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&[DATE] &[TIME] - &[FILENAME]

deflection of a cantilever beam with variable depth

beam depth profile h1 1 m * : h2 2 m * : l 6 m * : 31 sys

beam depth profile as a polyline polyline 0 h1 h2 h1 - h2 l h2 32 mat :

polyline

number of points nb 50 :

digitisation h l nb 1 - / : X h 0 nb 1 - (range * : 21 sys

X transpose m 0 0.1 0.2 0.4 0.5 0.6 0.7 0.9 1 1.1 1.2 1.3 1.5 1.6 1.7 115 mat

beam profile beam.depth Y nb 1 matrix : a 1 nb range Y a el polyline 1 col polyline 2 col X a el m / linterp m * : for Y 31 line :

X beam.depth augment

beam.depth transpose m 1 1.1 1.2 1.4 1.5 1.6 1.7 1.9 2222222222222121 mat

beam inertia (assuming width =1) beam.inertia Y nb 1 matrix : a 1 nb range Y a el beam.depth a el (3^12 / : for Y 31 line :

X beam.depth augment X beam.inertia augment 2 1 sys

beam.inertia transpose m 3^0.1 0.1 0.2 0.2 0.3 0.3 0.4 0.5 0.6 0.7 0.7 0.7 0.7 0.7 114 mat

moment in a cantilever with udl, and load per unit length=1 moment.udlcantilever Y nb 1 matrix : a 1 nb range Y a el l X a el - (2^2 / : for Y 31 line :

X beam.depth augment X moment.udlcantilever augment 2 1 sys

moment.udlcantilever transpose m 2^18 17.3 16.6 15.9 15.2 14.5 13.9 13.2 12.6 12 11.4 10.8 112 mat

derivative of the beam slope, assuming E=1 beam.d2 Y nb 1 matrix : a 1 nb range Y a el moment.udlcantilever a el beam.inertia a el / : for Y 31 line :

X beam.depth augment X beam.d2 20 / augment 2 1 sys

beam.d2 transpose 1 m / 216 146.6 103 74.5 55.1 41.6 31.9 24.8 19.5 18 17.1 16.2 15.4 14.6 13.8 13 12.2 11.5 10.8 10.1 120 mat

elastic modulus E 50 :

summation of a series Y summation YS nb 1 matrix : s 0 : a 2 nb range s s Y a 1 - el Y a el + 2 / + : YS a el s : 21 line for YS 41 line :

beam.slope h beam.d2 E / summation * :

X beam.depth augment X beam.d2 20 / augment X beam.slope augment 3 1 sys

beam.slope transpose 0 0.4 0.7 1 1.1 1.2 1.3 1.4 1.5 1.5 1.5 1.6 1.6 1.7 1.7 1.7 1.8 1.8 1.8 1.8 120 mat

beam.deflection h beam.slope summation * :

X beam.depth augment X beam.d2 20 / augment X beam.slope augment X beam.deflection augment 4 1 sys

beam.deflection transpose m 00 0.1 0.2 0.3 0.5 0.6 0.8 1 1.2 1.3 1.5 1.7 1.9 2.1 2.4 2.6 2.8 3 3.2 120 mat