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Beam Diagrams & Loading Types Cheat Sheet
A printable reference covering support reactions, point loads, distributed loads, shear force, bending moment, and beam diagrams for grades 11-12.
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Beam diagrams show how a structure is supported and how forces act along its length. This cheat sheet helps engineering students identify common supports, loading types, and the diagrams used to analyze beams. These skills are essential for checking whether a beam can safely carry loads without excessive shear force or bending moment. A clear reference makes it easier to connect free-body diagrams with real structural behavior. The main ideas are support reactions, load modeling, shear force diagrams, and bending moment diagrams. Supports such as pins, rollers, and fixed ends create different reaction forces and moments. Loads may be concentrated at one point or spread over part of the beam. Shear changes based on applied loads, and bending moment changes based on the area under the shear diagram.
Key Facts
- A roller support provides one reaction force perpendicular to the surface, usually written as Ry.
- A pin support provides two reaction force components, usually written as Rx and Ry, but it does not resist moment.
- A fixed support provides three reactions in 2D, which are Rx, Ry, and a reaction moment M.
- Static equilibrium requires sum of Fx = 0, sum of Fy = 0, and sum of M = 0 for a beam at rest.
- A point load causes an immediate jump in the shear force diagram equal to the load magnitude.
- A uniformly distributed load w over length L has an equivalent point load W = wL acting at the midpoint of the loaded length.
- The slope of the bending moment diagram equals the shear force, so dM/dx = V.
- The change in bending moment between two points equals the area under the shear force diagram, so delta M = area under V diagram.
Vocabulary
- Beam
- A structural member that carries loads mainly by resisting shear force and bending moment.
- Support Reaction
- A force or moment produced by a support to keep a beam in equilibrium.
- Point Load
- A force applied at a single location on a beam, often shown as a downward arrow.
- Distributed Load
- A load spread over a length of the beam, usually measured in force per unit length such as N/m.
- Shear Force Diagram
- A graph showing the internal shear force at each position along a beam.
- Bending Moment Diagram
- A graph showing the internal bending moment at each position along a beam.
Common Mistakes to Avoid
- Treating a pin support like a fixed support is wrong because a pin cannot resist a reaction moment.
- Placing the equivalent force of a uniformly distributed load at the wrong location is wrong because W = wL acts at the center of the loaded length, not always at the beam center.
- Forgetting sign conventions in shear and moment diagrams is wrong because inconsistent signs can make diagrams disagree with equilibrium equations.
- Drawing a sloped shear diagram under a point load is wrong because a point load creates a vertical jump in shear, not a gradual change.
- Ignoring units when converting distributed loads is wrong because kN/m multiplied by m gives kN, and using mismatched units changes reaction and moment values.
Practice Questions
- 1 A simply supported beam is 6 m long with a 12 kN point load at the center. Find the vertical reactions at the two supports.
- 2 A 4 m beam carries a uniformly distributed load of 3 kN/m over its full length. Find the equivalent point load and its location.
- 3 A cantilever beam has a 5 kN downward point load at its free end, 2 m from the fixed support. Find the vertical reaction and fixed-end moment at the support.
- 4 Explain why a bending moment diagram is usually curved under a uniformly distributed load but straight between point loads.