Equilibrium of Coplanar Force Systems Examples

Twelve worked applications progress from support modeling and load replacement to complete reaction calculations, member restrictions, and stability checks.

Identify Support Reactions

A planar beam has a pin at AA and a roller on a horizontal surface at BB. List the idealized reactions.

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Replace a Uniformly Distributed Load

An 8.00 m8.00\ \text{m} beam carries 10.0 kN/m10.0\ \text{kN/m} uniformly over its full span. Determine the equivalent concentrated load.

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Replace a Triangular Distributed Load

A 6.00 m6.00\ \text{m} load varies linearly from zero at the left to 30.0 kN/m30.0\ \text{kN/m} at the right. Find its resultant and location from the left.

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Simple Beam with an Eccentric Point Load

A 10.0 m10.0\ \text{m} simply supported beam has a pin at AA, roller at BB, and a 50.0 kN50.0\ \text{kN} downward load 2.00 m2.00\ \text{m} from AA. Determine the reactions.

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Simple Beam under a Full-Span UDL

A 6.00 m6.00\ \text{m} simply supported beam carries 12.0 kN/m12.0\ \text{kN/m} over the full span. Determine the vertical support reactions.

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Cantilever with a Point Load

A 3.00 m3.00\ \text{m} cantilever is fixed at AA and carries a 20.0 kN20.0\ \text{kN} downward load at its free end. Determine AxA_x, AyA_y, and MAM_A using counterclockwise positive.

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Check a Two-Force Member

A slender bar is pin-connected only at its two ends and has no intermediate load or applied couple. What can be stated about its end forces if it is in equilibrium?

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Detect an Unstable Support Arrangement

A horizontal rigid beam rests on two smooth rollers, both on horizontal surfaces, and has no other support. Is it stable against a horizontal load?

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Beam with an Inclined Roller Support

A 4.00 m4.00\ \text{m} beam has a pin at AA and a roller at BB resting on a surface inclined 30.0∘30.0^\circ above horizontal. A 10.0 kN10.0\ \text{kN} downward load acts at midspan. Determine the reaction components. The roller reaction is normal to the surface.

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Simple Beam under a Trapezoidal Load

A 6.00 m6.00\ \text{m} simply supported beam carries a linearly varying load from 4.00 kN/m4.00\ \text{kN/m} at AA to 10.0 kN/m10.0\ \text{kN/m} at BB. Determine the support reactions.

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Use the Three-Force-Member Concurrency Restriction

A rigid bracket is acted on by exactly three nonparallel forces and no applied couple. The lines of action of two forces intersect at point PP. What restriction must the third force satisfy for equilibrium?

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Fixed Support with an End Force and Applied Couple

A 3.00 m3.00\ \text{m} cantilever is fixed at AA. Its free end carries a 15.0 kN15.0\ \text{kN} downward force and an 8.00 kN⋅m8.00\ \text{kN}\cdot\text{m} clockwise applied couple. Determine the reactions using counterclockwise moment as positive.

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Reaction Direction

A roller reaction is normal to its supporting surface, not automatically vertical. On an inclined surface, resolve the normal reaction into global components only after defining its true direction.