INFLUENCE OF SPLICES ON THE STABILITY BEHAVIOUR OF COLUMNS AND FRAMES

Transcription

1 SDSS Ro 00 STABTY AND DUCTTY OF STEE STRUCTURES E. Batsta, P. Vellasco,. de ma (Eds.) Ro de Janero, Brazl, September 8-0, 00 NFUENCE OF SPCES ON THE STABTY BEHAVOUR OF COUMNS AND FRAMES Pedro D. Smão*, **, Ana M. Grão Coelho*, *** and Frans S. K. Bjlaard*** * nsttute of Computers and Systems Engneerng of Combra, Portugal (NESC-Combra) e-mal: pedro@dec.uc.pt, a.m.grao@clx.pt ** Department of Cvl Engneerng, Unversty of Combra, Portugal *** Department of Structural and Buldng Engneerng, Delft Unversty of Technology, The Netherlands e-mal: f.s.k.bjlaard@tudelft.nl Keywords: Column Stablty, GBT, agrange Multplers, Raylegh-Rtz method, Splced Columns. Abstract. The paper presents a study on the nfluence of splce connectons on the stablty behavour of compressed steel columns. The column s modelled as two ndependent prsmatc parts connected by a rotatonal sprng at the splce locaton and rotatonal and extensonal sprngs at the column ends to represent the effect of the adjacent structure. The general behavour s characterzed usng a polynomal Raylegh-Rtz approxmaton substtuted nto the potental energy functon, n combnaton wth the agrange s method of undetermned multplers, and based on ths model the crtcal load s found. The load-carryng capacty s analysed wth respect to the followng varables: () locaton and rotatonal stffness of the splce, () change n the column secton seral sze and () column end-restrants stffness coeffcents. A nonlnear regresson model s developed to predct smple relatonshps between the crtcal load and the relevant column characterstcs. NTRODUCTON n structural engneerng practce and due to manufacturng, transportaton and/or handlng restrants, ndvdual steel elements are usually fabrcated wth a maxmum length of meters. Durng erecton of a steel frame and where the element length s nsuffcent, splces are provded to form a sngle and longer element. Desgners often use the splces for changng cross-sectons, n vew of a more economcal and ratonal desgn. n steelwork constructon, column splces are located at a convenent dstance for erecton and constructon above floor beam level and have to be desgned () to jon lengths n lne, () to transmt forces and moments between the connected member parts and () to mantan contnuty of strength and stffness through the splce to safeguard the robustness of the structure []. Column splces are usually dsregarded n determnng the dstrbuton of moments and forces n the structure and when the desgn of the columns tself s beng consdered, assumng that the splce s provdng full contnuty n stffness and strength of the column. Ths practce s questonable as the splces most tmes do not provde ths contnuty. So, the splces may adversely affect the overall frame behavour, from a stffness and strength pont of vew. Prevous research pertanng to the load-carryng capacty of splced columns ncludes nvestgatons by ndner [], Snjder and Hoenderkamp [3] and Grão Coelho et al. [4,5,6]. ndner [] carred out expermental and numercal tests on dfferent column splce types and hghlghted the exstence of eccentrctes at the splce. An adjusted bucklng curve for columns havng contact splces at column mdheght was later proposed ndner []. Snjder and Hoenderkamp [3] conducted a seres of expermental tests to analyse the nfluence of end plate splces on the load-carryng capacty of slender columns. These 69

2 Pedro D. Smão et al. tests were used to make desgn recommendatons for column splces. Grão Coelho and co-authors [4,5] further extended ths work to produce a relatvely smple yet reasonably accurate engneerng method for predctng the crtcal behavour of splced columns n steel frames. Ths paper s a follow-up study to ths research. The current work presents a generalzed energy formulaton of a framed splced column n sway and non-sway frames (Fgure ). For analytcal modellng, a framed column s represented by means of extensonal and rotatonal restrants at the ends A and B. The splce s modelled as a rotatonal sprng at pont C. The potental energy functonal of ths system uses a Raylegh-Rtz approxmaton of the relevant deformaton modes of the column. Ths formulaton s presented together wth the method of agrange multplers to deal wth the constrants at the column splce. Elastc bucklng analyss s carred out to fnd the crtcal load of the system. The concept of end fxty factor C [7] s successfully appled and the sgnfcance of ths factor n smplfyng the analyss of results s emphaszed. Smple relatonshps between C and the relevant characterstcs of the column and splce are derved to a pont where the crtcal load can be readly determned by hand or by computer. x,u N Ed B Member : E, C Member : E, A z, w ENERGY FORMUATON Fgure : The framed splced column system.. Bendng and axal stran energy The deformaton of a prsmatc member under the acton of loads s characterzed by axal elongaton (mode ) and bendng deformatons (mode ). n the context of a smplfed Generalzed Beam Theory (GBT) strategy [8,9], the dsplacement functons are assumed to be as follows: Axal dsplacements:,, u x y z u y, z f x u u z u z n w 0 w w 0 and u () w Transverse dsplacements:,, w x y z w y, z f x and () 60

3 Pedro D. Smão et al. whereby z j s the dstance between pont j and the neutral axs, u(y, z) and w(y, z) are pre-establshed modal dsplacement patterns defned along the member cross-secton, and f u (x) and f w (x) are modal ampltude functons for warpng and transverse dsplacements, respectvely. For any mode of deformaton the ampltude functons for axal and transverse dsplacements are related n the form [8,9]: d fw x fu x f w x f (3) dx The extensonal stran of a column segment of length dx s readly defned as: u w x x x (4) Expanson n Taylor seres, neglectng hgher-order terms, yelds the followng knematc relaton: u w k k k l k l x u f w w f f (5) x x k k l From Hooke s law (consttutve relaton), the longtudnal stress s then gven by: j j x Ex E u f w w f f (6) j The nternal stran energy of the member, U m, s then equal to [5]: EA E EA EA 4 Um x x d f dx f dx f f dx f dx 8 (7) where denotes the member s volume, A the cross sectonal area and the moment of nerta.. Stran energy stored n the sprngs The energy stored n the lnear sprngs (Fgure ) s gven by the followng expressons: E Rotatonal sprng at end A: U a Ka a ka f (8) x0 E Rotatonal sprng at end B: U b Kb b kb f (9) x E Rotatonal sprng at splce: U c Kcc kc f f 0 (0) x x E Extensonal sprng at end B: U b K b b k 3 b f () x where K and K are rotatonal and extensonal sprng constants, respectvely, and k are sprng coeffcents that are defned n non-dmensonal form..3 Work done by load The fnal component of energy to be dentfed s the work done by the load. For a centrally loaded column, the potental energy of the external loadng s gven by: N f ().4 Potental energy functonal Ed x The total potental energy of the complete structure s a summaton of U m (for each ndvdual member and ), U a, U b, U c, U b mnus : V Um, Um, Ua Ub Uc U b (3) Ths functonal s subjected to the followng knematc constrants that ensure contnuty at the splce: 6

4 Pedro D. Smão et al. u, C u, C 0 f f 0 G a Ed x x 0 j, N 0 (4) w, C w, C 0 f f 0 G a Ed x x 0 j, N 0 We now wsh to fnd a statonary value of a functonal subjected to some subsdary condtons or constrants G k ( a j,n Ed ). The problem s easly tackled by usng the approach proposed by agrange [0]. The technque s to form a modfed potental energy expresson: m j, k, Ed j, Ed k k j, Ed (5) V a N V a N G a N where k are the agrange multplers. For the splced column, the modfed potental energy functonal may be wrtten as: k x x 0 x x0 V U U U U U f f f f.5 Crtcal bucklng load m a b c b Eq. (6) s a functonal representng the total potental energy of the physcal system. The advantage of ths method les n the fact that the problem wth constrants can be treated n exactly the same manner as though t was free. Thus, for the system to satsfy equlbrum V has to be statonary. The calculus of varatons s then used to fnd the statonary pont of the functonal. Exact solutons can be obtaned usng the method of egenvalue analyss. Ths s not a practcal method to solve the characterstc equatons of the dfferental equatons. Approxmate methods such as the Raylegh-Rtz method seem a very attractve alternatve to an otherwse complex problem. Essentally, n ths method, the modes of deformaton of the system are defned by means of assumed dsplacement functons that satsfy the geometrc boundary condtons of the system. As a result, the total potental energy reduces from a functonal to a gradent potental functon that depends on a fnte set of dscrete generalzed coordnates a j and k and the scalng factor N Ed. Thus, ordnary calculus can be used to obtan solutons drectly. n the context of the Raylegh-Rtz method, the ampltude functons are approxmated by polynomals n the form: n f aj j j (6) (7) These polynomals form a set of coordnate functons that satsfy the knetc boundary condtons of the problem and are orthonormal functons that enable fast convergence of the method [9]. The coordnate functons are gven by: Member :, 5 x, 3 x 5 3x4x, 3 3, 7 6x0x 5x , 9 0x60x 05x 56x , 5x40x 40x 504x 0x , 3 x80x 60x 50x 30x 79x , 5 8x504x 350x 940x 3860x 096x 3003x (8) 6

5 Pedro D. Smão et al. Member :, x, 3 x,, 3 x 5 6x6x 3, 3 4, 7 x30x 0x , 9 0x90x 40x 70x , 30x0x 560x 630x 5x , 5 4x40x 680x 350x 77x 94x Equlbrum of the system s obtaned by renderng statonary the total potental functon wth respect to the generalzed coordnates a j and k. The soluton that emerges from the unloaded state, the fundamental path (FP) s a functon of N Ed. A sldng set of ncremental coordnates q j and q k s then defned by the followng equatons []: aj aj FP NEd qj and k k FP NEd qk (0) A new energy functon W s now ntroduced []: W q,, Ed,, j qk N V aj FP qj k FP qk NEd () A global numberng for coordnates q j and q k (q l ) can now be adopted. The equlbrum and stablty condtons hold good for ths transformed energy functon W. n ths new N Ed q l space, the fundamental path s defned trvally by q l = 0.The crtcal ponts along the fundamental path are now those ponts that render zero the determnant of the total potental energy Hessan matrx along the fundamental path: HFP HFP,0 NEd H FP, () The relevant states of crtcal equlbrum are dentfed va a local lnear egenvalue equaton H FP q = 0, q representng the local egenvector []. By substtutng the forms n Eq. (), we thus obtan the crtcal state dentty: HFPq HFP,0 NEdHFP, q 0 (3) Ths analyss yelds the crtcal bucklng load of the splced column, N cr that can be expressed n terms of an end fxty factor C [7]: N C E (4) 3 NUMERCA RESUTS cr The purpose of ths numercal study s to ascertan the effect of the followng varables on the general equlbrum response: () splce locaton ( = ), () rato between second moment of area of lower and upper column members ( = / ), () splce rotatonal stffness (k c ) and (v) end-restrants stffness coeffcents (k a, k b and k b ). Results are ndependent from the column length. These propertes are vared parametrcally as shown n Table. Table : Parameters for regresson analyss Parameter Range of parameter selected Splce locaton 0. to 0.9, = 0. Rato between second moment of area to 3, = 0.5 Non-dmensonal stffness coeffcents 0, 0.5, 0.5,, 3, 5, 7.5,0, 5, 0, 35, 50, 75, 00 (9) 63

6 Pedro D. Smão et al. These results form a comprehensve analytcal database. We can now generate a contnuous functon that approxmates the values of the end fxty factor wthn the doman of analyses and wth a mnmum error. The analyss results are then used to develop a multple regresson model to approxmate the end fxty factor C C ft (,, k a, k b, k c, k b ) from the data n the database, by means of pecewse approxmatons. n developng the regresson model, the relatonshp between the dependent varable and each ndependent varable s studed separately, whle all other ndependent varables are kept constant. The dependent varable s approxmated by a contnuous functon that s lnear n terms of a set of regresson coeffcents, whch are determned by enforcng the method of least squares that mnmzes the sum of the squares of the resduals. Approxmatng (or coordnate) functons are then selected for each ndependent varable. The multple regresson model s formed as the product of the ndvdual coordnate functons. k a = 0.0, k b = 00, k b = 5 k c = 0.0,, 5, 0, 0, 00 k a = 00, k b = 00, k b = 0.0 k c = 0.0,, 5, 0, 0, 00 =, = 0.5, k b = 0.0 k c = 0.0,, 5, 0, 0, 00 =, = 0., k b = 5 k c = 0.0,, 5, 0, 0, 00 =, k a = 00, k b = 00 = 0., 0.3, 0.5, 0.7, 0.9 =, k a = 0.0, k b = 00 = 0., 0.3, 0.5, 0.7, 0.9 Fgure : Varaton of C wth the relevant propertes. 64

7 Pedro D. Smão et al. Formng the regresson model as a product allows the effect of each ndependent varable to be examned separately and facltates the process of selectng sutable coordnate functons for the ndvdual ndependent varables. Some key results are llustrated graphcally n Fgure. The graphcs suggest that:. The response C vs. can be approxmated by a quadratc functon.. Typcal C vs. behavour s characterzed by a monotonc ncreasng functon that can be generally approxmated by a smple lnear relatonshp. 3. The degree of rotatonal and extensonal end restrant s an essental parameter for the computaton of C. The shape of the curves C vs. k (k k a, k b ), C vs. k c and C vs. k b s best descrbed by an arctangent functon. The end fxty factor s then predcted by means of an expresson n the form: k C b ft,, k b, ka, kb, kc C CC3 C4 C5arctan C 6 (5) k a k b k c C7arctan C9arctan Carctan C 8 C 0 C where C are regresson coeffcents. As expected n developng a predctve regresson model, many models were tred, analysed and assessed for accuracy and effectveness. The fnal model presented here evolved out of several attempts to develop conventonal (nonlnear) regresson models by means of smple mathematcal functons. The overall character of the response s well captured and the number of regresson coeffcents s kept small n order to provde a compact procedure for the smplfed method. The accuracy of the model s measured by means of the R-Squared value (R ). The R-Squared gves the fracton of the varaton of the response that s predcted by the model. A good model ft yelds values of R-squared close to unty. Nonlnear regresson analyss s performed wth the Mathematca software []. Regresson coeffcents are determned for the splced column usng pecewse approxmatons dependng on the nature of the segments that comprse the above relatonshp. The doman of analyses of the sprng coeffcents s dvded nto three ntervals: G for k ]0,3], G for k ]3,5] and G3 for k ]5,00]. Table sets out the computed regresson coeffcents and values for the R-Squared factor are also gven. 4 CONCUDNG REMARKS AND SCOPE FOR FURTHER WORK The paper has presented an applcaton of the total potental energy method to the bucklng behavour of a splced column n sway and non-sway frames. Ths s a varatonal problem,.e. the fndng of thestatonary pont of a functonal, wth addtonal condtons at the splce that s solved by means of the agrange s method of undetermned multplers. The Raylegh-Rtz procedure has been used to reduce ths varatonal problem wth constrants to a mere dfferentaton that form a set of algebrac equatons of equlbrum. These equatons are solved by usng an algebrac manpulator and the crtcal load s calculated. The bucklng response was found to be partcularly senstve to the followng varables: () splce locaton, () rato between second moment of area of lower and upper column members, () splce rotatonal stffness and (v) end-restrants stffness coeffcents. The sgnfcance of each of these varables has been assessed. A parametrc study was devsed and the results were then used to develop regresson equatons for predctng the crtcal load of the system va the concept of end fxty factor C. The work outlned above affords some bass to produce desgn gudance on column splces. The authors attempt to set up sound desgn crtera regardng the requrements for stffness and strength of column splces. Expermental and numercal fnte element studes focusng on the bucklng response are also necessary n order to valdate the predctve expressons. t should be noted that the nvestgated confguraton was rather lmted to a partcular case. The dervaton has been carred for two-dmensonal frames and only unaxal bendng behavour has been consdered. Future work wll ncorporate b-axal bendng and torson effects n the desgn of splces. n addton to ths, the nfluence of splces on the overall stablty behavour of frames wll also be nvestgated. 65

8 Pedro D. Smão et al. Range k a G k b G k a G k b G k a G k b G3 Table : Parameter table. k a G k b G k a G k b G k a G k b G3 k a G3 k b G k a G3 k b G k a G3 k b G3 C C C C C C C C C C C C R REFERENCES [] CEN (European Commttee for Standardzaton), EN Eurocode 3: Desgn of steel structures Part -8: Desgn of jonts, Brussels, 005. [] ndner, J., Old and new solutons for contact splces n columns. Journal of Constructonal Steel Research, 64, , 008. [3] Snjder, H.H., Hoenderkamp, J.C.D., nfluence of end plate splces on the load carryng capacty of columns. Journal of Constructonal Steel Research, 64, , 008. [4] Grão Coelho A.M., Bjlaard F.S.K., Requrements for the desgn of column splces, Stevn Report , Delft Unversty of Technology, 008. [5] Grão Coelho A.M., Smão P.D., Bjlaard F.S.K., Stablty desgn crtera for steel column splces. Journal of Constructonal Steel Research, 66, 6-77, 00. [6] Grão Coelho A.M., Bjlaard F.S.K., Smão P.D., Stablty desgn crtera for steel column splces n non-sway frames. Proceedngs of the fourth nternatonal conference on structural engneerng, mechancs and computaton SEMC 00, (accepted for publcaton). [7] Smtses, G.J., An ntroducton to the elastc stablty of structures, Kreger Publshng Company, Malabar, 986. [8] Schardt, R., Verallgemenerte Technsche Begetheore, Sprnger, Berln-Hedelberg, Germany, 989. [9] Smão, P.D., Post-bucklng bfurcatonal analyss of thn-walled prsmatc members n the context of the Generalzed Beam Theory, Ph.D. thess, Unversty of Combra, Portugal, 007 ( [0] Rchards, T.H., Energy methods n stress analyss, Ells Horwood, Chchester, UK, 977. [] Thompson J.M.T., Hunt G.W., A General Theory of Elastc Stablty, John Wley & Sons, ondon, UK, 973. [] Mathematca 6, Wolfram Corp., Champagn, USA,