Solving beam deflection problems using the moment-deflection approach and using the Euler-Bernoulli approach

Transcription

1 Solving beam deflection problems using the moment-deflection approach and using the Euler-Bernoulli approach by Nasser M. Abbasi November 9 Links PDF file Mathematica notebook Introduction These are problems in beam deflection showing how to use Mathematica to solve them. Problem 1 This is problem 9-3, page 551, from bok Problem Solvers, strength of materials and mechanics of materials by REA. I show here how to solve this problem using Mathematica. Start by setting up the moment deflection equation for the Euler beam E I y''x Mx, this equation is found for both halves of the beam resulting in solutions, y 1 x and y x. On the right side, boundary condition is that y L and on the left side y 1, then we need an additional boundary conditions, for this, use the continuity conditions at L where we set y 1 L y L and y' 1 L y' L and now we have the complete solutions. For the left half of beam, it is easily found that Mx w L 8 x and for the right half Mx w l x 8 w x l Find y 1 x Clear,, w, L, eq1, eq, y1, y, M1, M

2 euler_beam.nb M1 w L 8 x; eq1 y ''x M1; sol1 First DSolveeq1, yx, x; y1 yx. sol1; y1 y1. C1 c1, C c c1 c x L w x3 48 Use first boundary condition y 1 First Solvey1. x, c1, c; y1 y1. Solve::svars: Equations may not give solutions for all "solve" variables. c x L w x3 48 Find y x M w L 8 x w x L ; eq y ''x M; sol First DSolveeq, yx, x; y yx. sol; y y. C1 c3, C c4 c3 c4 x L w x 16 5 L w x3 48 w x4 4 Use boundary condition y L First Solvey. x L, c3, c4; y y. ; y Collecty, c4 Solve::svars: Equations may not give solutions for all "solve" variables. c4 L x L w x 16 5 L w x3 48 w x4 4 We see from the above, that we have additional constants of integrations to solve for. c and c4. To solve for these, we use the continuity conditions at L sol First Solve y y1. x L, Dy, x Dy1, x. x L, c, c4 c 7 L3 w 384, c4 L3 w 384

3 euler_beam.nb 3 All the integration of constants now found, we can now plot moment diagrams, shear diagram, and deflection diagrams y1 y1. sol 7 L3 w x L w x y y. sol L w x 16 5 L w x3 48 w x4 4 L3 w L x 384 Assign some values for E, I, L and w and do the plots. PLot the deflection L 1; values 1 6,, w 1; Ploty1 HeavisideThetaL x. values, y HeavisideThetax L. values, x,, 1, Frame True, FrameLabel "yx", None, "x", "beam deflection"..5 yx

4 4 euler_beam.nb Plot the bending moment PlotM1 HeavisideThetaL x. values, M HeavisideThetax L. values, x,, 1, Frame True, FrameLabel "Mx", None, "x", "Bending moment" Mx Plot the shear diagram PlotEvaluateDM1, x HeavisideThetaL x, DM, x HeavisideThetax L. values, x,, 1, Frame True, FrameLabel "Vx", None, "x", "shear diagram" 1 Vx Solving the same problem directly from the Euler-Bernoulli 4 th order ODE The above approach (using the Moment-deflection ODE) is a standard approach to solve deflection beam problems. However, we can also use the 4th order Euler beam equation direclty as follows. One needs to make sure that the load on the RHS of this ODE is the load per unit length only, i.e. w in this problem. Notice the use of the unit step function since w only acts on half the beam.

5 euler_beam.nb 5 Clear,, w, L, eq, y eq y ''''x w HeavisideThetax L ; sol First DSolveeq, y, y L, y '', y ''L, yx, x; y yx. sol L4 w x DiracDeltaL 96 L 3 w x DiracDeltaL 64 L w x 3 DiracDeltaL 6 L 4 w HeavisideTheta L 1 L3 w x HeavisideTheta L 144 L w x HeavisideTheta L 48 L w x 3 HeavisideTheta L 4 L3 w x HeavisideTheta L 48 L w x3 HeavisideTheta L 6 L 4 w HeavisideTheta L x 48 L3 w x HeavisideTheta L x 144 L w x HeavisideTheta L x 19 L w x3 HeavisideTheta L x 96 w x 4 HeavisideTheta L x L5 w x DiracDelta L 3 L4 w x DiracDelta L L 3 w x 3 DiracDelta L L5 w x DiracDelta L L3 w x 3 DiracDelta L Assign some values for E, I, L and w and do the plots. PLot the deflection L 1; values 1 6,, w 1; Ploty. values, x,, 1, Frame True, FrameLabel "yx", None, "x", "beam deflection"..5 yx We see that we obtain the same result as earlier.

6 6 euler_beam.nb Plot the bending moment PlotEvaluate DDy, x, x. values, x,, 1, Frame True, FrameLabel "Mx", None, "x", "bending moment" Mx Plot the shear diagram PlotEvaluate DDDy, x, x, x. values, x,, 1, Frame True, FrameLabel "Vx", None, "x", "shear diagram" 1 Vx Problem In this problem. I set up a cantilever beam and first solved it using the Java beam applet that can be accessed here and then I solved this problem using Mathematica below, showing that I get the same answer as shown

7 euler_beam.nb 7 Clear,, w, L, eq, y, sol eq y ''''x w HeavisideTheta L x; The boundary conditions to use are: yl, y'', y''', y'l sol Simplify First DSolveeq, yl, y '', y ''', y 'L, yx, x; y yx. sol Out[8]= w 48 L4 64 L 3 x 16 x 4 8 L L x 5 L 4 x HeavisideTheta L L 3 7 L 8 x HeavisideTheta L L4 HeavisideTheta L x 8 L3 x HeavisideTheta L x 4 L x HeavisideTheta L x 3 L x3 HeavisideTheta L x 16 x4 HeavisideTheta L x

8 8 euler_beam.nb Assign same values as shown above for E, I, L and w and do the plots. In[17]:= L 1; values 8 1 6,, w 1; Ploty. values, x,, 1, Frame True, FrameLabel "yx", None, "x", "beam deflection" Print"Max deflection is ", y. values. x N, " inches" Out[19]= yx Max deflection is.1966 inches Plot the bending moment In[3]:= PlotEvaluate DDy, x, x. values, x,, 1, Frame True, FrameLabel "Mx", None, "x", "bending moment" Print"Max moment is ", DDy, x, x. values. x 1 N Out[3]= Mx Max moment is 37.5

9 euler_beam.nb 9 Plot the shear diagram In[5]:= PlotEvaluate DDDy, x, x, x. values, x,, 1, Frame True, FrameLabel "Vx", None, "x", "shear diagram" 1 Out[5]= Vx