This cheat sheet covers common cantilever and beam deflection formulas used in introductory structural, mechanical, and civil engineering courses. It helps students quickly match a loading case to the correct maximum deflection, slope, and bending moment relationship. These formulas are essential for checking stiffness limits, serviceability, and preliminary beam design.
The sheet assumes small deflections, linear elastic material behavior, and constant flexural rigidity EI.
Key Facts
- For a cantilever beam with an end point load P, the end deflection is delta_max = P L^3 / (3 E I) and the end slope is theta_max = P L^2 / (2 E I).
- For a cantilever beam with a uniformly distributed load w over the full length, the end deflection is delta_max = w L^4 / (8 E I) and the end slope is theta_max = w L^3 / (6 E I).
- For a simply supported beam with a central point load P, the maximum midspan deflection is delta_max = P L^3 / (48 E I).
- For a simply supported beam with a uniformly distributed load w over the full span, the maximum midspan deflection is delta_max = 5 w L^4 / (384 E I).
- Flexural rigidity is E I, where E is Young's modulus and I is the second moment of area about the neutral axis.
- The Euler-Bernoulli beam equation for constant EI is E I d^4y/dx^4 = w(x), where y is deflection and w(x) is transverse load intensity.
- Deflection scales strongly with span length because point-load deflection often scales with L^3 and distributed-load deflection often scales with L^4.
- Consistent units are required, so if E is in Pa, I is in m^4, loads are in N or N/m, and length is in m, deflection is in m.
Vocabulary
- Deflection
- Deflection is the transverse displacement of a beam from its original unloaded position.
- Slope
- Slope is the rotation of the beam's elastic curve, commonly written as theta = dy/dx.
- Flexural rigidity
- Flexural rigidity is the product E I, which measures a beam's resistance to bending deformation.
- Second moment of area
- The second moment of area I describes how a cross section's area is distributed relative to its neutral axis.
- Cantilever beam
- A cantilever beam is fixed at one end and free at the other end.
- Simply supported beam
- A simply supported beam is supported by a pin and a roller so it can rotate at the supports but cannot freely translate vertically.
Common Mistakes to Avoid
- Using the wrong load case is incorrect because an end point load, center point load, and uniform load have different deflection constants and different locations of maximum deflection.
- Mixing units is wrong because E, I, load, and length must be in a consistent unit system for the resulting deflection to have the correct length unit.
- Forgetting that I depends on orientation is wrong because rotating a rectangular beam changes the second moment of area and can greatly change deflection.
- Applying small-deflection beam formulas to large deflections is wrong because Euler-Bernoulli formulas assume geometry changes are small enough that linear theory remains valid.
- Confusing maximum moment with maximum deflection is wrong because the largest bending moment and largest displacement do not always occur at the same location.
Practice Questions
- 1 A cantilever beam has L = 2.0 m, E = 200 GPa, I = 8.0 x 10^-6 m^4, and an end load P = 500 N. Find the maximum deflection using delta_max = P L^3 / (3 E I).
- 2 A simply supported beam has L = 4.0 m, E I = 1.2 x 10^7 N m^2, and a central point load P = 6.0 kN. Find the maximum midspan deflection.
- 3 A cantilever beam with L = 3.0 m carries a uniform load w = 800 N/m and has E I = 2.5 x 10^6 N m^2. Find the free-end deflection using delta_max = w L^4 / (8 E I).
- 4 If two beams have the same material, cross section, and load type, but one beam has twice the span length, explain why its deflection increases much more than twice.
Understanding Cantilever & Beam Deflection Formulas
The formulas come from a chain of physical effects inside the beam. A transverse load creates shear force and bending moment. Bending moment makes one side of the beam shorten while the opposite side stretches.
Between them is a neutral axis, where the length changes very little. This pattern of tension and compression causes curvature. Greater bending moment gives greater curvature.
Greater stiffness resists curvature. Engineers first find the reactions at supports, then build shear and moment diagrams.
Those diagrams show where the beam is most heavily stressed and where its shape changes most rapidly. Deflection is found by connecting curvature across the full length of the member.
Support conditions matter as much as the load. A fixed end prevents vertical movement and rotation. A pin prevents vertical movement but permits rotation.
A roller usually prevents vertical movement in one direction while allowing horizontal movement. These restrictions determine the beam shape. A cantilever has its largest bending moment at the fixed support, even though its largest vertical movement is usually at the free end.
A simply supported beam has zero deflection at both supports, but it can rotate there. Students should sketch the expected bent shape before using any table. A sketch helps catch a formula chosen for the wrong support arrangement or load location.
The cross section has a large effect on stiffness. The second moment of area measures how much material lies away from the neutral axis. Material near that axis contributes relatively little to bending resistance.
Material placed farther away contributes much more. This is why deep beams and I shaped sections can be very stiff without being solid blocks. Increasing beam depth is often more effective than making a beam wider.
The material matters too. Steel, aluminum, timber, and plastics have different Young's modulus values.
A stiff material with a well shaped cross section bends less under the same loading. In practice, designers balance stiffness, strength, weight, cost, and available space.
Long spans need special attention. If a point load case follows a length cubed pattern, doubling the span can increase deflection by about eight times. For a full distributed load case with a length to the fourth power pattern, doubling the span can increase deflection by about sixteen times.
This explains why a shelf, diving board, floor joist, crane arm, or machine bracket may feel flexible even when it is not close to breaking. Deflection calculations are usually about acceptable movement and alignment, not only failure. Check units at every step, especially when dimensions are given in millimetres but material properties use metres.
The standard beam model becomes less reliable for very short deep beams, large bending, changing cross sections, cracked concrete, loose connections, or loads that move over the beam. Those cases may need a more detailed model or experimental testing.