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lecture_7a - Torsion Properties for Line Segments and...

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Y x2 y2 x1 area3 x0 x1 x1 Torsion Properties for Line Segments and Computational Scheme for Piecewise Straight Section Calculations Closed Thin walled Sections the new material consists of the "corrections for ω and Q ω A = enclosed area definition of 4 A 2 J = J 1 ω c = h c ds J s 1 ds = 2 A s 1 ds as b 2 A t b t 1 ds d c = h c 2 A t ds = d ω c 0 1 ds 0 t t 0 0 b 1 circ_integral = ds = X i ( 2 Y i ( 2 + ) ) = l i if we define l i = 0 t i t i t i segment X i ( 2 Y i ( 2 + ) ) i length: we now need calculation of the enclosed area A in this expression area of triangle determined by two points and the origin: area = area1 + area2 area3 y area x0,y0 x1,y1 y x0,y0 x1,y1 y area1 area2 x0,y0 x1,y1 y area3 x0,y0 x1,y1 0,0 x 0,0 x 0,0 x 0,0 1 1 1 area1 := 2 y1 area2 := (y0 + y1) (x0 ) area3 := 2 y0 ( ) 2 area := area1 + area2 1 1 1 area simplify y0 x1 + 2 y1 x0 area_2_pts_origin := (y1 x0 y0 ) 2 2 area between three points: x0,y0 x1,y1 x2,y2 area 1 1 area := (y1 x0 y0 x1) (y1 x2 x1) 2 2 1 1 area := (y1 x0 y0 x1) + (y2 x1 y1 ) 2 2 1 etc ......... area_enclosed := 2 ( X i i 1 X i 1 Y i ) + + i 1 notes_15a_tor_prop_clsd_calc.mcd x
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for a straight line segment ρ c = constant ∆ω c = ρ c L 2area_enclosed l i and is linear along line circ_integral t i
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