Instability Analysis of Damaged Pile Due to Static or Dynamic Overload

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1 Geomaterals, 0,, Publshed Onlne October 0 ( Instablty Analyss of Damaged Ple Due to Statc or Dynamc Overload P. N. Jk *, J. U. Agber, Cvl Engneerng Department, Unversty of Agrculture, Makurd, Ngera Electrcal Engneerng, Unversty of Agrculture, Makurd, Ngera Emal: * peterjk@yahoo.com Receved June 5, 0; revsed August 6, 0; accepted August 6, 0 ABSTRACT Instablty of a damaged ple due to a statcally or dynamcally appled overload s studed n ths work usng the fnte element method. A damage parameter from such a ple s calculated usng fracture mechancs concepts. The parameter s used to modfy the beam element at the cracked or damaged locaton. Sol samples were obtaned from the ste of the ple and were subjected to laboratory tr-axal tests to obtan shear strength parameters c and. Other sol parameters such as Young s modulus E and Posson s rato were also obtaned from the tr-axal tests. These were used to calculate shear strength and sub-grade modulus k for the sol. The parameters, E, and k were later used together wth the damage parameter n the fnte element smulaton of the strength of the damaged ple usng Egen value analyses. The layered sol modulus s approxmated by takng the mean value and s denoted by K f. The dscrete element matrces are assembled nto a system Egen-value equaton, the soluton of whch provdes the stablty or nstablty loads for the damaged ple. The results obtaned for a ple wthout damage, that s, when 0, are n good agreement wth those publshed n the lterature. It has also been found that hgher sol resstance s needed to support the damaged ple. It s concluded that the proposed model s a good canddate for use n the analyss and repar of damaged ples due to earthquake overload by sol stablzaton methods. Keywords: Damaged Ples; Earthquake Overload; Instablty Analyss; ayered Sol Support. Introducton Recently, earthquakes and Tsunams have caused severe damages to some nfrastructures around the world such as the quake n Hat and the Tsunamn Japan. Some of these nfrastructures suffer damage but do not collapse, partcularly f the nfrastructure s a ple beng supported by the surroundng layered sol. Some of these nfrastructures on ple supports that suffer damage nclude nuclear power statons (such as those n Japan), brdge pers and offshore platforms (see Smth []). In recent years, Burgess [,3] has drawn attenton to the possblty of nstablty occurrng durng nstallaton of slender ples. He assumes the ground shear resstance always act tangentally to the ple, rather than n a fxed (vertcal) drecton. Ths means that the nstablty can be of the wanderng or flutter varety rather than classcal bucklng and requres a dynamc analyss to solate the loads at whch loss of stablty occurs. A descrpton of the mechancs of non-conservatve nstablty and some closed form solutons for smple columns are to be * Correspondng author. found n epholz [4]. The addton of constant lateral (Wnkler) stffness was studed by Smth and Herrmann [5]. Dervng dscrete beam element matrces, Smth [6] has studed the onset of nstablty of undamaged ples n unform ground assumng both conservatve and nonconservatve cases. The purpose of the present artcle s to study the behavour of damaged ples under conservatve loads and supported by layered sols.. Calculaton of ocal Stffness Due to Damage A strp or a rectangular plate representng the damaged porton of the ple has a wdth b and thckness h wth a transverse crack c. It s well known that the complance dervatve of such a strp s related to the stran energy release rate G as n Jk [7]: Q df G () da where Q s the drvng force, F s the local flexblty due to crack, A s the area of the crack surface. The energy Copyrght 0 ScRes.

2 P. N. JIKI, J. U. AGBER 5 release rate G s also related to the stress ntensty of the cracked strp as: G K () 8 n whch μ s the shear modulus for the materal of the strp; 3 4 for plane stran and s Posson s rato. From Equatons () and (), we have: Q df K (3) da 8 For the present work, we work wth the unt wdth of crack. Then the area of the crack A s gven as n Jk [7]: Then: A c c (4) df df (5) da dc Substtutng Equaton (5) nto Equaton (3), we have: Q df K (6) dc 8 From Equaton (6) the complance dervatve becomes: df K (7) dc 4Q The condton for c 0 mples no crack flexblty. We ntegrate Equaton (7) to get the complance F as n Jk [7]: c F K cdc 4Q (8) 0 For mode due to bendng moment M, we have ( Jk [7], Anfants and Dmarogonas [8]): where et c K cy h c c Y h h (9) 6M, Q M (0) bh 3 c c c h h h 4 () Usng Equatons (9)-() nto Equaton (8) and ntegratng, we have: n whch 9 c F h h () c chc c c F d h Y 0 h h h 3 c c h h 4 5 c c h h 6 7 c c c h h h 9 0 c c h h The relatonshp between stffness and flexblty s: K T 8 (3) (4) F Then the non-dmensonal stffness parameter s: Kh T.0 (5) EI c 6 h Then the damaged stffness coeffcent for the ple s wrtten as n Jk [7]. EI (6) 3. Dscretzaton of the Dfferental Equatons Prevous work has typcally nvolved dscretzaton of the complete ple (or column) n terms of Fourer coeffcents, for example. A more modern approach, through whch exstng fnte element programs can be utlsed, s to dscretze locally over a small, typcal element, e.g. Fgure. For flexural modes the usual dscretzaton nvolves the transverse dsplacements and rotatons of the end nodes: y N N N N (7) 3 4 In ths equaton the well-known shape are gven by N x x x 3 3 N x x N3 3 3x x 3 N4 3 3 x x 3 3 functons N (8) Consder, for example, one of the examples analysed by Burgess [3]. Ths nvolves lnearly varyng sol stffness Copyrght 0 ScRes.

3 6 P. N. JIKI, J. U. AGBER Crack Crack c node node (a) (b) Secton propertes E, I, A, ρ, α Secton propertes E, I, A, ρ, α Fgure. (a) Typcal dscre te flexural lne element; (b) Enlarged detals of cracked element k and sde shear wth depth. The rates of change of k and are m k and m respectvely, and for equal ele- ment lengths th e condtons n the -th element are as shown n Smth [6]. Whle both Burgess [3] and Smth [6] model undamaged ples on unform ground under conservatve and non-conservatve forces, our present model consders the behavour of damaged ples on layered sol and subjected to conservatve forces only. It has been found n the lterature that damage to structural components n form of cracks can alter the statc strengths of these components as reported n Jk [7], Capuan and Wlls [9] and can also alter the vbraton characterstcs of these components such as mass and natural frequences as reported n Anfants and Dmarogonas [8]. Indeed our model modfes the conservatve equaton by Tmoshenko and Gere [0] wth our proposed pa- y rameter as: d EI Q y" m x y d xy k mkxy Ay (9) In Equaton (9) the prmes represent spatal, and the dots temporal dfferentaton. From the thrd and fourth terms n Equaton (9) t can be observed that the selfadjontness s a feature usually found n conservatve systems. Dscretzaton n terms of the four undetermned nodal parameters of the element s completed by applyng the Galerkn s process n the usual way. The resultng nformaton s tabulated, term by term, as Table. For completeness the relevant matrces are lsted n Appendx, although some are well known, KM and MM beng the stffness and mass matrces, for example see Smth [6]. Whle our model has consdered the behavour of a damaged ple on a layered sol, t also has ncreased dffculty n obtanng samples from dfferent layers of the layered sol for the calculaton of shear strength, sub-grade modulus k, Young s modulus E, Posson s rato as well as bearng capacty parameters such as Nc, Nq, N for the sol. These parameters are often obtaned from tests or are predcted usng the fnte element analyss n conjuncton wth exstng equatons n the lterature as n Jk et al. []. Assumng harmonc moton, the conservatve Equaton (9) dscretzes nto element egenvalue equaton EI KM QK KS MKQ K fmm M kkry AMMy (0) n whch K f s the average subgrade modulus whch s gven for the three sol layers shown n Fgure as K k k k 3 f () Element assembly to make up the complete ple follows n the usual way. Equaton (0) represents an unsymmetrcal egenvalue problem whch can predct two types of nstablty, dependng on the combnatons of the physcal parameters Q, K, etc. Ether one of the values can become negatve (classcal bucklng nstablty) or two real values can converge and n the lmt become equal. Thereafter a further perturbaton leads to complex con- jugate values of (Futter nstablty). In the conservatve case (symmetrcal egenvalue problem) only classcal bucklng can occur. Heren we assume the boundary terms that result from ntegraton by parts n the Galerkn process to be small and are dsregarded. Therefore the 3 Copyrght 0 ScRes.

4 P. N. JIKI, J. U. AGBER 7 usual conservatve boundary condtons apply to the present work as n Bolotn []. After assembly, the egenvalue equaton becomes K G M0 () For bucklng we have: K G For vbraton we have: 0 (3) K M 0 (4) 4. Results and Dscusson Usng the above matrces and the calculated damage pasolved wth twelve rameter α, many examples have been beam elements or more per ple. Ths has resulted n all cases n nstablty predctons that are n good agreement wth publshed work by Smth [6] when we set our crack or damage parameter to zero. For example, Fgures 3-5 show nstablty predctons for ples n a layered sol subjected to conservatve loads only. In our work here, we have approxmated the layered sol by takng the mean sub-grade modulus to represent sol strength (Wnkler model) needed to support the damaged ple. Therefore n Fgures 3-5, when the crack parameter s zero, our model approaches that by Smth [6]. As the crack parameter ncreases from 0. to 0.3, the stablty of the ple reduces and hgher sol modulus s needed to keep the ple n a vertcal and stable poston. Table. From Smth [6] Dscrete element matrx defn- tons. Terms n d.c. Galerkn ntegral Element matrx y NN KM y NN K x y P N d wherep x N KQ y NN d x MM xy NxN KR d xy NxN KS d x y NxN KT k P = appled load Fgure. Dstrbu ton of sol resstances. Copyrght 0 ScRes.

5 8 P. N. JIKI, J. U. AGBER (Smth[6]) Fgure 3. Effect of damage on ple bucklng wth zero sol modulus. (Smth[6]) Fgure 4. Effect of damage on ple bucklng wth 00% normalsed sol modulus. (Smth[6]) Fgure 5. Effect of damage on ple bucklng wth 00% normalsed sol modulus. Copyrght 0 ScRes.

6 P. N. JIKI, J. U. AGBER 9 5. Conclusons From the fndngs of the present study, we conclude as follows: ) Ple damage due to an earthquake can be evaluated usng a damage parameter whch s calculated usng fracture mechancs concepts. ) The results of the present study show that when a ple s damaged, more sol resstance s needed to support the damaged ple. Ths fndng s very helpful n a stuaton where the damaged ple cannot be replaced. In ths case the sol or the ground s strengthened or moblsed to support the damaged ple. 3) Exstng geotechncal methods of ground or sol mprovements can be used to moblse support for the damaged ple. In layered sols, all or some of the weaker sol layers can be strengthened by the use of sol stablzaton methods that currently exst. REFERENCES [] I. M. Smth, Transent Phenomena of Offshore Foundatons, In: O. C. Zenkewcz, R. W. ews and K. G. Stagg, Eds., Numercal Methods n Offshore Engneerng, John Wley and Sons, ondon, 978. [] I. W. Burgess, The Stablty of Slender Ples durng Drvng, Geotechnque, Vol. 6, 976, pp do:0.680/geot [3] I. W. Burgess, Analytcal Studes of Ple Wanderng durng Installaton, Internatonal Journal for Numercal and Analytcal Methods n Geomechancs, Vol. 3, No., 979, pp do:0.00/nag [4] H. epholz, Stablty Theory, Academc Press, New York, 970. [5] T. E. Smth and G. Herrmann, Stablty of a Beam on an Elastc Foundaton Subjected to a Follower Force, Journal of Appled Mechancs, Vol. 39, No., 97, pp do:0.5/ [6] I. M. Smth, Dscrete Element Analyss of Ple Instablty, Internatonal Journal for Numercal and Analytcal Methods n Geomechancs, Vol. 3, No., 979, pp do:0.00/nag [7] P. N. Jk, Bucklng Analyss of Pre-Cracked Beam- Columns by apunov s Second Method, European Journal of Mechancs A/Solds, Vol. 6, No. 3, 007, pp do:0.06/j.euromechsol [8] N. Anfants and A. D. Dmarogonas, Stablty Analyss of a Column wth a Sngle Crack Subjected to Follower and Vertcal oads, Internatonal Journal of Solds and Structures, Vol. 9, No. 4, 983, pp do:0.06/ (83) [9] D. Capuan and J. R. Wlls, Wave Propagaton n Elastc Meda wth Cracks. Part. Transent Nonlnear Response of a Sngle Crack, European Journal of Mechancs A/Solds, Vol. 6, 997, pp [0] S. P. Tmoshenko and J. M. Gere, Theory of Elastc Stablty, McGraw-Hll, New York, 96. [] P. N. Jk, J. U. Agber and N. N. Osadebe, Fnte Element Evaluaton of Bearng Capacty Parameters for Sols n the Unversty of Agrculture, Makurd, Ngera, Indan Journal of Innovatons and Development, Vol., No. 3, 0, pp. -6. [] V. V.Bolotn, Nonconservatve Problems of the Theory of Elastcty, Pergamon Press, Oxford, 963. Copyrght 0 ScRes.

7 0 P. N. JIKI, J. U. AGBER Appendx. Relevant Matrces from Smth [6]. 4 SYMM 4 KM = SYMM KR SYMM 30 KS SYMM MM = KP SYMM KT Copyrght 0 ScRes.