# Beam Deflection

**Topics:**Beam, Second moment of area, Vermiform appendix

**Pages:**13 (3819 words)

**Published:**April 24, 2012

Lab Group - 1st Mondays, Late: Jesse Bertrand, Ryan Carmichael, Anne Krikorian, Noah Marks, Ann Murray Report by Ryan Carmichael and Anne Krikorian

E6 Laboratory Report – Submitted 12 May 2008 Department of Engineering, Swarthmore College

Abstract:

In this laboratory, we determined six different values for the Elastic Flexural Modulus of a 4-by10 (100” x 3.50” x 9.46”) Parallam wood-composite test beam. To accomplish this, we loaded the beam at 1/3 span with 1200 psi in five load increments in both the upright (9.46 inch side vertical) and flat (9.46 inch side horizontal) orientations. We then used three different leastsquare methods (utilizing Matlab and Kaleidagraph) on the data for each orientation to fit the data, resulting in the following:

E: Upright Orientation Units Method One Method Two Method Three

E: Flat Orientation 10 ksi 103 ksi

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0.981 ± 0.100 1.253 ± 0.198 1.065 ± 0.247

1.880 ± 0.046 2.080 ± 0.083 1.881 ± 0.106

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Purpose:

The purpose of this lab is to determine the flexural elastic modulus of a Parallam woodcomposite beam by examining its behavior when simply supported and under flexural stress, and to analyze deflection data using different least-squares methods to fit theoretical deflection curves.

Theory:

In theory, a beam’s deflection can be mapped by the governing equation of beam flexure: EI d2y/dx2 = M(x), where E is the elastic modulus, I is the second moment of inertia about the neutral axis of the beam (the value of which changes significantly according to orientation), y is deflection, and M(x) is bending moment in the beam. This equation requires that several assumptions be made about the beam: 1) Geometric Assumption: the beam must be a straight, prismatic member with at least one axis of symmetry. 2) Material assumption: the beam must be linear, elastic, isotropic, and homogeneous, and the modulus of elasticity in tension must equal the modulus of elasticity in compression. 3) Loading Assumption: the beam must be loaded in pure moment in a plane of symmetry.

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4) Deformation Assumption: plane sections before bending must remain in plane after bending. Making these assumptions, we can apply the general equation for beam flexure to our experiment. Assuming we are using point loads or can model our setup with point loads, we can then use singularity functions to determine that the bending moment of the beam is: 2/3 P*x – P 1

Where P is the load applied with the UTM, L is the length of the beam, and x is the distance from the origin (defined as the end closest to the applied load). From this we get: M(x) = EI d2y/dx2 = 2/3 P*x – P 1 Taking an integral of both sides with respect to x yields: where c1 is a constant. Taking another derivative yields: where c2 is a constant. Rearranging we get: . EI dy/dx = P/3 * x2 – P/2 * 2 +c1

y * EI = P/9 * x3 – P/6 * 3 +c1x + c2

y = Px3/9EI – P/6EI * 3 +c1x/EI + c2/EI

To solve for the constants we need to make two more assumptions: that when x=0 and when x=L there will be no deflection (i.e. y=0). Using these assumptions, we can plug into our previous equation and use algebra to determine that c1 = -5PL2/81 and that c2 = 0. This gives us:

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y = P/EI (x3/9 - 3 /6 – 5L2x/81)

This is the theoretical beam deflection equation for the lab. Then, to ease calculations, we make the previous equation non-dimensional by multiplying both sides by EI/PL3, which yields: yEI/PL3 = (x/L)3/9 - 3/6 – 5/81 (x/L) We define this dimensionless quantity as: (x/L)3/9 - 3/6 – 5/81 (x/L) = !theoretical where: !theoretical = f(x/L) Similarly, we define: ymeasured * EI/PL3 = !measured.

If the beam were to behave as a theoretical beam, then !theoretical would equal !measured. E is defined as the slope of the stress-strain curve in the elastic region. However, there is no perfect way to measure stress and strain in the loaded beam. As a result, to determine E one must make some...

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