Teoria e applicazioni pratiche

Name: Teoria e applicazioni pratiche
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Documento completo delle spiegazioni teoriche e delle dimostrazioni degli argomenti trattati. Sono riportati anche tutti gli esercizi svolti a lezione con relative soluzioni numeriche e …

Esame Advanced Structural Design

Facoltà Ingegneria civile ambientale e territoriale

Dal corso del Prof. Biondini Fabio

Università Politecnico di Milano

A.A. 2016-2017

69 pagine

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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.

RC plates slide 04 - 05 - 06 - 07 - 00

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

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

Materials slide 02 - 03

Behavior 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.

Discontinuity regions slide 41

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 (Slide 01)

We can define thin plates when:

e 1:10
L/a 80 100
Negligible shear deformability

They are constrained by shear perpendicularly, bending moment M, axial N, and they react immediately developing flexure (in particular, twitted effect makes the plate stiffer & produces a spreaded tension of the same geometric intensity).

We define:

Midplane: Plane equidistant from the bounding faces of the plate in the undeformed configuration
Moderate: Transformation of midplane after loading (1 theorem workplan)

Reference system x, y, z origin in a corner of the midplane. Thin plates are loaded by transverse load (out of plane) normally to midplane which cause bending.

We assume:

Only transverse load
Deflections are small so no shear deformation

Kinematic Model

Straight normal (Navier Thompson)

A straight segment normal to the midplane, while remains straight and unstretched, is always but not necessarily normal to the surface. Therefore deformation occurs (shear warping is neglected even if transverse shear deformation is allowed). "A sheet metal normal to the plane medium of the first straight of width unstretched but not necessarily normal to the surface plane medium to box this is verification of the deforming."

Distorsioni di taglio sono trascurabili anche se deformazioni di taglio nel piano sono consentite interamente dall'equivalente normale corrente in un asse. Nota per abilitare il solido: displacements => 0α total displacementsα1 Midplane aximum.

Aspect ordinate normali:

θ_x (x-z) = ∂w/∂x
θ_y (y-z) = ∂w/∂y
ρ_x = ∂²w/∂x² (x-y)
ρ_y = ∂²w/∂y² (y-x)
ρ_xy = ∂²w/∂x∂y

We also note the COMPATIBILITY EQUATIONS:

_ε_α = _u_α/x_α + 1/2 (_w/x_α)²
_ε_β = _u_β/x_β + 1/2 (_w/x_β)²
_u_α _w/x_β + 1/2 (_w/x_β)²

Since _T_xy == 0 z= 0 then deformation... => new fields.

Unaltered Length

From these equations => new field:

_Ρ_x = γ_xz = ∫^h/2_−h/2 _T_zx dz
_Ρ_y = γ_yz = ∫^h/2_−h/2 _T_zy dz

Rotation of the Section Not Deflection

_Ρ_x = k_x _P_x_P₂ _−kγ3/3 step ⛷ because it requires simple integrations:

If (−) (−) once see a) right b)

Straight Normal (Kinematic law)

It is more intuition theore Malakasi. Prima Important because it completely neglect these cases/conditions _xy = 0 = equals to \uniaxial?

Hypothesis for Beam

a - no straight segment normal to the surface and remains straight of unlimited length
b - the mid-surface after application occurs; therefore, transverse shear deformation is neglected.

E NOTTURNO ALUIA SUPERFICIE TEDDY ADD DEFORMATION if case massimo deformation applied. (UNALTERED LENGTH) A = −12 E_xyz + (>1) case requires. And I can observe that.

Anteprima

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