Engineering & Technology · Statics, Stress & Deformation
Free-body diagrams, equilibrium and support reactions
A beam can look motionless while carrying several forces. To explain what holds it there, imagine a boundary around the beam alone. Every support and attached object outside that boundary can act on it. The sketch becomes a force model when those interactions are shown explicitly.
Use Forces, Energy & Simple Systems and Material Structure & Properties first. Resolve horizontal and vertical components and use simple ratios and powers. The chapter defines each new mechanics symbol before applying it; 3–4–5 triangles avoid a trigonometry prerequisite in the worked truss. All dimensions, loads, material parameters, limits and records are constructed teaching inputs, not equipment specifications, permissible loads or testing instructions.
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Replace supports with the correct interactions and solve a complete planar force and moment balance.
Draw interactions on one chosen body
A free-body diagram isolates a body from its surroundings and replaces each contact by the force or moment it can transmit in the adopted model. Include gravity and imposed loads. Do not place both members of an action–reaction pair on the same isolated body: the force of the support on the beam belongs here; the opposite force belongs on the support’s own diagram.
Choose axes before adding components. Here x points right and y up; a counterclockwise turning effect is positive. A force has a line of action as well as a magnitude. Its moment about a point is force multiplied by perpendicular distance to that line, with a sign for turning sense. A horizontal force on the beam centreline has no moment about a point on that same line.
Support symbols encode restrictions
In our planar idealisation, a frictionless pin prevents translation in two directions but permits rotation. Replace it by unknown Ax and Ay, with no reaction moment. A smooth roller on a horizontal surface provides one vertical normal reaction. A fixed support would have a different model, including a resisting moment. These are descriptions of constraints, not appearances of real hardware.
Draw an unknown in an assumed direction. A negative solved value reverses that assumed direction; it is not a negative amount of force. Check whether the physical contact permits the result. A support that can only push cannot supply an inferred pull without some other connection. Unknown counts can guide calculation, but geometry and stability still need examination.
Balance translation and rotation
For a rigid body remaining at rest in a plane, the sums of x forces, y forces and moments must each be zero. Satisfying only the vertical sum can leave an unbalanced horizontal force or rotation. Taking moments about the pin removes both pin-force components from that equation because their lines pass through the chosen point.
After solving, substitute into all balances. Change the moment centre for a useful additional check. Equilibrium determines reactions for this model; it does not establish stresses, deflections, joint adequacy or stability under another load. Those questions need information beyond the external force picture.
Solve the supported-beam record
Use the figure’s 2 m span, 120 N downward load at 0.5 m and 40 N beam weight at 1 m. A 30 N horizontal load acts right on the centreline. Moments about A give 2By−120×0.5−40×1=0, so By=50 N upward. Vertical balance gives Ay=120+40−50=110 N upward. Horizontal balance gives Ax=−30 N when right is positive, meaning 30 N left.
About B, the left reaction produces 220 N·m clockwise, while the downward loads produce 180+40=220 N·m counterclockwise. Both checks agree. The beam is treated as rigid for reaction calculation; this is not a claim that a real beam remains undeformed or can carry the loads.
Try the reasoning
Why is there no pin reaction moment in this model?
Show the answer and reasoning
The ideal frictionless pin permits rotation. A real connection may require a different model if it restrains rotation.
Can vertical force balance alone determine all reactions?
Show the answer and reasoning
No. The horizontal balance and independent moment balance are also required.
What does Ax = −30 N mean?
Show the answer and reasoning
The force acts 30 N opposite the assumed positive x direction: to the left.
What to carry forward
A useful free-body diagram names the body, interactions, geometry and support assumptions before any equation is solved.
References
Teaching resources and research supporting this lesson’s explanations and examples.
Daniel Baker and William Haynes · Engineering Statics2 sources