Equilibrium of Coplanar Force Systems

Learning Objectives

  • Construct complete free-body diagrams for isolated planar rigid bodies.
  • Replace idealized rollers, pins, and fixed supports with the correct reaction components.
  • Apply the three independent equations of planar rigid-body equilibrium.
  • Replace common distributed loads with equivalent concentrated resultants at the correct locations.
  • Recognize two-force and three-force members and use their geometric restrictions.
  • Distinguish equilibrium solvability from structural stability and identify obvious external indeterminacy.

Static Equilibrium

A state in which a rigid body's translational and angular accelerations are zero. For a planar rigid body, the resultant external force and resultant external moment must both vanish.

Planar Rigid-Body Equilibrium

The three independent scalar equations available for a general rigid body in two dimensions.

∑Fx=0,∑Fy=0,∑MO=0\sum F_x=0, \qquad \sum F_y=0, \qquad \sum M_O=0

Variables

SymbolDescriptionUnit
FxF_xHorizontal force componentN
FyF_yVertical force componentN
MOM_OMoment about any chosen point ON·m

Equilibrium Equations Require a Correct Model

A numerical solution is meaningful only if the free-body diagram includes the correct external loads, support reactions, geometry, and sign convention. Omitting or inventing a reaction can produce algebra that balances while representing the wrong physical system.

Free-Body Diagram

A diagram of an isolated body or system boundary showing all external forces and couples acting on that isolated system, together with the dimensions and coordinate directions needed for equilibrium.

Constructing a Free-Body Diagram

  1. Choose the body or subsystem to isolate.
  2. Remove surrounding bodies and replace each interaction with the appropriate external reaction or contact force.
  3. Add all known applied forces, couples, and equivalent distributed-load resultants.
  4. Show force directions, points or lines of action, dimensions, and a coordinate system.
  5. Label unknown reactions consistently before writing equilibrium equations.
  6. Check that no internal action of the isolated system has been drawn as an external force.

How to Use This Workflow

Use the workflow to move from physical isolation to a verified equilibrium solution. If the solved reactions are inconsistent with the support geometry or contact assumptions, return to the free-body diagram before accepting the algebra.

Planar Rigid-Body Equilibrium Workflow
Planar Rigid-Body Equilibrium WorkflowStart equilibrium analysis → Isolate the body or subsystem; Isolate the body or subsystem → Replace supports and contacts by reactions; Replace supports and contacts by reactions → Distributed load present?; Distributed load present? — Yes → Replace by equivalent resultant at centroid; Distributed load present? — No → Apply force and moment equilibrium; Replace by equivalent resultant at centroid → Apply force and moment equilibrium; Apply force and moment equilibrium → Reactions and geometry physically admissible?; Reactions and geometry physically admissible? — Yes → Verify with an independent equilibrium check; Reactions and geometry physically admissible? — No → Review FBD, support directions, and load locations; Review FBD, support directions, and load locations — Revise → Isolate the body or subsystem; Verify with an independent equilibrium check → Report reactions and assumptions

Start equilibrium analysis → Isolate the body or subsystem; Isolate the body or subsystem → Replace supports and contacts by reactions; Replace supports and contacts by reactions → Distributed load present?; Distributed load present? — Yes → Replace by equivalent resultant at centroid; Distributed load present? — No → Apply force and moment equilibrium; Replace by equivalent resultant at centroid → Apply force and moment equilibrium; Apply force and moment equilibrium → Reactions and geometry physically admissible?; Reactions and geometry physically admissible? — Yes → Verify with an independent equilibrium check; Reactions and geometry physically admissible? — No → Review FBD, support directions, and load locations; Review FBD, support directions, and load locations — Revise → Isolate the body or subsystem; Verify with an independent equilibrium check → Report reactions and assumptions

  • Start equilibrium analysis: terminator
  • Isolate the body or subsystem: process
  • Replace supports and contacts by reactions: process
  • Distributed load present?: decision
  • Replace by equivalent resultant at centroid: process
  • Apply force and moment equilibrium: process
  • Reactions and geometry physically admissible?: decision
  • Review FBD, support directions, and load locations: process
  • Verify with an independent equilibrium check: process
  • Report reactions and assumptions: terminator

Idealized Planar Supports

For a member in a two-dimensional model:

  • A smooth roller or rocker supplies one reaction normal to the supporting surface.
  • An ideal pin or hinge supplies two force components, commonly AxA_x and AyA_y, but no reaction moment.
  • A fixed support supplies two force components and a reaction couple, commonly AxA_x, AyA_y, and MAM_A.

These are analytical idealizations. Real connection behavior may be semi-rigid, nonlinear, or three-dimensional and must be modeled accordingly in design practice.

Choosing Efficient Equilibrium Equations

Moment equilibrium may be taken about any point. Choosing a point through which one or more unknown reaction lines pass eliminates those forces from the moment equation because their lever arms are zero.

After solving reactions, use an unused equilibrium equation or a second moment center as an independent arithmetic check whenever practical.

Interactive Exploration

Move loads along the beam and change their magnitudes. Observe how the support reactions redistribute while the total vertical force and moment remain balanced.

Simply supported beam reactions

Concept and model scope

Solve pin and roller reactions for a point load plus a full-span UDL.

Governing model: ΣFy = 0; ΣMA = 0

Every physical dimension shown by this studio is derived from the same state used by the solver. Readability-scaled force arrows preserve direction and application point.

Simply supported beam equilibrium. Out-of-span point loads remain visibly out of span instead of being silently clampedP 80 kNAy 98 kNBy 82 kNL 10 mGeometry-faithful engineering diagram with automatic fit-to-content framing.
Controls
Span

Span

Span is part of the same engineering state used by the diagram and solver.

10.0 m
Point load

Point load

Point load is part of the same engineering state used by the diagram and solver.

80 kN
Point-load position

Point-load position

Point-load position is part of the same engineering state used by the diagram and solver.

4.00 m
Full-span UDL

Full-span UDL

Full-span UDL is part of the same engineering state used by the diagram and solver.

10 kN/m
Engineering model scope

Category

Two-dimensional rigid-body equilibrium

Idealization

Ideal supports supply only their permitted reactions; deformation is neglected.

Acceptance check

Check ΣFx = 0, ΣFy = 0, ΣM = 0, plus contact, uplift, sliding, or tipping limits where relevant.

Equivalent Resultants of Distributed Loads

For a one-dimensional distributed load w(x)w(x), the equivalent concentrated force equals the area under the loading diagram, and its line of action passes through the centroid of that area.

Common cases are:

  • uniform load ww over length LL: W=wLW=wL acting at the midpoint;
  • triangular load increasing from zero to w0w_0: W=w0L/2W=w_0L/2 acting one-third of LL from the high-intensity end;
  • trapezoidal load: decompose it into a rectangle and triangle or integrate directly.

General Distributed-Load Resultant

Computes the magnitude and line of action of an equivalent load.

W=∫abw(x) dx,xˉ=∫abxw(x) dxWW=\int_a^b w(x)\,dx, \qquad \bar{x}=\frac{\int_a^b xw(x)\,dx}{W}

Variables

SymbolDescriptionUnit
WWEquivalent concentrated loadN
w(x)w(x)Distributed-load intensityN/m
xˉ\bar{x}Resultant location from the chosen originm

Two-Force Member

A member subjected to external forces at only two points and no external couple. If in equilibrium, the two forces are equal, opposite, and collinear, so the member carries only axial force in the idealized model.

Three-Force Member

A rigid body subjected to exactly three forces and no applied couple. If the forces are not parallel and the body is in equilibrium, their lines of action must be concurrent.

Determinacy and Stability

For one general planar rigid body, only three independent equilibrium equations are available. If the number and arrangement of external reaction unknowns exceed what those equations can determine, the body is externally statically indeterminate and additional deformation or compatibility relations are required.

Counting unknown reactions is only a screening test. Stability depends on reaction geometry as well as count. A body can have three reaction components yet still be unstable if the reaction lines cannot resist an admissible rigid-body motion.

For assemblies such as trusses and frames, internal member forces and additional equilibrium equations must also be considered; the simple statement "r=3r=3 means determinate" is not a general structural criterion.

Do Not Equate Indeterminacy with Safety

Static indeterminacy can provide redundancy, but redundancy alone does not guarantee robustness or prevent progressive collapse. Safe structural behavior depends on member strength, ductility, connection detailing, load paths, deformation compatibility, and the governing design provisions.

Stable, Unstable, and Neutral Equilibrium

Equilibrium describes a current force state; stability describes the response to a small disturbance.

  • In stable equilibrium, a small displacement tends to produce restoring behavior.
  • In unstable equilibrium, a small displacement tends to grow.
  • In neutral equilibrium, a displaced system can remain in a nearby equilibrium configuration.

Elementary rigid-body equilibrium equations alone do not quantify all stability phenomena. Buckling and geometric instability require additional mechanics developed in later courses.

Key Takeaways
  • Planar rigid-body equilibrium requires zero resultant horizontal force, vertical force, and moment.
  • A complete free-body diagram is the foundation of a valid equilibrium solution.
  • Roller, pin, and fixed supports contribute different idealized reaction components.
  • Distributed loads are replaced by forces equal to their load-diagram areas acting through the corresponding centroids.
  • Two-force and three-force members obey useful geometric restrictions that can simplify analysis.
  • Reaction counting helps screen determinacy, but reaction geometry and the structure type must also be checked for stability and solvability.

References