIMENSIONE MA NON NECESSARIAMENTE NORMALE UNA SUPERIFICIE MEDIA DOPO CHE SI È VERIFICATA LA DEFORMAZIONE."

DISTORSIONI DI TAGLIO SONO TRASCURABILI MENTRE LE DEFORMAZIONI DI TAGLIO PURA PIANO SONO CONSENTITE ☞ EQUIVALENTE ATTENZIONE → deformazione ammissibile potente"

dispacements x:

• (n)
• (γ₂)

α – actual deformation

α – Midplane – deformes

MIDPLANE:

• P(αx, γx) = P'(αx)
• P'(βx, αy)

Displacement diagram domain:

• βx = - η
• βy = - α

(1) We also obtain the compatibility equations:

ε_z = 1/E₃ (σ_z - E₂ α₃ΔT)

- Because not all (β_ij) = 0

(3) Plane

μ = 2 (λ + μ)

β_x = γyz - 2∂w / ∂x

β_y = γxz - 2∂w / ∂y

β_z = γxy

(4) a = b corresponds to shear angle.

5! o (o → -) θ + γ

2.1 Straight normal (kinematic + force)

If a straight segment normal to the surface remains straight it will remain “ normal” to the deflected surface; therefore, transverse shear deformation is neglected.

ε_zz = yz = 0

ε_xy = (2∂v / ∂x) = 0

_{βxx, βyy, βxy & βyx = 0}

ε_xx = -(κ_y + κ_z x) , ε_yy = -(κ_x + κ_z y)

And it can observe that

κ = - νκ_xx

^Constants

ε_xx = R_z /E₁ - α₁₁

4) τ_z = 0

Since τ is parallel to ground,

5) Internal actions: τ _{² = 0}

Extra Definitions

1. Homogeneous linear elastic material
2. Linear variation
3. Mx = - 12 t etc.
4. H = 2∫k dz

(4.4) PARABOLIC SHEAR STRESS PROFILE

At z = 0,

At z = ±h/2,

MUST EVALUATE SHEAR FORCE x, y TO SATISFY EQUILIBRIUM IN z ‑ DIRECTION

with the

Signature of the

PARABOLIC PROFILE Act in static

Then mean(E)

Shear distrubution

from(5) maxx maxy

(i) Now proceed over due by writing equilibrium equation.

local body is curved as along with

Now substitute the equation of , with the eq. now available:

Equation

Equation (M)

(4.5) INCOMPRESS and ISOTROPIC LINEAR ELASTIC MATERIAL

Young's Modulus

Shear Modulus