Elastic Equations

631 days ago by BSkaggs

This worksheet is intended to be a quick way to calculate the deflection of a specific cross section of a beam via its elastic equation.  Throughout the worksheet E represents the material's Modulus of Elasticity and I represents the Moment of Inertia.  A note of caution is that it is the user's responsibility to insure that the corrects units are being used in order to obtain accurate results.

Cantilever beam of length (L) with a concentrated force (P) located at its free end. 

@interact def elasticCurve(L = input_box(default = 1), P = input_box(default = 1), E = input_box(default = 1), I = input_box(default = 1),Point_of_Interest = input_box(default = 0)): y = (P/(6*E*I))*(x^3-3*L*x^2) eplot = plot(y,(x,0,L)) d = (P/(6*E*I))*((Point_of_Interest)^3-3*L*(Point_of_Interest)^2) print'Deflection at Point_of_Interest is: ',d.n(digits = 4) show(eplot) 
       

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Cantilever beam of length (L) with a uniformly distributed load directed downward along its entire length of magnitude w.   

@interact def elasticCurve2(L = input_box(default = 1), w = input_box(default = 1), E = input_box(default = 1), I = input_box(default = 1),Point_of_Interest = input_box(default = 0)): y = (-w/(24*E*I))*(x^4-4*L*x^3+6*L^2*x^2) e2plot = plot(y,(x,0,L)) d = (-w/(24*E*I))*(Point_of_Interest^4-4*L*Point_of_Interest^3+6*L^2*Point_of_Interest^2) print'Deflection at Point_of_Interest is: ',d.n(digits = 4) show(e2plot) 
       

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Cantilever Beam of length (L) with a moment (M) located at its free end. The moment is in a direction that results in the free end of the beam moving down. 

@interact def elasticCurve3(L = input_box(default = 1), M = input_box(default = 1), E = input_box(default = 1), I = input_box(default = 1),Point_of_Interest = input_box(default = 0)): y = -M*x^2/(2*E*I) e3plot = plot(y,(x,0,L)) d = -M*Point_of_Interest^2/(2*E*I) print'Deflection at Point_of_Interest is: ',d.n(digits = 4) show(e3plot) 
       

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Simply-supported beam of length (L) with a concentrated force (P)  at its midpoint.   Due to the symmetry of the situation, a cross section at equal distances from the midpoint will have equal deflections.  Therefore, the calculations used in this code are only valid for Points_of_Interest less than or equal to one-half the entire length.
@interact def elasticCurve4(L = input_box(default = 1), P = input_box(default = 1), E = input_box(default = 1), I = input_box(default = 1),Point_of_Interest = input_box(default = 0)): y = (P/(48*E*I))*(4*x^3-3*L^2*x) e4plot = plot(y,(x,0,L/2)) d = (P/(48*E*I))*(4*Point_of_Interest^3-3*L^2*Point_of_Interest) print'Deflection at Point_of_Interest is: ',d.n(digits = 4) print'Plot only displays first half of beam' show(e4plot) 
       

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Simply-supported beam of length (L) with a uniformly distributed load along its entire length of magnitude w. 
@interact def elasticCurve5(L = input_box(default = 1), w = input_box(default = 1), E = input_box(default = 1), I = input_box(default = 1),Point_of_Interest = input_box(default = 0)): y = (-w/(24*E*I))*(x^4-2*L*x^3+L^3*x) e5plot= plot(y,(x,0,L)) d = (-w/(24*E*I))*(Point_of_Interest^4-2*L*Point_of_Interest^3+L^3*Point_of_Interest) print'Deflection at Point_of_Interest is: ',d.n(digits = 4) show(e5plot) 
       

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Simply-supported beam with a length (L) and a moment (M) located above the roller on the right end of the beam.  A positive value for this moment is in the counterclockwise direction about the roller. 

@interact def elasticCurve6(L = input_box(default = 1), M = input_box(default = 1), E = input_box(default = 1), I = input_box(default = 1),Point_of_Interest = input_box(default = 0)): y = (-M/(6*E*I*L))*(x^3-L^2*x) e6plot = plot(y,(x,0,L)) d = (-M/(6*E*I*L))*(Point_of_Interest^3-L^2*Point_of_Interest) print'Deflection at Point_of_Interest is: ',d.n(digits = 4) show(e6plot) 
       

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Cantilever beam of length (L) with a roller support located beneath its free end.  The free end with roller is located at the right end of the plot.  The beam is loaded with a uniformly distributed load throughout its entire length of magnitude w.   

@interact def elasticCurve7(L = input_box(default = 1), w = input_box(default = 1), E = input_box(default = 1), I = input_box(default = 1),Point_of_Interest = input_box(default = 0)): y = -((w*x^2)/(48*E*I))*(3*L^2-5*L*x+2*x^2) e7plot=plot(y,(x,0,L),rgbcolor = 'green') d = ((w*Point_of_Interest^2)/(48*E*I))*(3*L^2-5*L*Point_of_Interest+2*Point_of_Interest^2) print'Deflection at Point_of_Interest is: ',d.n(digits = 4) show(e7plot) 
       

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