Frame element resists external loads or disturbances by developing internal axial forces, shear forces, and bending moments.

Transcription

1 CE7 Structural Analyss II PAAR FRAE EEET y 5 x E, A, I, Each node can translate and rotate n plane. The fnal dsplaced shape has ndependent generalzed dsplacements (.e. translatons and rotatons) noled. Frame element ressts external loads or dsturbances by deelopng nternal axal forces, shear forces, and bendng moments. The way we wll fnd the nternal generalzed force (.e. force or bendng moment) s. generalzed dsplacement (.e. dsplacement or rotaton) relatons s as follows: We wll dsplace one degree of freedom at a tme whle holdng all other degrees of freedom fxed. The force system requred to make ths partcular dsplacement pattern happen wll tell us how nodal forces and moments are related to nodal dsplacement and rotatons. We wll assume that the frame element s long and slender. Ths allows us to happly gnore the resstance deeloped due to shear strans. In other words, we wll consder the element to be resstng by axal stranng and by flexure. et s start. Case ) u u whle other dsplacements and rotatons =. and from statc equlbrum ote that axal deformatons n the frame element do not cause shear forces or bendng moments. So,

2 Case ) u u whle other dsplacements and rotatons =. and Case ) whle other dsplacements and rotatons =. Case ) whle other dsplacements and rotatons =.

3 Case 5) whle other dsplacements and rotaton =. Case ) whle other dsplacements and rotaton =.

4 PAAR FRAE EEET y 5 x E, A, I, ember Stffness atrx n ocal Coordnates K local

5 Frame Element Analytcal odel Relaton between member end dsplacements and rotatons and nternal forces, moments, dsplacements, and rotatons elsewhere n the element Prsmatc member wth homogeneous, lnear elastc, sotropc materal propertes AXIA u : dsplacement of the beam element n x-drecton. : dsplacement of the beam element n y-drecton. : rotaton of the beam (about z-axs) ( u u) ux ( ) u x Axal deflected shape can ary lnearly; axal stran can only be constant FEXURA and SHR At any gen locaton d & : shear & moment E : modulus of elastcty I : moment of nerta For a beam wth no concentrated forces or moments, the dfferental equatons of equlbrum are d d q q : unformly dstrbuted transerse load Usng the moment-curature relatonshp, d d q 5

6 For the beam element (model) under consderaton, q. Therefore, d Integrate once, d C wherec s a constant of ntegraton, mplyng that shear force s constant across the element. Contnue ntegratng, d Cx C d Cx CxC C x C x C xc where C, C, C andc are the four unknown constants of ntegraton. We shall make use of the four boundary condtons to fnd these four unknown constants. The boundary condtons are () ( ) d x Substtute and fnd C ( ) ( ) C ( ) ( ) C C Substtute to fnd and, ( ) ( ) ( x)( ) ( ) ( ) x

7 Deflectons n the transerse drecton s gen by x x x ( ) x x x x x x x = The functons n parentheses n front of the dsplacements and rotatons are called the shape-functons. They relate the end dsplacements and rotatons (the only ones our model can see) to the deflected shape suggested (hence, only approxmate) by our dealzed frame model. Recall that, longtudnal dsplaced shape s gen by ( u u) ux ( ) u x. To fnd the slope at any gen pont, smply take the derate of the expresson for aboe wth respect to x,.e., x s d. 7