PRATICAL STATIC CALCULATION METHOD FOR ESTIMATING ELASTO-PLASTIC DYNAMIC RESPONSES OF SPACE FRAMES

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1 th World Congress on Computatonal Mechancs (WCCM XI) 5th European Conference on Computatonal Mechancs (ECCM V) 6th European Conference on Computatonal lud ynamcs (EC VI) E. Oñate, J. Olver and A. Huerta (Eds) PRATICAL STATIC CALCULATION METHO OR ESTIMATING ELASTO-PLASTIC YNAMIC RESPONSES O SPACE RAMES KOICHIRO ISHIKAWA epartment of Archtecture and Cvl Engneerng Unversty of uku unkyo 3-9-, , uku-sh, Japan e-mal: shkawa@u-fuku.ac.jp Key Words: Truss walls, use type ball jont connectons, Steel bolts, Alumnum alloy struts, System trusses, Predcton statc method of sesmc responses. Abstract. The study deals wth alumnum alloy spatal truss walls wth fuse type jont connectons subjected to horzontal earthquake waves. The fuse type connecton prevents the brttle collapse nduced by the member bucklng and the tensle fracture of the weld connecton. The proposed method usng the property of the energy conservaton can statcally predct the horzontal maxmum sesmc response dsplacement at the top of the wall. INTROUCTION Ths study deals wth response and damage-controlled structures such as alumnum alloy lattced wall structures nduced by the plastc elongaton of the steel bolt among the jont connecton. The truss wall structure system s composed of several elements such as ball jonts, strut members and ther connectons. The spatal truss wall structure has two collapse mechansms such as the progressve member bucklng due to a compresson stress and the tensle fracture of the web connecton due to a tenson stress. In our prevous paper, we presented that the progressve member bucklng nduces the brttle collapse of the structure. Ths means that the structure looses the horzontal load carryng capacty, soon after the load reaches the maxmum capacty. On the other hand, t could be known that an axal plastc elongaton ductlty of the steel brngs about the response control of a structure due to the absorbng energy. The purpose of ths study s to nvestgate the energy absorbng mechansm wthn the jont connecton bolt tensle collapse type structure system caused by the bolt plastc elongaton due to an axal tenson stress. As far as the one degree of freedom equvalent model s concerned, the modelng procedure of the smplfed one degree of freedom vbraton system s proposed n order to predct the earthquake elasto-plastc responses of the lattced wall usng the earthquake response spectrum and the property of the energy conservaton. In the modelng, the calculaton method of predctng the equvalent one degree vbraton system such as the horzontal effectve stffness K and the horzontal strength y due to the steel connectng bolt yeld state s presented. And the frst horzontal natural vbraton perod T can be also calculated by usng the proposed equatons. The estmaton results obtaned by usng

2 Kochro Ishkawa. the smplfed vbraton model are verfed to be n close agreement wth the elasto-plastc dynamc analyss results of the full numercal analyss model. Structural engneers can statcally estmate the maxmum dynamc horzontal dsplacement of the spatal truss wall structures subjected to earthquake motons by means of the proposed practcal statc calculaton method. MOELING O SYSTEM TRUSS MEMERS g. llustrates a truss system composed of the strut, the hubs, the collars and the end plugs, whch are extruded alumnum alloy (606-T6). And the bolts are made of Cr-Mo Steel quenched and annealed. As far as the load transfer mechansm of the jont connecton, a compresson load s transferred through the collar and a tensle load s transferred through the steel bolt as shown n g.. In ths study, the system truss member s modeled by composng of two elements such as both the jont connecton elements and the strut member element n g.3. H U O L T S T R U T E N P L U G C O L L A R gure : Truss system gure : Load transfer mechansm Z JOINT CNNECTION JOINT CONNECTION Y STRUT MEMER X gure 3: Components of truss system

3 Kochro Ishkawa. 3 3 STINESS MATRICES O TRUSS MEMER OR PRESICE ANALYSIS g.4 shows the three axal sprng elements such as the two jont connectons and the strut. The stffness matrx s ntroduced to reduce the all unknown dsplacements at the nodes,, and to express the dsplacements {u, v, w} and {u, v, w} at the both ends. The u, v and w are dsplacements wth respect to the x, y and z drectons n g.4, respectvely. Node gure 4: Truss elements of truss system Jont Connecton ( ) : The relatonshp between the axal dsplacement and force of the jont connecton ( ) can be gven by Eq. () Strut Member ( ) : The relatonshp between the axal dsplacement and force of the strut member ( ) can be gven by Eq. (). S T K Jont Connecton ( ) : The relatonshp between the axal dsplacement and force of the jont connecton ( ) can be gven by Eq. (3). K R T Where s the ncremental dsplacement vector, s the nternal force vector and s the external load vector at the nodes. 3. Reduced member stffness matrces The reduced member stffness matrces expressed by the node dsplacements and stresses at the both ends can be derved from the above Eq. (), () and (3) as follows. C A T L T K () () (3) (4) x y z

4 Kochro Ishkawa. A T C (5) 4 HYSTERETIC MOELS O COMPONENT PARTS O SYSTEM TRUSS 4. Strut member restorng force characterstc model The system truss composes of the tubular strut member, the ball jonts and the jont connecton bolts. We present a modelng procedure of the strut member enough to predct the elasto-plastc bucklng behavor. The strut member s modeled to be buckled due to a compresson axal force and be fractured at the welded parts due to a tenson axal force. g.5 shows the hysteress curves used for the dmensonless slenderness rato dvded by the dvdng slenderness rato. The maxmum compressve stress σ cr of the member s calculated by usng Eq.(6). A bucklng length s needed to calculate a slenderness rato. The slenderness rato of the member bucklng load s taken to be the effectve slenderness rato consderng the jont connecton restrant. /.0 : / 0.5( / ) cr /.0 : / cr ( / ) (6) E / / 8. Where s the basc strength σy, λ s the member slenderness rato and Λ s the dvdng slenderness rato, whch s taken to be 8 n the study. The yeld stress σ y and Young s modulus E are set at 0Mpa and 70Gpa, respectvely. The member bucklng occurs at the maxmum compressve stress σ cr due to a compresson axal stress and the weld fracture occurs at 0.7σ y due to a tenson axal stress. 4. Jont connecton restorng force characterstc model The jont connecton composes of hubs, collars and end plugs n g.. g. shows the axal force transfer mechansm under a compresson and a tensle stress, respectvely. g.6 shows the steel bolt hysteress model of the relatonshp between the axal force and dsplacement. 4

5 Kochro Ishkawa..0 σ/σy 3 4 ε/εy YIEL STRESS y AXIAL TENSION ORCE γ RK RK3 RK AXIAL ISPLACEMENT YIEL ISP gure 5: Hysteress model of strut member gure 6: Hysteress model of steel bolt 5 STATIC ELASTO-PLASTIC ANARYSIS O SPATIAL TRUSS WALL The statc elasto-plastc analyss of the spatal truss walls wth the fuse type connecton s carred out to verfy the ductle collapse mechansm nduced by the tensle yeld of the steel bolt n the connecton. The above mentoned hysteress models of the elements are used to obtan the relatonshp between the stress and the stran of the elements n the step by step pushover analyss. 5. Confguraton and mechancal propertes of statc analyss model The global confguraton of the analyss model s shown n g.7. The structure has the m x m grd and the m depth of the wall type. The wall has 4.5m x 3.75m geometry. As far as the boundary condton s concerned, all the bottom nodes of the wall are pn support. All the top nodes of the wall can move for the horzontal axs (Y axs) and are restraned for the vertcal and out of plane axes (Z and X axes), as shown n g.7. The member propertes such as the sectonal area A (mm ), the radus of gyraton (mm), the length L (mm) and the slenderness rato λ are shown n Table and. Table : Mechancal propertes of alumnum struts ( L : member length, λ : member slenderness rato, Λ : member dvdng slenderness rato, A : member sectonal area, : radus of gyraton ) Λ=8 ALUMINUM STRUT L (mm) λ λ/λ A (mm ) (mm) Chord Φ50x0 x: 858 y: 483 x: 37.4 y: 9.9 x: 0.46 y: Web Φ50x

6 Kochro Ishkawa. Table : Mechancal propertes of steel connecton bolts ( A: sectonal area, L : length ) STEEL OLT A (mm ) L (mm) Mnmum Elongaton (%) Standard Strength (N/mm ) Yeld Axal Stress (kn) Chord Φ Web Φ X, Y and Z drectons are restraned. Y and Z drectons are restraned. 5. Elasto-plastc analyss of the truss wall gure 7: Analyss model of double layer truss wall g.8 shows the relatonshp between the horzontal load P and the horzontal dsplacement at the node of the analyss of the double layer truss wall n g.7. The regon from the orgn pont to the turnng pont results n a lnear elastc relatonshp. Ths s a reason why all of the members such as the chords and the webs reman n the elastc regon. The stress of the steel bolt reaches the yeld stress 35N/mm at the turnng pont of the horzontal load P=408(kN). The yeld elongaton of the truss wall appears beyond the turnng pont. The member bucklng doesn t occur to the termnal pont as shown n g.8. Ths means that the brttle collapse s avoded by means of the fuse type elongaton such as the tensle yeld elongaton of the steel bolt. 6

7 Load (kn) Kochro Ishkawa. Estmaton yeld dsplacement y =.3mm (kn).66 (mm) Estmaton yeld load y =43.4 kn splacement (mm) gure 8: Statc analyss result of relatonshp between the horzontal load and dsplacement 5.3 Practcal calculaton methods of the yeld load y and the correspondng dsplacement y nduced by the yeld of the steel bolt n the connecton The calculaton methods of predctng the horzontal resstant capacty, such as the horzontal dsplacement and load at the top level of the wall are proposed by means of the vrtual work theory. And the yeld load y and the correspondng dsplacement y nduced by the yeld of the steel bolt n the connecton are obtaned by means of the proposed calculaton method. Consder the wall type structures subjected to the plane horzontal load. The shear deformaton may be domnant n comparson wth the bendng one. Consequently, t can be assumed that the web members equally resst the shear force. Under the assumpton, the shear stress per half column of a grd s gven by p = / ( m ) The axal stress of the web member s gven by N = p / α. The yeld horzontal shearng load y s ntroduced n the followng equaton. Y = m α N Y (8) Where N Y s the steel yeld stress. Usng the prncple of the vrtual work, the horzontal dsplacement y correspondng to the web jont connecton bolt yeld s gven by (7) 7

8 Kochro Ishkawa. y = s + n y l S m ES AS n m E y A l (9) Where m s the grd number along the shear force drecton, n s the grd number along the orthogonal drecton to the shear force drecton. E s, l s and A s are the Young s modulus, the member length and the sectonal area of the strut member, respectvely. E, l and A are the Young s modulus, the member length and the sectonal area of the jont connectng bolt, respectvely. α s the cosne of the web to the lateral drecton. s s the component nduced by the strut elongaton and s the component nduced by the jont connecton bolt elongaton subjected to the horzontal load y. y brngs about the yeld of the jont connecton steel bolts. 6 YNAMIC ELASTO-PLASTIC ANALYSIS O SPATIAL TRUSS WALL As far as the numercal analyss method s concerned, the hysterc materal behavor and the geometrc non-lnearty are consdered n the dynamc analyss of the truss structure under the horzontal (X drecton) motons such as El Centro NS. And the peak ground acceleratons of the three earthquake motons are taken to be 5.6 (m/sec ). The Newmark β method s used n the numercal ntegraton method. Snce t has been known that a case of β=/4 wll be uncondtonally stable for most nonlnear problems, β=/4 s used n ths study. The Raylegh dampng s used and both of the frst and second dampng factors are taken to be 0.0. The detals of the analyss method are descrbed n the presented paper []. 6. Confguraton and mechancal propertes of dynamc analyss model The global confguraton of the analyss model s shown n g.9. The structure has the m x m grd and the m depth of the wall type. The wall has 6m x 8m geometry. As far as the boundary condton s concerned, all the bottom nodes of the wall are pn support. All the top nodes of the wall can move for the horzontal axs (Y axs) and are restraned for the vertcal and out of plane axes (Z and X axes), as shown n g.9. The member propertes such as the sectonal area A (mm ), the radus of gyraton (mm), the length L (mm) and the slenderness rato λ are shown n Table 3and Sesmc coeffcent q y of steel bolt yeld state and maxmum response dsplacement The frst natural perod of the prototype of the model s set to be 0.9second for a shearng shape mode of carryng out the egenvalue analyss. And the lattced wall structure has a shear resstance mechansm, because of the hgh bendng stffness n comparson wth the shear stffness for the aspect rato. Therefore, t s assumed that the web members equally resst the shear force. Under the assumpton, the analyss models are desgned to be the tensle yeld collapse mode of the steel bolt among the connecton. The statc sesmc resstant coeffcent q y of the lattced wall due to the web bolt yeld state s also determned varyng the yeld stress of the steel bolts before carryng out the dynamc analyss. 8

9 Kochro Ishkawa. The purpose of the elasto-plastc dynamc analyss s to calculate the maxmum response horzontal dsplacement at the top of the wall, and verfy the collapse mechansm. Ths study deals wth realzng a damage-controlled structure nduced by a ductle falure nduced by the plastc elongaton of the steel bolt among the jont connecton. It s one of the sgnfcant matters that the lattced wall structure collapse mode s not caused by brttle falure types such as a sequental member bucklng, before the ductle collapse occurs. 6.3 rst natural perod T E for horzontal mode shape The frst horzontal natural perod s gven by M T E (0) K Where M s the mass of the sngle degree of freedom vbraton system and K s the horzontal stffness of the wall. K s the horzontal stffness rgdty gven by y m K () y n l S l ES AS E A (a) X and Y plane (b) Y and Z plane (c) X and Z plane X, Y and Z drectons are restraned. Y and Z drectons are restraned. gure 9: Analyss model of double layer truss wall for dynamc analyss 9

10 Kochro Ishkawa. Table 3: Mechancal propertes of alumnum struts ( L : member length, λ : member slenderness rato, Λ : member dvdng slenderness rato, A : member sectonal area, : radus of gyraton ) Λ=8 ALUMINUM STRUT L (mm) λ λ/λ A (mm ) (mm) Chord Φ80 x x: 560 x: 6. y: 560 y: 6. x: 0.33 y: Web Φ50x Table 4: Mechancal propertes of steel connecton bolts ( A: sectonal area, L : length ) STEEL OLT A (mm ) L (mm) Mnmum Elongaton (%) Standard Strength (N/mm ) Yeld Axal Stress (kn) Chord Φ Web Φ Results of dynamc elasto-plastc analyss The elasto-plastc dynamc analyses are carred out usng the full model as shown n g.9 subjected to El Centro NS earthquake moton. In the analyses, a sgnfcant parameter such as the sesmc resstance coeffcent q y of the analyss model s set varyng the yeld stress of the steel connecton bolt as shown n g.. And the vbraton characters such as the natural perod T for the n-plane horzontal drecton and the dampng factor h are taken to be 0.9 second and 0.0 for the all models, respectvely. The purpose of the analyss s to fnd out the steel bolt mechancal propertes n order to make certan the ductle ultmate state nduced by the bolt yeld mechansm. g.0 shows the relatonshp between the sesmc resstant coeffcent q y and the horzontal maxmum response dsplacement at the top of the wall. In the fgure, a dotted lne s drawn usng the property of the energy conservaton and the elastc earthquake response spectrum by the equvalent sngle degree of freedom model. The detal of the property s descrbed the followng secton. And the mark are plotted the horzontal maxmum response dsplacements correspondng the sesmc resstance coeffcent q y of the lattced wall structures subjected to El Centro NS. The lattced wall structures wth q y rangng from ductlty factor (Eq. 4) μ= to.7 brng about the steel bolt yeld state by the peak ground acceleraton. It s seen that the predcton method usng the property of the energy conservaton shows a good agreement wth the numercal full scale analyss results. 0

11 Yeld Strength of Structure qy=y/mg Kochro Ishkawa μ = μ =.7.4. C C C Property of energy conservaton max(mm) gure 0: Sesmc coeffcent of yeld state and maxmum response dsplacement 7 CONCEPT O EUIVALENT SINGLE EGREE O REEOM VIRATION MOEL AN PROPERTY O ENERGY CONSERVATION 7. Property of energy conservaton by the sngle degree of freedom vbraton equvalent model Consder a relatonshp between the yeld strength and the maxmum response dsplacement δ of the sngle degree of freedom vbraton model n g.. δ N of the elastoplastc structure s calculated by usng the assumpton of the property of the energy conservaton between the trangular part of the elastc stran energy and the trapezodal part as shown n g.. A broken lne n g. s drawn usng the above mentoned property. L Y δ δ Y δ L δ N =μδy Maxmum Response sp. gure : Property of energy conservaton

12 Yl-axs Kochro Ishkawa. MASS M st NATURAL PERIO T M K 0.9 ELASTIC ISP. RESPONSE SPECTRUM RIGIITY y m K K y n ls ES AS l E A 0.6 SEISMIC COEICIENT O YIEL STATE 0.3 q y = y /Mg Property of energy conservaton エネルギー一定則 HORIZONTAL STRENGTH UE TO OLT YIEL X b-axs y N y m gure : Concept of modelng for predctng the horzontal maxmum response dsplacement of the wall g. shows a concept of predctng the maxmum horzontal response dsplacement of the ductle collapse type truss wall nduced by the jont connecton bolt elongaton subjected to earthquake motons by usng the equvalent sngle degree of freedom vbraton model and the property of the energy conservaton as follows. rst, the elastc earthquake maxmum dsplacement δ L n g. s gven by the response spectrum usng the equvalent natural perod T E (Eq.0) and the equvalent rgdty K (Eq.). Second, the elasto-plastc earthquake maxmum dsplacement δ N nduced by the connecton steel bolt yeld ultmate state can be estmated the followng equatons usng the property of energy conservaton (g.). The horzontal shear force Y can be calculated by usng Eq.(8) correspondng to the steel bolt yeld stress N Y. δ Y can also be calculated by Eq.(9). Y L N Y Where L s the elastc horzontal shear force correspondng to the sesmc response dsplacement spectrum δ L. L s derved by means of K(Eq.) multpled by δ L. 7 CONCLUSIONS The study has nvestgated the two ultmate state types of the double layer truss wall structures such as the jont connecton bolt yeld of the web member, and the web member bucklng by the dynamc elasto-plastc analyss. As a result, the proposed method usng the property of the energy conservaton can statcally predct the horzontal sesmc maxmum dsplacement at the top of the wall n the case of the steel bolt yeld type collapse mechansm. REERENCES [] Ishkawa, K. and Kato, S. Elastc-plastc ucklng Analyss of Retcular ome subjected to Earthquake Moton. Internatonal Journal of Space Structures, Mult-Scence Publshng Co. Ltd, (3/4), (997). δ L ()