Bridge Theory and Design - Appunti

Name: Bridge Theory and Design - Appunti
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Appunti del corso Bridge Theory and Desgin presi durante il corso.GRADE: 30/30Syllabus:1.Beam and Slab decks modelled as grillages 2.Transversally rigid decks3.Box-girder bridges4.Prestressing …

Esame Teoria e progetto di ponti - bridge design

Facoltà Ingegneria civile ambientale e territoriale

Dal corso del Prof. Biondini Fabio

Università Politecnico di Milano

A.A. 2022-2023

65 pagine

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Estratto del documento

BEAM AND SLAB DECKS MODELLED AS GRILLAGES

MASSONNET METHOD

* Problem: ORTHOTROPIC GRILLAGE

two opposite sides SIMPLY SUPPORTED and two ends FREE
loaded by SINUSOIDAL LINE LOAD

* the NODAL EQUILIBRIUM is obtained considering a GRILLAGE DECK composed of a SLAB and of LONGITUDINAL & TRANSVERSE BEAMS

* the BENDING MOMENTS in beams & crossbeams are:

M_x = -E_Ep ^∂²w/_∂x²
M_y = -E_Je ^∂²w/_∂y²
M_xy = -G_Jp ^∂²w/_∂x∂y
M_yx = -G_Je ^∂²w/_∂x∂y

* HP: BEAMS & CROSSBEAMS are CLOSELY SPACED

we can model the discrete system as a CONTINUOUS ONE
this can be done by SPREADING STATIC QUANTITIES & STIFFNESS on the relative distances

=> in this way we can substitute M&T with DISTRIBUTED MOMENTS & SHEARS per UNIT WIDTH (m ≥ t)

Beam and Slab Decks Modelled as Grillage

- Massonnet Method -

Problem: Orthotropic Grillage

two opposite sides simply supported and two ends free
loaded by sinusoidal line load

The nodal equilibrium is obtained considering a grillage deck composed of a slab and of longitudinal & transverse beams

The bending moments in beams & crossbeams are:

M_x = -E_p ∂w2 / ∂x2; M_y = -E_Je ∂w2 / ∂y2; M_xy = -G_p ∂w2 / ∂x∂y; M_yx = -G_Je ∂w2 / ∂x∂y;

HP: Beams & Crossbeams are Closely Spaced

We can model the discrete system as a continuous one

This can be done by spreading static quantities & stiffness on the relative distances

In this way we can substitute M & T with distributed moments & shears per unit weight (m & t)

MIX = -Ep _bi ^∂²w_∂x² - βp ^∂²w_∂x² ;

mXY = -Gp _bi ^∂²w_∂x∂y - γp ^∂²w_∂x∂y

my = -EJE ^∂²w_∂y² - JE ^∂²w_∂y² ;

myX = -GJE ^∂²w_∂x∂y - γE ^∂²w_∂x∂y (x)

-Sp ; JE × (JE+γp) : stiffness × unit length

! mXY can be different from myX ➔ different cross-section of the beams in long. and Transv. direction

* SHEAR FORCES can be obtained by means of EQUILIBRIUM:

◎for beams we use the beam distance x for the transverse beams we use the transverse beam distance

tx = ∂mx_∂x + ∂ mYX_∂y ; ty = ∂my_∂y + ∂mxy_∂x (∆)

➔ from the VERTICAL EQUILIBRIUM we can write:

∂tx_∂x + ∂ty_∂y = -p(x; yJ) (º)

* by substituting (∆) and (x) in (º) we obtain the EQUILIBRIUM EQUATION for ORTHOTROPIC GRILLAGE

∫p ^∂⁴w_∂x⁴ + ( γp + γE ) ^∂⁴w_∂x²∂y² + βE ^∂⁴w_∂y⁴ = ρ(x; yJ) .

by calling:

2α=E+ρEρ(αϵ[0;1])

α0: no torsional contribution

∂ 4w/∂4x+2d ∂∂w x∅2y+E∂ x∂4y y=

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