Lab 7 Exercise: Beam Bending Under 3-Point Loading
Bend testing (also flexural testing) is commonly performed to measure the flexural strength and modulus of all types of materials and products. Here we apply the load perpendicular to the longitudinal axis of the specimen to compare beams of different x-sectional geometries. This is important because for a given application one must choose a material and a beam geometrysuch that it does not bend significantly when supporting the loads required for the application. This experiment uses a universal testing machine and a three point bend fixture to bend the beam and acquire load vs. deflection data needed to calculate the properties of the beam.
Test Procedure:
1. Produce beams of different x-sectional geometry for comparison. Measure the samplegeometry carefully, including span length (L) between supports and calculate moment of inertia about z.
2. Adjust the support span length determined by your sample size (here L will be 28cm).
3. Place the test beam on the 3 point bend fixture and begin the test, collecting load vs. deflection data.
4. End the test after bending to 5% strain or until the sample breaks.
5. Answer the Questions below.
Questions to Answer:
1) Under 3-point bending, what is the maximum shear force (V) and bending moment (M) for a point load applied in the middle of the beam? Show free body diagrams and equilibrium equations (solve for all reactions). Draw shear force and bending moment diagrams for V and M (be sure to label max/min values as a function of P and L)!(25 pts)
2) Calculate the moment of inertia (Iz) and section modulus (S) for the two simple beam configurations below (measure the width and thickness of each board for composite beam). Then calculate Iz and S for YOUR beam –be sure to show all steps using the parallel axis theorem (see lecture notes and Ch 10)! (25 pts)
Vertical
Orientation
Horizontal
Orientation
Your Beam
3) From the class data collected, fill out Chart below (Pmax, δmax, and m for ALL Beams; remaining columns fill out just for the 1 horizontal, 2 vertical, and YOUR beam.(20 pts)
Beam Flexural Strength Pmax (N) δmax
(m) m (slope, N/m) Vmax
(N) Mmax
(Nm) ymax
(m) Iz
(m4) S
(m3) σxmax
(MPa) εf
(m/m)
1horizontal
2vertical
Beam 3
Beam 4
Beam 5
Beam 6
4) From the tabulated data above, determine which has the best properties in terms of flexural strength(Pmax) and flexural stiffness proportional to slope m.Did the result surprise you? How did the results correlate with S?(10 pts)
5) According to the flexure formula in Chapter 5, the normal stress is given by:
where M is bending moment, y is distance along specimen height, and I is moment of inertia. When/where is this a maximum (on the beam’s length and x-section)? Comparethe stress withstood by each beam. (10 pts)
6) Calculate the maximum flexural strain for each (prior to failure) using the formula:
where δ is deflection at the beam center (prior to failure) and L is the span length of the beam supports and h is the height of the beam.Which beam deformed the most? Rank them from highest to lowest. (10 pts)
*Bonus: Calculate Young’s modulus E using the formula: . Does this vary widely for each of the beams? Does this match literature values for Balsa wood? Why might there be variation or discrepancy?
