Advanced Computatonal echancs 18 Chapter 4 Applcaton to Plane rgd frame structure 1. Dscusson on degrees of freedom In case of truss structures, t was enough that the element force equaton provdes onl the relaton between the aal force and the etenson Δ see Table 4-1. However, n case of rgd frames, we have to consder three components as element forces, even f the analss s two-dmensonal. One of them s aal force and the other two are edge moments at both ends of an element. Furthermore, n case of three-dmensonal analses, we need components as shown n Table 4-1. Elemen t Dmen -son Table 4-1 Frame elements and degrees of freedom Element Forces Deformaton of Degrees of element freedom at a node Sze of the stffness matr Truss -D Aal force Etenson translaton 4 4 3-D Aal force Etenson 3translaton Beam -D Aal force Edge moments at both ends Etenson Edge deflecton angle at both ends translaton 1rotaton 3-D Aal force Edge moments at both ends, havng components Torsonal moment Total= Etenson Edge deflecton angle at both ends, havng components Torsonal angle Total= 3translaton 3rotaton Total= 1 1. Compatblt Equaton The relaton between etenson and nodal dsplacement at both ends of an element: Ths relaton becomes as same as eq.-1 n case of analses for truss structures. = 4-1 + fg.4-1 Elongaton and dsplacement 4-1
Advanced Computatonal echancs 18 et, we have to derve the relaton between edge deflecton angles and rotatonal dsplacements at both ends. See the fgure. And when, Rgd bod rotaton of the element = Edge deflecton angles =, Total amount of rotaton from ntal to current at both ends = r,r ou can derveθ andθ b eas geometrc. r r And f we defne the element drectonal vector before deformaton as; [,=[ -,- ] the element drectonal vector after deformaton becomes as follows; [ +-, +- ] = [ +, + ] Then, we calculate the vector product b two these drectons. sn r Drecton of the element After deformaton Drecton of the element Before deformaton r fg.4- Rgd bod rotaton and deflecton We can recognze that and are ver small. So, the followng assumptons are approved. sn,ad Then, can be rewrtten as follows; Therefore, the edge deflecton angles θ,θ becomes; r r r r As a result, the compatblt equaton of plane rgd frame structure s epressed as follows; r 1 4-1 r 4-
Advanced Computatonal echancs 18 3. Element force equaton The element force equaton provdes the stffness relaton of aal force etenson and edge moments edge deflecton angle as shown n the fgure. The support condton should be set nsde the local coordnate sstem n order to be sure that ever element force s ndependent each other. EA 4EI EI EI 4EI eq4-3 uz. Derve the element force equaton of eq.4-3 from two dfferent was to calculate, as follows; Usng Castglano's prncple. 4-3
Advanced Computatonal echancs 18 4-4 4. Equlbrum equaton When we descrbe nodal forces at both ends n a global coordnate, whch has two aes of horzontal and vertcal, we can draw the fgure below left one. In smultaneous procedure, when we descrbe element forces under a statcall determnate support condton n ths case, we can thnk about smple beam., we can draw the other fgure below rght one. Three components of [,,] derved b eq.4-3 are ndependent each other. B the wa, n the rght fgure becomes as follows, because t s regarded as reacton forces under the support condton. odal forces Element forces Therefore, translaton components of nodal forces are epressed b cosne vectors of and ; Consequentl, the equlbrum equaton can be wrtten as follows; 1 1 eq.4-4
Advanced Computatonal echancs 18 4-5 5. Total stffness equaton B combnaton of eq.4-, eq.4-3 and eq.4-4, we can wrte the stffness equaton on a element as follows. Epand ths b ourself once at least! r r EI EA 4 1 1 1 1 1 1 1 1 4 1 1 1 1 1 1 1 1 eq.4-5 ote: In addton, n equaton above, Clause 1 n a parenthess of rght-hand sde becomes same as a stffness equaton n case of truss elements. Same as case of truss elements n Chapter, we can epand a stffness equaton ever element nto a total stffness equaton b prncple of superposton. Assgnment 4.1 Compose a program to analze RIGID FRAE STRUCTURE WITH UTI EEETS b the dsplacement method, and draw a bendng moment dagram of the desgnated rgd frames. It s suggested that easer wa s to be modf the program n Assgnment 5. odels for analss are desgnated as follows; oung s odules: E=Gpa Area of cross secton: Column part: Beam part: agntude of load P=1k,Thckness of plate = 1.5cm Sze of rgd frame: Heght: H=5m, Span B=4m The frst level heght of two levels of rgd frame =H/ ou have to submt: 1 Programs Input dataif ou used S-ECE 3 Bendng moment dagrams drawn b outcome from computaton P P P P.4m.4m.4m.4m