1. Purpose and description#

The objective of this macro-command is to build a formula for applying pressure fields to the walls of large reservoirs in the context of seismic verification by equivalent mechanical calculation according to Eurocode 8.

1.1. Total pressure#

Equation (1.1) gives the expression for this total pressure:

(1.1)#\[ P_ {total} = P_ {hydro} +\ text {sign} (\ text {sign} (\ sign}) (\ cos (\ theta))\ sqrt {\ bigl (\ left (P_ {ir} +P_ {if}\ right) ^2+P_ {c1}} ^2\ sign} (\ sign} (\ sign} (\ sign) (\ cos (\ theta))\ sqrt {\ bigr))\ sqrt {\ bigr)\ sqrt {\ bigr)}\ left (P_ {ir} +P_ {if}\ right) ^2+P_ {c1} ^2} ^2\ bigr)}\ pm 0.4\ left (P_ {vr} + P_ {vf}\ right) ^2+P_ {c1} ^2\]

where \(\theta\) is the azimuth of the point where the pressure is applied.

Depending on the Newmark combination chosen \(\pm\) becomes \(+\), it’s the « PC+ » combination, or \(-\), it’s the « PC- » combination of the keyword NEWMARK.

1.2. Hydrostatic pressure \(P_{hydro}\)#

The hydrostatic pressure is given by:

\[\]

: label: eq-phydr

P_ {hydro} =rho_l gleft (h-Zright)

where \(\rho_l\) is the density of water, \(g\) is the acceleration due to gravity, \(H\) is the height of water in the tank, and \(z\) is the distance from the point to the bottom of the tank.

1.3. Rigid impulsive pressure \(P_{ir}\)#

The rigid impulse pressure is given by:

(1.3)#\[ P_ {ir} = c_0 (z)\ rho_l H ZPA_h\ cos (\ theta)\]

where \(ZPA_h\) is the zero-period acceleration read on the input spectrum in one of the horizontal directions and

\[\]

: label: eq-c0

c_0 (z) =frac {8} {pi^2}sum^ {infty} _ {n=0}frac {(-1) ^n} {(2n+1) ^2}varepsilon_ncosBigl ((2n+1)frac {pi} {2}frac {z} {H} {H}Bigr)

with

(1.4)#\[ \ varepsilon_n =\ frac {I_1\ biggl [(2n+1)\ frac {\ pi} {2}\ frac {R} {H}\ biggr]} {I'_1\ biggl [(2n+1)\ biggl [(2n+1)\ frac {\ pi} {2}\ frac {R} {H}\ biggr]}\]

\(I_1\) refers to the first order modified Bessel function and \(I'_1\) its first derivative. \(R\) is the radius of the tank.

When constructing the formula, we limit ourselves to the first 20 terms in the expression for \(c_0\).

1.4. Flexible impulsive pressure \(P_{if}\)#

The flexible impulse pressure is given by:

\[\]

: label: eq-pif

P_ {if} =rho_l Hcos (theta)Psi (z)sum^ {infty} _ {n=0} d_ncosbiggl ((2n+1)frac {pi} {2}frac {pi} {2} {2}frac {z} {2}frac {z} {2}frac {z} {H}biggr)left (Sa (f_ {hfh}) - ZPA_hright)

where \(Sa(f_{hfh})\) is the acceleration read on the input spectrum in one of the horizontal directions at the pulsive mode frequency n (4% spectrum) and

\[\ Psi (z) =\ frac {\ int_ {0} {0} ^ {0} ^ {H}\ sin\ biggl (\ frac {z\ pi} {2H}\ biggr)\ Biggl [\ frac {\ rho_s t (z)}}} {\ rho_l H}} {\ rho_l H}\ sin\ biggl (\ frac {z\ pi} {2H}\ biggr)\ Biggl [\ frac {\ rho_s t (z)}}} {\ rho_s (z)}} {\ rho_l H}} +\ sum^ {\ infty} _ {n=0} b'_n\ cos\ biggl ((2n+1)\ biggl ((2n+1)\ frac {z\ pi} {2H}\ biggr)\ biggr)\ Biggr)\ Biggr] dz} {\ int_ {0} ^ {H}\ sin\ biggl (\ frac {z\ pi} {2H}\ biggr)\ Biggl)\ Biggl [\ biggr)\ Biggl [\ frac {z\ pi}\ Biggl)\ Biggl [\ frac {z\ pi} {2H}\ biggl)\ Biggl [\ frac {z\ pi} {2H}\ biggl) +\ sum^ {\ infty} _ {n=0} _ {n=0} d_n\ cos\ biggl ((2n+1)\ frac {z\ pi} {2H}\ biggr)\ biggr)\ Biggr] z}\]

with \(\rho_s\) the density of the steel, \(t\) the thickness of the shell at the point of application and

\[b'_n =\ frac {8} {\ pi^2}\ frac {(-1) ^n} {(2n+1) ^2}\ varepsilon_n\]
\[d_n =\ frac {4} {\ pi}\ frac {\ frac {1} {H}\ int_ {0} ^ {H}\ sin\ biggl (\ frac {z\ pi} {2H} {2H}\ biggr)\ biggr)\ cos\ biggr)\ biggr)\ biggr)\ biggr} {2H}\ biggr) z} {2n+1}\ biggr)\ biggr)\ biggr)\ biggr)\ biggr)\ biggr)\ cos\ biggl ((2n+1)\ cos\ biggl ((2n+1)\ frac {z\ pi} {2H}\ biggr) dz} {2n+1}\ biggr No_n\]

For the infinite sum of eq-pif, we limit ourselves to the first term (therefore the first impulsive mode) which is preponderant.

1.5. Convective pressure \(P_{c1}\)#

Convective pressure is given by:

\[\]

: label: eq-pc1

P_ {c1} = 0.837rho_l Rcos (theta)frac {coshbiggl (frac {1.841z} {R}biggr)} {coshbiggl (frac {1.841H}theta)frac {1.841H} {R}biggl (frac {1.841H}} {R}biggr)} Sa (f_ {c1})

where \(Sa(f_{c1})\) is the acceleration read on the input spectrum in one of the horizontal directions at the convective mode 1 frequency (take the spectrum to 0.5%).

1.6. Rigid vertical pressure \(P_{vr}\)#

The rigid vertical pressure is given by:

\[\]

: label: eq-pvr

P_ {vr} =rho_l ZPA_v (h-Z)

where \(ZPA_v\) refers to the zero-period acceleration of the input spectrum in the vertical direction.

1.7. Flexible vertical pressure \(P_{vf}\)#

The rigid vertical pressure is given by:

\[\]

: label: eq-pvf

P_ {vf} = 0.815 frho_l Hcosbiggl (frac {zpi} {2H}biggr)Bigl (Sa (f_ {vf})) - ZPA_vBigr) ZPA_v

with:

\[f = 1.078 + 0.274\ ln\ biggl (\ frac {H} {R}\ biggr)\ text {si} 0.8<\ frac {H} {R}\]
\[f = 1\ text {otherwise}\]

\(Sa(f_{vf})\) refers to the acceleration read on the input spectrum in the vertical direction at the vertical impulse frequency (4% spectrum).

1.8. Free surface pressure \(P_{slib}\)#

This pressure is supplied directly by the user and is added to the total pressure. It represents the pressure above the water level. For a point on the shell located above the water level, the total pressure is equal to the pressure at the free surface.

1.9. Calculation of pressures on the bottom of the tank#

Let \(r\) be the radius in cylindrical coordinates of the point of application of pressure. Hydrostatic and vertical pressures have no dependence on \(r\).

1.9.1. Adaptation of rigid impulsive pressure#

To take into account the radius \(r\) less than the radius \(R\) of the tank on the bottom, we modify the expression for \(\varepsilon_n\) (present in equation eq-c0) to \(\varepsilon_n(r)\). (1.4) then becomes:

\[\]

: label: eq-epsnr

varepsilon_n ({color {red} {r}}) =frac {I_1biggl [(2n+1)frac {pi} {2}frac {color {red} {r} {r}} {r}} {r}}} {H}biggr]} {I’_1biggl [(2n+1)frac {pi} {2}frac {R} {H}biggr]}

1.9.2. Flexible impulse pressure adaptation#

The radius of the point of application is taken into account by assuming a linear dependence. So we add factor \(r/R\) to equation eq-pif, which gives:

\[\]

: label: eq-pifr

P^ {r} _ {if} (r) =frac {r} {R} P_ {if}

1.9.3. Adaptation of convective pressure#

The radius of the point of application is taken into account by adding a factor using the first Bessel function \(J_1\). In this case, equation eq-pc1 becomes:

\[\]

: label: eq-pc1r

P_ {c1} = 0.837rho_l Rcos (theta)frac {coshbiggl (frac {1.841z} {R}biggr)} {coshbiggl (frac {1.84gl (frac {1.841H}} {theta)frac {1.841H} {R}biggr)} Sa (f_ {c1}) {color {red}biggl)} {coshbiggl (frac {1.841H}} {frac {1.841H} {R}biggr)} Sa (f_ {c1}) {color {red}biggl)} {coshbiggl (frac {1.841H}} {bigl (1.841frac {r} {r} {R}bigr)} {J_1 (1.841)}}

1.9.4. Non-flat bottoms#

The use of the formula created on a tank with a non-flat bottom is not prohibited but no specific consideration has been given or any validation of this type of use.

1.10. Use of the formula produced#

The purpose of the created formula is to be provided to AFFE_CHAR_MECA/FORCE_COQUE_FO. It is then necessary to provide a CARA_ELEM for the thickness and a CHAM_MATER for the density of the material constituting the reservoir. This formula can also be evaluated directly for testing or verification purposes. It is then necessary to call the formula with 5 parameters, first of all the 3 coordinates of the point of application, then the thickness of the reservoir at this point and finally the density of the material at this point.

The formula will produce an error message if:

  • the \(z\) coordinate of the point to be evaluated is less than Z_ FOND (see Opérandes)

  • the radius \(r\) of the point to be evaluated is more than 1% greater than RAYON.

In the other cases, a result will be obtained, the user must therefore ensure that his mesh corresponds to the geometry of the defined reservoir.

In practice, the formulas adapted for the bottom are used in all cases, because it is not known a prima facie whether the point to be evaluated belongs to the shells or to the bottom of the tank. But it’s the same when \(r\) is equal to or very close to \(R\).