Two-Dimensional and Three-Dimensional Rigid-Body Equilibrium

Rigid-Body Equilibrium

$Learning Objectives$

Replace common two-dimensional and three-dimensional supports with correct reactions.
Construct complete free-body diagrams for rigid bodies.
Assemble and solve force and moment equilibrium equations.
Detect uplift, instability, singularity, and underconstrained systems.
Interpret fixed, cable, link, contact, bearing, and ball-and-socket constraints.

Rigid-body support reactions and equilibrium

A pin supplies two planar reaction components and a roller supplies one normal reaction. The isolated body must satisfy both force and moment equilibrium.

$Governing principle and sign conventions$

A rigid body is in static equilibrium only when both translation and rotation are prevented by balanced external effects. In 2D, use $+x$ right, $+y$ up, and positive counterclockwise moments. In 3D, use a right-handed system and moment vectors defined by the right-hand rule.

Rigid-body equilibrium

Three planar or six spatial scalar equations.

\sum\mathbf F=\mathbf0,\qquad \sum\mathbf M_O=\mathbf0

\text{2D: }\sum F_x=0,\ \sum F_y=0,\ \sum M_O=0

\text{3D: }\sum F_x=\sum F_y=\sum F_z=\sum M_x=\sum M_y=\sum M_z=0

$Support reactions$

2D roller or smooth contact: one force normal to the surface.
2D pin: two force components.
2D fixed support: two force components and one couple moment.
Cable or short link: one force along its axis; a cable is tension-only.
Ball-and-socket: three force components and no reaction moments.
Journal bearing: reactions depend on bearing axis and whether thrust is restrained.
Fixed support in 3D: three force components and three couple moments.

$Automatic reaction and validation workflow$

STEP-BY-STEP

Isolate the body and replace each support with only the reactions it can physically supply.
Replace distributed loads by equivalent resultants before assembling equations.
Choose a moment center that eliminates the most unknown reactions.
Compare unknown count with the number of independent equilibrium equations.
Solve and check matrix rank, residual, reaction signs, and contact conditions.
Report determinate, indeterminate, unstable, singular, or uplift conditions honestly.

Two-dimensional rigid-body equilibrium

$Guided example: simply supported beam$

A $60\text{ kN}$ point load acts $4\text{ m}$ from the pin on an $8\text{ m}$ beam. Taking moments about the pin gives $B_y(8)=60(4)$ , so $B_y=30\text{ kN}$ . Vertical equilibrium gives $A_y=30\text{ kN}$ .

$Common 2D misconceptions$

Giving a roller two reaction components.
Omitting the reaction couple at a fixed support.
Treating a negative contact reaction as compression rather than uplift or separation.
Assuming three force components imply equilibrium without checking moments.
Calling a mechanism stable because the number of unknowns equals three.

Controls

10.0 m

80 kN

4.00 m

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.

$Interpretation question$

How do the two reactions change as the point load moves toward one support?

Controls

5.00 m

60 kN

8 kN/m

0 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.

$Interpretation question$

Why must a fixed support develop a reaction moment even when the applied load is purely vertical?

Controls

50 kN

30 °

130 °

6.0 m

3.0 m

2.0 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.

$Interpretation question$

Why must the three lines of action be concurrent unless all three forces are parallel?

Controls

100 kN

20 kN

40 °

130 °

10.0 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.

$Interpretation question$

What happens when the tie cable becomes nearly parallel to the boom?

Controls

400 kN

80 kN

4.00 m

3.00 m

0.50

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.

$Interpretation question$

Can a body be safe against sliding but unsafe against overturning?

Three-dimensional rigid-body equilibrium

Spatial equilibrium provides six scalar equations, but a unique solution also requires that the support-reaction coefficient matrix has full column rank. A system with fewer independent equations than unknowns is underconstrained or indeterminate; coincident or dependent support directions may be singular even when the counts match.

$Guided example: fixed spatial bracket$

A downward $10\text{ kN}$ force acts at $\mathbf r=\langle2,0,1\rangle\text{ m}$ . The fixed wall reaction force is $\langle0,0,10\rangle\text{ kN}$ . Since $\mathbf r\times\mathbf F=\langle0,20,0\rangle\text{ kN·m}$ , the wall reaction moment is $\langle0,-20,0\rangle\text{ kN·m}$ .

$Common 3D misconceptions$

Treating the six equations as automatically independent.
Adding reaction moments to a ball-and-socket support.
Ignoring thrust restraint when modeling a bearing.
Reporting a unique solution when the support matrix is singular.
Omitting a force moment because the force is parallel to one coordinate axis.

Controls

50 kN

15 m

20 m

12 m

75 kN

0 °

Engineering model scope

Category

Three-dimensional rigid-body equilibrium

Idealization

Ideal spatial supports and connections are represented by their permitted reaction components.

Acceptance check

Check all six equilibrium equations, matrix rank, unilateral contact, and tension-only constraints.

$Interpretation question$

Why can a three-guy arrangement become singular or require a negative cable tension for some load directions?

Checking 3D rendering support…

Controls

20 kN

15 kN

3.00 m

1.00 m

2.00 m

0 °

Engineering model scope

Category

Three-dimensional rigid-body equilibrium

Idealization

Ideal spatial supports and connections are represented by their permitted reaction components.

Acceptance check

Check all six equilibrium equations, matrix rank, unilateral contact, and tension-only constraints.

$Interpretation question$

Which offset controls the largest wall reaction moment for a vertical sign weight?

Controls

90 kN

0.50 m

2.00 m

Engineering model scope

Category

Three-dimensional rigid-body equilibrium

Idealization

Ideal spatial supports and connections are represented by their permitted reaction components.

Acceptance check

Check all six equilibrium equations, matrix rank, unilateral contact, and tension-only constraints.

$Interpretation question$

What does a negative leg reaction mean about the load position relative to the support polygon?

Checking 3D rendering support…

Controls

60 kN

210 °

-45 °

6.0 m

4.0 m

45 °

Engineering model scope

Category

Three-dimensional rigid-body equilibrium

Idealization

Ideal spatial supports and connections are represented by their permitted reaction components.

Acceptance check

Check all six equilibrium equations, matrix rank, unilateral contact, and tension-only constraints.

$Interpretation question$

Why is matching the number of unknowns and equations necessary but not sufficient?

Checking 3D rendering support…

Controls

300 kN

40 m

250 kN

15 m

100 kN

30 m

0 °

Engineering model scope

Category

Three-dimensional rigid-body equilibrium

Idealization

Ideal spatial supports and connections are represented by their permitted reaction components.

Acceptance check

Check all six equilibrium equations, matrix rank, unilateral contact, and tension-only constraints.

$Interpretation question$

How do jib reach and wind height affect different components of the base moment vector?

Rigid-Body Equilibrium Workflow

Model supports and unilateral contacts correctly, use equilibrium-matrix rank to detect instability, distinguish determinate from indeterminate systems, then solve and verify force and moment equilibrium.

Isolate the rigid body → Replace supports, contacts, and cables by physically permitted reactions; Replace supports, contacts, and cables by physically permitted reactions → Add applied forces, couples, distributed-load resultants, and dimensions; Add applied forces, couples, distributed-load resultants, and dimensions → Choose 2D or 3D equilibrium and count scalar reaction unknowns n; Choose 2D or 3D equilibrium and count scalar reaction unknowns n → Assemble the support-equilibrium matrix A with m = 3 (2D) or 6 (3D) rows; Assemble the support-equilibrium matrix A with m = 3 (2D) or 6 (3D) rows → rank(A) = m, so all rigid-body equilibrium modes are restrained?; rank(A) = m, so all rigid-body equilibrium modes are restrained? — No → Support geometry is underconstrained, singular, or only load-case stable; rank(A) = m, so all rigid-body equilibrium modes are restrained? — Yes → Reaction unknowns n > m?; Reaction unknowns n > m? — Yes → Statically indeterminate: add compatibility and deformation relations; Reaction unknowns n > m? — No → Solve ΣF = 0 and ΣM = 0; Solve ΣF = 0 and ΣM = 0 → Unilateral reactions physically admissible (N ≥ 0, cable T ≥ 0)?; Unilateral reactions physically admissible (N ≥ 0, cable T ≥ 0)? — No → Revise contact, cable, or support state; Unilateral reactions physically admissible (N ≥ 0, cable T ≥ 0)? — Yes → Independent force and moment residuals are within tolerance?; Revise contact, cable, or support state → Replace supports, contacts, and cables by physically permitted reactions; Independent force and moment residuals are within tolerance? — Yes → Rigid-body equilibrium solution accepted; Independent force and moment residuals are within tolerance? — No → Correct FBD, signs, units, equations, or algebra; Correct FBD, signs, units, equations, or algebra → Replace supports, contacts, and cables by physically permitted reactions

Isolate the rigid body: terminator
Replace supports, contacts, and cables by physically permitted reactions: process
Add applied forces, couples, distributed-load resultants, and dimensions: process
Choose 2D or 3D equilibrium and count scalar reaction unknowns n: process
Assemble the support-equilibrium matrix A with m = 3 (2D) or 6 (3D) rows: process
rank(A) = m, so all rigid-body equilibrium modes are restrained?: decision
Support geometry is underconstrained, singular, or only load-case stable: terminator
Reaction unknowns n > m?: decision
Statically indeterminate: add compatibility and deformation relations: terminator
Solve ΣF = 0 and ΣM = 0: process
Unilateral reactions physically admissible (N ≥ 0, cable T ≥ 0)?: decision
Revise contact, cable, or support state: process
Independent force and moment residuals are within tolerance?: decision
Correct FBD, signs, units, equations, or algebra: process
Rigid-body equilibrium solution accepted: terminator

Key Takeaways

Support reactions must match the actual kinematic constraints.
Planar rigid bodies provide three independent equilibrium equations; spatial rigid bodies provide six.
Negative contact reactions indicate uplift or loss of contact.
Equation count and matrix rank must both be checked before claiming a unique solution.
Fixed supports can supply force and moment reactions; cables cannot supply compression.

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