Engineering question: Why does one beam bend more than another, even if they carry the same load?
By the end, you should explain beam deflection, make relative predictions and solve a straightforward central-load problem.
Finish additional problems or the fair-testing plan after class, as directed.
Imagine a wood strip spanning two desks. It sags when you place a weight in the middle. That movement is deflection.
Beam deflection is the displacement of a beam from its original position under load. It is usually measured in millimeters (mm).
A beam can bend and then return to its original position when unloaded. Elastic deflection is not automatically failure; excessive deflection can still prevent a design from functioning.
Stiffness: How strongly does a part resist deformation? A stiffer beam bends less under the same load.
Strength: How much stress can a part withstand before yielding, breaking or otherwise failing?
Important: Less deflection does not automatically prove that a beam has sufficient strength or is safe.
Downward point load at the center, measured in newtons (N).
Distance between the two supports, not the total stock length.
Maximum vertical displacement at the midpoint for this symmetrical load.
They keep the beam from falling while permitting the rotations assumed by a simply supported model.
A beam resting on two supports is not the same as a shelf rigidly mounted to a wall.
For today's numerical formula, the beam rests on supports at its two ends with one load at its midpoint. This is the idealized three-point bending setup.
Under a centered load and a symmetric beam, the upward support reactions are equal: RA = RB = P/2.
One end is fixed and the other is free. Under a load at its tip, maximum deflection is at the free end. Its formula is different from the simply supported case.
Do not mix formulas: first identify the supports, then the load arrangement.
Start with the formula-based example, then watch how beam orientation affects the stiffness. The third video is a deeper extension; its calculus derivations are not required in this lesson. If embedding is blocked at school, use the direct YouTube links.
Mark Mattson demonstrates how to select and apply standard beam-deflection formulas. Focus on the simply supported central-load cases.
Pause and do: Identify P, L, E, I and the result's units in one example.
Open video on YouTube ↗The Efficient Engineer explains why two beams with the same material and area can bend by very different amounts when oriented differently.
Pause and do: Draw the same rectangular strip flat and on edge, labeling which direction is stiffer.
Open video on YouTube ↗Extension: how engineers analyze different bending and deflection cases. The calculus methods shown go beyond the required POE calculations.
Open advanced video on YouTube ↗Why are boundary conditions important? Why does increasing EI make beams deflect less? Which of the methods in the video looks easiest to apply to a familiar problem?
The formulas below apply to a straight, prismatic, uniform, linearly elastic beam under small deflections and the support/loading assumptions shown. We use mm and N throughout.
P = applied central force (N)
L = distance between supports (mm)
E = elastic modulus, or material stiffness (N/mm²)
I = second moment of area of the cross-section (mm⁴)
δ = midpoint deflection (mm)
b = width of the section (mm), perpendicular to the bending direction.
h = depth in the direction the beam bends (mm).
The height is cubed. That makes the orientation of a rectangular beam especially important.
Units check: (N · mm³) ÷ [(N/mm²) · mm⁴] = mm.
| Change one variable | Deflection becomes | Why? |
|---|---|---|
| Double the force P | 2× as much | Directly proportional to load |
| Double the span L | 8× as much | L is cubed (2³ = 8) |
| Double material modulus E | ½ as much | E is in the denominator |
| Double width b; keep h fixed | ½ as much | I doubles |
| Double height h; keep b fixed | ⅛ as much | I increases 8× |
Each row is an idealized one-variable-at-a-time prediction. Changing multiple factors requires recomputing the formula.
A rectangular beam spans 400 mm between simple supports and carries a 30 N point load at its center. For practice, use an assumed modulus E = 10,000 N/mm². Its cross-section is 20 mm wide and 10 mm high.
For this first calculation, use the flat orientation: b = 20 mm, h = 10 mm. The second orientation is for comparison.
Use the same units for every term before substituting.
Reasonableness check: The value is small compared with a 400 mm span. It describes elastic behavior only—not whether the beam will fail.
Flat: b = 20 mm, h = 10 mm → I = 1,666.67 mm⁴ → δ = 2.40 mm.
On edge: b = 10 mm, h = 20 mm → I = 6,666.67 mm⁴ → δ = 0.60 mm.
Important distinction: Rotating the 20 × 10 mm rectangle makes I four times larger, reducing deflection to one-quarter. Doubling height without changing width would instead make I eight times larger.
More material lies farther from the cross-section's horizontal centroidal axis, increasing the second moment of area I. A higher EI means lower predicted deflection under the same force and span. The material and cross-sectional area are unchanged when the strip is rotated.
Change one slider at a time. The model calculates midpoint deflection for a simply supported, center-loaded rectangular beam. Use the graph to see trends—not the actual physical scale of bending.
Held constant: width b = 20 mm; modulus E = 10,000 N/mm². All quantities are assumed idealized teaching values, not a certified material specification.
At these settings, the calculated midpoint deflection is 2.40 mm.
You want to compare the same wooden ruler held flat and on edge between two supports. What must stay the same to make your comparison meaningful?
Do not use body weight, climb on structures or load school fixtures. Perform any physical testing only with instructor-approved materials and loads.
| Variable type | Example |
|---|---|
| Independent | Cross-section orientation: flat vs. on edge |
| Dependent | Measured center deflection (mm) |
| Controlled | Same material, piece, span, force, load location and measurement method |
Before the test: Which orientation do you predict will bend less? Use the I formula to justify it.
For a wide, thin rectangle rotated upright, the effective section height increases. The second moment of area about the bending axis is larger, so it usually bends less under otherwise identical conditions.
Write your prediction and any calculations in your 2.3 worksheet or notebook before opening each answer.
Beam B is the same as Beam A in all respects except its support span is twice as long. How many times greater is its theoretical deflection under the same centered load?
8 times greater. Since δ ∝ L³, doubling L produces 2³ = 8 times the deflection.
What happens if the central load doubles and E also doubles, while span and cross-section remain unchanged?
No change in the calculated deflection. Doubling P doubles the numerator, but doubling E doubles the denominator, so the effects cancel.
Using the worked example, calculate I for a 10 mm-wide, 20 mm-high section. How does its predicted deflection compare with the flat 20 × 10 mm section?
I = (10)(20³)/12 = 6,666.67 mm⁴. This is 4× the flat section's I, so its deflection is 0.60 mm, one-quarter of the original 2.40 mm.
For a simply supported beam: P = 20 N, L = 300 mm, E = 9,000 N/mm², b = 15 mm and h = 12 mm. Find I and δ.
I = (15)(12³)/12 = 2,160 mm⁴. δ = (20)(300³)/[(48)(9,000)(2,160)] = 0.579 mm (about 0.58 mm).
Can you use δ = PL³/(48EI) directly for a shelf fixed at one end and loaded at the free tip? Explain.
No. The formula assumes two simple supports and a centered point load. The wall-mounted shelf is a cantilever with different boundary conditions and a different formula.
A beam under a small weight bends 3 mm but returns to its original shape when the weight is removed. Has it necessarily failed?
No. Reversible deflection is consistent with elastic behavior. The amount could still be unacceptable for a design that has a strict stiffness/deflection limit.
These external links are optional study resources. The learning and worked exercises above are sufficient to complete the lesson.
Readable explanation of beam deformation, why engineers study deflection, and additional methods.
The Efficient Engineer — Deflection of Beams ↗Visual explanation of the second moment of area and why cross-section orientation changes stiffness.
The Efficient Engineer — Area Moment of Inertia ↗University-level reference showing the central point-load formula and how beam deflection is modeled.
Engineering LibreTexts — Beam Displacements ↗Explore I for rectangles and other cross-sections after you understand the rectangular formula.
The Efficient Engineer — Cross-Section Calculator ↗Complete the POE 2.3 Beam Deflection classwork and homework worksheet (or teacher-directed notebook equivalent). The guided resources and self-check answers are study aids; turn in your own calculations and reasoning.
Due date: Refer to the Daily Agenda for the current deadline. Follow the submission method provided by your teacher.
