CE 312 Lecture 26: Beam/Frame Deflections IV, Virtual Work Tips/Tricks (2026.10.19)

Overview

Gregory Michaelson presents four ways to simplify virtual-work calculations for beam and frame deflections: exploit symmetry, place coordinates at free ends, cut sections toward the coordinate origin, and use integrals with a zero lower bound. He demonstrates the methods with a symmetric variable-stiffness beam and a full example that finds 0.419 in downward deflection at C and 0.141° counterclockwise rotation at E.

Key takeaways

For a symmetric beam, symmetry can halve the integration work only when geometry, loading and boundary conditions, flexural stiffness, and the point of interest are all symmetric.
Placing the coordinate origin at a cantilever's free end can eliminate fixed-support reactions from section equations and simplify moment functions.
A section cut toward the coordinate origin generally avoids distant loads and reactions, reducing polynomial expansion while preserving the same internal moment.
Using a separate coordinate system for each integration region can replace nonzero lower bounds with zero, simplifying hand evaluation of polynomial antiderivatives.
Virtual-work unit conversions differ by response: use 1728 for deflection when moments use feet and EI uses inches, and use 144 × 180/π for rotation in degrees.
The full example combines these techniques to calculate 0.419 in downward deflection at C and 0.141° counterclockwise rotation at E.

Chapters

0:00 Course Updates: Homework 6, MASTAN2, and Exam Preparation

Attendance grades are current and homework has been graded through 6.1; homework 6.2 is due Wednesday, while 6.3 and 6.4 are due Friday.
Homework 6.3 requires a screenshot showing MASTAN2 installed, and unusually large deflection or rotation results on 6.2 may be correct.
The second exam is scheduled for Monday; the class plan is virtual-work tips, frame deflections, shear deformation, and a Friday review.

1:45 Virtual Work Setup and Four Strategies for Easier Beam Calculations

Beam and frame deflections use integrals of the real and virtual moment functions, M and m, divided by flexural rigidity EI.
When moment functions use feet and EI uses inches, apply a factor of 1728 for deflection; rotation uses 144 and 180/π to convert radians to degrees.
Michaelson's four strategies are exploiting symmetry, locating the coordinate origin at a free end, cutting sections toward the origin, and using zero lower integration bounds.

5:13 Exploit Symmetry to Reduce Virtual-Work Integrals

The example is a simply supported beam with an 8 kip midspan load and variable moment of inertia, so both moment functions and EI change across the span.
Without symmetry, the calculation requires four integrals to account for different moment-function and EI regions.
When geometry, loading, flexural stiffness, and the point of interest are symmetric, calculate the two integrals on one half of the beam and double their sum.
The symmetry method reduces repeated algebra and integration, especially for real structures such as bridges with variable depth.

13:11 Place the Coordinate Origin at a Cantilever's Free End

For a 15 ft cantilever carrying 2.5 kip/ft, placing the origin at the free end gives compact moment functions: M = -1.25x² and m = -x for the tip-deflection calculation.
With the origin at the fixed end, section equations must include support reactions and produce longer polynomial expressions.
Both coordinate choices yield the same physical answer, but the free-end origin makes the virtual-work algebra substantially simpler.

18:11 Cut Sections Toward the Coordinate Origin to Simplify Equilibrium

For the same 15 ft cantilever, a section cut looking toward the origin requires only the nearby distributed load and gives a compact moment equation.
Looking away from the origin introduces the fixed-end reactions and terms involving 15 − x, creating additional polynomial expansion.
Either side of a section cut gives the same internal moment; deriving both can serve as a check, but the origin-facing side usually requires less algebra.

24:13 Use Multiple Coordinate Systems for Zero-Bound Integrals

For an overhanging beam with a pin and an inset roller support, separate integration regions are unavoidable, but each region can use its own coordinate origin.
Integrating a polynomial from 0 to a distance avoids evaluating the antiderivative at a nonzero lower bound.
A 46 ft beam with a support 30 ft from one end can use ranges 0 to 30 and 0 to 16 instead of 0 to 30 and 30 to 46.
When a coordinate axis is flipped, a point-slope equation retains the slope magnitude but reverses its sign.

29:40 Complete Example: Deflection at C and Rotation at E

The example uses three real-moment regions, A to B, B to D, and E to D; coordinates start at free ends A and E to simplify the equations and integration bounds.
For vertical deflection at C, a unit load produces two nonzero virtual-moment regions in the middle; the end regions contribute zero, giving 0.419 in downward.
For rotation at E, a unit moment produces zero moment from A to B, a linear function from B to D, and a constant unit moment from E to D.
The rotation calculation uses 144 × 180/π with the summed integrals and gives 0.141° counterclockwise.

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.