INCI 4022 STRUCTURAL ANALYSIS II

Transcription

1 INCI 0 STRUCTURAL ANALYSIS II PART I DEINITIONS Node: Itersectio of two or more members, supports, member eds Degrees of freedom (at a ode): Number of possible displacemets for a ode The umber of degrees of freedom depeds o the type of structure or example, there are two degrees of freedom at each ode of a truss, two degrees of freedom at each ode of a beam, three degrees of freedom at each ode of a frame Degrees of freedom (of a structure): Total umber of degrees of freedom of a structure ree degree of freedom: Degrees of freedom free to displace There is othig exteral to restrai its movemet Restraied degrees of freedom: Degrees of freedom with restraits to displace There is somethig that restrais the movemet, or a o-zero displacemet is imposed Nodal displacemets: Displacemets alog degrees of freedom Nodal forces: orce applied alog a degree of freedom Nodal displacemet vector {}: A colum idicatig the displacemets of all degrees of freedom Nodal forces vector {}: A colum idicatig forces actig i all degrees of freedom Stiffess matrix: A matrix relatig the odal displacemets ad odal forces {}[]{}, which upo expasio becomes: To calculate ay elemet, like, we expad the secod row: Let, ad 0, the ; That is is the force applied to the DO for a uite displacemet of DO, keepig zero the displacemets at the other DOs

2 I geeral, ij is the force to be applied at the DO i for a uit displacemet of DO j, keepig zero the displacemets of the other DOs Elemet (member) stiffess matrix: Matrix relatig odal forces ad displacemets i a elemet 6 Truss elemet rame elemet Local coordiates (x, y): Coordiate system alog the elemet Axis x is directed alog the member axis from start to ed odes Axis y is obtaied by rotatig x axis 90 degrees couterclockwise Displacemets ad forces i local coordiates are δ ad Q respectively Global coordiates (X, Y): Axis X is horizotal, axis Y is vertical Nodal displacemet ad forces ca be expressed i either local or global coordiates Displacemets ad forces i global coordiates are ad respectively Wheever possible, local ad global coordiates will be differetiated i the case Lower case is for local, cap case will be for global

3 Y Global coordiates X y x Local coordiates Coordiate trasformatio matrix [T]: Trasforms displacemets or forces from global to local coordiates The same matrix trasforms both displacemets ad forces, sice they are both vectors {δ} [T]{} {Q} [T]{} {} ad {} are the odal displacemets ad forces i global coordiates {δ} ad {Q} are the odal displacemets ad forces i local coordiates δ δ δ δ θ A importat property of the coordiate trasformatio matrix [T] is that: [T] - [T] T, that is its iverse is equal to its traspose Stiffess matrix i local coordiates: If the odal displacemets ad forces are expressed i local coordiates for a elemet, the matrix that relates them is called stiffess matrix i local coordiates [k] {Q} [k] {δ} Numberig: Numberig of odes: Nodes ca be umbered i ay way

4 Numberig of elemets: Elemets are also umbered i ay fashio Orietatio of elemets: or each elemet, the start ad ed ode must be specified Ay ode ca be the start ode A arrow is placed o each elemet with the head orieted to the ed ode Numberig of the degrees of freedom: or coveiece, first umber the free degrees of freedom ad the the restraied degrees of freedom Solvig the structural equatio: {} []{δ} The matrix is partitioed i kow ad ukow forces ad displacemets, which have bee coveietly arraged through the umberig system r f rr rf fr ff r f f ukow displacemets r kow displacemets p kow forces r ukow forces Solvig the matrix equatio above: f ff f + fr r r rf f + rr r f ff - ( f - fr r ) Member forces: or practical purposes, it is coveiet to kow the member forces i local coordiates {Q} [T] {} [T] [] {} Also {Q} [k] {δ} [k] [T] {}

5 Assemblage of structural stiffess matrix: Each elemet ij of the structural stiffess matrix is the cotributio of ij of each idividual structural members The followig rules ca be see for a elemet ij of the structural stiffess matrix a If i ad j are located i the same ode, ij is the cotributio of all members coectig i that ode b If i ad j are located i odes joied by a member, ij will come from that member oly c or all other cases: ij 0 EXAMPLE: or the elemet, members,, ad will cotribute or elemet 6, member will be the oly cotributor or elemet 0 RELATION BETWEEN STINESS MATRIX IN LOCAL AND GLOBAL COORDINATES {Q} [k] {δ} [T] {} [k] [T] {} {} [T] - [k] [T] {} sice [T] - [T] T {} [T] T [k] [T] {} sice {} [] {} [] [T] T [k] [T] The elemet siffess matrix [k] ad the coordiate trasformatio matrix [T] deped o the type of structure: truss, beam, frame, etc We will ow determie these matrices startig with a truss elemet