EXPERIMENTAL STUDIES OF FORCES ON PILES

In the design of a pile struoture exposed to surface waves of a given height and period, some of the faotors involved in the problem and studied herein are the sise, shape and spacing of the piles and the moment distribution on uniform and non-uniform piles. Theoretical and experimental investigations have shown that the force exerted by surface waves on a pile consists of two components — a drag foroe and an inertia foroe. The drag foroe is proportional to the fluid density, the projected area and the square of the fluid particle velocity. The inertia force, inoluding the virtual mass, is proportional to the fluid density, the volume of the object and the fluid particle acceleration. The virtual mass is the apparent increase of the displaced mass of fluid necessary to account for the increase in foroe resulting from the acceleration of the fluid relative to the object. This factor is included in the coefficient of mass term in the foroe calculations.


INTRODUCTION
In the design of a pile struoture exposed to surface waves of a given height and period, some of the faotors involved in the problem and studied herein are the sise, shape and spacing of the piles and the moment distribution on uniform and non-uniform piles.Theoretical and experimental investigations have shown that the force exerted by surface waves on a pile consists of two components -a drag foroe and an inertia foroe.The drag foroe is proportional to the fluid density, the projected area and the square of the fluid particle velocity.The inertia force, inoluding the virtual mass, is proportional to the fluid density, the volume of the object and the fluid particle acceleration.The virtual mass is the apparent increase of the displaced mass of fluid necessary to account for the increase in foroe resulting from the acceleration of the fluid relative to the object.This factor is included in the coefficient of mass term in the foroe calculations.
The experimental and analytical approaches to the pile problem presented in this paper have been based on the total moment about the bottom of the pile and the moment distribution over the length of the pile.In order to calculate a theoretical moment it is necessary to obtain from the experimental results two empirioal coefficients -a drag coefficient and a mass coefficient (Morison, O'Brien, Johnson and Schaaf, 1950).The theoretical equations of total moment corresponding to the orest, trough, and still-water level positions along the surface wave are used to compute these coefficients from the measured total moments at the same positions.Using these coefficients and the theory, a comparison to experimental results is made by comparing the maximum moments, the phase relationships of maximum moments to the surface wave orest, and oomparing the calculated and measured total moment time histories.A comparison of the coefficients obtained by these experiments to other published coefficients obtained in different manners, some being steady-flow values, shows that the results herein are of the right order of magnitude but have considerable variability.'"Further investigation of the problems would clarify the reasons for the scatter of the ooeffioients.
Using the experimentally determined ooeffioients, the moment distributions on uniform diameter and variable diameter round piles were computed and compared to the measured distributions.The computed results are shown to predict the moment distribution with reasonable accuracy for design purposes.
x ' ' ' Errors occurred in Chapter 28, "Design of Piling" in the Proceedings, First Conference on Coastal Engineering and are oorreoted in the Appendix of this Chapter.
The effects of site, shape and spacing of piles were obtained experimentally* Sheltering and mutual interference effects were found for piles arranged in rows or oolumns.Results are presented in comparative form as moment ratios with respeot to a single cylindrical pile.Center piles in rows of piles aligned parallel to the wave crests showed maximum moments that were higher than those for a single isolated pile.Ihe moment depended upon the relative clearances.Moments on piles arranged in oolumns parallel to the direction of the wave travel showed a sheltering effect on the oenter piles in the oolumns with moments less than those for a single isolated pile.
Moments on piles such as an H -section and a flat plate section were larger than those for cylindrical piles of the same projected area.

THEORETICAL CONSIDERATIONS
The dynamic force on an object in fluid moving with a steadystate velocity relative to the object is given by the expression The coefficient of mass must be determined experimentally.This total force does not include any hydrostatic forces.The system under consideration is essentially in a balanced hydrostatic field.
A pile, extending vertically in a fluid in motion due to osoillatory waves, is in a non-uniform flow field with respeot to time and to the submerged pile length.Consider a pile at any instant of time.Equation (3) must be written in the differential form and integrated over the pile length in order to obtain the total resultant foroe on the pile.In Equation (3) the area A is D dS and the displaced volume V m is (wD^/i) dS.Thus, the differential foroe on the pile is given by the expression dF^C^D^ + C^iSf ^)ds This study is based on the total moment about the bottom of the pile, or the total moment contributed by the wave motion above any level, Si, above the bottom.This moment is given by the expression

EXPERIMENTAL STUDIES OF FORCES ON PILES
In order to simplify the calculations of the first few experiments made, it was assumed that the wave elevation above or below mean water level contributed little to the total moment about the bottom; that is, 77 at the surface was small compared to d. Hence in Equation ( 7 9) for the total moment contains sine and cosine terms which are functions of the angular position, 9» Thus, a phase angle is indicated which depends upon the relative magnitude of the sine and oosine terms, the wave equations ( 5) and ( 6) are referenced at a wave crest at time t s 0. The phase angle, /3 , of the maximum moment in relationship to the wave crest is determined by differentiating Equation ( 9) with respect to 9 and setting the results equal to seroj thus, The phase angle of Equation (15) shows that the maximum moment usually does not occur when a wave crest passes a pile.When the pile is in water which is shallow compared to the wave length (d/L small), the phase angle approaches zero.When the pile diameter is small compared to the wave height (D/H small) the phase angle also approaches zero.The phase angle approaches 90° for piles in deep water (d/L large) or for large piles in small waves (D/n" large).
Measured moment-time histories on the pile and wave surfaoe-time histories at the pile are used to determine CQ and Cg from Equation (9).Two variables are involved which necessitate seleotion of two times with the corresponding two moments.The solution is simplified if the selected times are zero (crest or trough at the pile) and the one-quarter or three-quarter wave length time (surfaoe profile at the mean water level).These times result in Sin 9 s 0, and Cos Q s 0, respectively.Thus, the selected points reduoe Equation (9) to two equations, eaoh with but one unknown, C*. and Cj. , respectively.
The moment distribution on a non-uniform pile, that is a pile whioh consists of various lengths of different diameters (Fig. 1) results from a summation of the moments contributed by eaoh section.The solution of Equation (9) for this system is given by the expression,
The conditions imposed upon Equation ( 16) in order to perform the summation resulting effeot on Equations (LI) to (14) are summarized for the conditions illustrated in Figure 1

( Sinh ip )
where S 8 is the elevation to the water surface above the bottom* The ealoulation of an explfeoit expression pig, 1 for the phase angle, similar to Equation (15) when considering the ohange in surface elevation is impossible so that it beoomes necessary to plot a graph of equations or use approximate methods to obtain the phase angle of the total moment with respect to the wave orest (See Fig. 10).
In order to evaluate the total moment exerted on a pile subjeoted to a known wave oondition, the ooeffioients Cjg and Cj£ must be known* Measurements of the moment time history of piles subject to known wave conditions enable evaluation of Cp and C M « The established coefficients then can be used to prediot moments on piling for any pile and imposed wave conditions subject to the limitations and approximations of the analysis which leads to Equations (8), ( 9) and (15).The drag coefficient, (?D, in Equations ( 8), ( 9) and ( 15) is 00mparable in significance to the steady state drag coefficient of Equation (1).Thus, comparisons may be made between the drag coefficients which result from measurements on piling subjeet to the periodic motion of wave action and those reported in the literature for the same geometrical systems in a steady state fluid stream.The steady state drag coefficients are functions of the Reynolds number, Equation (2), in addition to the geometrical shape* In periodic motion the Reynolds No. varies from zero to a maximum* The maximum influence of the motion of a wave past a pile occurs near the surface in the regions of the highest velooity.Hence, the crest particle velocity is assumed to be most nearly representative of the velooity to be used in the Reynolds number* This results from Equation (5) with 3 » d and 0 » 0.

EXPERIMENTAL INVESTIGATIONS ON MODEL PILES
Experiments were designed to measure the moment history on piles of oonstant and variable diameter about hinge points in the piles when subjected to wave action* The wave shape was measured simultaneously to determine the height, velooity, and period of the wave at the pile.The wave length is related to the velocity and period as followsi from the measurements of the variables, the coefficients OQ and C M were obtained from Equation ( 9).Onoe having determined the coefficient, then evaluation of the moments was possible for a given pile subjected to known wave action* Experiments were oonduoted in the wave channel at the University of California (Morison, 1950a(Morison, , 1950b(Morison, , 1950c Tests on single oiroular pilest Moments were measured on single piles hinged at the bottom as well as at various elevations, (Fig* 2).In one instance a 1 inoh diameter pile was hinged only at the bottom and subjected to a large range of wave conditions.In another series of tests, piles of 1/2, 1 and 2 inches in diameter were subjected to three different wave conditions, and the moments were obtained at hinge points located at various elevations to obtain moment distributions.A summary of test conditions is presented in Table 1, and a summary of the test results is given in Table 2 Same results were obtained for a pile placed in breaking waves.The departure of actual conditions from the assumed conditions as stated in the development of Equation ( 9) was too great to justify use of this equation in the interpretation of results in breakers.The results showed maximum moments produoed by a breaker or incipient breaker greatly in excess of the forces corresponding to the orbital motion described by Equation ( 9) The ooeffioients as determined for any one wave oondition were used with Equation (9) to compute the oomplete moment history over the oyole from one wave oreet to the next.A typioal comparison is shown in Fig. 5. Moment distribution comparisons were made for piles 2, 3, and 4 (Table 1).Equation ( 9 4).The H -section was oriented in three different directions as shown in the table.All piles were subjected to the same wave conditions as indicated in Table 3.   6 along with the wave characteristics.One feature of the interpretation of the equations from whioh the coefficients of mass and drag were computed is evident in the results shown in Table 6.When the phase angle is small, the mass ooeffioient is evaluated from moments whioh are near the point of zero moment* Small experimental errors become significant and reduoe the reliability of the value of the mass coefficient.The mass coefficients for the oiroular pile and the flat plate pile are small as compared to those reported in Table 2.These low ooeffioients are not representative*

Effeot of mutual interference of pilest
The one-inoh oiroular and flat plate piles were arranged in rows parallel to the wave direction and in columns perpendioular to the wave direction (see Fig. 11).Three piles were used in eaoh case with moment measurements made on the center pile (Fig. 12).Spaoings between the piles were ^D, D and ijD, where D is the pile diameter.Results are shown in Table 7. %e ratio of the maximum moment on the oenter pile of the oolumn or row to the maximum moment on a single pile shows the results of interference effects* The wave conditions used were the same as listed in Table 3.The results show that, at spaoings of less than l£ D in the row arrangement, interference effeots are noticeable.Higher moments are experienced by the oenter pile as contrasted to a single pile.The blocking effeot of adjacent piles increases the foroe and resulting moment on an individual pile.The blocking effect deoreases as the spacing between piles increases.For the limited range of the tests, the blocking effeot is oonoluded to be negligible for spaoings of l£ D or greater.7^ in that moments were less than those represented by a single pile* The maximum spacing at which the sheltering effect is negligible was not reached in these tests.

Forces on orose members;
The measurement of the horizontal force on oross-menbers was madeon a foroe balance apparatus.The cross-member was mounted on a rod which was pivoted near its center and restrained by calibrated springs at one end (Fig. 13).The submerged part of the rod was shielded from the wave action so that a tare test, without the oross-member attached)showed only about one-peroent deflection.The foroe and the wave characteristics were recorded in the same manner as in the case of the single piles.Three lengths (2-£, 5 and 10 inches) of cross-members were used so that the end effects could be determined.
The measurement of the vertical foroe on cross-members was made directly by a oalibrated spring system.The oross-member was placed at the end of a vertical rod that was attached to springs (Fig. 14).The submerged part of the rod was shielded and held in guides near the orossmember.A tare test showed less than one-peroent deflection.The wave characteristics were measured ij-feet in front and l£ feet behind the oross-member with a reference measurement of the wave orest being made directly above the cross-member* The foroe and wave characteristics were recorded simultaneously on the same oscillograph reoord.The same wave conditions were reproduoed as those used for the measurement of the horizontal forces on the cross-members.In both the tests of the horizontal and of the vertical foroes, the same wave conditions were used for the horizontal and inclined members at the 1/3 and 2/8 positions of water depth.
The horizontal foroe per unit length on a cross-member (Tables 8 and 9) indicated that the orientation of the oross-member is not critical for model studies.The test showed also that the end effects are not appreciable.The vertical foroe per unit length on a cross-member (Table 10) indicated some effeots due to orientation.The magnitudes of the forces were about half those for the horizontal direction.

FIELD PILE TESTS
The model tests, as desoribed above, yielded a considerable amount of information on the moments and foroes on piles subjected to a wide range of wave conditions and depths of immersion.The limited size of the model system introduces a possible soale effect in the direct application of the model results to predict prototype behavior.

!
The wave height history was obtained from a recording pressure actuated diaphragm type wave meter which was located approximately two feet above the sand bottom and adjacent to the pile* Two auxiliary graduated piles were placed seaward of the measuring pile, 3he measuring pile and bracing structure also were painted with alternate black and white bands, each one foot high.Motion pictures taken from the beach reoorded the surface profile of the waves as they passed the pile.A clock was suspended in the field of view of the camera to provide timing intervals between successive frames of the film, The wave velooity at the pile was obtained from the distance between the seaward auxiliary pile and the measuring pile (19.8 feet), and the time interval of the wave crest travel between these points.The motion pictures also reoorded wave heights at the measuring pile.Trough and orest elevations of each wave were obtained from the intersection of the water profile with the graduated vertioal piles.The record from the wave meter also gave wave heights and periods* Analysis of data* Die analysis as presented previously in this paper includes the two resistance terms that contain Cp and CJJ, and also the phase angle relationship, /3, between the two resistance terms.In the analysis of the field pile results, the timing accuracy was not precise enough to determine the time comparison between the water surface profile and the moment history.
The data and results were obtained for waves in various conditions depending on the stage of the tide, s ome data were obtained with the pile in a foam line shoreward of the breaker.Other data were obtained with the pile in the smooth unbroken swell seaward of the breaker.The data have been segregated with respect to the wave oondition at the pile into the following groupst (1) foam line} (2) foam line immediately shoreward of the breaker point} (3) breaker} (4) sharp peaked swell at incipient breaks (5) sharp peaked swell immediately seaward of the breaker point; and (6) swell some distance seaward of the breaker point.The data are summarized in Table 11.
The wave force, which is actually a distributed force extending from the ocean bottom to the water surface, was reoorded as an equivalent force at the calibration point.By multiplying the reoorded force by the calibration-point lever-arm (9 feet 8 inches) the total moment of the wave force about the bottom hinge was determined.nhen the maximum foroe exists (approximately at the time the wave orest passes the pile), the oentroid of the wave foroe was assumed to be looated near the mean height of the wave.This location of the oentroid was estimated by considering the horizontal oomponent of the partiole motion as observed in model studies.By computing the wave foroe at the mean wave height, as defined above, the data were found to be reasonably consistent.The values obtained from the computation indicate that waves of a given site and hhape will exert the same force at the oentroid independent of water-level changes over the range encountered in the tests, although the moment about the hinge point varied considerably due to variation of the effeotive lever arm as the water depth changed.A graph of the wave foroe at mean wave height is shown in Fig. 15.OU0 feature becomes apparent in reviewing the data that permits a comparison between the model results and the field test results.x he majority of the field test conditions were obtained with samll ratios of the pile diameter to the wave height, and with small ratios of the water depth at the pile to the wave length.Under these conditions, the phase angle as given by Equation (15) approaches zero and the maximum moment of Equation ( 9

Fri
is in non-steady motion past an object, the acceleration or deceleration of the fluid in the vicinity of the object produces a foree component.Adding this foroe due to the fluid inertia to the friotional force, the total force is given by the expression ( be integrated if CD, Cy, and u, and du/dt are known as funotions of time (t), or the phase angle, and of the position S. Taking S • (d + y +77) where d s depth of still water, y a depth below the mean water surface to the mean particle position (measured negatively downward), and 77* vertical particle displacement about the mean position, and assuming that the horizontal particle velooity is sero when 77 • 0, then the horizontal velooity and acceleration of the fluid in wave action are given by the expressions (Stokes, 1901/T, angular position of partiole in its orbit measured counter-clockwise from the crest position at time t * 0. The coefficients C D and C M depend upon the state of the fluid motion with respect to the object motion* Little is known about either of the coefficients in aooelerated systems.As a first approximation they are considered as constant with respect to time and position to enable integration of Equation (4).Thus, C D and CJJ become overall coefficients.

Fig. 3 .
Fig. 3. Total moment about the bottom of a single circular pile.
Computed and measured time history of total moment on circular, H-section, and flat plate piling.
Fig. 11.Arrangement of piling, for tests on mutual interference.
the right side of Equation (22) were measured and OD then oomputed.CQ is a drag coefficient whioh depends upon the state of the disturbance of the wave motion due to the movement of the wave past the pile.For shallow-water waves, the velooity distribution from the orest of the wave to the bottom is a function of the ratio of wave height to water depth, and is essentially independent of the wave length or period.The resulting moment on the pile, and henoe C D , are functions of this ratio, H/d.Ihe results are shown in Fig. 16 on this basis, with segregation of the results aooording to wave type* The field pile results were obtained for wave conditions of d/L less than 0.06, with the majority of the waves characterized by d/L less than 0.03.interference effects of rows of oiroular piling, while limited in soope, indicated that for clearances greater than li pile diameters the interference effects are negligible.Moments on oenter piles of a row are inoreased as compared to moments on an isolated pile for spaoings less than 1JS pile diameters* Moments on oiroular piles arranged in columns are decreased as compared to moments on an isolated pile* Ho limits were determined at which the moment became independent of the spacing.with a greater number of wave conditions on oiroular piles, H-sections, flat plates and various other objects are needed in order to establish the relationship of the coefficients of drag and mass to the wave characteristics.

Table 1 . Summary of test conditions on oirculax piles
.
fr-ELEMENT NO.

Table 11
Test data on field pile EFt.