notes_16_torsion_prop

notes_16_torsion_prop - Lecture 6 - 2003 Torsion Properties...

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Lecture 6 - 2003 Torsion Properties for Line Segments and Computational Scheme for Piecewise Straight Section Calculations this consists of four parts (and how we will treat each) A - derivation of geometric algorithms for section properties (cover quickly for sense of approach) B - derivation of first moment approach (for info - not covered) C - computational routine resulting from A (demo a few examples - routine available in lab) D - computational routine resulting from B (routine available in lab) sourced from section 6.1 to 6.3 of Kollbrunner, Curt Friedrich, Torsion in structures; an engineering approach TA417.7.T6.K811 1966, and geometry. starting point: a line defined by two points, x1,y1 and x0,y0 this assumes X.cg and Ycg known y y 0 y 1 y 0 = line passing through two points xx 0 x 1 x 0 y 1 y 0 ( y 1 y 0 ) x 1 x 0 x 1 x 0 y = x 1 x 0 x + y 0 x 0 ( x 1 x 0 ) or . .. x = y 1 y 0 y + x 0 y 0 y 1 y 0 consider calculation of increment of moment of inertia (relative to centroid) b dx x b t x 1 2 ⋅ y dx y 2 t ds ds = () s = length = y 2 t ds = 0 cos α cos α 0 cos α x 0 x 1 2 x 1 x 0 y 1 y 0  simplify 3 y 0 2 + y 1 y 0 + y 1 2 ( x 1 x 0 ) y 1 y 0 x + y 0 x 0 x 1 x 0  dx factor 1 x 0 b t x 1 0 cos α y 2 dx = t ( x 1 3 x 0 ) y 1 2 y 2 t ds = cos α 2 + y 0 y 1 + y 0 x 0 I x = ts 1 s 0 ) y 1 2 ( 2 + y 0 y 1 + y 0 3 similarly (by the symmetry of the expression for the line above): I y = 1 s 0 ) x 1 2 ( 2 + x 0 x 1 + x 0 3 cross moment of inertia b t x 1 t y 1 I xy = xy t ds = cos α xy dx = 0 sin α xy dy x 0 y 0 1 notes_15_torsion_prop_calc.mcd
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x 1 x 1 y 1 y 0 y 1 y 0  xy dx = x 1 x 0 x + y 0 x 0 x 1 x 0  x dx x 0 x 0 x 1 y 1 y 0 y 1 y 0 simplify 1 x 1 x 0 x + y 0 x 0 x 1 x 0 x dx factor 6 ( x 1 x 0 ) ( 2x 1 y 1 + y 0 x 1 + x 0 y 1 + 2y 0 x 0 ) x 0 t x 1 t x 1 x 0 I xy = cos α () ( ( 6 + x ( 0 y 1 + x 1 y 0 x 0 = 6 cos α 1 y 1 + x 0 y 0 ) + x 0 y 1 + x 1 y 0 = ts 1 s 0 )  1 y 1 + x 0 y 0 ) ...  we could calculate Iyx using the same relationship but we know it is = Ixy I yx = I xy to evaluate the warping relationships: start with line passing through two points and obtain normal form of line y y 0 y 1 y 0 y 1 y 0 ( y 1 y 0 ) = xx 0 x 1 x 0 or . ..
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notes_16_torsion_prop - Lecture 6 - 2003 Torsion Properties...

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