CE 312 Lecture 26b: Hints on HW 6.2, P1 (2026.10.20)

Overview

Gregory Michaelson develops the two real moment functions for CE 312 Homework 6.2, Problem 1 by cutting the beam and taking moments of the left-hand free body. A 1 kip/ft distributed load gives the shared term −0.5x², while crossing the 30-kip point load creates a second expression valid from x = 10 to 30; continuity at x = 10 provides a check on the equations.

Key takeaways

A concentrated load at x = 10 divides the beam into two regions with different left-side free-body diagrams, requiring two real moment functions.
A uniform load of 1 kip/ft over a length x is replaced by a resultant of x kips acting x/2 from the origin, producing a moment contribution of 0.5x².
The 30-kip load contributes a moment based on its distance from the section cut, x − 10, not the full coordinate x.
Using the stated sign convention, the piecewise functions are M₁(x) = −0.5x² for 0 ≤ x ≤ 10 and M₂(x) = −0.5x² − 30(x − 10) for 10 ≤ x ≤ 30.
Because there is no concentrated moment at x = 10, the bending-moment diagram must be continuous there; both expressions give −50 kip-ft at that point.

Chapters

0:00 Why the Beam Needs Two Moment Functions

Move a section cut from x = 0 to x = 30 while analyzing the left side of the beam.
The loading changes after the cut passes the 30-kip point load at x = 10, so the beam requires separate sections over 0 to 10 and 10 to 30.
Michaelson cautions that the resulting deflections may be relatively high, which is not by itself a reason to reject the final answer.

2:20 Section 1: Replace the 1 kip/ft Load with a Resultant

For section 1-1, the cut lies between x = 0 and x = 10, and the left free body contains only the distributed load.
Over a length x, the 1 kip/ft load has a resultant of 1.0x kips located x/2 from the origin.
Taking moments at section 1-1 gives M₁(x) = −0.5x², valid for 0 ≤ x ≤ 10.

4:00 Section 2: Include the 30-kip Load and Check Continuity

For section 2-2, x is still measured from the origin, but the cut is beyond the point load, so x > 10.
The distance from the 30-kip load at x = 10 to the cut is x − 10; taking moments gives M₂(x) = −0.5x² − 30(x − 10), valid for 10 ≤ x ≤ 30.
At x = 10, both functions equal −50 kip-ft, confirming that the moment diagram has no jump where the two regions meet.
Plotting the expressions in Excel is another suggested way to check that they match the hand-drawn moment diagram.

Summary, takeaways, and chapters were generated by AI from the video's transcript and may contain errors. The video belongs to its creator, Gregory Michaelson.