Beam Deflection

Beam Deflection Lecture Notes Introduction Engineers must look for better ways to build structures. Less material typically means that structures will be lighter and less expensive. Knowing the moment of inertia for different shapes is an important consideration for engineers as they strive to make designs lighter and less expensive. Equipment • • • • • • • 2x4 (preferably straight, free of knots and imperfections) Dial calipers or a ruler with 1/32 divisions 2 - 1 foot lengths of 2x4 for use as supports Tape measure Permanent marker Floor scale Cinder block (Concrete Masonry Unit) Procedure You will determine the weight of one of your classmates using nothing more than a standard 2x4 and a measuring device. This activity will provide you with a better understanding of Moment of Inertia and how it can be used to determine the strength of beams. Preliminary lab calculations to determine beam Modulus of Elasticity 1. Calculate beam Moment of Inertia Position the beam as shown below. 2. Measure the span between the supports. Record your measurement below. Total Span (L) = 96 in. 3. Measure the distance between the floor and the bottom of the beam. Pre-Loading Distance ( 𝐷𝑃𝐿 ) = 8.5 in. (measured from middle) 4. Position a volunteer (V1) to stand carefully on the middle of the beam. Have a person on either side of the beam to help support the volunteer. Measure the distance between the floor and the bottom of the beam. Applied Load Distance ( 𝐷𝐴𝐿 ) = 7.2 in. (measured from middle) 5. Calculate the maximum beam deflection (MAX). △𝑀𝐴𝑋 = DPL – DAL △𝑀𝐴𝑋 = 1.3 in. 6. Determine the weight of volunteer (𝑉1) using the classroom floor scale. Volunteer weight (F) 150 lb 7. Calculate your beam’s Modulus of Elasticity (it is important to know that each beam will have its own specific Modulus of Elasticity) by rearranging the equation for beam maximum deflection to isolate (E). Show all work. Note: An object’s Modulus of Elasticity is a material-based property and stays the same regardless of orientation. Calculate volunteer ( 𝑽𝟐 ) weight 8. Position the beam as shown below. 9. Measure the span between the supports. Record your measurement below. Total Span (L) = 96 in. 10. Measure the distance between the floor and the bottom of the beam. Pre-Loading Distance ( 𝐷𝑃𝐿 ) = 9.5 in. (measured from top) 11. Position a second volunteer ( 𝑉2 ) to stand carefully in the middle of the beam. Have a person on either side of the beam to help support the volunteer. Measure the distance between the floor and the bottom of the beam. Applied Load Distance ( 𝐷𝐴𝐿 ) = 8 in. (measured from top) 12. Calculate the maximum beam deflection ( △𝑀𝐴𝑋 ). △𝑀𝐴𝑋 = 𝐷𝑃𝐿 – 𝐷𝐴𝐿 △𝑀𝐴𝑋 = 1.5 in. 13. Calculate volunteer ( 𝑉2 ) weight by rearranging the equation for maximum deflection to isolate (F). Show all work. Determining Beam Deflection 14. Using the information you collected and calculated in steps 1 – 14, calculate the max deflection of the beam if volunteer (V2) is positioned to stand on the beam in a vertical orientation. 15. Verify your calculated max deflection answer and work to your instructor by having volunteer ( 𝑉2 ) carefully stand in the middle of the beam. Place a person on either side of the beam to help support the volunteer. Measure the distance between the floor and the bottom of the beam. Calculated deflection: .276 Measured deflection: 11.25-10.75=.5 Practice Problem 16. Complete the chart below by calculating the cross-sectional area, Moment of Inertia, and beam deflection, given a load of 250 lbf, a Modulus of Elasticity of 1,510,000 psi, and a span of 12 ft. Show all work in your engineering notebook. Conclusion 1. Using Excel, create a Deflection vs. Moment of Inertia graph. What is the relationship between moment of inertia and beam deflection? Moment of inertia and beam deflection are inversely proportional, which means that their product stays the same (the graph will look like a hyperbola). 2. How could you increase the Moment of Inertia (I) of a beam without increasing its crosssectional area? Keep the area the same while you make the height bigger and the width smaller. The ratio of height to width should always be the opposite of what it is.