Stability of slender columns on an elastic foundation with generalised end conditions

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1 INGENIERÍ E INVESTIGCIÓN VOL. No., DECEMBER - 01 (4-40) Staility of lender column on an elatic foundation with generalied end condition Etailidad de columna eelta ore fundación elática con condicione de apoyo generalizada J. D. ritizaal-ochoa 1 BSTRCT Slender column lateral taility under compreive axial load i preented, with uninhiited, partially inhiited and totally inhiited end ide-way, including the effect of emi-rigid connection and a uniformly ditriuted lateral elatic foundation (Winkler' type) throughout it entire pan. The propoed claification of primatic column on an elatic foundation and the correponding taility equation are general and relatively imple to apply, yielding exact reult when compared to other analytical method. The uckling load wa otained y making the determinant of a 4 x 4 matrix equal to zero for column having ide-way uninhiited or partially inhiited at oth end, and of a x matrix for column having ide-way inhiited at one or oth end. The effect of emi-rigid connection on the uckling load of five claical column cae i fully dicued and the reult compared to thoe ariing from other analytical method. Keyword: racing, uckling, column, elatic foundation, pile, emi-rigid connection, taility. RESUMEN Se preenta de una manera cláica la etailidad lateral de columna eelta ajo carga axiale de compreión; con deriva en lo extremo deinhiida, inhiida parcialmente y totalmente inhiida, incluyendo lo efecto de la conexione emirrígida y una fundación elática lateral y uniformemente ditriuida (tipo Winkler) a lo largo de toda u luz. La claificación propueta en la columna primática, ore fundación elática y la ecuacione correpondiente de etailidad on generale y relativamente imple de aplicar; oteniéndoe reultado exacto cuando e comparan con otro método analítico. La carga de pandeo e otiene igualando a cero el valor del determinante de una matriz de 4 x 4, para columna con deriva lateral deinhiida o parcialmente inhiida en amo extremo, y de una matriz de x para columna con deriva lateral inhiida en uno o amo extremo, repectivamente. Lo efecto de la conexione emirrígida ore la carga de pandeo, de cinco cao de columna cláico, on dicutido y lo reultado on comparado con lo de otro método analítico. alara clave: arriotramiento, pandeo, columna, fundación elática, pila, conexione emirrígida y etailidad. Received: May 19th 01 ccepted: Octoer 15th 01 Introduction 1 The taility and econd-order tatic analyi of column laterally upported on an elatic foundation in generalied end condition are of importance whit tructural engineering. noted y Hetenyi (1967), there are many application for uch analyi, uch a the contruction of pile foundation, railroad track, ridge, thin revolution hell (preure veel, oiler and container) and large-pan cylindrical hell and dome. For intance, a ymmetrically loaded longitudinal element of a cylindrical tue regarding it axi can e treated a a eam on an elatic foundation. Thu, the propoed approach can e ued for tudying a wide range of prolem related to elatic taility and econd-order tatic analyi, eing of particular importance in tructural and geotechnical engineering, pecifically in oil-tructure interaction prolem dealing with pile, drilled haft, caion and/or pier ujected to a comination of axial, moment and lateral load. Hetenyi (1967) outlined the claical procedure for reolving the elatic taility of primatic column having hinged and clamped connection on an elatic foundation preenting the exact olution for particular cae, uch a column having free-free, hingedhinged and clamped-clamped end condition. Timohenko and Gere (1961) ued a imilar approach to reolving other cae of interet, uch a a column under a uniformly-ditriuted axial load on an elatic foundation. Wet and Mafi (1984) determined the eigen-value for column on elatic upport uing an initial-value numerical method. Razaqpur (1986) preented the tiffne and equivalent joint load matrice for a column finite element reting on a Winkler-type elatic foundation uing exact hape func- 1 J. Dario ritizaal-ochoa. MS, hd, Univerity of Illinoi at Champaign-Urana, US. ffiliation: Full rofeor, Dept. of Civil Engineering, Univeridad Nacional de Colomia, Medellin, Colomia. jdariti@unal.edu.co. How to cite: ritizaal-ochoa, J. D., Staility of lender column on an elatic foundation with generalied end condition., Ingeniería e Invetigación, Vol., No., Decemer 01, pp

2 RISTIZBL-OCHO tion. Cheng and antelide (1988) derived the differential equation, tiffne coefficient and fixed-end force of a column on an elatic foundation, including ending and hear deformation. ng and Wang (1990), a well a Wang, Xiang and Kitipornchai (1991) ued finite element for tudying the uckling of column on an elatic foundation. More recent paper on the taility of eamcolumn reting on an elatic foundation, including two and three parameter foundation model, are availale in the technical literature, uch a thoe y Struther and Jayaraman (010), Morfidi (007, 010), vramidi and Morfidi (006) and Onu (006). 5) End rotation are partially inhiited y flexural pring a and located at and B, repectively; and 6) The econd-order analyi in the next ection i intended to e in the mall deflection range (commonly referred to a a "linearied" approach). It hould e pointed out that reult are valid a long a the lateral deflection of the column remained mall. ritizaal-ochoa (1994, 1996 and 1997) ha preented the effect of emi-rigid connection on the taility and econd-order analyi of column and framed tructure. reiza-hurtado et al., (005) evaluated econd-order tiffne matrix and loading vector with generalied end condition regarding a Timohenko column on a Winkler foundation. More recently, roleda-monalve et al., (008) and Zapata-Medina et al., (010) have tudied hear effect on elatic taility and the dynamic of eam-column on an elatic foundation. However, a complete et of taility equation for column laterally upported on a continuou elatic foundation having generalied end condition ujected to axial end load are not availale in the technical literature. Thi pulication main ojective wa thu to preent complete claification and correponding taility equation for primatic column ujected to end axial load, including the effect of emirigid ending connection, racing tiffne at oth end and a uniformly ditriuted elatic upport (Winkler type). The uckling load wa otained y making the determinant of a 44 matrix equal to zero for column having ide-way uninhiited or partially inhiited at oth end, and of a matrix for column having ide-way inhiited at one or oth end. The preent paper i retricted to an elatic tatic analyi of a ingle lender column, with three different lateral racing type, leading to manageale analytical olution. The effect of hear deformation on the memer uckling capacity are neglected. The importance of hear effect on the tatic and dynamic repone of hort column or column with low hear tiffne [reiza-hurtado et al (005), roleda- Monalve et al., (008), and Zapata-Medina et al. (010)]. Structural model umption primatic element i conidered that connect point and B (Fig. 1). B conit of a column 'B' and lumped flexural connector ' and BB' at the top and ottom end, repectively. It i aumed that: 1) 'B' i made from a homogeneou linear elatic material having a modulu of elaticity E and continuouly upported along it pan y a uniformly ditriuted elatic foundation (Winkler type) of magnitude k; ) The centroid axi of the column i initially a traight line in which the elatic centroid coincided with the hear centre of the cro ection. Initial geometrical imperfection in the column were not conidered; ) The column i uject to end compreive axial load applied along it centroid axi and alo ujected to overturning moment and hear force Ma, Va and M, V at end and B, repectively; 4) End lateral way i partially inhiited y pring Sa and S located at and B, repectively; Figure 1. Model of a eam-column reting on an elatic foundation k with ide-way partially inhiited y end racing tiffne Sa and S and rotationally retrained y end flexural pring a and Flexural connector ' and BB' had tiffne a and (with unit in force-ditance/radian), repectively. The ratio Raa/(EI/L) and R/(EI/L) were denoted a the tiffne indice of the flexural connection, where I = column moment of inertia aout the principal axi in quetion and L = column pan. The tiffne indice varied from zero (i.e. Ra = R = 0) for hinged connection to infinity (i.e., Ra = R = ) for rigid connection or clamped end. The main difference etween the claical olution preented in thi paper and that availale in the pertinent technical literature (Timohenko and Gere, 1961; Hetenyi, 1967; Scott, 1981) i that the olution for the model hown in Fig. 1 included the effect of emi-rigid connection and could e ued for oth taility and econd-order analyi of column on an elatic foundation ujected to any end load comination. For convenience, the following two parameter were introduced: a= 1/(1+/Ra); and = 1/(1+/R) (1a-) where a and were called fixity factor. For perfectly hinged connection, oth fixity factor and tiffne index R were zero; ut, for perfectly rigid connection, fixity factor wa 1 and the tiffne index wa infinity. Since fixity factor in the elatic range can only vary from to +1 (while tiffne index R may vary form to +), it wa more convenient to ue it in the analyi of tructure having emi-rigid connection, a uggeted y Monforton INGENIERÍ E INVESTIGCIÓN VOL. No., DECEMBER - 01 (4-40) 5

3 STBILITY OF SLENDER COLUMNS ON N ELSTIC FOUNDTION WITH GENERLISED END CONDITIONS (196). Factor in equation (1a-) came from the lope-deflection analyi of a eam having emi-rigid connection at oth end. Deriving lateral equilirium equation The econd-order governing of a differential equation for a primatic column laterally upported on an elatic foundation k ujected to compreive axial load at oth end a well a to overturning moment and hear Ma, Va and M, V at and B, repectively, i a follow: 4 d y d y EI k y 0 () 4 Eq. () i the equation governing a column laterally upported on an elatic foundation uject to compreive axial load at oth end in deformed condition. It olution i known a econd-order analyi (ee Chen and Lui, 1987, p. ). For a column having ide-way partially inhiited at oth end (Fig. a), the olution to Eq. () ha to e uject to the following four oundary condition: 1) t x= L ) t x= 0 ) t x= L 4) t x= 0 EI M EI a M a EI S y V EI S a y V a (a) () (c) (d) The olution to Eq. () depend on the relative value of compreive regarding 4k EI, a explained y Hetenyi (1967). There were two type of olution in the compreive range (i.e., for 0), a follow: for 0 4 k EI y = (C1e x Ce -x ) cox + (Ce x C4e -x ) inx (4) k where: ; and for 4 k EI where: k y= C1 cox + C cox + C inx + C4 inx (5) EI k and ; EI EI EI k EI EI The four contant C1, C, C and C4 in Eq. (4) and (5) i determined from the four natural oundary condition given y Eq. (a)-(d). Thee i reduced to Eq. (6a)-(6d) after ome algeraic manipulation: C1 + 1 C + 1 C + C4 = (1)M/(EI/L) 1 C1 + C + C + 4 C4 = (1a)Ma/(EI/L) (6a) (6) 1 C1 + C + C + 4 C4 = V/EI 41 C1 + 4 C + 4 C + 44 C4 = Va/EI where ij coefficient were a follow: I) for 0 4 k EI = e L + L(1) inl + + L(1)( )col 1= e L L(1) inl + L(1)( ) col 1= e L + L(1)( ) inl + + L(1) col = e L L(1)( ) inl L(1) col (6c) (6d) (7a) (7) (7c) (7d) 1= al(1a)( ) (7e) = al(1a)( ) (7f) = al(1a) (7g) 4= a+l(1a) (7h) 1= e L EI( ) inl + (S)EI( ) col = e L EI( ) inl + (S + )EI+( ) col = e L (S)EI( ) inl EI( ) col 4= e L (S+)EI + ( )inlei( ) col (7i) (7j) (7k) (7l) 41= (Sa+)EI + ( ) (7m) 4= (-)EI ( ) (7n) 4= 44= EI( ) (7o) k where: ; and II) for 4 k EI k = inl + L(1) col (8a) 1= inl + L(1) col (8) 1= col L(1) inl (8c) = col L(1) inl (8d) 1= L(1a) (8e) = L(1a) (8f) = a (8g) 4= a (8h) 1= (SEI)coL ( EI) inl (8i) = (SEI)coL ( EI) inl (8j) = (SEI)inL + ( EI) col (8k) 4= (SEI)inL + ( EI) col (8l) 6 INGENIERÍ E INVESTIGCIÓN VOL. No., DECEMBER - 01 (4-40)

4 RISTIZBL-OCHO 41= 4 = SaEI (8m) 4= ( EI) (8n) 44= ( EI) (8o) where: EI k and ; EI EI EI k EI EI For column having ide-way totally inhiited at and partially inhiited at B (Fig. ), the fourth natural oundary condition given y Eq. (d) or (6d) ha to e utituted y forced oundary condition y= 0 at or imple C1= C, which, when introduced into the other three natural oundary condition given y Eq. (6a)-(6c), ecame: ( 1)C1 + 1 C + C4 = (1)M/(EI/L) (9a) (1 )C1 + C + 4 C4 = (1a)Ma/(EI/L) (1 )C1 + C + 4 C4 = V/EI (9) (9c) For column having lateral ide-way totally inhiited at oth end and B (Fig. c), the third natural oundary condition given y Eq. (c) or (9c) ha to e utituted y forced oundary condition y = 0 at B. Thi condition ecame: for 0 4 k EI (e L 1) col C1 + e L inl C+ inl C4= 0 (10) for 4 k EI (col col) C1 + inl C+ inl C4 = 0 () ropoed taility equation ccording to the type of end ide-way jut decried, three major uckling mode i ditinguihed in column having emi-rigid connection and eing laterally upported on a continuou Winkler foundation. They were column having lateral way which wa uninhiited or partially inhiited at oth end (Fig. a), totally inhiited at one end and partially inhiited at the other end (Fig. ) and/or totally inhiited at oth end (Fig. c). The correponding taility equation are lited and explained elow for eay reference: Type 1: Column with ide-way uninhiited or partially inhiited at oth end For column in which ide-way at and B i partially inhiited y pring Sa and S (Fig. a), repectively, the taility equation conited of the eigen-value olution to the 44 determinant given y Eq. (1): (1) Type : Column having ide-way inhiited at one end and partially inhiited at the other end For column in which lateral way at i totally inhiited and partially inhiited at B y lateral pring S (Fig. ), the taility equation conit of the eigen-value olution to the determinant given y Eq. (1): (1) Type : Column with lateral ide-way inhiited at oth end For column having lateral way totally inhiited at oth end (Fig. c), the taility equation conit of the eigen-value olution to the determinant given y Eq. () or (15): for 0 4 k EI : 1 1 ( e L 1) col 1 e L in L 4 in L 0 () for 4 k EI : 1 1 col col 1 in L 4 inl 0 (15) Figure. Beam-column claification according to the ide-way allowed at each end: a) ide-way uninhiited or partially inhiited at oth end; ) ide-way inhiited at and partially inhiited at B; and c) lateral ideway inhiited at oth end The ij coefficient in Eq. (1), (1), () and (15) i given y expreion (7) or (8), depending on the magnitude of compreive load when compared to 4 k EI (the value of elatic uckling load of an infinitely long column, a explained y Hetenyi (1967), p. 16). It may e noticed that the uckling modal hape for any of the three type of uckling could e determined directly uing either Eq. (4) or Eq. (5) once the value of the uckling load had een determined from the correponding characteritic equation lited aove (i.e. Eq. (1)-(15)] and correponding ij coefficient given y either Eq. (7) or (8) calculated). Thi olution wa identical to that for any tandard eigen-value prolem. INGENIERÍ E INVESTIGCIÓN VOL. No., DECEMBER - 01 (4-40) 7

5 STBILITY OF SLENDER COLUMNS ON N ELSTIC FOUNDTION WITH GENERLISED END CONDITIONS Claical column cae Variation in the critical load of a column upported y a uniformly ditriuted elatic foundation of tiffne k under different oundary condition (five cae hown in Fig. a-e), including the effect of emi-rigid connection, were tudied. The critical load for different value of k L 4 /(EI), a uggeted y Hetenyi (1967), and for five different fixity factor ranging from hinged to clamped condition, are lited in Tale 1-5 for each cae hown in Fig. a-e, repectively. The value lited in thee tale were calculated to five ignificant figure (for comparion with thoe otained y Hetenty' formulae) uing the propoed model and Eq. (1)-(15). Tale 1. Buckling load/( EI/L )Column with ide-way uninhiited at oth end, hinged at and retrained againt rotation at B a= 0 and , , Tale. Buckling load/( EI/L ) Column with ide-way uninhiited and retrained againt rotation at oth end a= Figure. Five claical eam-column cae: a) ide-way uninhiited at oth end, hinged at and retrained againt rotation at B; ) ide-way uninhiited and equally retrained againt rotation at oth end; c) ideway inhiited and rota-tionally retrained at and hinged at B; d) ideway inhiited at oth end, rotationally retrained at and hinged at B; and e) ide-way inhiited and equally retrained againt rotation at oth end The value lited in Tale 1 and for = 0 (i.e. hinged) were verified with thoe y Hetenyi (1967) [Eq. 15 and Fig. 7, p. ], yielding identical reult, wherea thoe in Tale compared very well to the implified formula cr= k EI for infinitely long column and any value of rotational retraint at B reported y Hetenyi (1967). Similarly, value lited in Tale 4 and 5 for = 0 (i.e. hinged) and = 1 (i.e. perfectly clamped) were verified with thoe y Hetenyi (1967) [Eq. 17 and 18, and Fig. 8 and 9, repectively, page 5-7], identical reult eing yielded too. The value in Tale 4 alo compared very well to the implified formula cr= 4 k EI for infinitely long column and any value of end rotational retraint reported y Hetenyi (1967). Tale 1 indicate that for column having ide-way uninhiited at oth end and rotational retraint at one end, it wa only effective in column on elatic foundation having k 5EI/L 4. For larger value of k, increaed uckling load regarding rotational retraint wa rather mall (le than 0% for value of k 900EI/L 4 and reaching increae lower than % for k 500EI/L 4 ). Tale indicate that critical load increaed over 500% for k 5EI/L 4 y perfectly clamping oth end a compared to perfectly hinged end. For larger value of k, the uckling load increaed y jut over 100% , , Tale indicate that for column having ide-way inhiited at one end, rotational retraint at the ame end wa only effective in column on an elatic foundation where k 0. For 5EI/L 4 k400ei/l 4 the increae in uckling load wa le than 1% and for value of k 900EI/L 4 the increae in uckling load wa le than 1%. Tale 4 and 5 indicate imilar trend to thoe oerved in Tale 1 and for column having ide-way inhiited at oth end. Rotational retraint at one or oth end were only effective in column on an elatic foundation where k 5EI/L 4. For larger value 8 INGENIERÍ E INVESTIGCIÓN VOL. No., DECEMBER - 01 (4-40)

6 RISTIZBL-OCHO of k, increaed uckling load regarding rotational retraint wa not ignificant. It wa oviou that it wa more effective to retrain oth end imultaneouly againt rotation than jut one end. The value lited in Tale 1, and 4 varied linearly with k L 4 /(EI), having very mall variation in cr with the degree of flexural retraint, making rotational condition of econdary importance in long column. Tale. Buckling load/( EI/L ) Column with ide-way inhiited and rotationally retrained at, and free and hinged at B 0 a 1 and = , , Summary and concluion n analytical method ha een preented for evaluating the critical uckling axial load of column having ide-way uninhiited, partially inhiited and totally inhiited and laterally upported y a uniformly ditriuted elatic foundation (Winkler' model). The propoed column claification and correponding taility equation were general and relatively imple to apply, yielding exact reult when compared to other linear elatic analytical method. The propoed formulae can alo e applied to the taility of column having rigid, emi-rigid and imple connection, with or without elatic upport. Semi-rigid connection effect were explicitly included. The end fixity factor were elected to conider the effect of emi-rigid connection in taility analyi, ince they were practical and convenient. The uckling load i otained y making the determinant of a 44 matrix equal to zero for column having ide-way uninhiited or partially inhiited at oth end, and of a matrix for column having end ide-way inhiited at one or oth end. The effect of lateral racing one or oth end of the column have een preented. The uckling load of five claical cae wa preented and compared to thoe y other analytical method to demontrate the ue and relative implicity of the propoed column claification and correponding taility equation. Tale 1-5 can e ued directly in taility analyi of column having emirigid connection on an elatic foundation. The propoed method and it equation can e programmed, facilitating calculation and efficient computer coding and avoiding cumerome procedure. It hould e emphaied that the propoed model i jut an approximation of the real load tranfer mechanim on pile, drilled haft, caion and pier. It neglect the oil-pile friction kin contriution to load capacity and i unale to calculate vertical ettlement and tree in foundation oil which are generally the controlling factor during deign. Tale 4. Buckling load/( EI/L ) Column with ide-way inhiited at oth end, rotationally retrained at and hinged at B 0 a 1 and = , , Tale 5. Buckling load/( EI/L ) Column with ide-way inhiited and equally retrained againt rotation at oth end a= , , The analytical reult indicated that the effect of a uniformly ditriuted lateral upport of magnitude k ecame more ignificant in column having free end than in column having oth end retrained againt rotation and ide-way. The uckling load of a column, laterally upported y a very oft elatic foundation, could e utantially increaed y either retraining it end againt rotation or y racing oth end againt ide-way. INGENIERÍ E INVESTIGCIÓN VOL. No., DECEMBER - 01 (4-40) 9

7 STBILITY OF SLENDER COLUMNS ON N ELSTIC FOUNDTION WITH GENERLISED END CONDITIONS Reference roleda-monalve, L. G., Zapata-Medina, D. G. and ritizaal- Ochoa, J. D., Timohenko eam-column with generalized end condition on elatic foundation: Dynamic-tiffne matrix and load vector., Journal of Sound and Viration, Vol. 10, 008, pp reiza-hurtado, M., Vega-oada, C., ritizaal-ochoa, J. D., Second-Order Stiffne Matrix and Loading Vector of a Beam- Column with Semirigid Connection on an Elatic Foun-dation., J. Engrg. Mech., SCE, Vol. 11, No. 7, 005, pp ritizaal-ochoa, J. D., K-Factor for Column in any Type of Contruction: Nonparadoxical pproach., J. Struct. Engrg., SCE, Vol. 10, No. 4, 1994, pp ritizaal-ochoa, J. D., Braced, partially raced and un-raced column: complete et of claical taility equation., Int. J. Struct. Engrg. and Mech., Vol. 4, No. 4, 1996, pp ritizaal-ochoa, J. D., Story taility of raced, partially raced and unraced frame: claical approach., J. Struct. Engrg., SCE, Vol. 1, No. 6, 1997, pp ritizaal-ochoa, J. D., Elatic taility of eam-column with flexural connection under variou conervative end axial force., J. Struct. Engrg., SCE, Vol. 1, No. 9, 1997, pp vramidi, I. E., Morfidi, K., Bending of eam on three-parameter elatic foundation., Int. J. Solid Struct., Vol. 4, 006, pp Chen W. F., Lui, E. M., Structural Staility: Theory and Implementation, Englewood, New Jerey, TR rentice Hall, 1987, pp.. Cheng, F. Y., antelide, C.., Static Timohenko Beam-Column on Elatic Media., J. Struct. Engrg., SCE, Vol. 4, No. 5, 1988, pp Hetenyi, M., Beam on elatic Foundation., 8th printing, nn ror, Michigan, The Univerity of Michigan re, 1967, Chapter VII. Monforton, G. R., Wu, T. S., Matrix analyi of emi-rigidly connected frame., J. Struct. Engrg., SCE, Vol. 89, No. 6, 196, pp. 1-4.Morfidi, K., Exact matrice for eam on three-parameter elatic foundation., Computer & Structure, Vol. 85, No , 007, pp Morfidi, K., Viration of Timohenko eam on three-parameter elatic foundation., Computer & Structure, Vol. 88, No. 5-6, 010, pp Onu, G., Finite Element on Generalized Elatic Foundation in Timohenko Beam Theory., J. of Engineering Mechanic-SCE, Vol., No. 9, 008, pp Razaqpur, D. G., Stiffne of eam-column on elatic foundation with exact hape function., Computer & Structure, Vol. 4, No. 5, 1986, pp Struther,., Jayaraman, G., Elatic taility of column on partial elatic foundation under utangential loading., J. of Sound and Viration, Vol. 9, No. 18, 010, pp Scott, R. F., Foundation nalyi., Englewood Clifft, NJ., rentice- Hall, Timohenko, S.., Gere, J. M., Theory of elatic taility., nd. Ed., New York, N.Y., McGraw-Hill, 1961, pp Wang, C-M., Xiang, Y., Kitipornchai, S., Buckling of retrained column with hear deformation and axial hortening., J. of Engrg. Mech., SCE, Vol. 7, No. 9, 1991, pp Wet, H. H., Mafi, M., Eigenvalue for column on elatic upport., J. Struct. Engrg, SCE, Vol. 0, No. 6, 1984, pp Zapata-Medina, D. G., roleda-monalve, L. G., ritizaal- Ochoa, J. D., Static Staility Formula of a Weakened Timohenko Column: Effect of Shear Deformation., J. of Engineering Mechanic-SCE, Vol. 16, No. 1, 010, pp INGENIERÍ E INVESTIGCIÓN VOL. No., DECEMBER - 01 (4-40)