BASIC COASTAL ENGINEERING Part 7 ppsx

Moored Xoating structures Rubble mound structures, both massive structures and rubble mound veneers to protect embankments Vertical-faced rigid structures Some structures may combine the types mentioned above. An example would be a vertical faced concrete caisson placed on a submerged rubble mound platform that provides a stable base against wave attack and bottom scour. There are two primary concerns in the design of any coastal structure. One is the structural aspects which address the stability of the structure when exposed to design hydrodynamic and other loadings. The other is the functional aspects which focus on the geometry of the structure to see that it satisWes the particular design function(s) such as keeping the wave heights in the lee of the structure reduced to an acceptable level or helping to retain a suYciently wide beach at the desired location. This chapter deals primarily with the Wrst concern, but ad- dresses some aspects of the second concern, which are also covered in the next chapter. For rigid structures such as piles, vertical-faced walls, and large submerged structures, our focus is on determining the loadings on the structure. This leads to the analysis for design stresses which is a classical civil-structural engineering concern. On the other hand, for a structure such as a rubble mound breakwater, our concern is to determine the stone unit size (and related structure component sizes) required to withstand attack by a given design wave and water level. 7.1 Hydrodynamic Forces in Unsteady Flow Water particle motion in a wave is continually unsteady Xow. When this un- steady Xow interacts with a submerged solid body a force is exerted on the body owing both to the particle Xow velocity and the Xow acceleration. The Xow velocity causes a drag force F d to act on the submerged body owing to frictional shear stress and normal pressure that is typically given by F d ¼ C d 2 rAu 2 (7:1) For streamlined bodies such as an airfoil, A in Eq. (7.1) would be the surface area, but for a blunter body (e.g., circular and rectangular cross-sections) A is the cross-sectional area projected in the direction of Xow. In Eq. (7.1) u is the Xow velocity approaching the body, r is the Xuid density, and C d is a drag coeYcient that depends on the body’s shape, orientation to Xow, surface roughness, and the Xow Reynolds number. 196 / Basic Coastal Engineering For a circular cylinder the Reynolds number R ¼ uD=v where D is the cylinder diameter and v is the Xuid kinematic viscosity. For a given body shape, orienta- tion, and surface roughness the drag coeYcient depends primarily on the Rey- nolds number. This dependence has been determined experimentally for a variety of body shapes and typical values are presented in most Xuid mechanics texts. Hoerner (1965) gives a thorough compilation of drag coeYcients and related information for various shapes. When Xow accelerates past a body the Xow velocity and thus the Reynolds number and to some extent the drag coeYcient are continually changing. Thus, since u and C d are both variable in accelerating Xow the drag force (Eq. 7.1) can vary signiWcantly. Consider a wave passing a vertical cylindrical pile. The water particle velocity at any point on the pile continually changes with time and at a given instant the particle velocity varies along the pile. This produces a very complex drag force pattern on the pile. The accelerating Xow causes an additional force on the submerged body beside that given by Eq. (7.1). This acceleration or inertial force has two components. One component arises because an accelerating Xow Weld must have a pressure gradient to cause the Xow to accelerate. This pressure gradient causes a variable pressure around the body’s surface which produces a net force on the body. Also, when Xow accelerates past a body an added mass of Xuid is set into motion by the body. (If, for example, a body that is initially at rest in a still Xuid is accelerated to some particular velocity, the surrounding Xuid that was initially still is also set into motion. A force is required to accelerate that additional mass of Xuid. Conversely, when Xow accelerates past a still body there is an added mass that produces a force on the body.) This second component of inertial force is a function of the Xuid density and acceleration, and the body shape and volume. Thus, when there is unsteady Xow, the total instantaneous hydrodynamic force F on the body can be written F ¼ C d 2 rAu 2 þ ð A p x dA þ krV du dt (7:2) The second term on the right, where p x is the pressure acting on the body in the Xow direction and dA is the diVerential area on which the pressure acts, is the inertial force owing to the accelerating Xow Weld pressure gradient. The third term on the right is the added mass term. In this term V is the volume of Xuid displaced by the body so rV would be the displaced Xuid mass. The dimension- less coeYcient k is the ratio of a hypothetical Xuid mass having an accelera- tion du/dt to the actual mass of Xuid set in motion (by the body) at its true acceleration. The pressure Weld term in Eq. (7.2) can be written in a more usable form by realizing that this pressure Weld creates a force that is capable of accelerating a mass of Xuid having the same volume as the body but at a rate du/dt. Thus, Coastal Structures / 197 ð A p x dA ¼ rV du dt So Eq. (7.2) becomes F ¼ C d 2 rAu 2 þ (1 þk)rV du dt (7:3) In potential Xow, k has the values given below for various shapes and Xow orientations: Sphere k ¼ 0:50 Cube-Xow normal to a side k ¼ 0:67 Circular cylinder-Xow normal to axis k ¼ 1:00 Square cylinder-Xow normal to axis k ¼ 1:20 Additional k values can be obtained from Sarpkaya and Isaacson (1981). In a real Xuid, Xow patterns past the body and thus the value of k would also depend on the body’s surface roughness, the Reynolds number, and the past history of the Xow. Typically, 1 þ k is called the coeYcient of mass or inertia C m and Eq. (7.3) is written F ¼ C d 2 rAu 2 þ C m rV du dt (7:4) Thus, for potential Xow past a circular cylinderC m ¼ 2:0, but for real Xow past this cylinder C m values less than 2.0 are common. Equation (7.4), when applied to wave forces on submerged structures, is commonly called the Morison equation after Morison et al. (1950), who Wrst applied it to the study of wave forces on piles. For an in-depth discussion of the Morison equation see Sarpkaya and Isaacson (1981). It is important to note that the application of Eq. (7.4) to determining wave forces on submerged structures requires that the structure be small compared to the wave particle orbit dimension so that the assumed Xow Weld past the struc- ture is reasonably valid (see Section 7.3). Also, if the submerged structure extends up to near or through the water surface, wave-induced Xow past the structure will generate surface waves which cause an additional force on the structure not given by the Morison equation. 7.2 Piles, Pipelines, and Cables Marine piles, pipelines, and cables constitute a class of long cylindrical structures that must be designed to withstand the unsteady Xow forces from wave action. There may also be steady current-induced drag forces on these structures. Electrical cables laid along the sea Xoor are somewhat similar to underwater pipelines from the stability point of view, but cables are typically less than 15 cm in diameter while some pipelines such as municipal waste outfall lines can be up 198 / Basic Coastal Engineering to 3 m in diameter. Marine cables are also used to moor ships, buoys, and Xoating breakwaters. Piles for piers, oVshore drilling structures, dolphins, and navigation aids are usually vertical or near vertical, but some of these structures can have horizontal and inclined cylindrical members for cross bracing. Pile diameters can vary from less than a meter for piers up to a few meters for the legs of some deep water oil drilling structures. For a circular cylinder with its axis oriented in a horizontal y or vertical z direction and wave propagation normal to the axis, the force F s per elemental length ds of the cylinder can be written F s ¼ F ds ¼ C d 2 rDu 2 þ C m r pD 2 4  @u @t (7:5) The particle velocity u would be given by Eq. (2.21), and the local acceleration @u @t given by Eq. (2.23) is used in place of the total acceleration du/dt. Use of the local acceleration yields reasonable results for most cases but particularly for waves of low steepness in deeper water. Note that the u 2 term should be computed as ujuj. The water particle acceleration lags the particle velocity by 908 so the drag F d and inertia F i components of F s at a given point along the cylinder will vary through the wave cycle as shown in Figure 7.1. To construct Figure 7.1 it was assumed that C d and C m remain constant through the wave cycle and structure/ wave conditions are such as to cause the peak inertia and drag forces to be equal. In any given wave/structure situation the peak total force occurs at some point F s =F d + F i F d F i 0 2π π kx η η, F s , F d and F i Figure 7.1. Surface elevation and drag, inertia, and total forces versus phase position— for equal peak drag and inertia components. Coastal Structures / 199 along the wave between the wave crest or trough and the still water line, the exact position depending on the values of C d and C m , the wave height and period, the water depth, and the cylinder diameter. At the wave crest and trough the total force is all drag force and at the still water positions along the wave the total force is all inertia force. Inserting the relationships for wave particle velocity and acceleration into Eq. (7.4), diVerentiating F with respect to the phase (kx À st), and setting the result equal to zero yields sin (kx) p ¼ 2C m V sinh kd C d AH cosh k(d þ z) (7:6) where time is set to zero so (kx) p represents the position along the wave where the peak force occurs. Equation (7.6) also indicates the relative magnitudes of the drag and inertia components of the total force. If the drag component is larger the total peak force occurs closer to the wave crest and trough, and if the inertia component is larger the total peak force occurs near the still water line position on the wave. Since the volume of the cylindrical segment is a function of D 2 whereas the projected area of the segment is only a function of D, Eq. (7.6) shows that the ratio D/H indicates where along a wave the peak force will occur and the relative size of the drag and inertia force components. For a given wave, as the structure size increases the inertia force tends to become more dominant and vice versa. For common wave conditions, the total wave force on cables is essentially all drag force whereas the total force on large structures such as submerged oil storage tanks is eVectively due solely to inertia eVects. For piles either compon- ent may dominate, with drag forces being largest at lower D/H ratios and vice versa. Often, structures are attacked simultaneously by waves and a current moving at some angle to the direction of wave propagation. The total drag force on the structure is due to the combined eVects of the current and wave particle veloci- ties. The wave characteristics are somewhat modiWed by the current, so the exact nature of the resulting force on a cylinder is diYcult to determine. The usual design procedure is to vectorally add the current and wave particle velocities and use the resulting velocity component in the drag term of the Morison equation. It was indicated in the previous section that the drag and inertia coeYcients are generally a function of the structure shape, orientation to Xow, and surface roughness, as well as the Reynolds number and the prior history of the Xow. For cylindrical structures in waves it is common to introduce another independent parameter X/D where X is the distance that a particle moves as it passes the cylinder, i.e., essentially the particle orbit diameter normal to the cylinder axis. This parameter indicates how well the Xow W eld develops around the structure in the wave-induced reversing Xow past the structure. 200 / Basic Coastal Engineering The particle Xow velocity in a wave at the structure can be represented by u ¼ u m sin 2pt T  where u m is the maximum horizontal particle velocity (i.e., u under the wave crest). Then X equals the average Xow velocity past the structure times the wave period or X ¼ u m 2p  T so X D ¼ 1 2p u m T D  The term in parentheses (u m T=D) is known as the Keulegan–Carpenter number KC, which is proportional to X/D and is also commonly used in deWning values of C d and C m . Note that since u m is proportional to pH=T, KC is inversely proportional to D/H so either KC or D/H has been used in drag and inertia coeYcient investi- gations. Generally, for KC > 25 drag forces dominate and for KC < 5 inertia forces dominate. Most pile and pipeline structures will have several modes of resonant oscilla- tion that may be excited by wave action. This may come about by one of the resonant modes being approximately equal to the incident wave period. Or, if H/ D is suYciently large so Xow adequately envelops a structure a vortex Weld may develop in the lee of the structure and cause oscillatory forces that act normal to the direction of Xow. The frequency of the vortices shed by the structure may also excite a resonant mode of the structure. The period of vortex shedding T e is given by the Strouhal number: S ¼ D T e u where u is the Xow velocity past the structure having a diameter D. S for the common range of prototype pile Reynolds numbers varies from 0.2 to 0.4. For example, given typical values of D ¼ 1:0m, u ¼ 1: 5m=s, and S ¼ 0:3, T e ¼ 2:2 s. Thus, with an incident wave period that is signiWcantly greater than 2.2 s, Xow past a pile could last long enough for a few cycles of vortex shedding to occur. Coastal Structures / 201 Piles Equation (7.5) gives the force per unit length acting on a cylinder. This can be integrated from the mud line to the water surface to obtain the total force on a vertical circular cylindrical pile as a function of time, i.e., F ¼ ð h Àd C d 2 rDu 2 þ C m r pD 2 4 @u @t  dz where dz is a unit length of the pile. If we assume that C d and C m are constant, that the water particle velocity and acceleration are given by the small-amplitude wave theory, that the integration is carried out up to the mean water surface elevation, and that the pile location is conveniently taken to be at x ¼ 0, the integration yields F ¼ C d 8 rgD H 2 n cos 2 ( Àst) þ C m rg pD 2 8 H tanh kd sin ( Àst) (7:7) where n is the ratio of the group to phase celerities. Note that cos 2 ( Àst) should be computed as cos ( À st)jcos À(st)j. This wave-induced force causes a moment on the pile around the mudline given by M ¼ Z h Àd C d 2 rDu 2 þ C m r pD 2 4 @u @t  (d þ z)dz Integration, with the same assumptions yields M ¼ C d 8 rgDH 2 n cos 2 ( À st)d 1 2 þ 1 2n þ 1 2 þ 1 Àcosh 2kd 2kd sinh 2kd  ! þ C m 8 rgpD 2 Hd tanh kd sin ( Àst)1þ 1 Àcosh kd kd sinh kd ! (7:8) Equations (7.7) and (7.8) yield the force and moment as a function of time. The peak force and moment will occur at some point along the wave depending on the wave and pile characteristics, as discussed above. A procedure for the direct determination of the peak force and moment is given by the U.S. Army Coastal Engineering Research Center (1984). This peak force and moment procedure also employs Dean’s stream function wave theory rather than the small-ampli- tude wave theory because, for most design situations, waves of great height are being employed as the design wave. Note that the drag and inertia terms in the moment equation are simply the drag and inertia forces times the water depth 202 / Basic Coastal Engineering times the term in brackets. The term in brackets is simply the fraction of the depth up from the mudline at which the force acts. An important aspect of the calculation of wave forces and moments on a pile is the selection of values for C d and C m . One could use the theoretical value of 2.0 for C m and, after calculating a Reynolds number using some representative water particle velocity, determine C d from a typical steady Xow plot of drag coeYcient versus Reynolds number. Remember that C d and C m vary both with depth at an instant and with time throughout the wave cycle. The value selected for use in Eqs. (7.7) and (7.8) should provide results that are representative of peak load and moment conditions. A number of experiments have been conducted both in laboratory wave tanks and in the Weld to determine design values for the drag and mass coeYcients. In the laboratory either monochromatic or spectral waves can be used and wave characteristics can be controlled to yield results for the desired range of wave conditions. Also, the experimental setup, which usually involves a wave gage to measure the incident wave and an instrumented pile to measure the time- dependent wave loading, is easier to install, access, and control. However, a major drawback to laboratory experiments is that they must generally be conducted at reduced scale so Reynolds numbers are usually orders of magni- tude smaller than those found in the Weld. Field experiments are much more diYcult and expensive to carry out and there is no control over the incident wave conditions that will occur. Also, particularly for the larger waves, the wave conditions will be quite irregular and currents may also be acting on the pile. By either approach, when a record of the water surface time history and the resulting time-dependent load on the pile as a wave passes are obtained, a very serious problem arises as to how to evaluate these data to determine drag and mass coeYcients. Particle velocities and accelerations are needed to employ the Morison equation to determine C d and C m . So a wave theory must be selected to calculate particle velocities and accelerations from the wave record. DiVerent theories, as we have seen, yield diVerent results so the resulting values obtained are dependent on the wave theory used. Given a wave record, an associated wave load record, and a selected wave theory for calculating particle velocities and accelerations, a variety of ap- proaches have been used to determine C d and C m values, i.e. 1. At the wave crest and trough the total force is all drag force (see Figure 7.1) so this value can be used to directly determine a value for the drag coeYcient from the drag term in the Morison equation. The measured force, when the wave is at the still water line, is all inertia force so this condition can be used in turn to determine the coeYcient of mass from the inertia term in the Morison equation. Coastal Structures / 203 2. The Morison equation can be used at any two points along the wave record such as the points of maximum and zero force to solve for C d and C m from two simultaneous equations. 3. A least-squares Wtting procedure can be employed with the Morison equa- tion serving as the regression relationship. The value 1 T Ã Z T Ã o F À C d 2 rDu 2 À C m r pD 2 4 @u @t  dt is minimized where T Ã is the length of the wave and wave force records, F is the time-dependent measured wave force, and the particle velocity and acceleration at each point along the record are calculated from the measured wave proWle. This would yield best Wt values of C d and C m for the entire wave and wave force records. The results of many of these Weld and laboratory experiments are referenced in Woodward–Clyde Consultants (1980), Sarpkaya and Isaacson (1981), and U.S. Army Coastal Engineering Research Center (1984). These results typically show a great deal of scatter in the resulting drag and mass coeYcient values. Based on evaluations of the available laboratory and Weld experimental results Hogben et al. (1977) and U.S. Army Coastal Engineering Research Center (1984) have recommended design values for C d and C m . For the latter, a Reynolds number R is calculated using the maximum water particle velocity in the wave (i.e., at the crest and water surface). Then for: R < 10 5 , C d ¼ 1:2 R between 10 5 and 4 Â10 5 ,C d decreases from 1.2 to 0.6–0.7 R > 4 Â 10 5 , C d ¼ 0:6 0:7 R < 2:5 Â 10 5 , C m ¼ 2:0 R between2:5 Â 10 5 and5 Â10 5 , C m ¼ 2:5 R=5 Â 10 5 R > 5 Â 10 5 , C m ¼ 1:5 The Wnal values selected for design will also depend on the conWdence the designer has in the selected design wave and the related factor of safety desired. 204 / Basic Coastal Engineering Example 7.2-1 A vertical cylindrical pile having a diameter of 0.3 m is installed in water that is 8 m deep. For an incident wave having a height of 2 m and a period of 7 s, determine the horizontal force on the pile and the moment around the mudline when the pile is situated at the halfway point between the crest and still water line of the passing wave. Solution: For T ¼ 7s, L o ¼ 9:81(7) 2 =2p ¼ 76:5 m. Then, by trial we can solve Eq. (2.18) for the wave length at a water depth of 8 m L 76:5 ¼ tanh 2p(8) L yielding L ¼ 55:2 m and k ¼ 2p=55:2 ¼ 0:1138. Then, from Eq. (2.21) the maximum horizontal particle velocity is u ¼ p(2) 7 cosh (0:1138)(8) sinh (0:1138)(8) (1) ¼ 1:24 m=s and the Reynolds number is R ¼ 1:24 (0:3) 0:93 Â10 À6 ¼ 4 Â10 5 Using the recommended values for C d and C m given above yields C d ¼ 0:72 C m ¼ 1:8 From Eq. (2.38) n ¼ 1 2 1 þ 2(0:1138)8 sinh 2(0:1138)8  ¼ 0:803 We can now calculate the force and moment from Eqs. (7.7) and (7.8) where for our point along the wave st ¼ 3p=4. From Eq. (7.7) F ¼ 0:72 8 (9810)(0:3)(2) 2 (0:803)(0:707) 2 þ 1:8(9810)p(0:3) 2 (2) 8 (0:7213)(0:707) ¼ 851 þ 637 ¼ 1488 N Coastal Structures / 205 [...]... the structure face Table 7. 1 Suggested KD Values Trunk Unit Quarrystone—smooth rounded Quarrystone—rough angular Riprap Dolos Tetrapod Tribar Head Breaking Nonbreaking Breaking Nonbreaking 1.2 2.0 2.2 15.8 7. 0 9.0 2.4 4.0 2.5 31.8 8.0 10.0 1.1 1.6 — 8.0 4.5 7. 8 1.9 2.8 — 16.0 5.5 8.5 U.S Army Coastal Engineering Research Center, 1984 220 / Basic Coastal Engineering Table 7. 2 H/H (zero damage) as a... 1.11 1.10 1.14 1.19 1. 17 1.25 1.14 1.20 1. 27 1.24 1.36 1. 17 1.29 1. 37 1.32 1.50 1.20 1.41 1. 47 1.41 1.59 1.24 1.54 1.56 1.50 1.64 1. 27 Adapted from U.S Army Coastal Engineering Research Center, 1984 The stability coeYcient can be increased if a certain percent damage is to be allowed for the design wave condition ModiWed allowable wave height values can be determined from Table 7. 2 which gives the ratio... submerged pipe weight in seawater is: " # p(0:53)2 p(0:61)2 (9 870 ) À (9 870 )(1:025) ¼ À1 67 N=m 613 þ 4 4 So, without the collars the pipe would Xoat to the surface The submerged weight of the collars is: 10,000 10,000(9 870 )(1:025) À ¼ 1910 N=m 3 (2:4)(9 870 )(3) Thus, the submerged weight of the system is 1910 À 1 67 ¼ 174 3 N=m From Lyons (1 973 ) for a sandy sea Xoor we can estimate the static coeYcient... 4s 6s L ¼ 6.3 m 24.0 m 44.4 m For a breakwater width of 6.4 m the respective W/L values and Ct values from Figure 7. 3 are: T¼2s 4s 6s W/L ¼ 0.98 0. 27 0.14 Ct ¼ 0:24 0 .79 0.91 The values for 2 and 6 s were determined by a slight extrapolation of the curve in Figure 7. 3 214 / Basic Coastal Engineering The Ct values indicate that the catamaran is very eVective for a 2 s wave but quite ineVective for a... have been conducted both for steady Xow in Xumes and for wave action in wave tanks (Brown, 19 67; Beattie et al., 1 971 ; HelWnstine and Shupe, 1 972 ; Brater and Wallace, 1 972 ; Grace and Nicinski, 1 976 ; Parker and Herbich, 1 978 ; Knoll and Herbich, 1980) For summary discussions of these investigations see Grace (1 978 ) and Herbich (1981), who also discuss various other aspects of marine pipeline design These...206 / Basic Coastal Engineering and, from Eq (7. 8)  ! 1 1 1 1 À 3:169 þ þ 2 2(0:803) 2 2(0:1138)8(3:0 07) ! 1 À 1:444 þ 6 37( 8) 1 À ¼ 6550Nm (0:1138)8(1:041) M ¼ 851(8) The force calculation shows that the drag and inertia components of the wave force are about equal at the point considered so the plot of forces would be somewhat similar to that shown in Figure 7. 1, indicating that the... experiments The basic problem is to determine the required weight W of an armor unit having a particular 218 / Basic Coastal Engineering shape and speciWc gravity S (in air), when exposed to waves of a given height and when placed on a structure face having a given slope u From a rough analysis of the forces involved one can deWne a stability number for an armor unit Ns ¼ H=(S À 1)(W =gs )1=3 (7: 9) where... structure face slope Combining Eqs (7. 9) and (7. 10) yields the classic Hudson equation for armor unit stability W¼ gs H 3 KD (S À 1)3 cotu (7: 11) Most laboratory studies to evaluate KD have used waves having a constant period and height For irregular waves, some designers use the signiWcant height for H in Eq (7. 11) For conservative design, the U.S Army Coastal Engineering Research Center (1984) recommends... 4. 87 m and a draft of 1. 07 m, tested in water 7. 6 m deep (Hales, 1981) Breakwater 2 is a catamaran breakwater having pontoons 1. 07 m wide with a 1.42 m draft and a total width of 6.4 m (Hales, 1981) The water depth for the catamaran was also 7. 6 m Breakwater 3 is a tire breakwater consisting of four Goodyear modules for a total width of 12.8 m tested at a water depth of 3.96 m (Giles and Sorensen, 1 979 )... Ocean 2 1 A -1 ne sto sto e ne A - ton s B - stone CD - stone Figure 7. 4 Typical rubble mound breakwater cross-section Harbor 216 / Basic Coastal Engineering Runup deflector Gravel 1 3 ap Ripr MLW Filter cloth Embankment Cutoff wall Figure 7. 5 Typical revetment cross-section The proWle of a typical shore revetment is shown in Figure 7. 5 The armor layer would be a wider graded stone riprap rather than . st ¼ 3p=4. From Eq. (7. 7) F ¼ 0 :72 8 (9810)(0:3)(2) 2 (0:803)(0 :70 7) 2 þ 1:8(9810)p(0:3) 2 (2) 8 (0 :72 13)(0 :70 7) ¼ 851 þ 6 37 ¼ 1488 N Coastal Structures / 205 and, from Eq. (7. 8) M ¼ 851(8) 1 2 þ 1 2(0:803) 1 2 þ 1. action in wave tanks (Brown, 19 67; Beattie et al., 1 971 ; HelWnstine and Shupe, 1 972 ; Brater and Wallace, 1 972 ; Grace and Nicinski, 1 976 ; Parker and Herbich, 1 978 ; Knoll and Herbich, 1980). For. structure. 200 / Basic Coastal Engineering The particle Xow velocity in a wave at the structure can be represented by u ¼ u m sin 2pt T  where u m is the maximum horizontal particle velocity