Matrix_Methods

# Matrix_Methods - Matrix Methods(Notes Only MAE 316 Strength of Mechanical Components C Heeter Tran 1 Matrix Methods Stiffness Matrix Formation

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Matrix Methods (Notes Only) MAE 316 – Strength of Mechanical Components C. Heeter Tran Matrix Methods 1

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tiffness Matrix Formation Stiffness Matrix Formation ` Consider an “element”, which is a section of a beam with “node” at each end a node at each end. ` If any external forces or moments are applied to the beam, there will be shear forces and moments at each end of the element. ` Sign convention – deflection is positive downward. L 12 M 1 M 2 V 1 V 2 x y (+v) Matrix Methods 2 Note: For the element, V and M are internal shear and bending moment.
tiffness Matrix Formation Stiffness Matrix Formation ` Integrate the load-deflection differential equation to find expressions for shear force, bending moment, slope, and deflection. 0 4 4 = x v d EI dx V c dx v d EI = = 1 3 3 M c x c dx v d EI = + = 2 1 2 2 v 2 θ EI c x c x c dx dv EI = + + = 3 2 1 2 2 3 x x Iv Matrix Methods 3 4 3 2 1 2 6 c x c c c EIv + + + =

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tiffness Matrix Formation Stiffness Matrix Formation ` Express slope and deflection at each node in terms of tegration constants c nd c integration constants c 1 , c 2 3 , and c 4 . I c v v 4 1 ) 0 ( = = EI EI c dx dv 3 1 ) 0 ( = = θ + + + = = 4 3 2 2 3 1 2 2 6 1 ) ( c L c L c L c EI L v v + + = = 3 2 2 1 2 2 1 ) ( c L c L c EI L dx dv ote: nd eflection and slope) are the same in the element as for the Matrix Methods 4 Note: ν and θ (deflection and slope) are the same in the element as for the whole beam.
tiffness Matrix Formation Stiffness Matrix Formation ` Written in matrix form 1 0 0 0 I 1 1 2 1 0 1 0 0 θ v c c EI EI 2 2 4 3 2 2 3 1 2 6 v c c EI EI L EI L EI L 0 1 2 EI EI L EI L Matrix Methods 5

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tiffness Matrix Formation Stiffness Matrix Formation ` Solve for integration constants. 2 2 I I I I 1 1 2 3 2 3 2 1 2 6 4 6 6 12 6 12 θ v EI EI EI EI L EI L EI L EI L EI c c = 2 2 2 2 4 3 0 0 0 v EI L L L L c c 0 0 0 EI Matrix Methods 6
tiffness Matrix Formation Stiffness Matrix Formation ` Express shear forces and bending moments in terms of the constants. 1 1 ) 0 ( c V V = = 2 2 2 3 1 2 1 3 1 6 12 6 12 θ EI v EI EI v EI V + + = L L L L 2 1 ) 0 ( c M M = = 2 2 2 1 1 2 1 2 6 4 6 EI v EI EI v EI M + + = L L L L 1 2 ) ( c V L V = = 6 12 6 12 EI v EI EI v EI V + = 2 2 2 3 1 2 1 3 2 L L L L 2 1 2 ) ( c L c M L M = = 4 6 2 6 EI EI EI EI Matrix Methods 7 2 2 2 1 1 2 2 L v L L v L M + + =

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tiffness Matrix Formation Stiffness Matrix Formation ` This can also be expressed in matrix form. 1 1 1 1 2 2 2 6 4 6 6 12 6 12 M V v L L L L L L EI θ 2 2 2 2 2 2 3 4 6 2 6 6 12 6 12 M V v L L L L L L L `
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## This note was uploaded on 07/07/2011 for the course MAE 315 taught by Professor Wu during the Spring '08 term at N.C. State.

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Matrix_Methods - Matrix Methods(Notes Only MAE 316 Strength of Mechanical Components C Heeter Tran 1 Matrix Methods Stiffness Matrix Formation

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