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Wave Load Calculations, Methodology & ResultsNavarre Beach Fishing Pier Design

Input Parameters

In order to calculate an appropriate bed elevation for the pier, the pier is divided into eight loadsections. These sections are spaced approximately 300 feet apart between the end of the pierand the shore. At each load section, a bed elevation was determined from the December 2006beach profile at monument R-210 surveyed by Morgan & Eklund. Based on the maximumdifference between historic beach profiles at the location, a scour value was determined andsubtracted to the December 2006 bed elevation value. To be conservative, areas with less thanfive feet of observed scour were given a scour value of five feet.

Loadsection

(ID)Location

December 2006 BedElevation (ft, NAVD88)

Scour (ft,R-210)

Maximum BedElevation (ft,

NAVD88)

1 15+00 -18.4 6.0 -24.4

2 12+00 -14.4 5.0* -19.43 9+00 -15.4 5.0 -20.4

4 6+00 -7.4 5.0 -12.4

5 4+00 4.6 10.0 -5.4

6 2+50 9.6 13.0 -3.4

7 1+00 12.6 7.0 5.6

8 -1+00 7.6 5.0** 2.6

Table 1. Load sections, depths and scour for force calculations.*observed scour of 3 feet**observed scour of 0 feet

To calculate the water depth, the storm surge elevation of 7.4 feet (NAVD88) associated with the20 year wave event (BSRC, Okaloosa West profile) is applied, giving a total water depth. Thisdepth is used to calculate the height of the depth-limited breaking wave that will be applied to thepier according to Equation 1,

(1) bb hH =

where Hb is the breaking wave height, hb is the breaking wave depth, and =0.7.

Wave period is determined as the largest period that can propagate into the depth at the loadsection, up to twelve seconds. This is explained in more detail in the wave hydrodynamicssection below.

Section IDBreaking Wave

Height (ft) Wave Period (s) Water Depths (ft)

1 22.3 11 31.8

2 18.8 10, 10, 11 26.8

3 19.5 10, 10, 11 27.8

4 13.9 8, 8, 9 19.8

5 9 7, 7, 8 12.8

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6 7.6 6, 7, 7 10.8

7 1.3 0, 0, 4 1.8

8 3.4 4, 5, 5 4.8

Table 2. Breaking wave heights and total water depths (bed+scour+surge) for each sectionduring a 20-year storm event.

Wave Hydrodynamics

The wave kinematics for each wave height, depth, and period scenario are calculated usingstream function wave theory, a fully nonlinear theory that is valid from deep water up to near-breaking waves. The specific code utilized is translated from a Fortran routine written by Dr. JohnChaplin of Southampton University. As seen in Table 2, the wave periods vary with water depth.In all cases, these periods were the maximum that the program could compute in the associatedwater depth. The output of this program is the wave free-surface height, particle velocities andaccelerations, which are used to calculate forces on the pilings. Figure 1 shows an example ofthe results obtained using this routine.

Figure 1. Example output of stream function wave program.

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Pile Force Calculations

With the wave kinematics determined, the forces on the pile induced by the wave can becalculated. The pilings in this case are square and 2 feet in diameter. In all cases, the wave isassumed to be breaking. To find the maximum force, the kinematics at the wave crest will beused in the calculations.

Morrisons equation (Equations 2a-2c) is used to calculate the pile forces. It consists of two parts;a drag force FD and an inertia force FI.

(2a) ID FFF +=

(2b) uuDCF DD 2

1=

(2c)

dt

duDCF

MI

2=

In the above equations, is the density of seawater, CD is the drag coefficient, D is the pilediameter, u is the water particle velocity, CM is the inertia coefficient, and du/dt is the total waterparticle acceleration. Table 4 summarizes the constants in the above equations.

Constant Description Value

Density of seawater 1026 kg/m3

CM Inertia coefficient 2.5

CD Drag coefficient 2 (5*)

D Pile diameter 2 ft / 0.61 m

Table 3. Constants in Morrisons equation. *The drag coefficient is multiplied by 2.5 above thestill water level, SWL, in the leading side of the wave crest to account for slamming forces fromwave breaking.

Morrisons equation results in a distribution of wave force per unit vertical length of pile from thesand water interface/mudline to the wave crest. An example of this distribution is shown in Figure2. The discontinuity at the SWL is the point above which breaking effects are considered and thedrag coefficient is increased.

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Figure 2. Force per unit vertical length at location 15+00 for a 20yr storm event and 2ft diameterpile.

The total resultant force, moment arm, and moment can be calculated from this force distributionusing Equations 3-5.

(3)

=

h

total FdzF

(4) hzFdzF

lhtotal

+=

1

(5) totalFlM

1=

In the above equations, Ftotal is the resultant force, h is the total water depth, is the free surfaceelevation (in this case, the crest elevation), z is the vertical axis, l is the moment arm, and M is thetotal moment.

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This force can also be calculated in two dimensions along the pier, so that a framework of pilingscan be analyzed. Figure 3 is an example of this presentation at Station 15+00. The diagrams forStations 12+00, 9+00, 6+00 and 4+00 are attached.

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Figure 3. Force per unit vertical length at location 15+00 for a 20 year storm event and 2ft diameter pile, displaye

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Hand-check of Pile Force Calculation

The results of equations 3-5 are used to hand-check the results of using the stream functionmethod for calculating force against Equation 6.7 on p. 143 found in Basic Coastal Engineeringby R.M. Sorensen, presented here as Equation 6.

(6)

2

22

22

2

22

))(cos())(sinh(

))(2sinh()(2

16

)sin()sinh())(sinh(

4

tkh

hkhk

T

LCDH

tkh

hkTHLCDF

Df

Mf

+++

++=

In Equation 6, f is the density of seawater, H is the wave height, L is the wavelength, T is thewave period, k is the wavenumber, and is the wave angular frequency. Other variables are thesame as previously described. This equation is based on small-amplitude, linear wave theoryand does not include the effects of wave breaking. Table 4 compares results of Equation 6 withthe results using stream function theory. Breaking is not accounted for in the stream functionmodel.

Stream function (lb) 41000

Sorensen (lb) 34400

Difference (%) 19

Table 4. Comparison of Stream function method and Sorensen equation at section 1 of pier a20-year storm.

The force calculated by the stream function method is consistently about 20% higher than that ofSorensens method. This can likely be explained by the fact that nonlinear waves (as calculated

by stream function theory) have higher velocities above the SWL than linear waves. In addition,Sorensens equation assumes a small-amplitude linear wave, which is not applicable with thelarge shallow water waves in this case. Hand-checking shows that the stream functioncalculations are of the right magnitude and are reliable.

Additional Considerations

Breaking Wave Bores

For the interior piles, it is possible that a bore from a wave that breaks on the outer piles couldhave a more powerful effect than the breaking wave associated with that pile. Therefore, anbroken wave decay analysis was conducted to determine if wave bores are a concern and shouldbe factored into the force calculations. Dally, Dean, and Dalrymple (1985) developed a model of

breaking wave dissipation, which for linear, shallow-water theory reduces to Equations 7a-7c.

(7a)

= 2

5

22

1

22

1

2)(

ddHddx

dHd for H > Hstable

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(7b) 0)( 2

1

2

=dx

dHdfor H < Hstable

(7c) dHstable

=

In the above equations, H is the wave height, d is the water depth, x is the axis of wavepropagation, is an empirical decay coefficient with a value of approximately 0.15, and is anempirical coefficient with a value of approximately 0.4. Using this model, it can be determined if abroken wave bore has a greater height than the depth-limited wave at a section, and its forcingneeds to be determined. Figure 4 shows the result of this model using the 20 year wave breakingat section 1 and propagating along the pier.

Figure 4. Model result of the 20 year, section 1 wave breaking and propagating along the pier.

Figure 4 shows that the breaking wave height (in blue) is smaller than the depth-limited waveheight (in red). The results are similar for all scenarios; thus, a broken wave bore has a lesser

effect on the pier than the design conditions already investigated, and no further forcecalculations are necessary.

Uplift forces on horizontal members

Uplift forces imposed by waves are of great importance to the horizontal members of the pier,including the pile cap, pier deck, and handrails. Uplift can be described with two components:

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the slowly-varying force and the impact force. Equations 8a and 8b define these two forces interms of a pressure.

(8a) )( memsv hp =

(8b) svim pp 4=

In the above equations, psv is the slowly-varying pressure, pim is the impact pressure, is thespecific weight of seawater, is the wave crest elevation, and hmem is the height of the member(pile cap, deck, etc). For the cases that the height of the pier member is above the crest height ofthe wave, uplift is not a concern. During the 20-year storm, the only section of the pier affectedby wave uplift forces is at section 15+00 where the bottom of the cap receives minimal upliftforces as shown in Table 5.

Uplift forces (lb/ft^2)

Section Slowly-Varying Impact

1 (15+00) 103 414

Table 5. Calculated uplift forces during a 20-year storm.

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