Calcolo dell’Azione Sismica su un muro in presenza di falda

Calcolo dell’Azione Sismica su un muro in presenza di falda

Muro in zona II

Accelerazione al suolo (adimensionale): a

_s

= 0 . 25 [-]

Tipologia del suolo (sabbie): S = 1 . 35 [-]

Coefficiente “r” per terreni saturi: r = 1 . 00 [-]

Parametri geotecnici del terreno:

Peso del terreno: γ

_t

= 18 . 00 [kN/m

³

]

Peso del terreno secco: γ

_td

= 16 . 00 [kN/m

³

]

Angolo di attrito interno: φ = 30 [°]

Angolo tra terreno e muro: δ = 15 [°]

Falda:

La falda si trova a -2.00 [m] dal piano di campagna sul quale grava un carico uniformemente distribuito pari a 4.00 [kN/m

²

]

Calcolo delle azioni nella zona di terreno non interessato dalla falda:

Coefficiente sismico orizzontale: 0 . 3375

00 . 1

35 . 25 1 . r 0 a S

k

_h

=

_s

⋅ = ⋅ = [-]

Coefficiente sismico verticale: k

_v

= 0 . 5 ⋅ k

_h

= 0 . 5 ⋅ 0 . 3375 = 0 . 1687 [-]

Angoli di rotazione del cuneo di spinta:

2887 . 1687 0 . 0 1

3375 . 0 k 1 tan k

v

A h

=

= +

= +

θ θ

_A

= 16 . 107 [°]

4060 . 1687 0 . 0 1

3375 . 0 k 1 tan k

v

B h

=

= −

= −

θ θ

_B

= 22 . 096 [°]

Calcolo dei coefficienti di spinta attiva (Mononobe e Okabe):

°

=

−

<

=

β 0 30 16 . 107 13 . 893

( )

( ) ( ) ( ) ( ) ( )

( ) ( )

548 . 0 0 90 sen 15 107 . 16 90 sen

107 . 16 0 30 sen 15 30 1 sen 15 107 . 16 90 sen 90 sen 107 . 16 cos

107 . 16 30 90

K sen

₂

2

2 A

.

MO

=

+

⋅

−

−

−

−

⋅ + +

⋅

−

−

⋅

⋅

−

= + [-]

°

=

−

<

=

β 0 30 22 . 096 7 . 904

( )

( ) ( ) ( ) ( ) ( )

( ) ( )

729 . 0 0 90 sen 15 096 . 22 90 sen

096 . 22 0 30 sen 15 30 1 sen 15 096 . 22 90 sen 90 sen 096 . 22 cos

096 . 22 30 90

K sen

₂

2

2 B

.

MO

=

+

⋅

−

−

−

−

⋅ + +

⋅

−

−

⋅

⋅

−

= + [-]

Calcolo delle azioni nella zona di terreno interessato dalla falda:

Coefficiente sismico orizzontale: 0 . 3375

00 . 1

35 . 25 1 . r 0 a S

k

_h

=

_s

⋅ = ⋅ = [-]

Coefficiente sismico verticale: k

_v

= 0 . 5 ⋅ k

_h

= 0 . 5 ⋅ 0 . 3375 = 0 . 1687 [-]

Angoli di rotazione del cuneo di spinta:

5774 . 1687 0 . 0 1

3375 . 0 10 18

16 k

1 tan k

v h w t

td

A

=

⋅ +

= −

⋅ + γ

− γ

= γ

θ θ

_A

= 30 . 01 [°]

8120 . 1687 0 . 0 1

3375 . 0 10 18

16 k

1 tan k

v h w t

td

B

=

⋅ −

= −

⋅ − γ

− γ

= γ

θ θ

_B

= 39 . 08 [°]

Calcolo dei coefficienti di spinta attiva (Mononobe e Okabe):

°

−

=

−

>

=

β 0 30 30 . 01 0 . 01

( )

( ^{30 . 01} ) ^sen ( ) ^{90 sen} ( ^{90 30 . 01 15} ) ^{1 . 633}

cos

01 . 30 30 90

K

_MO.A

sen

²₂

=

−

−

⋅

⋅

−

= + [-]

°

−

=

−

>

=

β 0 30 39 . 08 9 . 08

( )

( ^{39 . 08} ) ^sen ( ) ^{90 sen} ( ^{90 39 . 08 15} ) ^{2 . 141}

cos

08 . 39 30 90

K

_MO.B

sen

² ₂

=

−

−

⋅

⋅

−

= + [-]

Calcolo dell’andamento delle pressioni a tergo del muro:

Altezza totale del muro: H

_tot.m

= 4 . 54 [m]

Altezza del paramento del muro: H

_m

= 4 . 54 − 0 . 6 = 3 . 94 [m]

Spessore della suola di fondazione: H

_fond.

= 0 . 60 [m]

Sovraccarico uniformemente distribuito: q = 4 . 00 [kPa]

Profondità della falda: z

_w

= − 2 . 00 [m]

Tensioni orizzontali totali con +k

v

:

( ^z

1

⁰ ) ^{q K}

MO.A

( ^{1 k}

v

) ^cos ( ) ^{0 . 3 4 . 00 0 . 548} ( ^{1 0 . 1687} ) ^cos ( ) ^{15 0 . 7423}

h

= = ζ ⋅ ⋅ ⋅ + ⋅ δ = ⋅ ⋅ ⋅ + ⋅ =

σ [kPa]

( ⁼ ) ^{= ⋅} [ ^{ζ ⋅ + γ ⋅} ] ^⋅ ( ⁺ ) ^⋅ ( ) ^δ

σ

_h

z

₂

2 K

_MO.A

q

_td

z

₂

1 k

_v

cos

( ^z

²

² ) ^{0 . 548} [ ^{0 . 30 4 . 00 16 2 . 00} ] ( ^{1 0 . 1687} ) ^cos ( ) ^{15 20 . 54}

h

= = ⋅ ⋅ + ⋅ ⋅ + ⋅ =

σ [kPa]

( ⁼ ) ^{= ⋅} [ ^{ζ ⋅ + γ ⋅} ] ^⋅ ( ⁺ ) ^⋅ ( ) ^δ

σ

_h

z

₂

2 K

_MO.A

q

_td

z

₂

1 k

_v

cos

( ^z

2

² ) ^{1 . 633} [ ^{0 . 30 4 . 00 16 2 . 00} ] ( ^{1 0 . 1687} ) ^cos ( ) ^{15 61 . 20}

h

= = ⋅ ⋅ + ⋅ ⋅ + ⋅ =

σ [kPa]

(

3

)

MO.A

[

td 2

(

t w

)

3

] (

v

) ( )

w w

h

z = 2 . 54 = K ⋅ ζ ⋅ q + γ ⋅ z + γ − γ ⋅ z ⋅ 1 + k ⋅ cos δ + γ ⋅ z

σ

h

( ^z

3

= ^{2 . 54} ) = ^{1 . 633} ⋅ [ ^{0 . 30} ⋅ ^{4 . 00} + ¹⁶ ⋅ ^{2 . 00} + ( ¹⁸ − ¹⁰ ) ⋅ ^{2 . 54} ] ⋅ ( ¹ + ^{0 . 1687} ) ⋅ ^cos ( ) ¹⁵ + ¹⁰ ⋅ ^{2 . 54} = ^{124 . 06}

σ [kPa]

Tensioni orizzontali totali con -k

v

:

( ^z

1

⁰ ) ^{q K}

MO.B

( ^{1 k}

v

) ^cos ( ) ^{0 . 3 4 . 00 0 . 729} ( ^{1 0 . 1687} ) ^cos ( ) ^{15 0 . 7024}

h

= = ζ ⋅ ⋅ ⋅ − ⋅ δ = ⋅ ⋅ ⋅ − ⋅ =

σ [kPa]

( ⁼ ) ^{= ⋅} [ ^{ζ ⋅ + γ ⋅} ] ^⋅ ( ⁻ ) ^⋅ ( ) ^δ

σ

_h

z

₂

2 K

_MO.B

q

_td

z

₂

1 k

_v

cos

( ^z

2

² ) ^{0 . 729} [ ^{0 . 30 4 . 00 16 2 . 00} ] ( ^{1 0 . 1687} ) ^cos ( ) ^{15 19 . 44}

h

= = ⋅ ⋅ + ⋅ ⋅ − ⋅ =

σ [kPa]

( ⁼ ) ^{= ⋅} [ ^{ζ ⋅ + γ ⋅} ] ^⋅ ( ⁻ ) ^⋅ ( ) ^δ

σ

_h

z

₂

2 K

_MO.B

q

_td

z

₂

1 k

_v

cos

( ^z

2

² ) ^{2 . 141} [ ^{0 . 30 4 . 00 16 2 . 00} ] ( ^{1 0 . 1687} ) ^cos ( ) ^{15 57 . 08}

h

= = ⋅ ⋅ + ⋅ ⋅ − ⋅ =

σ [kPa]

(

3

)

MO.B

[

td 2

(

t w

)

3

] (

v

) ( )

w w

h

z = 2 . 54 = K ⋅ ζ ⋅ q + γ ⋅ z + γ − γ ⋅ z ⋅ 1 − k ⋅ cos δ + γ ⋅ z

σ

h

( ^z

3

= ^{2 . 54} ) = ^{2 . 141} ⋅ [ ^{0 . 30} ⋅ ^{4 . 00} + ¹⁶ ⋅ ^{2 . 00} + ( ¹⁸ − ¹⁰ ) ⋅ ^{2 . 54} ] ⋅ ( ¹ − ^{0 . 1687} ) ⋅ ^cos ( ) ¹⁵ + ¹⁰ ⋅ ^{2 . 54} = ^{117 . 41}

σ [kPa]

Spinta idrodinamica:

70 . 12 54 . 2 10 3375 . 12 0 z 7 12 k

E

_wd

= 7 ⋅

_h

⋅ γ

_w

⋅

²_w

= ⋅ ⋅ ⋅

²

= [kN/m]

Analisi del paramento in calcestruzzo armato:

La condizione più gravosa per il dimensionamento del paramento murario è quella in cui k

v

risulta

avere segno positivo.

Caratteristiche di sollecitazione:

Area di armatura necessaria per sostenere il momento flettente:

86 . 4 16 . 37 36 9 . 0

20430 f

d 9 . 0 A M

y Ed

s

=

⋅

= ⋅

⋅

= ⋅ [cm

²

]

Sono necessari 5Ø22 ogni metro di paramento.

Determinazione della resistenza a taglio della sezione del paramento murario:

1125 . 60 0 . 1

18 . 0 18 . C 0

c c .

Rd

= =

= γ [-]

745 . 360 1 1 200 d 1 200

k = + = + = [-]

00528 . 360 0 1000

1900 d

b A

w sl

1

=

= ⋅

= ⋅

ρ [-]

( ) ( )

3 Ed

1 3 w

1 ck 1 c

. Rd 1 c .

Rd

C k 100 f b d 0 . 1125 1 . 745 100 0 . 00528 25 1000 360 167 V

V = ⋅ ⋅ ⋅ ρ ⋅ ⋅ ⋅ = ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ = < [kN]

4033 . 0 25 745 . 1 035 . 0 f k 035 . 0

v

_min

= ⋅

²³

⋅

_ck²¹

= ⋅

²³

⋅

²¹

= [MPa]

Ed w

min 2 c .

Rd

v b d 0 . 4033 1000 360 145 . 22 V

V = ⋅ ⋅ = ⋅ ⋅ = < [kN]

E’ necessario predisporre un’orditura a taglio nei primi 40 [cm] in prossimità dell’incastro con la fondazione:

Ed 2

ywd st

w .

Rd

0 . 9 360 373 . 9 2 242 V

400 4 5 12 2 f d 9 . s 0

V A ⋅ ⋅ ⋅ ⋅ = >

⋅

⋅ π

=

⋅

⋅

⋅

⋅

= [kN]