A Theoretical Model On Coupled Fluid-structure Impact Buckling

Coupled fluid-structureand boundaryu,(W)impact buckling:0. Zhang et al.condition: O,P,(b,t) p,,,(6g)where u,, P,, and Pa are the air velocity in the x-direction, pressure (absolute), and density, respectively;V(t) and u, (x,t) are the falling plate velocity and the freesurface water velocity in the x-direction, respectively;h, is the air layer thickness. Other symbols are referredto in the nomenclature.The water motion equation”dudv(74clx - OdYauaup -- uan v(audY -CCU’b)axa2P a2P2 2 -Q(7c)P P[(32 2(?3(3 (32](7d)with initial condition:P(X,Y,O)u(x,y,O) pm,and boundary 0, G,Y,O) 0(7e)condition:on CGa4xw)apcby, 0 0, u(O,y,t) o,axaxx 0 0x 0(7f)on CDP(x,h,t) P,,u(x,h,t) u,, u(x,h,t) v,(Q)on DE Pa,,,,, u(x,h,t) us, u(x,h,t) usW&f)(7h)on GFafw,oay -pg,u(x,O,t) 0, v(x,O,t) 0y o(79In the above expressions, u and u are the watervelocities in x and y directions, respectively; P and p arethe water pressure (absolute) and density, respectively;u, and v, are the free-surface water velocities in x and ydirections, respectively. Other symbols are referred toin the nomenclature.Peak impact and treatmentafter peak impactIt has been indicated’, that the hydrodynamic slamming pulse is essentially a semi-sine wave before thecolumn’s postbuckling and plastic collapse take place.A typical record for the slamming pulse from the experiment is shown in Figure 3 where E,, tp, and to are theslamming peak value, peak slamming time, and slamming duration, respectively. The experiment demonstrates that E, rises with increasing slamming height andr. varies with column slenderness: the smaller theslenderness,the shorter the to. For a prescribedslenderness, however, to is basically unchanged provided that the column does not display postbuckling andplastic collapse in one impact.For a theoretical study, however, when does thepeak impact occur and how can it be determined? Theanswer to this question may lead to a simplified treatment for the present model analysis. From the studiesby Chuang,’ Verhagen,” and Koehler,” the peak impact is considered to occur at the instant the deformedwater free surface makes contact with the edge of theplate. According to this peak impact condition, ?,, inFigure 3 coincides with the time the plate’s edgetouches the deformed water free surface. On the otherhand, equations (6a)-(7k) are valid until the deformedwater free surface makes contact with the plate’s edge.Thus, (6a)-(7k) constitute the mathematical description of the air and water motion in the phase of 0 5 t 5 tp.To obtain the decay phase of the slamming pulse, themathematical description of the air and water motionafter the peak impact is needed. However, this description is not always necessary. It is known from the experiment that there is no distortion phenomenon in theslamming pulse shape before the column’s postbucklingand plastic collapse take place. Therefore, if only thebuckling criterion is required rather than all the criteriaon EFw.w)ax - ww)wx,y,oaxayx L x LVj)0x LMoreover, on the air-water interface, the free-surface water velocity u, is related to the air velocity u,.This relationship isu, 0.198, ujl.20p .533p,on.133 L333x0.20]/p0.667(7k)28Appl.Math. Modelling,1993, Vol. 17, JanuaryFigure 3.A typical fluid-structureslammingpulse.

Coupled fluid-structureincluding the postbuckling and plastic collapse, a simplified treatment for the mode1 analysis is possible sincethe decay phase of the slamming pulse can be estimatedin this case from the rise phase. The simplificationprocedure is (a) to divide the motion of the mode1 intotwo phases: 0 5 t 5 tp and t tp; (b) for 0 5 t 5 t,,, todescribe the motion with equations (l)-(7k); (c) fort tp, to define the motion of the column remaining with thedescription in the section outlining the structural motion equation except the initial condition, but the slamming pressure in this phase is treated as a prescribedload and estimated from the preceding phase (i.e., therise portion of the pulse). In this simplification, themotion description of the air and water in t t,, isomitted, and the column’s motion is considered to beindependent of the air and water behavior.In this paper, we restrict our attention to the bucklingcriterion and the relationship between the peak impacttime and the column slenderness. Therefore, we use thesimplification procedure in this analysis.Solutionof mathematicalequationsIt is obvious from the mathematical description givenpreviously that the column, air, and water motion equations are three sets of coupled ones. It is seen thatequation (4) contains an unknown pressure p,, which isrelated to the water-plate hydrodynamic slamming. Onthe other hand, the air equation (6d) involves an unknown quantity V, the falling velocity of the plate.Between the air and water, their behaviors are coupledthrough the air equation (6d), which includes the freesurface water velocity u, and the water equation (7g),which contains the air pressure p, and air velocity u,.Because of the coupling among the three sets ofmotion equations, they are difficult to solve directly. Inwhat follows, we first introduce a decomposition procedure for these equations. Once these equations are decomposed by the procedure, their solutions can be obtained by some appropriate numerical methods.Decomposition of the coupled equationsBefore the decomposition ofthe coupled equations isstarted, it is useful to review the coupling characteristics. It is found from the air description that except (6d ),which gives a relationship between h,, V, and u,, theremaining equations have no explicit connection withthe water and structural equations. Thus if h, can bepredetermined independently of the other quantities,the air description will be separated from the water andstructural equations. Moreover, since there is no directconnection between the structural and water equations,these two sets of equations can be treated separatelyafter the air description is decomposed.The above discussion suggests a possible decomposition procedure for the coupled equations. The keypoint of the procedure is to give an appropriate estimation for the air thickness h, in advance. Usually thisis difficult. However, when the coupling characteristicsare combined with a special iteration technique, thedifficulty can be surmounted.impact buckling:0. Zhanget al.Below, we start the decomposition procedure. Wefirst divide the time domain 0 % t 5 tp by the discretetimepointstj tj , At(j 1,2,. . .J),whereAt t,lJ is the time step length from tj , to tj and J is aprescribed number for the total time steps. Then weassume that the quantities at the time tj ,, such ash,(x,tj ,),V(ti-,), and q(x,t, ,),are known. Ourgoal is to decompose the coupled equations defined atthe time tj. The decomposition procedure consists of thefollowing steps:Step 1. Estimate an appropriate value for h, (x,tj) fromequation (6d) and h,(x,ti ,), V(t, ,), andu,(x,tj-,I.This value is used as hz, the firstapproximation of h, (x,tj).Step 2. Substitute hO,into the remaining air equationsand separate them from the water and structuralones.Step3. Solve the separated air equations for UHand e,which are the first approximations for u,(x,tj)and p, (x,t,), respectively.Step 4. Substitute hz, uz, and P”, into the water andstructural equations respectively;thus thesetwo sets of equations are decomposed.Step5. Solve the decomposed structural equations forOllC,‘L&‘,“N“, ‘Nb, and V”, which are the firstapproximationsof u’([,t,),uh([,tj), N”([,t,),Nb( ,tj), and V(t,), respectively.Step 6. Solve the decomposed water equations for u”,IJ’, uf, and P”, which are the first approximationof u(x,tj), u(x,tj), and p(x,tj), respectively.Step 7. From (6d) and the current values of hO,,P, anduz, calculate the updated air thickness h:.Step 8. Identify whether or not the inequality (hi - h:( y holds true, where y is a positive value lessthan unity. If the inequality is true, add nexttime step to fj, then repeat steps l-7. Otherwisereplace hO,with h:, then continue steps 2-7 untilthe inequality operates.The above decomposition procedure is shown inFigure 4. It is seen from the figure that the procedure isin fact a staggered iteration technique.Solution of the decomposed equationsWe now treat the three sets of decomposed equations. From the decomposed equations (6a) and (6b), itis found that if they are expressed in a proper finitedifference form, an explicit solution scheme for I(, andpOcan be obtained. This is also true for the decomposedequations (7a) and (7b). Moreover, once u and u arecalculated from their explicit solution schemes, thesource term Q in (7 ) can be estimated independently ofthe pressure p(x,y,tj), and the pressure equation becomes a linear one. Thus, the solution of the pressureequation also becomes quite easy.Based on the above discussion, a finite differencemethod is used in this paper to solve the decomposed airand water equations. The present method is identical tothat used by Koehler” in his analysis of the plate-waterhydrodynamic impact; therefore, its details are omittedhere.Appl. Math. Modelling,1993, Vol. 17, January29

Coupled fluid-structureimpact buckling:0. Zhang et al.geometric and material nonlinearities.7method is used in the present problem, thecalculate the tangential stiffness matrix [k].to Ref. 7, the tangential stiffness matrix [ k]is defined asWhen thecenter is toAccordingin this case{AF} [Z?,l{A8},{AF} {F)t eAr- {F)‘,{Aa} {@‘ eAt- (6)’(8)Where [ k] is the tangential stiffness matrix at the time t;{As} and {AF} are the nodal point displacement andinternal force increments, respectively, from the time tto t OAt, 8 and At being the Wilson’s parameter andthe time step length in the time integration scheme.By substituting the expression of {F} in equation (4)into equation (8) and using the following definitions[K% [ Lml* o,rKal uGJ* o,N% ml, ,(9)(11)the tangential stiffness matrix is derived in this case asa1 m Kl [&I(124where(a) [R,] {e * {E}, {F’} being the global structuralvect r of the finite element assemblage by {Fp J,JNl’ * L-p,* . &, 017 d5 and {c} being equal pact buckling.procedureforFor the solution of the decomposed equation (4), atangential stiffness method with the Wilson-8 time integration schemer2 i3 is utilized in this paper. This methodhas been adopted by one of the present authors to treat adynamically loaded stiffened plate subjected to both[ KZ’I[ KT1’ [ K%l[ f%‘l[K:l [Ki,l U&,1 [&,r,lin which the submatrices{F) 11Kyo . [&I), [ qil,*[ BmNL1l.l [ & ‘I0 *[B;],, 0, 0, *. . 01 (12b)(b) [ kJ {i; * {c}, {p} being equal to{ } [-1,o,o, o,p,,o,o, o] (13)(c) [K,] C, [K,]‘, e being the total finite elementnumber and [K,]’ being calculated by [ K:,,IT1(14)are defined as(15)In these expressions, [ DG], [ D ], and [ DT;“] are thecolumn’s membrane, bending, and membrane-bendingelastioplastic matrices, respectively.These matriceshave been defined in Ref. 7.Once [&,I is determined from equations (ll)-(19),the response of the decomposed nonlinear equation (4)is easily obtained by the solution of its linearized form as[ M]{ y ““’30Appl.Math.Modelling,1993,Vol. 17, January [k,]‘{A6} {Z?), eAr- {F)’(20)

Coupled fluid-structurewhere {R)‘ eAr is the nodal point external force at thetime t 8A t.30Impact criteria and numericalexamplesIn this section, we will analyze two numerical exampleswith the present theoretical model. One of the examplesis a calculation of the relationship between the peakimpact time r,, and the column’s slenderness ratio A; theother is a calculation of the column’s critical values. Thecommon parameters used in the two examples are givenbelowL 3.Om,h, 3.0m,a 0.4m,bIimpact buckling:0. Zhang et al.Peory.‘.expe&entcw OAmp 999.6kg/m3, p 1.19 x 10p9Ns/m3P afm 101.36 K Pa, pat,,, 1.20 kg/m3pair 1.77 x 10p5Ns/m20’M 1000.0 kg, m 80.0 kg, g 9.80m/s2L76406080/uuI20440slenderness ratio ,J (;I fi/ / h, )pS 7.8 x lo3 kg/m3, CQ 2.6264 x 108N/m2Figure 5. Relationshipslenderness ratio.E 2059.96 x lo8 N/m’, H’ 0.0Example 1. Calculation on the tp vs. A curveFour columns with a clamped boundary conditionand a rectangular cross-section are calculated in thisexample. The geometry of the columns is b, x h, x 1,where 1 0.3 m, 0.4 m, 0.5 m, and 0.6 m is the lengthof the columns, b, 14 x lop3 m and h, 8.0 x10e3 m are the cross-section width and thickness, respectively. The three columns with the lengths of0.4 m, 0.5 m, and 0.6 m are the same as those used inthe experimental study, namely the specimens s 11, s2 1,and s3 1 in Refs. 1 and 2. The initial condition of the columns is assumed to beU’([,O) S&O) tiQ,O) 0(21)L&&O) 16w, . ; . (l-2: ;)(22)Where w0 is the initial transverse displacement disturbance at the center, its value is taken as 1.O x lop3 m.This value is close to those of sl 1, s2 1, and s3 1.The initial air thickness is chosen in this example ash,(x,O) 0.2 m.For each of the considered columns, the peak impacttime tp is calculated from the present model. The resultsare plotted in Figure 5 against the slenderness ratio A.From this figure, it is observed that tp is approximatelydirectly proportional to A. This conclusion is in agreement with that of the experimental curve, which is alsopresented in the figure.Example 2. Calculation of the critical impact valuesThree impact criteria defining the structural elasticand inelastic behavior in fluid-structureimpact buckling have been suggested in Refs. 1 and 2. The threecriteria are buckling, plasticity, and plastic collapse.These criteria will be used in the numerical analysis. Forconvenience,betweenpeakimpacttimeandthey are restated as follows:buckling criterion-fora column with a prescribedslenderness, the fluid-structureimpact buckling isidentified as occurring in one impact if the peak axialcompressive strain E, reaches a critical value E,,,, atwhich the maximum bending strain IE&, grows to avalue equal to the axial loading magnitude. E,,,, iscalled critical buckling impact strain.plasticity criterion-fora column with a prescribedslenderness, the fluid-structureimpact plasticity isidentified as occurring in one impact if E, reaches(does not exceed) a critical value E,,, at which themaximum compressive-bendingresultantstrainJEW,,,grows to a value equal to the yield strain of thematerial. E,,, is called critical plasticity impactstrain.collapse criterion-fora column with a prescribedslenderness, the fluid-structureimpact plastic collapse is identified as occurring in one impact if E,reaches a critical value E,., beyond which further increases in drop height result in decreases inthe peak axial loading strain rather than in increases. E,,, is called critical plastic collapse impact strain.Below, we will use the buckling and plasticity criteriafor calculating the numerical results of the criticalbuckling and plasticity impact strains. The model cannot describe the column’s plastic collapse; thereforethe critical impact strain E,,, cannot be calculatedin this numerical analysis.The columns used in this example are also those ofsl 1, s21 and s31 in Refs. 1 and 2. The geometrical sizesof these columns are given in the previous example. Theinitial geometrical disturbance also takes the form ofAppl.Math. Modelling,1993, Vol. 17, January31

Coupled fluid-structureTable 1.Numericalimpact buckling:0. Zhang et al.results for critical imDact values.Critical buckling impact valueE,,,&strain)Critical plasticity impact valueQ,,&CL .0840.01000.0580.0600.0750.0(22), except that w,, is given by the real imperfectionsmeasured in the experiment.To obtain the critical impact values, each of thecolumns considered is calculated under different impacts that correspond to different drop heights. Thecalculation for each impact gives a maximum compressive-bending resultant strain IE\ ,maximum bendingForstrain IGL , and maximum compressive strain 1 1.different impacts, these results are drawn with twocurves: l&ax vs. I4 andvs. 1q.l.From these twohImaxcurves and the buckling and plasticity criteria, the numerical results of E,,,, and E,,, are determined. Thenumerical values of the three columns are presented inTable 1 in which the experimental values are also listedfor comparison. It is seen from the table that the discrepancies between the two sets of results range from29% to 40%. This agreement is acceptable, even quitegood, since in the present model we treat the air motionwith the one-dimensional theory whereas in the experiment the motion of the air is two-dimensional flow.ConclusionThis paper presents a theoretical model for a coupledfluid-structureimpact buckling phenomenon.Themodel is an impact system consisting of a small imperfection elastoplastic column with its upper end attachedto a large mass and its lower end to a flat plate, anincompressible viscous water field, and a one-dimensional inviscid air layer lying between the plate and thewater free surface. When the mass-column-plate unit inthe system perpendicularly impacts the water free surface from a certain drop height or in a certain fallingvelocity, the column will produce a transversely flexural vibration due to the axial compression induced bythe hydrodynamic slamming between the plate and thewater. The flexural vibration simulates a coupled buckling phenomenon occurring in some fluid-structure impact environments where structural buckling is causedby fluid-structureimpact and strongly depends on thehydrodynamic behavior of the fluid.The paper provides a mathematical description forthe model motion. In the description, the motion isdivided into two phases. The first phase corresponds to0 5 t 5 tp and the second phase to tp 5 t 5 to where tp isthe peak impact time. For 0 5 t 5 tp, the motion istreated as a coupling problem between the structuralbuckling and the hydrodynamicslamming. In thisphase, the virtual work principle and the Prandtl-Reussplastic theory are applied to the structural buckling, theone-dimensional continuity and momentum equations32Appl. Math. Modelling,1993, Vol. 17, Januaryof a compressible inviscid fluid are applied to the airmotion, and the two-dimensional Navier-Stokes equations of an incompressible viscous fluid are applied tothe water motion. This phase ends when the deformedwater free surface makes contact with the edge of theplate. In the second phase, the motion is treated as adecoupled structural buckling problem where the column is assumed to be loaded by a prescribed axialcompression.In numerical analysis for the first phase, the threesets of coupled equations are first decomposed by astaggered iteration method. The decoupled structuralequation is then solved by a nonlinear finite elementmethod with the Wilson-8 time integration scheme; thedecoupled air and water equations are solved by a finitedifference method. The paper calculates two numericalexamples. One example is a calculation of the relationship between the column’s slenderness and the slamming duration; the other is a calculation of the column’scritical impact values. The numerical results are compared with the experimental ones and the agreement isfound to be quite good.Nomenclaturecolumn’s membrane and bending displacementscolumn’s generalized membrane and bending strainscolumn’s membrane force and bendingmomentcolumn’s stress and yield stresscolumn’s length and thickness coordinatescolumn’s length and length in a finite elementcolumn’s slenderness and cross-sectionareacolumn’s elastic modulus and strain-hardening parametercolumn’s bulk and length densities, p,* -Apo t mass& mlM,half lengthvelocity oftionrectangularregionsair velocityatmosphericPCltWZand mass of flat plate in Fig. 1pz l/Mand width of flat plate in Fig. 1flat plate and gravity acceleracoordinatesin air and waterand pressure (absolute)pressure and densityunder

Coupled fluid-structureh,,hu,uUS7usP,P:;:LLW,air layer thickness and height of deformedwater regionwater velocities in x and y-directionsfree-surface water velocities in x and y-directionswater pressure (absolute) and densit

Fluid-structure impact buckling has two basic char- acteristics as revealed by the experimental study: (a) the structures may experience three critical states of buckling, postbuckling, and plastic collapse in one im- pact, each of the states corresponding to a particular