1. “SLENDER” BEAM THEORY L D y Bring: yardstick, dial calipers, string
Theory holds for slender beams and small deflections. “Slender” means L/D > 10. “Small Deflection” means y/L < 100. L
Bring: yardstick, dial calipers, string
Beam deflects a distance y from neutral for applied load D

2. Galileo’s 1638 loaded beam theory (he got the equations wrong)
This is Galileo’s famous illustration of a loaded beam from Dialogues on Two New Sciences (1638). Galileo found this beam could support twice the load at L/2 and that fracture resistance goes as h^3. Galileo did not analyze bending and completely ignored bending. That came with Hooke in Bernoulli and Euler developed bending theory further and Coulomb put it all together. Slender beam theory is one of the triumphs of mechanical engineering and is simple and effective.

3. WHAT MATTERS 1. Load 2. Elasticity (material)
3. Geometry (shape and length)
4. Boundary conditions (how beam is held)

4. Equation for cantilever beamP y L
load
geometry
elasticity
boundary conditions

5. Geometry y
I is moment of inertia, taken about neutral axis. Neutral axis is where beam has no strain. P h b D P
I = moment of inertia (“second moment), taken about neutral axis.
Neutral axis is where beam has no strain.
First moment about neutral axis is zero, integral(ydA) = 0. This is how neutral axis is defined.

6. Elasticity (material)
E is elasticity in stress/strain = (force/area)/(length-change/length)
Wood E is 1-2x10^6 psi
All steels have about same E. No point in using high-strength steel to minimize deflections
Note cost of plastic. That coupled with ability to mold is what makes plastic popular
See Appendix C (p993) of Norton for more material properties

7. Boundary conditions Free Simple Pin Clamp
Free: no constraints on motion or slope
Simple: allows horizontal motion, no vertical motion Pin: motion fixed, any slope
Clamp: no motion, zero slope
Can approximate most real situations by combination of these two.
Clamp

8. Yardstick expt P slope = y/P y y P Simply supported
Simply supported (stick can slide on supports)
E is elasticity of wood, what you have to find
For plot, plot all data points, then fit line to all. Don’t average first, let fit do the averaging
Plot all data from both runs, then fit line