Wave Calculations - PDF by lechuan

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```									©COMPUTERS AND STRUCTURES, INC., BERKELEY, CALIFORNIA JANUARY 2003

SAP2000

®

Technical Note

This Technical Note outlines the methodology used to calculate the wave load and wave wind load values.

Overview
The program calculates the force exerted by a wave at a particular location on a structural object using the following steps. Steps 1 and 2 apply only if the wave water particle velocities and accelerations are calculated from wave theory (i.e., the “From Selected Wave Theory” check box was checked on the Wave Characteristics form; see Defining Wave Loads for more information). Those steps are skipped when user-defined waves are specified. Note the wave loads are applied to only the portion of the structure that is above the mud line and below the wave surface. 1. Calculate the apparent wave period. 2. Calculate two-dimensional regular wave kinematics (water particle velocities and accelerations) using the selected wave theory. 3. Use the specified wave kinematics factor to modify the water particle velocities and accelerations. 4. Calculate the current profile using the specified current stretching method. Modify the current velocities using the specified current blockage factor. 5. Vectorially combine modified water particle velocities with the modified current velocities. 6. Determine the section dimensions (not including marine growth) based on the defined section properties or the wave overwrites.

Overview

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7. Determine the amount of marine growth, if any, on the considered object based on the defined marine growth parameters or the wave overwrites. 8. Determine the drag and inertia coefficients for the considered object based on the defined drag and inertia coefficient parameters or the wave overwrites. 9. Apply the Morison Equation to calculate the force exerted by the wave at a particular location on the object. 10. Calculate the buoyant forces acting on the object.

The program calculates the force exerted by a wave wind load at a particular location on a structural object using the following steps. Note that wave wind loads are only applied to the portions of the structure that are above the wave surface. 1. Calculate the design wind speed. 2. Determine the section dimensions (not including marine growth or ice) based on the defined section properties or the wave overwrites. 3. Determine the amount of marine growth, if any, on the considered object based on the defined marine growth parameters or the wave overwrites. 4. Determine the amount of ice, if any, on the considered object based on the wave overwrites. Note that if both marine growth and ice is specified at a location, only the marine growth value is used. 5. Determine the shape coefficient used for wind loads for the considered object based on the defined shape coefficient that applies to all elements or the wave overwrites. 6. Determine the wind load shielding factor, if any, on the considered object based on the wave overwrites. The wind load on the object is multiplied by this shielding factor. 7. Calculate the wind drag at a particular location on the object.

Overview

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Apparent Wave Period
The wave period input in the wave definition data does not include the effect of the current. The wave period used when calculating the wave water particle velocities and accelerations must include the effect of the current component in the direction of the wave. The wave period that includes the effect of the current is called the apparent wave period, Tapp. The apparent wave period is calculated by solving a system of three simultaneous nonlinear equations. Those equations, which are documented in Section C2.3.1b1 of the commentary of the API Recommended Practice (American Petroleum Institute 2000), are:

λ λ = + VI T Tapp
2 Tapp =

2πλ g tan (2πd / λ )

C2.3.1b1

VI =
where

4π / λ sinh(4πd / λ )

∫

 4π ( z + d )  U c ( z ) cosh   dz −d λ  
0

λ
T

= Wave length. = Wave period as input by user (not considering the current).

Tapp = Apparent wave period (considers the current). VI = Effective current speed in the direction of the wave. g z
= Acceleration due to gravity. = Elevation referenced to the storm water level (positive above storm water level).

Apparent Wave Period

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U c (z ) = Component of steady current profile at elevation z in the
wave direction and not multiplied by the current blockage factor.

d

= Storm water depth.

Wave Kinematics
Wave kinematics yield the wave water particle velocities and accelerations. The velocities and accelerations are calculated from a specified wave theory or they are user-defined. Regardless of which method is used to obtain the wave water particle velocities and accelerations, they are then modified by the wave kinematics factor, which is intended to account for wave directional spreading and irregularity in the wave profile shape. The modification consists of multiplying the horizontal velocities and accelerations by the wave kinematics factor. The vertical velocities and accelerations are not modified.

Current Profile
The user specifies the current profile (velocity and direction of current as a function of height) from the mud line to the storm water level. The user specifies that either a Linear or a Nonlinear current stretching method is used to stretch or compress the current to the wave surface at a particular location. The current velocity at a particular location determined from applying the current stretching technique is multiplied by the current blockage factor to obtain the current velocity that is combined with the wave velocity.

Linear Current Stretching
Linear current stretching is based on the following equation, which is found in Section 2.3.1b-5 of the API Recommended Practice (American Petroleum Institute 2000). The equation is solved directly for z'.

( z' + d ) = ( z + d )
where

d d +η

2.3.1b-5

Wave Kinematics

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z

= Elevation of the location where the water particle current velocity is desired referenced to the storm water level (positive above storm water level). = Elevation of the location in the user-specified current profile where the current velocity should be obtained referenced to the storm water level (positive above storm water level).

z'

η = Elevation of the wave surface directly above the water particle
referenced to the storm water level (positive above storm water level). d = Storm water depth.

Nonlinear Current Stretching
Nonlinear current stretching is based on the following equation, which is found in Section C2.3.1b-5 of the commentary of the API Recommended Practice (American Petroleum Institute 2000). The equation is solved iteratively for z'.

z = z' + η
where z

sinh (2π (z ' + d ) / λn ) sinh (2π d / λn )

C2.3.1b-5

= Elevation of the location where the current velocity is desired referenced to the storm water level (positive above storm water level). = Elevation of the location in the user-specified current profile where the current velocity should be obtained referenced to the storm water level (positive above storm water level).

z'

η = Elevation of the wave surface referenced to the storm water
level (positive above storm water level). d = Storm water depth.

η = Wave length.

Current Profile

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Morison Equation
The Morison equation is used to calculate the force exerted by the wave at a particular location on an object. The equation is given in Section 2.3.1b-10 of the API Recommended Practice (American Petroleum Institute 2000).

F = FD + FI = C D
where

w w dU AU U + C m V g dt 2g

2.3.1-1

F

= Hydrodynamic force per unit length acting normal to the object longitudinal axis.

FD = Drag force per unit length. FI = Inertia force per unit length. C D = Drag coefficient. w g A
= Weight density of water. = Gravitational acceleration. = Projected area normal to object axis per unit length. For pipes and circles this is the effective diameter of the object, including marine growth. For other section type, it is the dimension of the side of the rectangle that encloses the section (including marine growth, if any) that is normal to the direction of the load. = Displaced volume per unit length. For pipes and circles this is π D2/4 where D is the effective diameter of the object, including marine growth. For other section types it is the product of the dimensions of two adjacent sides of the rectangle that encloses the section (including marine growth, if any).

V

U

= Component of the water particle velocity acting normal to the axis of the object.

Morison Equation

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U = The absolute value of U. C M = Inertia coefficient.
dU = Component of the water particle acceleration acting normal to the axis of the object. dt

Buoyant Forces
Buoyant forces are only included when so indicated in the wave load definition. Buoyant forces are only applied to objects (or portions of objects) that lie above the mud line and below the wave surface. Buoyant forces consist of a uniform projected Z direction load applied to objects that are not vertical and concentrated compressive axial forces applied to the ends of all objects.

The magnitude of the uniform load is calculated as:

f z = wV
where fz = A uniform load in the projected Z direction.

w = Weight density of the water. V = Displaced volume per unit length of the object.

For pipes and circles the displaced volume V is calculated as V = π d2/4, where d is the diameter including marine growth, if any. For other sections V is calculated as V = bd, where b and d are the width and height of a rectangle that would enclose the section.

Concentrated Compressive Loads at Object Ends
The magnitude of the concentrated compressive axial load at each end of each object is calculated as:

P = wAc
where

Buoyant Forces

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P

= A concentrated compressive axial load.

w = Weight density of the water. Ac = Cross sectional area to which the load is applied. The magnitude cross-sectional area to which the load is applied depends on whether the object is flooded. All objects are assumed to not be flooded unless the are specifically indicated to be flooded in the wave overwrites. If the object is not flooded, for pipes and circles the cross sectional area Ac is calculated as Ac = π d2/4 where d is the diameter, including marine growth, if any. For other sections, Ac is calculated as Ac = bd, where b and d are the width and height of a rectangle that would enclose the section. If the object is flooded, the cross-sectional area Ac is taken equal to the area specified for the section property that is assigned to the object.

The wave wind loads are calculated based on Sections 2.3.2b-1 and 2.3.2c of the API Recommended Practice (American Petroleum Institute 2000).

Design Wind Speed
The design wind speed is calculated using the following equations that are taken directly from the API Recommended Practice.

 t u ( z , t ) = U ( z ) 1 − 0.41 I u ( z ) ln t  0 

   

2.3.2-1

where the one hour mean wind speed U(z) (ft/sec) at level z (ft) is given by:

  z  U ( z ) = U 0 1 + C ln   32.8   C = 0.0573 1 + 0.0457U 0
and where the turbulence intensity Iu(z) at level z is given by:

2.3.2-2

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 z  I u ( z ) = 0.06 [1 + 0.0131U 0 ]    32.8  Wind Drag Force

−0.22

2.3.2-3

The wind drag force is calculated using the following equation that is taken directly from the API Recommended Practice.

ρ F =   u 2 Cs A 2
where F ρ u = Wind force = Mass density of air (slugs/ft3) = Wind speed (ft/sec)

2.3.2-8

Cs = Shape coefficient A = Are of element (ft2)