Question

Tables of Second moment of inertia and mass moment of inertia are provided at the end of the assignment. Be sure that you are looking at the right table. Mass moments have units of mass length2, second moment of inertia have units of length Problem 1 (10 pts) A floor joist has a 1" hole drilled in it to accommodate a water pipe. (see Figure at left). Compare the 2nd moment of inertia of the beam about the x axis ify = 0 (the hole is in the middle of the beam) vs y 2". (Compare means to quantify the effect of the change, one way to do this is to find the percent increase/decrease of I with the change in y) 4" y 4* Section A

Answer 1

1 (a) y: 0 Hole is located centrally in middle of beam. " beam The cross-section of beam is divided into two parts. 1. Solid Rectangle of 2"X8" 2. Hole of " diameter Al moment of Inertia's second moment of inertia) are about x-axis. Ij = bode - 2x80 - 86.333 in4 I body type 0.1666 1n4 I C40) = I, - I2 = 85.333 -0.1666 = 85.1664 in 4 (b) y=2" the The axis of hole is a" away from Beam axis (1-axis). Moment of Inertia of Solid Rectangle, Tie bed - 2283 - 85.333 in4 Moment of Inertia of Hole Hollow Rectangle section) Iz = bd3 + Ah? (parallel axis theorem) Ix = bd3 tah² b=2" d=1" A-bd = 2x1 = 2 in2 h=y=2" Iz = 2x12 + 2(a)? = 8.1666 iny Icy=2) = I, -I, = 85,333 - 8.1666 = 77.166 in4 Iyzo - F0952x100 = Iy=2" % decrease in I with change iny from yoo to y =2", Tyzo 85.1664 - 77.166 85.1664 9.393% =