Static equilibrium conditions
A rigid body is in static equilibrium when its resultant force and resultant moment both vanish.
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Formula
F in N; M in N·m; sums are vector or signed scalar sums
Variables & units
| Symbol | Meaning | Unit |
|---|---|---|
| External force, including support reaction | N | |
| Moment of a force about reference point O | N·m |
LaTeX
\sum_i \vec F_i=\vec 0,\quad \sum_i M_{i,O}=0
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Derivation & background
These equations are special cases of Newton–Euler balance for a body at rest. In a plane they provide at most three independent equations; in three dimensions, at most six. Keep force directions, lever arms and positive rotation convention consistent. Equilibrium equations alone do not solve a statically indeterminate structure.
Exam blueprint
Validity range
These equations are special cases of Newton–Euler balance for a body at rest. In a plane they provide at most three independent equations; in three dimensions, at most six. Keep force directions, lever arms and positive rotation convention consistent. Equilibrium equations alone do not solve a statically indeterminate structure.
Derivation steps
A rigid body is in static equilibrium when its resultant force and resultant moment both vanish.
- 1State the model assumptions and list quantities with units.
- 2Substitute into ΣF = 0; ΣMo = 0; convert quantities to SI first.
- 3A beam with a symmetric 6 kN load and two symmetric supports has 3 kN reaction at each support. Moment balance about either support must also hold.
Rearrangements
Base equation for the target quantity
Preserve signs, boundary conditions and units when rearranging.
Task variant
Work through the numerical example for “Static equilibrium conditions” and check its unit and magnitude.
A beam with a symmetric 6 kN load and two symmetric supports has 3 kN reaction at each support.
Moment balance about either support must also hold.
Common mistakes
Substituting a related quantity that shares the same symbol.
Use variable definitions and check: F in N; M in N·m; sums are vector or signed scalar sums.
Exam context
- Find reactions and internal resultants in statically determinate structures.
Formula cluster
Engineering mechanics
A rigid body is in static equilibrium when its resultant force and resultant moment both vanish.
Worked example
A beam with a symmetric 6 kN load and two symmetric supports has 3 kN reaction at each support. Moment balance about either support must also hold.
Applications
Find reactions and internal resultants in statically determinate structures.
How do you calculate with Static equilibrium conditions?
Here is how to work through a typical Static equilibrium conditions (ΣF = 0; ΣMₒ = 0) task step by step:
- 1
Task
Work through the numerical example for “Static equilibrium conditions” and check its unit and magnitude.
Solution path
A beam with a symmetric 6 kN load and two symmetric supports has 3 kN reaction at each support.
Moment balance about either support must also hold.
Frequently asked questions about Static equilibrium conditions
How many equations does planar statics provide?+
A rigid body in a plane has at most three independent equilibrium conditions: horizontal-force sum, vertical-force sum and moment sum about a point. They determine all unknown reactions only for a statically determinate model. Additional unknown supports require deformation compatibility or another model. Before calculating, define the free body, positive axes and reference point. The sums must include every external force and moment, including support reactions. A negative result means the actual direction opposes the assumed one. Check force and moment balance separately at the end.
Why choose the moment reference point carefully?+
A suitable point can remove several unknown support reactions from the moment equation when their lines of action pass through it. This simplifies the algebra without changing the physical solution. A force with lever arm creates moment F·d; use the perpendicular distance to its line of action, not the slanted distance. Before calculating, define the free body, positive axes and reference point. The sums must include every external force and moment, including support reactions. A negative result means the actual direction opposes the assumed one. Check force and moment balance separately at the end.
When is a structure statically indeterminate?+
If there are more independent unknown reactions than available equilibrium equations, statics alone is insufficient. Additional conditions then come from deformation compatibility and material stiffness. A negative calculated reaction does not itself imply indeterminacy; it may simply mean the assumed direction was reversed. Before calculating, define the free body, positive axes and reference point. The sums must include every external force and moment, including support reactions. A negative result means the actual direction opposes the assumed one. Check force and moment balance separately at the end.
How do I check the result and units for Static equilibrium conditions?+
Write the quantity and unit from the variable list next to every value: F in N; M in N·m; sums are vector or signed scalar sums. Substitute only after converting to a consistent unit system, and state sign or reference direction explicitly. The example A beam with a symmetric 6 kN load and two symmetric supports has 3 kN reaction at each support. Moment balance about either support must also hold. is a calculation check: result, unit and order of magnitude must fit the described situation. Round after intermediate calculations so small contributions are not lost prematurely. Before calculating, define the free body, positive axes and reference point. The sums must include every external force and moment, including support reactions. A negative result means the actual direction opposes the assumed one. Check force and moment balance separately at the end.
Which model limit matters for Static equilibrium conditions?+
The background states this starting condition: These equations are special cases of Newton–Euler balance for a body at rest. In a plane they provide. That limits when the result is a valid model prediction. Compare the problem’s boundary conditions with this assumption before substituting values; real components, material properties or loads can introduce additional effects. The cited reference supports the relation used here but does not remove the need to check whether the specific case fits the model. Before calculating, define the free body, positive axes and reference point. The sums must include every external force and moment, including support reactions. A negative result means the actual direction opposes the assumed one. Check force and moment balance separately at the end.
Quanta exam set
Curated exam set for "Static equilibrium conditions":
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Which formula describes Static equilibrium conditions?
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How do you rearrange Static equilibrium conditions for Base equation for the target quantity?
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Which common mistake happens with Static equilibrium conditions?
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Subject references and further reading
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More formulas: Mechanics, strength and materials
Retain Static equilibrium conditions for exams
Create a curated FSRS exam set for ΣF = 0; ΣMₒ = 0: formula recall, variables, derivation, rearrangement, worked example, common mistakes and exam context.
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