Send Orders of Reprints t reprints@benthmscience.net 50 The Open Civi Engineering Journ, 0, 7, 50-57 Open Access Identifiction of Axi Forces on Stticy Indeterminte Pin-Jointed Trusses by Nondestructive Mechnic Test Emiio Turco* DADU, Università degi Studi di Sssri, 0704 Aghero, Sssri, Iti Abstrct: This work concerns the stress-stte identifiction, produced by ods or disoctions, of stticy indeterminte structures. In prticur, this pper des with the cse of pin-jointed trusses, since they re combintion of simpicity of the structur mode with very interesting technic ppictions. We present n pproch tht foows the min guideines of the fexibiity method to rete n ddition od nd the consequent nod dispcements. When sufficient number of nod dispcements re mesured, they produce system of equtions tht when soved furnishes compete reconstruction of the stress-stte to identify. In order to highight the potentiities nd the imits of the proposed pproch nd so to deinete its min chrcteristics, some simpe tests re discussed nd nyzed. Keywords: Inverse probems, stress-stte identifiction, nondestructive tests.. INTRODUCTION Knowedge of the stress-stte of mechnic prt, such s n engine, whee or ger, is significnt for the mechnic industry since the excess of stresses coud give rise to premture nd sudden brek. In order to design possibe reiforcements nd risk ssessments, n in-depth knowedge of the ctu stress-stte on structure is necessry in the fied of civi engineering. Another reserch fied where the identifiction of the ctu stress hs some reevnt ppictions is biomechnics: n enightening exmpe is reted to the determintion of rtery hypertension, s reported by []. These brief discussions contribute in highighting the importnce of these issues in the scientific community. It is known tht compete estimtion of the stress-stte is required for residu stresses, i.e. stresses corresponding to nu ods []. In the technic iterture, vrious pproches hve been reported in retion to the probem of identifying the ctu stte of stress in structure. These methods cn be subdivided into destructive nd nondestructive tests. The first css of methods incudes the tests requiring n hoe or cut on the seected prt. This cuses rextion of the stress fied nd consequent dispcements; the mesurement of such dispcements gives the dt required to reconstruct the pre-existing stress fied. In fct, it is enough to impose the opposite of the mesured dispcement fied nd derive the consequent stress fied, for exmpe the ppers of [] nd [4]. Destructive tests obviousy hve the disdvntge to prtiy or toty compromise the eement to test. On the other hnd, nondestructive tests to some extent deduce the stress fied by mesuring the effects induced by prescribed sign. Among others, the method bsed on the *Address correspondence to this uthor t the DADU, Università degi Studi di Sssri, 0704 Aghero, Sssri, Iti; Te: +9 079 970408; Fx: +9 079 97040; E-mi: emiio.turco@uniss.it X-ry diffrction is quite widespred but imited by the poor cpcity of penetrtion. Another we-estbished pproch is the utrsonic one, bsed upon the dependence between the wve veocity nd stresses. For both these methods, the book of [5] contins brief introduction nd sever references. Aso the rtice of [6] introduces the use of mgnetophotoesticity in this sme frmework. Both in the sttic nd dynmic fieds signs nd mesurements of mechnic type, forces nd dispcements, hve been considered, both in sttic nd dynmic fied. In the technic iterture [] is n ttempt to frme the probem furnishing some usefu nd firy gener guideines [7]. shows tht residu stresses re defined on the boundry when the Dirichet to Neumnn mp is known [8]. gives some usefu comments on the modeing of the residu stress suggesting the use of mode bsed on the Hrtig s w. These suggestions re so prtiy described in [9], which improves the mode redy presented in [7] showing the iniectivity of the mp. Another interesting work contined in [0] concerns the identifiction of pre-stressed cbe structures by dynmic tests. This work hs initiy portryed nondestructive stressstte identifiction strtegy ppicbe to re structur probems. The work provide detis bout simpicity nd technic ppiction of the pin-jointed trusses, which ow to expore the essenti spects of the probem without invoving hevy computtions. However, the formution deveoped for pin-jointed trusses cn be esiy extended to ny probem discretized by finite eements by using the mtrix nguge. The pper shows the significnce of the retion between n ddition sttic od nd the consequent dispcements on the exmined structure. Moving into the frmework of the force method, this retion is written using stndrd esticity hypotheses such s Hooke's w nd sm dispcements. 874-495/ 0 Benthm Open
Identifiction of Axi Forces on Stticy Indeterminte Pin-Jointed Trusses The Open Civi Engineering Journ, 0, Voume 7 5 Some brief considertion retive to the use of the Hrtig's w re so presented. Here, the choices in the design of the proposed gorithm were simpe in order to be esiy impemented on n ctu probem. This, for exmpe, ws the motivtion to use the ddition sttic ods with respect to those dynmic. After this brief introduction, Section contins brief discussion on the sttement of the underying identifiction probem nd discussion on the min toos used to buid the identifiction strtegy for stticy indeterminte pin-jointed trusses. Aso some remrks on the possibe use of Hrtig's w re briefy sketched. Section comprises of some resuts on simpe pin-jointed truss structures usefu to iustrte the min chrcteristics of the proposed pproch. Concuding remrks re eborted in Section 4.. IDENTIFICATION PROBLEM: STATEMENT AND TOOLS.. Probem Sketch It is considered n ssigned configurtion of pin-jointed truss, denoted s current, the structure is so considered in equiibrium under the extern ods. Axi forces on the truss re coected in the vector t nd cn be produced by ods nd disoctions such s therm vrition or ck of fit between the brs. The identifiction of vector t shoud be done by simpe procedure possiby dptbe to in situ tests. In cse the system is stticy determined the probem cn be soved through the soution of equiibrium equtions, obviousy. Conversey, if the truss is stticy undetermined it is necessry n d hoc identifiction procedure. Beginning from the current configurtion, n ddition od ws ppied nd the truss moves to new configurtion denoted s test configurtion. The increment of xi forces nd of nod dispcements were denoted by nd respectivey. The question is wheter it is possibe to identify the xi force vector by ssuming the dequte number of the dispcement increments of the vector s known? It hs been ssumed tht identifiction probem cn be treted s iner nd estic by considering the position of ech node in the current configurtion, nd so the ength of ech br, the mteri constnt, such s the Young's moduus, nd, finy, the cross-section re of ech br s known. There re, bsicy, two terntive wys to sove the identifiction probem s mentioned erier. The first foows the guideines outined by the stiffness method tht substntiy ssumes the nod dispcements s primry unknows. On the other hnd, the second method is the fexibiity method which ssumes the xi forces on the truss s primry unknowns. Here, the unknowns fctors re xi forces nd so the preferences wi be given to the second method. The potentiities of the fexibiity method ws expored to buid n gorithm be to sove our probem. In the next section, the identifiction probem wi be defined in more form wy by buiding too which wi generte the system of equtions nd soving those equtions the xi force vector wi be obtined... Toos Foowing the wy trced in [] nd more recenty in [], n ssemby of b brs nd j joints on which c kinemtic constrints prevent some dispcements hs been considered. Extern forces cting on uncostrined joints re coected in the -dimension vector where d is equ to or for two- nd three-dimension probem respectivey. The equiibrium mtrix inks the xi forces on the brs coected in the b-dimension vector with the od vector nd, consequenty, hs dimensions. Kinemtic comptibiity is ensured by the mtrix, see [], which inks the -dimension vector of nod dispcements to the b-dimension vector binding br eongtions () The retion between br eongtions e nd xi forces t cn be written by mens of the xi fexibiity mtrix F nd the vector of the imposed br eongtion coects br eongtions deriving for exmpe from therm vrition or ck of fit between brs. We cn write this constitutive retion s () Fexibiity mtrix is squre, with b-dimension, nd digon with the i-th term, in the cse we use Hooke's w, which is equ to nd depends on the ength, the cross-section re nd the Young's moduus of the i-th br. It hs been emphsized tht the bove retion is deduced from the St. Vennt's probem nd therefore is referred to the initi nd stress-free configurtion. Here the cse of stticy indeterminte nd kinemticcy determined pin-jointed trusses s defined in [] re considered. A structure is supposed to beong to this css if the equiibrium system of equtions hs infinite soutions for ny right hnd side for the equiibrium system (), whie the comptibiity system of equtions () hs unique soution for some prticur right hnd side, but otherwise hs no soution. This choice derives from the technic importnce of this css of structures. Systems of equtions (), () nd () cn be rrnged in vrious wys to find the unknown vector, nd of the so-ced direct probem, giving the weknown soution gorithm, such s the force method nd the dispcement method. For the stress stte identifiction probem defined bove the force method ppers to be the most promising. The rod trced by the force method works by ssuming the br xi forces s primriy unknowns. This method cn be synthesized strting from the representtion of br xi forces which stisfy, priori, the equiibrium equtions: (5) where soves the system of equtions nd the k-th coumn of, tht is, soves the homogeneous system of equtions. Using the constitutive equtions (), the br eongtion vector cn be expressed s (6) () (4)
5 The Open Civi Engineering Journ, 0, Voume 7 Emiio Turco nd, finy, the imposition of the kinemtic comptibiity in the form proposed in [] (7) produces the fin system of equtions (8) The sme resut cn be obtined by imposing the kinemtic comptibiity through the use of the virtu work theorem, precisey the virtu force theorem, see for exmpe []. The theorem of virtu forces cn so be used to evute nod dispcements by using the retion (9) where nd re, respectivey, the xi force vector nd the nod dispcement vector deriving from the soution of (8), whie is soution of the probem. It is enough to choose s s vector with the ony nonzero component equ to unity in correspondence with the dispcement to ccute nd obtin the formu (0) where corresponds through the equiibrium mtrix, to the prticur choice of. Writing (0) for two distinct soutions of (8) beed with nd we obtin () where to sim the nottion we hve used the positions nd. In the i-th digon term of the mtrix, see (4), there is the initi br ength tht is unknown in our probem whie the current br ength is known. For generic br nd two distinct configurtion nd, the eongtion increment is worth whie the br ength referred to stte is Combining the st two retions we obtin () () (4) nd so the corresponding digon term of the fexibiity mtrix referred to the current configurtion is (5) Finy, nod dispcement increments coected in the vector cn be written s (6) where is formed simpy by binding the corresponding to the considered nod dispcements. With these simpe toos cn be identified the xi forces in n ssigned configurtion, for exmpe in the current, which cn be reched under the ction of ods nd imposed eongtions. By imposing n ddition od on the structure produces both the increment of xi forces nd of nod dispcements. In gener, ony prt of this dispcement increments is mesurbe. We denote with the subset of mesured. Axi forces to identify nd their increments produced by the ddition od, cn be represented in such wy tht they stisfy the equiibrium equtions: (7) where the unknown quntities re coected in nd. Retion (6) cn be seen s system of equtions contining the unknowns nd whie is the known term... Remrks on the constitutive w We hve discussed bove simpe strtegy for the identifiction of the xi forces in pin-jointed trusses bsed on stndrd hypotheses of iner esticity. Here, s suggested in [8], some possibe modifictions to the constitutive w wi be considered briefy in order to modify (5) in ccordnce with the Hrtig's w insted of the cssic Hooke's w. Hrtig's w, see [4], cn simpy be written s (8) where the Young's moduus, defined s increment rtio between the stress nd the engineering strin, depends inery on through two constitutive coefficients:, id est the Young's moduus for, nd the dimension coefficient. The stress cn expicity be written s (9) from which the engineering strin cn esiy be deduced (0) Considering the bove two different sttes, denoted with nd, for generic br of the truss, simpe gebric mniputions permit to write the eongtion increment s () nd by using (0) () where the xi force nd its increment on the considered br re used. This retion inks the eongtion increments nd xi force increments for the considered br nd consequenty furnish the definition of the fexibiity coefficient for the br. The fexibiity coefficient for generic br cn so be derived by inerizing Hrtig's w s reported in [8]. Strting from the ineriztion () the combintion of () nd (0) gives (4) tht is the ink between nd which gin defines gin the fexibiity coefficient of the considered br. Hrtig's w or its inerized version produces ony different evution of the fexibiity coefficients nd its use in stress-stte identifiction gorithm shoud be evuted by compring costs nd benefits.. SOME RESULTS Here some resuts referred to simpe pin-jointed trusses re being discussed. The im is to expore the min fetures of the proposed identifiction strtegy nd to render the princip drwbcks evidenty. In the foowing, trusses formed by brs with the sme Young's moduus nd cross-section re hve been considered. Where necessry for numeric ccutions,
Identifiction of Axi Forces on Stticy Indeterminte Pin-Jointed Trusses The Open Civi Engineering Journ, 0, Voume 7 5 GP nd cm corresponding to n iron L- profie 40 x 5 (UNI 578-7) hve been ssumed... First exmpe: Two-Br Truss The current configurtion of the two-br pin-jointed truss sketched in Fig. () hs been considered s first exmpe. Fig. (). Two-brs truss probem: geometry of the current configurtion. If the ntur or stress-free ength of ech br is, on ech br there is n xi force. In order to identify the xi forces, strting from the current configurtion, horizont force on the centr node hs been ppied. The symmetry of the probem produces horizont dispcement which chrcterizes the test configurtion. Assuming the equiibrium of the xi forces in the current configurtion, both nd its increment, cn be represented s (5) since becuse there re no extern ods in the current configurtion. Vectors nd hve ony one component nd represent, for exmpe, the xi force nd its increment on the nd br. It is simpe to prove tht the sef-stress mtrix is equ to whie the vector is worth (6) (7) nd, finy, the mtrix ssumes the foowing form (8) Consequenty, the retion between the horizont dispcement increment of the centr node nd the stressstte, current nd its increments, is (9) nd cn be used to evute the unknown if it is observed tht for the symmetry nd tht the increment dispcement is, in our identifiction probem, mesured quntity nd therefore shoud be treted s known. It resuts (0) which, using the first of (5) to evute, produces the sme vue of the reference soution. First of the symmetry condition ws used in this cse s there ws ony one mesurbe dispcement. Secondy, n error free soution ws obtined since the position of the centr node is the sme both in the current nd ntur configurtions. In more gener exmpes beow, we hve the possibiity to evute the identifiction error when current nd ntur configurtions re different... Second Exmpe: Three-Br Truss The stress-free configurtion of the three-br truss is iustrted in Fig. (). If n imposed eongtion is ppied on the nd br, for exmpe deriving from n increment of temperture, this produces vertic dispcement of the free node nd xi forces on the brs. In order to simpify the ccutions, we choose for the imposed eongtion the vue () tht produces vertic dispcement of the free-node just equ to nd the xi force vector () The identifiction of this xi force vector foowing the proposed procedure strts from the ccution of the vector, of the sef-stress mtrix nd of the mtrix Fig. (b). It is convenient for these ccutions to consider the stticy determined schemes reported in Fig. (). More precisey, derives from the soution of the equiibrium equtions reted to the scheme reported in Fig. () nd resuts whie () derives from the soution of the equiibrium equ- e f o () (b),,,, Fig. (). Three-br truss probem: Figur geometry e. of Three-br the stress-free trussconfigurtion probem: geometry () nd current of the stress-free configurtion configu- with test od (b). f v
54 The Open Civi Engineering Journ, 0, Voume 7 Emiio Turco tions reted to the scheme reported in Fig. (b). (4) nd, finy, the mtrix cn be written by referring to the schemes reported in Fig. (c) nd (d) nd produces the foowing resuts (5) The dispcement increment vector retive to the ony free node cn be be ccuted s (6) being the term retive to -th br of the digon mtrix equ to (7) The use of xi force representtion (7) permits to ccute the expicit expressions of the horizont nd vertic dispcement increment of the free node. The resut is then (8) (9) nd using (7) the xi force vector to identify. It foows tht (40) where it ws noted tht for this probem it is necessry to use horizont force s test od in order to void nu vue of horizont dispcement. For the purpose of this work, reference soution for dispcement increments nd cn be ccuted by studying the probem reported in Fig. (). Using the stiffness method, the equiibrium imposed on the current configurtion, see gin Fig. (), immeditey produces the vues (4) Assuming tht both nd re mesured nd consequenty known, we cn evute the unknowns nd tht tke on the vues where the esticity coefficients of the i-th br re referred to the stress-free stte tht in this ccution is considered s known. Now, the identifiction error is defined s id ref (4) ref f o f v () (b) (c) (d) Fig. (). Three-br truss probem: uxiiry schemes for the identifiction probem. Figur e. Three-br truss probem: uxiiry schemes for the identifiction probem. Error Vrying Tbe. Three-br Test: Identifiction Rtio
Identifiction of Axi Forces on Stticy Indeterminte Pin-Jointed Trusses The Open Civi Engineering Journ, 0, Voume 7 55 where s is the ssumed vue of (), simpe ccutions show tht this error is ony dependent on the rtio. Tbe shows the identifiction error s the rtio increses in the cse of m, N nd N. It ws noted tht, in the rnge considered for the rtio, the identifiction error depends prcticy on iner wy from in such wy tht mkes the proposed identifiction strtegy ccurte... Third Exmpe: Six-Br Truss In the third test we consider the stress-free stte of the six-br pin-joined truss ws considered s reported in Fig. (4). An equiibrted sef-stress fied cn be chieved by incresing temperture on the -rd br. This produces dispcement of the 4-th node, depicted in Fig. (4b), denoted by the prmeter. It is very simpe to ccute the dispcement prmeter by using the fexibiity or the stiffness method. It ssumes the vue (4) where is the thermic dittion coefficient chrcteristic of the considered mteri. The corresponding vector of xi forces which needs to be identified is equ to (44) Our identifiction process uses the forces nd on the 4-th node s mechnic sign, see Fig. (4c), nd s mesured dispcement increments tht do not ssume the zero vue (48) Mtrix cn be ccuted by mking use of the uxiiry schemes reported in Fig. (6). It tkes the vue (49) In order to ccute (6) it is necessry to dd the expression of the digon eement of the fexibiity mtrix corresponding to the i-th br (50) Assuming the known the dispcements coected in the soution of (6) cn be chieved in the foowing two steps (5) This is necessry since the probem defined by (6) is noniner in the unknowns coected in nd so convenient for the prticur structure of mtrix. For the first system of (5), simpe ccution provides (5) (45) where nd re the horizont nd vertic dispcement of the k-th node, respectivey. As in the previous exmpes, both the vector of xi forces nd its increment cn be represented s (46) whie there re no conditions on nd The prticur structure of permits the foowing rerrngement In this cse, nd coect the xi forces nd their increments produced by the ddition forces, nd, on rd nd 4th br. Vector nd mtrix cn esiy be ccuted by referring to the schemes reported in Fig. (5) nd Fig. (5,b-c) respectivey. Using the prmeters nd, they ssume the vues which, when soved, gives the unknowns (5) (54) (47) where the foowing positions re used
56 The Open Civi Engineering Journ, 0, Voume 7 Emiio Turco (55) As in the exmpes iustrted bove, in order to compre the obtined resuts it is necessry to estimte the nod dispcements coected in. This cn be done by evuting the dispcements of the scheme reported in Fig. (4c) cused by the forces nd. By referring to the current configurtion nd in the frmework of the fexibiity method, xi force increments on the rd nd 4th br, see (8), cn be computed from the foowing system tht when it is soved it produces (56) (57) from which, using (5), cn be ccuted. Successivey, the requested dispcements cn be ccuted by mens of virtu work (9). Resuts deriving from the proposed identifiction pproch depends gin ony on the rtio. In the rnge it vries in prcticy iner fshion from. It is so interesting to study how the proposed identifiction pproch is ffected by the errors tht certiny exist in the considered identifiction probem. For exmpe, nod dispcements re obtined from mesurements nd thus re, certiny, ffected by errors. In order to simpy evute the effects of mesurement errors, we indicte with nd the re nd the mesured dispcements respectivey. In this cse the dt error prmeter cn impicity be defined s (58) Tbe reports for the identifiction error incresing dt error prmeter. In ddition to dt reported in the beginning of this section we hve ssumed m, N nd N. These resuts show, in n unmbiguous wy, the necessity of speci techniques to fiter the errors tht re wys present in the dt. A fitering strtegy such s tht proposed by Tikhonov [5] coud be usefu to sove ctu probem. 4. CONCLUDING REMARKS This pper hs sketched n identifiction procedure be to reconstruct ctu xi forces on pin-jointed truss. The proposed procedure is bsed on the resuts gthered through simpe mechnic test tht uses prescribed od nd the mesured dispcement induced. In the frmework of the fexibiity method, the ink between the xi forces to be identified nd the structur response to the test od is described in the context of stndrd iner esticity. In Section simpe numeric tests showed the min chrcteristics of the proposed procedure nd so highighted its performnces. When mesurement errors re tken into ccount, quick view of the identifiction of xi forces suggests tht n d hoc procedure is strongy necessry to fiter such errors s it is usu in the identifiction procedures. Tbe. Six-Br Truss Probem: Identifiction Error Vrying Dt Error Prmeter for. IDENTIFICATION OF AXIAL FORCES ON STATICALLY INDETERMINATE PIN-JOINTED TRUSSES 5 6 4 f v f o 4 5 () (b) (c) Fig. (4). Six-br truss probem: stress-free configurtion (), current configurtion (b) nd test od (c). Figur e 4. Six-br truss probem: stress-free configurtion (), current configurtion f v (b) nd test od (c). f o Fig. (5). Six-br truss probem: uxiiry schemes for the ccutions of (t) nd (s) () (b) (c)
Identifiction of Axi Forces on Stticy Indeterminte Pin-Jointed Trusses The Open Civi Engineering Journ, 0, Voume 7 57 Fig. (6). Six-br truss probem: uxiiry schemes for the ccution of mtrix. Figur e 6. Six-br truss probem: uxiiry schemes for the ccution ACKNOWLEDGEMENTS Some future deveopments re syntheticy isted beow:. In this work we hve considered ony stticy indeterminte nd kinemticy determined pin-jointed trusses, it ppers interesting to extend this pproch to prestressed mechnisms.. Ony simpe probems were considered here. Probems of medium nd rge dimensions, speciy from the technic point of view, require prticur cre nd specific strtegies to be computtiony efficient.. Athough the fexibiity method ppers to be more promising s frmework to identify the stress-stte, the cpbiities of the stiffness method shoud be dequtey expored. 4. Impementtions of more sophisticted constitutive ws s briefy described in Section, hve to be expored to understnd computtion costs nd dvntges. 5. Pin-jointed trusses were considered importnt due to their technic ppictions nd so for the simpicity of the mthemtic mode which used to study them. Viewing the procedure presented here in finite eement frmework, sm effort woud be dequte for the ntur extension of the resuts presented here. 6. As it is we known, the soution of n identifiction probem necessriy requires specific strtegies cpbe of fitering errors which cnnot be eiminted, e.g. those on the dt. Such errors occur from vrious cuses for e.g. inccurcy of the dt, pproximtion of the mthemtic mode nd sometimes due to the numeric mode nd rough ccurcy of the soution gorithm. Athough there re numerous gener purpose strtegies tht cn be used to reduce the forementioned errors but specificy designed procedure hs its own benefits s it cn de with specific probem. 7. Finy, it woud be interesting to expore the resuts obtined from geometricy noniner mode. CONFLICT OF INTEREST The uthors confirm tht this rtice content hs no conficts of interest. () (b) (c) (d) None decred. REFERENCES [] Y. C. Fung, A First Course in Continuum Mechnics, Prentice- H: USA, 994. [] A. Hoger, On the determintion of residu stress in n estic body, Journ of Esticity, vo. 6, pp. 0-4, 986. [] B. C. P. Burke, S. O. Kim nd K. S. Kim, Prti por decomposition inverse method ppied to determintion of intern stresses in n estic compex structures, Interntion Journ of Soids nd Structures, vo. 44, pp.00-00, 007. [4] F. Zhng, A. J. Kssb nd D. W. Nichoson, A boundry eement soution of n inverse esticity probem nd ppiction to determining residu stress nd contct stress, Interntion Journ of Soids nd Structures, vo. 4, no. 6, pp. 07-086, 997. [5] H. D. Bui, Inverse Probems in the Mechnics of Mteris: An Introduction, CRC Press: USA, 994. [6] A. Puro, Mgnetophotoesticity s prmetric tensor fied tomogrphy, Inverse Probems, vo. 4, pp. 5-0, 998. [7] R. L. Robertson, Boundry identifibiity of residu stress vi the Dirichet to Neumnn mp, Inverse Probems, vo., pp. 07-9, 997. [8] C. S. Mn, Hrtig's w nd iner esticity with initi stress, Inverse Probems, vo. 4, pp. -9, 998. [9] R. L. Robertson, Determining residu stress from boundry mesurements, Journ of Esticity, vo. 5, pp. 6-7, 998. [0] G. E. B. Tn nd S. Peegrino, Non-iner dynmic identifiction: An ppiction to prestressed cbe structures, Journ of Sound nd Vibrtion, vo. 08, no., pp. -45, 997. [] S.Peegrino, Anysis of prestressed mechnism, Interntion Journ of Soids nd Structures, vo.6. no., 9-50, 990. [] S.Guest, The stiffness of prestressed frmeworks: A unifying pproch, Interntion Journ of Soids nd Structures, vo. 4, pp. 84-854, 006. [] R. K. Livesey, Mtrix method of structur nysis, nd ed. Pergmon Press: Oxford, 975. [4] J. F. Be, The experiment foundtions of soid mechnichs, vo. I, Springer-Verg: Berin, 97. [5] A. N. Tikhonov nd V. Y. Arsenin, Soution of i-posed probems, John Wiey & Sons: New York, 977. Received: December 0, 0 Revised: December 8, 0 Accepted: December 8, 0 Emiio Turco; Licensee Benthm Open. This is n open ccess rtice icensed under the terms of the Cretive Commons Attribution Non-Commerci License (http://cretivecommons.org/icenses/ by-nc/.0/) which permits unrestricted, non-commerci use, distribution nd reproduction in ny medium, provided the work is propery cited.