3.5b Stress Boundary Conditions: Continued

Transcription

1 3.5b Stre Boundar Condition: Continued Conider now in more detail a urface between two different material Fig One a that the normal and hear tree are continuou acro the urface a illutrated. 2 1 Figure : normal and hear tre continuou acro an interface between two different material material 1 and material 2 Note alo that ince the hear tre i the ame on both ide of the urface the hear tree acting on both ide of a perpendicular plane paing through the interface between the material b the mmetr of tre mut alo be the ame Fig a. Figure : tree at an interface; (a) hear tree continuou acro the interface ( tangential tree not necearil continuou However again the tangential tree thoe acting parallel to the interface do not have to be equal. For eample hown in Fig b are the tangential tree acting in the upper material - the balance no matter what the magnitude of the tree. Decription of Boundar Condition ( a) ( The following eample bring together the notion of tre boundar condition tre component equilibrium and equivalent force. 8 Kell

2 Eample Conider the plate hown in Fig It i of width 2 a height b and depth t. It i ubjected to a tenile tre r preure p and hear tree. The applied tree are uniform through the thickne of the plate. It i welded to a rigid bae. r p b Uing the 2a Figure : a plate ubjected to tre ditribution ae hown the tre boundar condition can be epreed a: Left-hand urface: Top urface: Right-hand urface: ( a ) p ( a ) b ( r ( a a ( a ) ( a ) b Note carefull the decription of the normal and hear tree over each ide and the ign of the tre component. The tree at the lower edge are unknown (there i a diplacement boundar condition there: zero diplacement). The will in general not be uniform. Uing the given ae thee unknown reaction tree eerted b the bae on the plate are (ee Fig ) 81 Kell

3 Lower urface: ( ) a a ( ) Note the direction of the arrow in Fig the have been drawn in the direction of poitive ( ) ( ). () () Figure : unknown reaction tree acting on the lower edge For force equilibrium of the complete plate conider the free-bod diagram 3.5.2; hown are the reultant force of the tre ditribution. Force equilibrium require that F F bpt 2at t 2art t a a a a ( ) d ( ) d 2at 2art bpt bt bt Figure 3.5.2: a free-bod diagram of the plate in Fig howing the known reultant force (force on the lower boundar are not hown) For moment equilibrium conider the moment about for eample the lower left-hand corner. One ha 82 Kell

4 a d M bpt( b / 2) 2at( 2art( a) bt(2a) t ( ) If one had taken moment about the top-left corner the equation would read a a M bpt( b / 2) 2art( a) bt(2a) t a a a ( ) b d t a ( ) a d Problem 8. Conider the point hown below at the boundar between a wall and a diimilar material. Label the tre component diplaed uing the coordinate tem hown. Which tre component are continuou acro the wall/material boundar? (Add a upercript w for the tree in the wall.) 9. A thin metal plate of width 2 b height h and depth t i loaded b a preure ditribution p () along a a and welded at it bae to the ground a hown in the figure below. Write down epreion for the tre boundar condition (two on each of the three edge). Write down epreion for the force equilibrium of the plate and moment equilibrium of the plate about the corner A. p() 2a h A 2b 83 Kell

5 84 Kell