Khoslas Theory Some structures designed on Blighs theory gave trouble. Khoslainvestigatedthecausesbyinsertingsomepipesonthed/ssideof the weir through impervious apron. The pressures measured were not as calculated according to Blighs theory. Khosla proved that seeping water through permeable soil follow parabolic streamlines and not along the underside profile of the impervious floor as given by Bligh. Flow of water takes place according to Laplace equation, i.e. thepotential flow theory. Stream Lines Thestreamlinesrepresentthepathsalongwhichthe water flows through the sub-soil. Everyparticleenteringthesoilatagivenpointupstream of the work, will trace out its own path and will represent a streamline. Thefirststreamlinefollowsthebottomcontourofthe works and is the same as Bligh's path of Field creep. The remaining streamlines follow smooth curves transiting slowlyfromtheoutlineofthefoundationtoasemi-ellipse, as shown in above figure. Equipotential Lines(1)Treatingthedownstreambedasdatumandassumingnowateronthe downstreamside,itcanbeeasilystatedthateverystreamlinepossessesa head equal to h1 while entering the soil; and when it emerges at the down-streamendintotheatmosphere,itsheadiszero.Thus,theheadh1is entirely lost during the passage of water along the streamline.Further, at every intermediate point in its path, there is certain residual head (h)stilltobedissipatedintheremaininglengthtobetraversedtothe downstreamend.Thisfactisapplicabletoeverystreamline,andhence, therewillbepointsondifferentstreamlineshavingthesamevalueof residualheadh.Ifsuchpointsarejoinedtogether,thecurveobtainedis called anequipotential line. Every water particle on line AB is having a residual head h= h1 and on CD ishavingaresidualheadh=0,andhence,ABandCDareequipotential lines. Since an equipotential line represents the joining of points of equal residual head, hence if piezometers were installed on an equipotential line, the water will rise in all of them up to the same level. (2)Theseepagewaterexertsaforceateachpointinthedirectionofflowand tangential to the streamlines as shown in the below figure. This force (F) has an upward component from the point where the streamline turns upward. Forsoilgrainstoremainstable,theupwardcomponentofthisforceshouldbe counterbalanced by the submerged weight of the soil grain. This force has the maximum disturbing tendency at the exit end, becausethedirectionofthisforceattheexitpointisvertically upward, and hence full force acts as its upward component. For the soil grain to remain stable, the submerged weight of soil grain should be more than this upward disturbing force. The disturbing force at any point is proportional to the gradient of pressureofwateratthatpoint(i.e.dp/dt).Thisgradientof pressure of water at the exit end, is called the exit gradient. Inorderthatthesoilparticlesatexitremainstable,theupward pressureatexitshouldbesafe.Inotherwords,theexitgradient should be safe. Critical Exit Gradient Iftheupwarddisturbingforceonthegrainisjustequaltothe submergedweightofthegrainattheexit,theexitgradientis called critical exit gradient. To keep the structure safe against piping, the exit gradient should beequal to 1/4 to 1/5 ofthe critical exit gradient, i.e.afactor of safety equal to 4 or 5 is used. The submerged weight (Ws) of a unit volume of soil is given as : wherew = unit weight of water Ss = sp. gravity of soil particles n = porosity of the soil material ( ) ( ) 1 1 =s sS n w WThe upward disturbing force on unit volume (for unit area, dV = 1 x dl) of the grain For critical conditions to occur at the exit point, F = Ws (dh/dl is the critical exit gradient) For most of the river sand Ss 2.65 and n 0.4 Therefore, Critical exit gradient = (1-0.4)(2.65-1) = 0.99 1.0 ( )dldhwdlwh ddldpF . = = =( )( ) 1 1 . =sS n wdldhw( )( ) 1 1 =sS ndldhMaterialKhoslas Safe Exit Gradient Shingle1/4 to 1/5 Coarse sand1/5 to 1/6 Fine sand1/6 to 1/7 (3) Undermining of the floor starts from the downstream end of the d/s pucca floor, andifnotchecked,ittravelsupstreamtowardstheweirwall.Theundermining startsonlywhentheexitgradientisunsafeforthesubsoilonwhichtheweiris founded. It is, therefore, absolutely necessary to have areasonably deep vertical cut-off at the downstream end of the d/s pucca floor to prevent undermining. The depth of this d/s vertical cut off is governed by two considerations i.e. (i) maximum depth of scour; (ii) safe exit gradient. Whiledesigningaweir,downstreamcutofffromthemaximumscoureddepth considerationsis,firstofall,provided,andcheckedforexitgradient.Ifasafe valueofexitgradientisnotobtained,thenthedepthofcutoffisincreased.The depthofcutoffisalsogovernedandlimitedbypracticalconsiderations,asthe execution of very deep cutoff may be difficult or unpracticable at site. Aweirorabarragemayfailnotonlyduetoseepage(i.e.sub-surfaceflow)as statedbyBligh,butmayalsofailduetothesurfaceflow.Thesurfaceflow(i.e. when flood water flows over the weir crest) may cause scour, dynamic action; and inadditionwi1lcauseupliftpressuresinthejumptrough,(ifthehydraulicjump forms on the downstream). These uplift pressures must be investigated for various flow conditions. Themaximumupliftduetothisdynamicaction(i.e.for surfaceflow)shouldthenbecomparedwiththemaximum upliftundersteadyseepage(i.e.forsub-surfaceflow);and the maximum of the two chosen for designing the aprons and the floors of the weirs. Owing to the simplicity, Bligh's theory is still used for design of small works. Aminimumpracticalthicknessforthefloorandadeep verticalcutoffatthedownstreamendis,however,always provided, in addition to the requirements of Bligh's theory. However,onmajorworks,Bligh'stheoryshouldneverbe used,asitwouldleadtoexpensiveandunsafeerroneous designs.

Khoslas Conclusions 1. Outer faces of sheet piles are much more effective than inner faces and horizontal length of the impervious apron. 2. Intermediatesheetpilesifsmallerorequalinlengthare almost ineffective. Khoslas Theory of Independent Variables This is based on the assumption that the Potential flow theory can be applied to sub-soil flow. Darcys law,V = k.i = k. dh/dx Any system to which this theory is applicable must satisfy Laplace equation = f(x, y, z) = velocity potential = k h k = coefficient of permeability, h = residual head at any point within the soil For two-dimensional flow If H is the differential head acting on the hydraulic structure, Khosla introduced a hypothesis known as independent variables. 0222222= + +dzddyddxd 02222= +dyddxd 02222= +dyH ddxH dAccordingtoKhoslashypothesis,weirprofile/irrigationstructureshouldbe splitted into a no. of simple standard forms where analytical solution are available. The most useful standard forms are: (1)Astraightfloorofnegligiblethicknesswithasheetpileatsomeintermediate position.

(2) A straight horizontal floor of negligible thickness with sheet pile at (i) u/s end,(ii) d/s end (3)A straight horizontal floor depressed below the bed but with no vertical cutoff. Ingeneraltheweir/irrigationstructuresconsistsofcombinationofallforms mentionedabove.Theupliftpressurearedeterminedonlyatkeypoints,i.e.the intersection of the pile lines and the floor. The pressure variation between the key points is assumed to be linear. If, p = pressure at any point, H = total head = p/H x 100 i.e. pressure at any point expressed as the % of the total head. = b/d (1) Sheet pile not at end = b/d,1 = b1/d,2 = b2/d L1 = (1+ 12),L2 = (1+ 22) angles are in radians. 21 1222212 1o o+ + +=+=L L21 1222212 11o o+ +==L L|.|

\|=t1cos111E|.|

\|+=t1cos111C|.|

\|=t11cos1D(2) Sheet pile at end As, b1 = b,b2 =0 So, 2 = b2/d = 0, 1 = b1/d = b/d = |.|

\|=t2cos11E0 cos11=|.|

\|=tC|.|

\|=t1cos11DE C =1001D D =1001% 1001=E21 1222212 1o o+ + +=+=L L21 12o+ +=21 121 +=o(3) Depressed Floor Where = b/d ( )2'332o + =D E D D|.|

\|=t2cos11E|.|

\|=t1cos11D21 12o+ +='100'1DD =Exit Gradient (GE) The pressure gradient at the exit point is called exit gradient. For standard form consisting floor of length b, with a vertical cutoff of depth d, the exit gradient at its d/s end is given by the equation: = b/d In the equation, if d = 0, GE is infinite. It is therefore essential that a vertical cutoff should be provided at the d/s end of the floor. Tosafeguardagainstpiping(undermining),theexitgradientmustnotbe allowed to exceed a certain safe limit for different soils. t1dHGE =21 12o+ +=MaterialSafe Exit Gradient Shingle1/4 to 1/5 Coarse sand1/5 to 1/6 Fine sand1/6 to 1/7 Graphical Solution: Correction to be applied: 1. Correction for the thickness of the floor 2. Correction for the mutual interference of the piles 3. Correction for the sloping floor 1. Correction for the thickness of the floor (a) Pile at u/s end AtpointE:Nocorrection is required,as pressure atthis pointisnot going to interfere with pressure system of any other pile. At point C: t1 = thickness of floord1 = depth of u/s pile (b) Pile at d/s end ( ) additive C pointforCorrection111tdC D||.|

\| = 111C atpressure tdC DC||.|

\| + = ( ) e subtractiv E pointforCorrection221tdD E||.|

\| = 221E atpressure tdD EE||.|

\| = (c) Pile at intermediate point ( ) additive C pointforCorrection1tdC D|.|

\|= ( ) e subtractiv E pointforCorrection1tdD E|.|

\|= tdC DC C|.|

\|+ = = 1C atpressure1tdD EE E|.|

\| = = 1E atpressure12. Correction for the mutual interference of the piles C = correction to be applied in percentage of head D=depthofthepile,theinfluenceofwhichisrequiredtobedeterminedon the neighboring pile of depth d. (D is to be measured below the level at which interference is desired). d=depthofthepile,onwhichtheeffectofanotherpileofdepthDis required to be found out. b = distance between the two piles b = total length of the impervious floor |.|

\|+=bD dbDC'19Sign of the correction: +ve for the points in u/s direction i.e. against the flow and-ve for the points in d/s direction i.e. in the direction of flow. This equation does not apply to the effect of an outer pile on an intermediate pile if:intermediate pile is equal to or smaller in length than the outer pile andintermediate pile is at a distance less than twice the length of outer pile. Interferenceofanypileisonlyforthefacesoftheadjacentpileswhichlie towards the interfering pile, e.g. pile no.2 will interfere with d/s of pile no.1 only. Whereas u/s of pile no.2 will interfere with d/s of pile no. 1 and at the same timed/s of pile no. 2 willalso interfere with u/s of pile no.3. 3. Correction for the sloping floor Sign of the correction: +vefor down i.e. the ve slope ve for up i.e. the +ve slope Thecorrectionisapplicabletothekeypointsofthepilesfixedatthestartor end of the slope. In the above figure the correction is only applicable to point of the intermediate pile. The%correctiongivenintheabovetableistobemultipliedwiththeratio bs/b, where bs = horizontal length below the slope b = distance b/w two pile lines in b/w which the sloping floor exists.SlopeCorrection jn% of pressure 1 in 111.2 1 in 26.5 1 in 34.5 1 in 43.3 1 in 52.5 1 in 62.5 1 in 72.3 1 in 82.0 Design Procedure: (1) Find out the pressure at the key points. (2) Apply the corrections due to mutual interference. (3) Find out the corrected pressures. (4) Find out the thickness of floor at different points. (5) Find out the corrections due to thickness. (6) Find out the corrected pressures at key points. (7) Draw the hydraulic gradient line (H.G.L). (8) Compute the Exit gradient (GE). Example Determinethepercentagepressuresatvariouskeypointsinfollowing figure.Alsodeterminethe exitgradientandplotthe hydraulicgradient line for pond level on u/s and no flow on d/s. Solution. (1) For Upstream Pile Line No. (1) Total length of the floor = b = 57.0 m. Depth of u/s pile line = d = 154.00 - 148.00 = 6.0 m From curve Animperviousfloorofaweironpermeablesoilis16mlongandhas sheetpilesatboththeends.Theupstreampileis4mdeepandthe downstreampileis5mdeep.Theweircreatesanetheadof2.5m. Neglecting the thickness of the weir floor. Calculate the uplift pressures at the junction of the inner faces of the pile with the weir floor by using Khosla's theory.