Keywords: Three-degree of freedom, mathematical model, free vibration, axial motion, simulate.

Transcription

1 ISSN: ISO 900:008 Certiied International Journal o Engineering Science and Innovative Technolog (IJESIT) Volue, Issue 4, Jul 0 A Three-Degree o Freedo Matheatical Model Siulating Free Vibration o a Prisatic Reinorced Concrete Pile in Axial Motion Sule, Sauel Departent o Structural and Geotechnical Engineering, Universit o Port Harcourt, P.M.B. 5, Rivers State, Nigeria Abstract: In this paper, a three degree o reedo atheatical odel is orulated to siulate the ree vibration o a prisatic reinorced concrete pile in axial otion using lexibilit ethod o structural analsis. The reinorced concrete pile itsel is analzed as a lexural bea ith distributed ass and elasticit. The distributed ass is assued to be luped at three discrete points called the nodal points and the displaceents at the nodal points o the luped asses are obtained using slope and delection ethod o structural analsis ith the unit load serving as the transverse load. The luped asses are assued to undergo oscillation about their ean position in the direction transverse to the longitudinal axis o the eightless bea. The lexural lexibilit and axial lexibilit actor are assued to be dnaicall equivalent. Piles undergo axial otion under repeated loading ro ipact haer. The natural requencies o vibration or the three vibration odes ere copared ith the siulated values obtained ro literature to deonstrate the eectiveness o the orulated odels and ere ound to agree avorabl shoing that the orulated odel can be used to siulate the ree vibration o a prisatic reinorced concrete pile in axial otion. Keords: Three-degree o reedo, atheatical odel, ree vibration, axial otion, siulate. I. INTRODUCTION Piles are used as oundation eleents and are eploed here and hen heav structural loads are required to be transitted to the load bearing stratu o the soil. Vibration o piles due to repeated blos ro ipact haer is axial in nature and thereore, produces axial deoration hich results ro axial otion o the pile. The dnaic analsis o ree vibration o a prisatic reinorced concrete pile in both cohesive and cohesion less soils is a task o paraount iportance to structural engineers as it a lead to a ver dangerous phenoenon called resonance hen the natural requenc o an o the vibration odes is exceeded b the excitation requenc resulting ro repeated blos on the pile [] [4]. The prisatic reinorced concrete pile is a lexural bea ith distributed ass and elasticit and all structural sstes having distributed ass and elasticit posses ininite nuber o degrees o reedo [5] [6]. Degree o reedo is the nuber o independent geoetric paraeters that describe the positions o all asses or all possible displaceents o the structural sste at an oent in tie. For instance, or a cantilevered bea ith evenl distributed ass, an ininite nuber o coordinates are needed to deine the displaceent coniguration. I the bea ass is luped into to bodies the orce-displaceent properties o the cantilever reaining constant and the external orces causing the otion are applied at the to asses, then the delected coniguration o the bea at an oent in tie can be copletel deterined b coordinates ith 6 coordinates at each luped asses [7].The dnaic analsis o structural sstes ith ininite nuber o degrees o reedo is diicult and atheaticall cubersoe [8] [9]. In this paper, a atheatical odel is orulated to siulate the ree vibration o a prisatic reinorced concrete pile in axial otion using lexibilit and slope and delection ethods o structural analsis. The orulated odel is siple and straightorard. II. MATHEMATICA FORMUATION Consider a three-degrees o reedo eightless lexural bea subected to a sste o unit loads P, P and P at coordinates, and as shon in Fig.. The displaceents t), ( t) and ( ) at the respective coordinates due to asses, and are as ollos: t) ( ( t () 8

2 t) ISSN: ISO 900:008 Certiied International Journal o Engineering Science and Innovative Technolog (IJESIT) Volue, Issue 4, Jul 0 () ( ( t) () lexibilit actor resulting ro unit load i i An ith luped ass at ith nodal point. Fig : A - degree o reedo eightless bea odel. The generalized equation o otion considering the three luped asses, and is: i ( t) 0 (4) i The solution to equation (4) can be in the or: ( t) C cos t (5) C (,,) represent the aplitudes o displaceent due to, and respectivel. Fro equation (4), the orce o inertia due to ass, and in generalized or is given b: F (6) (,,) The second derivative o equation (5) ith respect to as is: '' ( t) C cos t (7) Substituting or F in equation (6) transors equation (6) to: C cos t (8) According to Osadebe [8], The aplitude o the ass C at ith nodal point is given b: (9) Thereore, C (0) natural requenc o vibration associated ith a particular vibration ode. P P P Substituting or (t) in equation (4) and using equation (0) transors equation (4) to: 84

3 ISSN: ISO 900:008 Certiied International Journal o Engineering Science and Innovative Technolog (IJESIT) Volue, Issue 4, Jul 0 cos 0 i t i (,,) cos t 0 () The generalized requenc equation no becoes: i i 0 At points, and using equation (), the olloing equations are obtained: 0 (4) 0 (5) 0 (6) Putting equations (4 6) in atrix or e have: et 0 (9) Then, ro equation (8) e have: Expansion o equation (0) gives 0,,..., n such that: () () () (7) (8) (0) III. STRUCTURA MODE The reinorced concrete pile behaves structurall as a lexural bea. Consider a lexural bea subected to - degrees o reedo luped asses as shon in Fig. and carring a point load p at the nodal points, and respectivel. 85

4 ISSN: ISO 900:008 Certiied International Journal o Engineering Science and Innovative Technolog (IJESIT) Volue, Issue 4, Jul 0 P / / / Fig : Derivation o lexibilit actors or the three locations o the luped asses Fro structural analsis, the generalized displaceent equation or lexural bea subected to a concentrated end load is given b: P( ) 6EI P EI P 6EI () location o a particular inertia ass at a particular nodal point Using unit load ethod iplies that P = at each nodal point. Equation () no transors to: ( ) i 6EI EI 6 EI () An exaple proble / IV. RESUTS AND DISCUSSION / / / / / EA= constant Fig (b): A lexural bea odel Fig (a): A Cantilever odel The lexibilit actors obtained using equation () are as ollos: 8,, E 8EI 8EI According to Maxell s Reciprocal Theore, 4 8EI,.5, 8EI 4 8EI The lexural lexibilit actor is assued to be dnaicall equivalent to axial lexibilit actor EI Putting the obtained axial lexibilit actors in atrix or using equation (0) e have:. 86

5 ISSN: ISO 900:008 Certiied International Journal o Engineering Science and Innovative Technolog (IJESIT) Volue, Issue 4, Jul (4) Multipling the irst, second and third ro b axial rigidit and dividing the sae ro, and b e have: Again, let Equation (5) no transors to: (5) (6) (7) Expansion o the deterinant in equation (7) and sipliing ields the polnoial: (8) The roots are: 0.66, 0.0, Considering igure (b) and ro statics, l, 4l, = ass per unit length. Using equation (6), l 87

6 ISSN: ISO 900:008 Certiied International Journal o Engineering Science and Innovative Technolog (IJESIT) Volue, Issue 4, Jul 0.58 rad / sec l rad / sec rad / s Table I: Coparison o the results o present odel ith those o the exact solution Natural requenc (rad/s) Present odel Exact solution [0] V. Discussion o Results and Conclusion The results o siulation o -degrees o reedo reinorced concrete pile in axial otion is as presented in Table I. Fro Table I, it can be seen that the present orulation ields results that agreed avorabl ith those o the exact solution. The disparit beteen the results o the present odel and those o the exact values a be due to odel errors. It can be seen that the present odel ielded results that are alost identical ith those o the siulated values shoing the eectiveness o the present odel in the analsis o dnaic response o a prisatic reinorced concrete pile in axial otion. ACKNOWEDGEMENT S. Sule thanks Engr. Eeka Naobakata, Engr. Teple Noor and Engr. D.B. Ee o the departent o Civil and Environental Engineering particularl or their helpul contributions. REFERENCES [] W.T. Thoson, Theor o vibration, rd ed., CBS Publishers, Ne Delhi, 988. [] J.. Huar, Dnaics o structures, Prentice Hall Inc., 990. [] G.B. Warburton, The dnaical Behavior o Structures, nd ed., Pergaon Press, 976. [4] J.H. Argris and H.P. Mlenek, Dnaics o Structures, North Holland,

7 ISSN: ISO 900:008 Certiied International Journal o Engineering Science and Innovative Technolog (IJESIT) Volue, Issue 4, Jul 0 [5]. Meirovitch, Analtical ethods in vibrations, The Macillan Copan, Ne York, 967. [6] R.R. Crag, Structural Dnaics. John Wile and Sons, Ne York, 98. [7].O. Onundi and F.A. Adenii, Application o Methods o Initial paraeters and Substitute Cantilever to Dnaic Analsis o a ulti-store steel raed building, Nig. Journal o Construction and Manageent, Vol., No., 000. [8] N.N. Osadebe, An iproved MDOF odel siulating soe sste ith distributed ass. Journal o Universit o Science and Technolog, Kuasi, Vol. 9, Nos., and, 999. [9] S. Sule, Approxiate ethod or the deterination o Natural Frequencies o A Multi-Degree o Freedo Sste, Nig. Journal o Tech., Vol. 0, No., pp 4-49, 0. [0] R.W. Clough, and J. Penzien, Dnaics o Structures, nd ed., McGra-Hill, Ne York,