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.

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.

Engineering simulation studio

Purpose-built 2D FBD

Simply supported beam reactions

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

correct supportsreaction arrowsdimensions

Span10 m

1–20

Point load80 kN

0–200

Point-load position4 m

0–20

Full-span UDL10 kN/m

0–50

Display units

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?

Engineering simulation studio

Purpose-built 2D FBD

Cantilever support reactions

Compute fixed-end reactions for a tip load, full-span UDL, and applied tip couple.

fixed supporttip couplesupport moment

Length5 m

0.50–12

Tip load60 kN

0–200

Full-span UDL8 kN/m

0–50

Tip couple (+CCW)0 kN·m

-200–200

Display units

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?

Engineering simulation studio

Purpose-built 2D FBD

Three-force rigid-body explorer

Construct the concurrency point and solve the remaining two force magnitudes.

actual rigid bodyconcurrencysolved direction

Known force F₁50 kN

1–150

F₁ direction30 °

-180–180

F₂ line direction130 °

-180–180

Point B x6 m

1–10

Point C x3 m

0–10

Point C y2 m

-5–5

Display units

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?

Engineering simulation studio

Purpose-built 2D FBD

Crane-boom equilibrium

Solve cable tension and pin reactions including boom self-weight.

mechanism geometryself-weightcable arm

Lifted load100 kN

1–300

Boom weight20 kN

0–150

Boom angle40 °

5–85

Cable-force direction130 °

-175–175

Boom length10 m

1–30

Display units

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?

Engineering simulation studio

Purpose-built 2D FBD

Sliding and overturning stability

Compare sliding, overturning, eccentricity, and contact-pressure limits.

middle thirdbase resultantuplift

Weight400 kN

1–1000

Horizontal load80 kN

1–500

Base width4 m

0.50–12

Load height3 m

0.50–20

Friction coefficient0.50

0.10–1

Display units

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.

Corrected spatial engineering studio

Geometry-linked 3D

Guyed communication mast

Prescribe one guy pretension, solve the other two from moment equilibrium, then recover base reactions.

Engineering view

Orbit freely or snap to a projection.

XYZ vertical

Wind load50 kNWind load height15 mGuy attachment height20 mAnchor radius12 mGuy 3 prescribed tension75 kNWind azimuth0 °

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?

Engineering simulation studio

True 3D

Wall-mounted spatial sign bracket

Compute all six fixed-support reactions for weight and wind applied at the sign centroid.

wall and bracketweight plus windsix reactions

Synchronized view

Orbit freely or snap to an engineering projection.

Preparing spatial engineering model…

Sign weight20 kN

1–100

Wind force15 kN

0–100

Centroid x offset3 m

-10–10

Centroid y offset1 m

-10–10

Centroid z offset2 m

0–20

Wind azimuth0 °

-180–180

Display units

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?

Corrected spatial engineering studio

Geometry-linked 3D

Three-legged supported platform

Solve three vertical reactions and compare the load position with the support triangle.

Engineering view

Orbit freely or snap to a projection.

XYZ vertical

Platform load90 kNLoad x0.5 mLoad y0.5 mSupport radius2 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?

Engineering simulation studio

True 3D

Spatial rigid body with mixed supports

Solve a ball-and-socket, short link, cable, and smooth-contact reaction system using six equations.

distinct supportsmatrix rankunilateral checks

Synchronized view

Orbit freely or snap to an engineering projection.

Preparing spatial engineering model…

Applied force60 kN

1–200

Load azimuth210 °

-180–180

Load elevation-45 °

-80–80

Body length6 m

1–12

Body width4 m

1–10

Cable elevation45 °

5–85

Display units

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?

Engineering simulation studio

True 3D

Tower-crane base-reaction explorer

Combine lifted load, counterweight, and elevated wind to compute the six fixed-base reactions.

jib and counter-jiblift and wind3D base moment

Synchronized view

Orbit freely or snap to an engineering projection.

Preparing spatial engineering model…

Lifted load300 kN

1–1000

Jib radius40 m

1–80

Counterweight250 kN

0–1000

Counter-jib radius15 m

1–40

Wind load100 kN

0–500

Wind load height30 m

1–100

Wind azimuth0 °

-180–180

Display units

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?

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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