515 18750 n force2 pna pntl 5000 25 2

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: tress elem. is shown. The elem. experiences a 10 C temp. difference. y t = .5 cm k(1,3) (e ) E = 15(10)6 N / cm2 [k ]= ? (e ) ] = ? [f Solution: µ = .25 α = 6(10−6 )/ C i(0,0) j(2,0) x [k(e )]= tA[B]T [ D][ B] Chapter 23 Page 24 K. Haghighi but ⎡bi 0 b j 0 bk 1⎢ [B] = ⎢ 0 ci 0 c j 0 2A ⎢ ci b i c j b j ck ⎣ where 0⎤ ⎥ ck ⎥ bk ⎥ ⎦ A=3(2)/2=3 cm2 . Find all b’s & c’s, subst. to get: 0 3 ⎡− 3 1⎢ [B] = ⎢ 0 − 1 0 6 ⎢−1 − 3 −1 ⎣ 0 0 0⎤ − 1 0 2⎥ ⎥ 3 2 0⎥ ⎦ Chapter 23 Page 25 K. Haghighi Also ⎡ ⎤ 0⎥ 0⎤ ⎢1 µ 6 ⎡1 14 E⎢ ⎥ = 15 10 ⎢1 4 1 [D] = 0 µ1 0⎥ ⎥ ⎥ 1 − .252 ⎢ 1 − µ2 ⎢ 1−µ ⎢0 0 3 8⎥ ⎢0 0 ⎥ ⎣ ⎦ ⎣ 2⎦ () or ⎡16 4 0⎤ [D] = 106 ⎢ 4 16 0⎥ ⎢ ⎥ ⎢ 0 0 6⎥ ⎣ ⎦ Chapter 23 Page 26 K. Haghighi then evaluate [B]T [D] and finally ⎡ ____ ⎤ ⎢ ____ ⎥ (see the ⎥ book!) k (e ) = tA[B]T [D][B] = 41667 ⎢ ⎢ ____ ⎥ ⎥ ⎢ ⎣ ____ ⎦ 6×6 The thermal force vector, [f (e )]= [B]T [D]{εT}tA Chapter 23 Page 27 K. Haghighi where ⎧αδT⎫ ⎧60⎫ ⎪ ⎪ −6 ⎪ ⎪ {ε T } = ⎨αδT⎬ = 10 ⎨60⎬ ⎪0⎪ ⎪0⎪ ⎩ ⎭ ⎩⎭ ⎧ − 600 ⎫ and finally ⎪ − 200 ⎪ ⎪ ⎪ (e ) = ⎪ 600 ⎪ f ⎨ ⎬ ⎪ − 200 ⎪ ⎪0⎪ ⎪ ⎪ ⎩ 400⎭ {} Chapter 23 Page 28 K. Haghighi Example: Calculate the equivalent set of conc. forces acting @ nodes 10, 12 & 18. pn = 5000 N / cm2 18 1.5 2 cm Surface stress and elements along an exterior boundary. (1) 12 t = 2.50 cm ( 2) 10 60 Chapter 23 Page 29 K. Haghighi Solution: 1 ⌠ T ⎧p x ⎫ Use ⎮ [N] ⎨ ⎬dΓ ⎩p y ⎭ ⌡ Γ ⎧0⎫ ⎪0⎪ ⎪⎪ tL jk ⎪p x ⎪ ⎪⎪ = ⎨p ⎬ 2 ⎪ y⎪ ⎪p x ⎪ ⎪⎪ ⎪p y ⎪ ⎩⎭ Chapter 23 Page 30 K. Haghighi 2 Use physical interpretation of 1 (easier & faster!) ∴ calc. total force acting on the side of the elem., then divide bet. the 2 connecting nodes equally. or and force(1) = PnA = PntL = 5000( 2.5)(1.5) = 18750 N force(2 ) = PnA = PntL = 5000( 2.5)( 2) = 25000 N Chapter 23 Page 31 K. Haghighi dividing these forces between the nodes; 9375 9375 18 4688 12 8119 10938 18 12 9375 12500 18 12500 10 (a) 21875 12 12500 1...
View Full Document

Ask a homework question - tutors are online