CELE Engineering Mechanics — Analysis of TrussesCheat Sheet
One-page cheat sheet for CELE Engineering Mechanics — Analysis of Trusses. Every formula, definition, and key fact you need for this chapter, condensed to a single printable page. Designed for the final review session before the CELE 2026.
Exam context
On the CELE 2026, the Engineering Mechanics subtest carries a "Core" weight in Professional Regulation Commission (PRC) — Board of Civil Engineering's pattern. Analysis of Trusses lands at position 3rd out of 8 in the standard review order. Target score is 70% weighted average, no sub-test below 50%, and roughly a meaningful share of items come from Engineering Mechanics on a typical CELE paper.
Analysis of Trusses - Cheat Sheet
Your last-minute revision companion for truss analysis — formulas, determinacy checks, method shortcuts, and exam traps. Master the two-force member concept, determinacy equation, and the method selection decision tree in 30 minutes.
Sections
Section Title
Truss Fundamentals & Assumptions
Important Facts
- Members are straight and assumed weightless; self-weight (if included) is split equally to end joints.
- All loads and reactions act only at joints, not along member spans.
- Pin connections are frictionless — no moment can be transmitted from one member to another.
- Every member is a two-force member because of joint assumptions; force must lie along the member axis.
- Sign convention: assume tension (+); negative result = compression (−).
- Truss must be rigid, properly arranged, and satisfy determinacy equation — all three are necessary.
Key Definitions
Term
Truss
Example
Bridge trusses, roof trusses, transmission tower bracing.
Definition
A structure of straight members joined at their ends by frictionless pins, forming a rigid framework; members carry pure tension or compression.
Term
Two-force member
Example
Every member in a truss under the pin-jointed, load-at-joints assumption.
Definition
A member carrying only two equal, opposite forces acting along its axis; resultant force is zero (equilibrium), resultant moment is zero.
Term
Tension (positive sign, +)
Example
Lower chord of a simply supported roof truss.
Definition
Member pulls on its joints; cross-section is under tensile stress.
Term
Compression (negative sign, −)
Example
Upper chord of a simply supported roof truss.
Definition
Member pushes on its joints; cross-section is under compressive stress.
Term
Zero-force member
Example
Vertical web member at a joint with two collinear members and no external load.
Definition
A member carrying no internal force under the given loading; present for geometric stiffness or other load cases.
Term
Pin joint
Example
Gusset plate connection in classical truss design.
Definition
Connection allowing rotation but transmitting force; no moment is transferred between members.
Diagrams To Know
- Two-force member free body diagram (two equal, opposite forces on axis)
- Joint equilibrium diagram (all member forces and reactions acting at a pin)
- Tension vs. compression force directions at a joint (pulls vs. pushes)
Formulas
Formula
m + r = 2j
Meaning
m = number of members; r = number of reaction components; j = number of joints
Watch Out
This equation is necessary but NOT sufficient — the truss can satisfy m + r = 2j and still be geometrically unstable if members are improperly arranged (concurrent, collinear groups without proper triangulation). Always verify rigid geometry.
When To Use
Check if a plane truss is statically determinate before attempting to solve.
Section Title
Determinacy of Plane Trusses
Important Facts
- Count reaction components: pin = 2 (horizontal + vertical); roller = 1 (perpendicular to surface); fixed = 3 (2 forces + 1 moment, rare in trusses).
- m + r = 2j is the minimum constraint count; proper triangulation from exterior pins ensures geometric stability.
- A truss can be determinate (m + r = 2j) in count but unstable if three or more members are concurrent at a joint or all lie in a straight line.
- Always draw the truss to scale and visualize before calculating; geometry rules.
Key Definitions
Term
Statically determinate truss
Example
Simple triangulated roof truss with 6 joints, 9 members, 3 reactions: 9 + 3 = 12 = 2(6) ✓
Definition
m + r = 2j and members are properly arranged in a rigid configuration; reactions and all member forces can be found using only equilibrium equations.
Term
Statically indeterminate truss
Example
Continuous truss over multiple spans or truss with redundant internal bracing.
Definition
m + r > 2j; more members or support constraints than required for rigidity; equilibrium alone is insufficient.
Term
Unstable/Mechanism
Example
Four joints forming a quadrilateral with only two diagonal members (no cross-bracing).
Definition
m + r < 2j; fewer constraints than needed; structure can move freely (mechanism, not a truss).
Diagrams To Know
- m + r vs 2j comparison diagram (determinate, indeterminate, unstable zones)
- Improperly triangulated truss (concurrent or collinear members) with m + r = 2j but unstable
- Simple triangle truss (determinate and stable geometry check)
Reactions Or Equations
Note
Indeterminate trusses require methods beyond equilibrium (flexibility method, stiffness method, etc.).
Equation
m + r > 2j → indeterminate (extra bracing or supports)
Conditions
Degree of indeterminacy = m + r − 2j (how many extra members or constraints)
Note
Structure will collapse or move under load; not acceptable as a truss.
Equation
m + r < 2j → unstable (mechanism, not a structure)
Conditions
Member count insufficient relative to joints
Note
This is the target for most exam problems.
Equation
m + r = 2j and proper arrangement → determinate (solvable by equilibrium alone)
Conditions
No concurrent or collinear member groups; members fan out from exterior supports
Formulas
Formula
At each joint: Σ F_x = 0, Σ F_y = 0
Meaning
Sum of horizontal forces = 0; sum of vertical forces = 0 (particle equilibrium at each pin joint)
Watch Out
Do NOT skip the support reactions step; you must know all external forces before analyzing joints. Assuming a member is in tension and getting a negative answer means compression — don't change your sign convention mid-problem.
When To Use
Solving for all member forces or to find forces at joints where only two unknowns exist.
Section Title
Method of Joints
Important Facts
- Start with support reactions (whole-structure equilibrium) FIRST: Σ F_x = 0, Σ F_y = 0, Σ M = 0.
- Begin analysis at a joint with at most 2 unknown member forces.
- Assume each unknown member is in tension (pulls the joint); if result is negative, the member is in compression (pushes the joint).
- Once a member force is found, carry it forward to adjacent joints; it is now known.
- Work systematically from known to adjacent joints until all member forces are found.
- Best for finding ALL member forces or when you need forces at several joints.
Key Definitions
Term
Joint equilibrium
Example
At joint C with three members and a 20 kN downward load: sum of vertical components of member forces = 20 kN upward.
Definition
Each pin joint is a particle with all member forces and external load (if any) in equilibrium; no rotation occurs at a pin.
Diagrams To Know
- Free body diagram of a single joint with all member forces as arrows along member axes
- Step-by-step joint progression diagram showing sequence of solution (starting joint → adjacent joints)
- Force parallelogram or polygon at a joint in equilibrium
Reactions Or Equations
Note
Magnitude is always positive; sign (+ or −) indicates tension or compression.
Equation
F_member = √(F_x² + F_y²) (magnitude, if components resolved separately)
Conditions
When member force is resolved into horizontal and vertical components
Formulas
Formula
Cut through ≤ 3 unknown members; apply Σ F_x = 0, Σ F_y = 0, Σ M_point = 0 to one side as a rigid body
Meaning
Select a section line that cuts through at most 3 members with unknown forces; isolate one side and treat it as a free body.
Watch Out
CRITICAL: Take moments about the intersection point of the other two cut members, leaving only one force unknown in the moment equation. If you take moment about the wrong point, you'll have two or three unknowns in one equation (unsolvable). Also, never cut more than 3 members with unknowns.
When To Use
Finding one, two, or three specific member forces deep in the truss without solving all members; faster than method of joints for a few interior members.
Section Title
Method of Sections (Ritter Method)
Important Facts
- Select a cut line that passes through the members of interest.
- Cut must separate the truss into two parts; choose the part with fewer unknowns or simpler geometry.
- Internal forces at the cut are drawn as external forces on the free body diagram (tension pulls the section away, compression pushes it together).
- Use moment equilibrium about a strategic point to eliminate variables.
- KEY STRATEGY: Take moments about the intersection of the two unknown members you want to eliminate; the third unknown (the one you want) stands alone in the moment equation.
- Faster than method of joints when only a few interior member forces are needed.
Key Definitions
Term
Section plane (cut line)
Example
A vertical or diagonal cut through 2–3 members in the middle of a bridge truss.
Definition
An imaginary line dividing the truss; the cut separates it into two parts, each treated as a rigid body in equilibrium.
Term
Free body diagram of section
Example
Left half of a truss after cutting through three members; internal forces at the cut are treated as external forces on this free body.
Definition
Diagram showing one side of the cut truss with all external loads, reactions, and internal member forces (at the cut) acting on it.
Diagrams To Know
- Section cut line through 3 members (diagonal cut showing which members are separated)
- Free body diagram of the left (or right) section with all external forces and internal forces at the cut
- Moment-center diagram showing the intersection point of the two members to eliminate
Reactions Or Equations
Note
The remaining cut member force can be solved directly from the moment equation.
Equation
Σ M_intersection = 0 (take moments about the intersection of two of the three cut members)
Conditions
Moment-center method; eliminates the two members passing through that point
Note
Usually only one force is required per problem; the moment strategy finds it instantly.
Equation
Σ F_x = 0 and Σ F_y = 0 (after finding the first unknown from moments)
Conditions
Use force equilibrium to find the remaining two unknowns if needed
Section Title
Zero-Force Members
Important Facts
- RULE 1: At a joint with exactly 2 non-collinear members and NO external load, both members are zero-force.
- RULE 2: At a joint with exactly 3 members, 2 collinear (along a straight line) and 1 non-collinear (the odd one out), and NO external load at that joint → the non-collinear member is zero-force.
- Identifying zero-force members early simplifies the analysis; remove them from the force-finding process.
- Zero-force members are NOT useless — they prevent buckling of other members and carry load under different load cases.
- Always check for zero-force conditions before applying method of joints or sections.
Key Definitions
Term
Zero-force member
Example
Vertical web in a panel with symmetric loading and no joint load at that panel.
Definition
A member carrying no internal force under the current loading; it remains for stiffness, other load cases, or to prevent buckling of adjacent members.
Diagrams To Know
- Joint with two non-collinear members and no load (both zero-force)
- Joint with three members: two collinear, one perpendicular; external load zero (perpendicular member is zero-force)
- Typical truss panel showing common zero-force member locations
Reactions Or Equations
Note
Both members are redundant for this load case.
Equation
At joint J: if F_load = 0 and members A, B non-collinear only → F_A = F_B = 0
Conditions
Exactly 2 members, no external load, members not along same line
Note
The odd member is zero; the collinear pair carries equal and opposite forces along the line.
Equation
At joint J: if F_load = 0 and members A, B collinear; member C non-collinear → F_C = 0
Conditions
Exactly 3 members, 2 collinear, 1 non-collinear, no external load
Formulas
Formula
Σ F_x = 0 (horizontal equilibrium)
Meaning
Sum of all horizontal external forces and reaction components = 0
Watch Out
Remember to include all applied loads and reaction components. A pin support has two reaction components (H and V); a roller has one (perpendicular to the support surface). Neglecting a reaction will lead to wrong answer in every joint.
When To Use
First step before analyzing joints or sections; finding horizontal reaction components.
Formula
Σ F_y = 0 (vertical equilibrium)
Meaning
Sum of all vertical external forces and reaction components = 0
Watch Out
Make sure the direction of the reaction is consistent with the support type. A roller cannot provide horizontal force; a pin can provide both.
When To Use
Finding vertical reaction components; must do before joint/section analysis.
Formula
Σ M_point = 0 (moment equilibrium about any point)
Meaning
Sum of all moments from applied loads and reactions, about any chosen point, = 0
Watch Out
Moment = force × perpendicular distance. Use the right-hand rule or sign convention (counterclockwise +, clockwise −, or vice versa — be consistent). A common error is forgetting to include the moment arm distance.
When To Use
Most efficient for finding one unknown reaction; take moments about the point where other unknowns act (they drop out).
Section Title
Reaction Calculation (Whole-Truss Equilibrium)
Important Facts
- ALWAYS solve reactions first (whole-structure equilibrium) before analyzing joints or sections.
- Three equilibrium equations available: Σ F_x = 0, Σ F_y = 0, Σ M = 0 (not all three may be independent for redundant structures).
- For a plane truss, usually 2 or 3 reaction components; simple support (pin + roller) → 3 reactions total.
- Strategy: Take moments about the pin support to find the roller reaction; then sum forces to find pin reactions.
- Check: Sum moments about a different point to verify reactions are correct.
Key Definitions
Term
Pin support (fixed joint)
Example
Support A at one end of a simply supported beam; pin can resist left/right and up/down motion but allows rotation.
Definition
Support preventing translation in two directions (horizontal and vertical); provides two reaction components.
Term
Roller support (simple support)
Example
Support B at the right end of a simply supported beam; roller can move along the support surface (or the surface can rotate beneath it) but resists vertical motion.
Definition
Support preventing translation perpendicular to the surface; provides one reaction component (normal to surface).
Term
Fixed support (built-in)
Example
Cantilever truss fixed into a wall.
Definition
Support preventing translation in two directions and rotation; provides two forces and one moment. (Rare in trusses; typically used in other structures.)
Diagrams To Know
- Pin support symbol (circle with lines/gusset) showing both H and V reaction components
- Roller support symbol (circle with triangle or roller wheel) showing perpendicular reaction
- Free body diagram of entire truss with all external loads and reaction arrows
Reactions Or Equations
Note
Roller is vertical (perpendicular to horizontal surface); takes vertical load only.
Equation
Simple support (pin at A, roller at B): R_A_x + R_B_x = 0 (or external horizontal loads), R_A_y + R_B_y = sum of downward loads
Conditions
Pin support at left, roller at right, typical for horizontal span
Formulas
Formula
θ = tan⁻¹(Δy / Δx)
Meaning
θ = angle of member from horizontal; Δy = vertical distance between end joints; Δx = horizontal distance between end joints
Watch Out
Be careful with the sign of Δy and Δx — use absolute values for angle magnitude. Always verify the angle lies in the correct quadrant relative to the member's orientation. Use tan⁻¹ (inverse tangent) carefully; verify with a diagram.
When To Use
Finding the angle when a member is inclined (not horizontal or vertical); needed to resolve member force into x and y components.
Formula
L = √(Δx² + Δy²)
Meaning
L = length of the member; Δx = horizontal span; Δy = vertical rise
Watch Out
This is just the Pythagorean theorem; always check your coordinate input. Errors here propagate to force component calculations.
When To Use
When member length is needed for stress calculations or to verify geometry.
Formula
F_x = F_member × cos(θ), F_y = F_member × sin(θ)
Meaning
F_x = horizontal component of member force; F_y = vertical component; F_member = magnitude of internal force in the member; θ = angle of member from horizontal
Watch Out
The direction of the component depends on the assumed direction of the member force (tension pulls away from the joint; compression pushes toward it). Verify the sign: for a tension force pulling diagonally up-right, both F_x and F_y should be positive in the expected directions.
When To Use
Resolving the internal member force into x and y components for joint equilibrium equations.
Common Values
Value
≈ 0.577
Symbol
tan(30°)
Quantity
tan(30°)
Value
≈ 0.866
Symbol
cos(30°)
Quantity
cos(30°)
Value
0.5
Symbol
sin(30°)
Quantity
sin(30°)
Value
1
Symbol
tan(45°)
Quantity
tan(45°)
Value
≈ 0.707
Symbol
cos(45°)
Quantity
cos(45°)
Value
≈ 0.707
Symbol
sin(45°)
Quantity
sin(45°)
Value
≈ 1.732
Symbol
tan(60°)
Quantity
tan(60°)
Value
0.5
Symbol
cos(60°)
Quantity
cos(60°)
Value
≈ 0.866
Symbol
sin(60°)
Quantity
sin(60°)
Section Title
Member Geometry & Angle Calculation
Important Facts
- Always work in a consistent coordinate system (x = horizontal right, y = vertical up).
- Member angles are usually measured from the horizontal (0° to 90° in typical truss panels).
- For a member sloping down-left to up-right: θ is positive; for a member sloping down-right to up-left: θ is measured as approaching 90° from the opposite direction (or use absolute angles and track signs carefully).
- Cos(θ) affects horizontal components; Sin(θ) affects vertical components.
- When multiple members meet at a joint, each must be resolved; the sum of all components in each direction must equal zero.
Diagrams To Know
- Right triangle showing Δx, Δy, L, and angle θ from horizontal
- Member force resolution diagram: F_member along the member axis, F_x and F_y components shown perpendicular
- Common angle triangles (3-4-5, 45-45-90, 30-60-90) with side ratios labeled
Reactions Or Equations
Note
Also: tan(θ) = 4/3 ≈ 53.13°. Many trusses use these or scaled versions (6-8-10, 9-12-15, etc.).
Equation
For a 3-4-5 right triangle member: cos(θ) = 3/5 = 0.6, sin(θ) = 4/5 = 0.8
Conditions
Common angles in roof trusses; memorize these ratios for speed in exams.
Note
Force components are equal; simplifies calculation for 45° members.
Equation
45° member (equal Δx and Δy): cos(45°) = sin(45°) ≈ 0.707, tan(45°) = 1
Conditions
Diagonal members in square or symmetric panels
Note
Useful for steep roof trusses.
Equation
30-60-90 triangle member: cos(30°) ≈ 0.866, sin(30°) = 0.5; cos(60°) = 0.5, sin(60°) ≈ 0.866
Conditions
Some roof and lattice trusses use these angles
Section Title
Sign Convention & Interpretation
Important Facts
- Assume all unknowns are tension (positive) when setting up equilibrium equations.
- If the solution is negative, the member is in compression; do NOT change your assumption mid-problem; the negative sign tells you the actual state.
- At a joint, a tension member pulls the joint along its axis; a compression member pushes the joint along its axis.
- When drawing a free body diagram of a joint, show member forces as arrows pulling (tension) or pushing (compression) the joint.
- The magnitude of the force is |F_member|; the sign (+ or −) indicates the state (tension or compression).
Key Definitions
Term
Tension (+)
Example
Lower chord of a simply supported roof — pulls on both ends.
Definition
Member pulls on both of its joints; internal stress in the member is tensile; member is elongated (or resists elongation).
Term
Compression (−)
Example
Upper chord of a simply supported roof — pushes on both ends; tends to buckle.
Definition
Member pushes on both of its joints; internal stress in the member is compressive; member tends to shorten (or resists shortening).
Diagrams To Know
- Tension member at a joint: force arrow points away from the joint (pulling)
- Compression member at a joint: force arrow points toward the joint (pushing)
- Free body diagram of a joint showing mixed tension and compression members
Reactions Or Equations
Note
Sign convention is consistent throughout the problem.
Equation
F_member(+) = Tension; F_member(−) = Compression
Conditions
Solving method of joints or method of sections
Section Title
Common Truss Types & Patterns
Important Facts
- Warren truss: diagonal members alternate up and down; all diagonals at the same angle (efficient in bending).
- Pratt truss: vertical members connected to top chord, diagonals slope inward and downward from top to bottom (efficient for long spans; compression diagonals resist buckling better).
- Howe truss: vertical members connected to bottom chord, diagonals slope outward and downward (compression verticals; tension diagonals).
- King-post truss: single vertical member from midspan of bottom chord to apex; two diagonals from top corners to the top of the king post (simple, short spans).
- Queen-post truss: two vertical members and more complex internal bracing (medium spans).
- For simply supported trusses: upper chord is in compression, lower chord is in tension under downward loads; diagonals alternate tension/compression depending on slope and loading.
Key Definitions
Term
Simple truss
Example
Roof truss built by successively adding triangular panels.
Definition
A truss formed by starting with a triangle and adding two new members and one new joint at a time, maintaining triangulation and avoiding concurrent or collinear member groups.
Term
Compound truss
Example
Multiple roof panels connected at a central ridge line.
Definition
Two or more simple trusses connected together (end-to-end or side-by-side) by a common joint or member.
Term
Complex truss
Example
Some high-rise bracing systems.
Definition
A truss that is neither simple nor compound; members intersect in complex patterns (rare in exams but possible in advanced structures).
Diagrams To Know
- Warren truss profile with alternating diagonal pattern
- Pratt truss profile showing vertical and diagonal layout
- Howe truss profile with opposite diagonal pattern
- King-post and queen-post truss sketches
Must Remember
- Every truss member is a two-force member: force acts along the axis, pulling (tension +) or pushing (compression −) on the joints. Assume tension; negative result = compression.
- Determinacy: m + r = 2j is necessary but NOT sufficient — the truss must also have proper geometry (no concurrent or collinear member groups without additional triangulation). Always verify both the count AND the rigid arrangement.
- Method of Joints: Start with support reactions (whole structure, then joint by joint). Begin at a joint with at most 2 unknowns; solve, then move to adjacent joint. Best for ALL member forces.
- Method of Sections: Cut through ≤3 unknown members; take moments about the intersection of the OTHER two cut members. The third unknown solves alone from the moment equation. Fast for a few interior members.
- Zero-Force Members: (1) Joint with 2 non-collinear members, no load → both zero. (2) Joint with 3 members (2 collinear, 1 odd), no load → the odd member is zero. Identify early; they simplify analysis.
- Member angles & components: θ = tan⁻¹(Δy/Δx); then F_x = F·cos(θ), F_y = F·sin(θ). Common angles: 3-4-5 (37° & 53°), 45-45-90 (45°), 30-60-90. Use these for mental math in exams.
- Sign convention: Always assume unknowns are POSITIVE (tension). If the solution is negative, that member is in compression. Do NOT change the sign assumption mid-problem; the negative sign tells you the state.
- Support reactions MUST be found FIRST (equilibrium of the whole structure: Σ F_x = 0, Σ F_y = 0, Σ M = 0). Without correct reactions, every joint will be wrong.
- Truss geometry: Simple truss = triangle + pairs of new members and joints (maintaining triangulation). Compound = simple trusses connected. Complex = neither (rare). Verify stability before solving.
- Exam pitfalls: (a) Skipping reactions; (b) Cutting more than 3 unknowns in sections; (c) Taking moment about the wrong point (leaving 2+ unknowns in the equation); (d) Confusing member angle (recompute from coordinates, don't reuse another panel's angle); (e) Sign error in components (check direction of the force arrow).
Last Minute Tips
- Draw the whole free body diagram of the truss with ALL external loads and reaction arrows before solving ANY member forces. Errors in reactions = errors in every joint. Take 1 minute, save 20 minutes of rework.
- For method of sections: immediately identify the intersection point of the two members you DON'T want to solve (the 'other two'), then take moments about that point. The third member force comes out in one line. This trick is an exam time-saver.
- Spot zero-force members BEFORE you start method of joints. Remove them from the diagram. If a web diagonal is zero-force, you have one fewer unknown per adjacent joint, cascading savings throughout the truss.
- Check determinacy (m + r = 2j) AND visually verify rigid geometry. A problem can state 'solve the truss' but be unsolvable if the member arrangement is concurrent or collinear without proper triangulation. Trust the visual check.
- On exam day, if a member force comes out unreasonably large (e.g., 100× the applied load), stop and re-examine: check the angle calculation, the reaction input, and the sign convention. Magnitude errors are usually geometry or arithmetic mistakes, not concept errors.
Comparison Tables
Rows
Values
- Finding all member forces or forces at multiple joints.
- Finding one, two, or three specific interior member forces.
Property
When to use
Values
- Joint with at most 2 unknowns.
- Select a section cut through ≤3 unknown members.
Property
Starting point
Values
- Joint by joint: draw FBD, apply Σ F_x = 0 and Σ F_y = 0, solve, move to adjacent joint.
- Cut the truss, isolate one side, apply Σ F and Σ M to the entire section as a rigid body.
Property
Procedure
Values
- 2 equations per joint (Σ F_x and Σ F_y); can have at most 2 unknowns at each step.
- 3 equations total (Σ F_x, Σ F_y, Σ M); can handle up to 3 unknowns in one cut.
Property
Equilibrium equations per step
Values
- N/A — only use force equilibrium at a joint.
- Take Σ M about the intersection of the other two cut members; the third unknown solves directly.
Property
Moment about which point?
Values
- Slow — must analyze every joint sequentially.
- Fast — cut once, solve in one or two steps.
Property
Speed for a few members
Values
- Faster — systematic, fewer decisions required.
- Slow — multiple cuts required; cumulative time.
Property
Speed for all members
Values
- High — an error at one joint carries forward to all subsequent joints.
- Lower — each cut is independent; errors confined to that section.
Property
Risk of propagating errors
Columns
- Aspect
- Method of Joints
- Method of Sections
Table Title
Method of Joints vs. Method of Sections
Rows
Values
- Mechanism (unstable)
- Structure collapses; incomplete constraints.
- Not solvable; unacceptable truss.
- Check the problem statement — likely an error. Verify member count and support type.
Property
m + r < 2j
Values
- Determinate and stable
- All reactions and member forces uniquely determined by equilibrium alone.
- Solvable by method of joints or sections.
- Proceed with analysis; use the method best suited to what you need (all forces vs. a few specific ones).
Property
m + r = 2j AND proper geometry (triangulated, no concurrent/collinear groups)
Values
- Geometrically unstable (internal mechanism)
- Truss satisfies count equation but can collapse internally; not rigid.
- NOT solvable by equilibrium alone; static analysis invalid.
- Reject the configuration; this truss will fail. Requires redesign of member arrangement.
Property
m + r = 2j BUT improper geometry (concurrent or collinear members)
Values
- Statically indeterminate
- More constraints than needed; redundant members or supports.
- Equilibrium equations alone insufficient; requires strain compatibility or flexibility/stiffness methods.
- Beyond scope of basic licensure exam; if this occurs, problem may be mislabeled. Verify problem statement.
Property
m + r > 2j
Columns
- Condition
- Outcome
- Solvability
- Exam Action
Table Title
Determinacy Check: Conditions vs. Outcome
Rows
Values
- Horizontal (R_x) and Vertical (R_y)
- 2
- Circle or ball-and-socket; prevents translation in x and y; allows rotation.
Property
Pin (hinge, fixed joint)
Values
- One component perpendicular to surface (usually vertical R_y for horizontal surfaces)
- 1
- Circle on lines or wheels; prevents motion perpendicular to support surface; allows sliding and rotation.
Property
Roller (simple support)
Values
- Horizontal (R_x), Vertical (R_y), and Moment (M)
- 3
- Prevents all translation and rotation. Rare in truss problems; common in cantilever beams.
Property
Fixed (built-in, cantilever)
Values
- None
- 0
- No constraint; used for load application points (joints), not support.
Property
Free end (no support)
Columns
- Support Type
- Reaction Components
- No. of Unknowns
- Symbol / Notes
Table Title
Support Types: Reactions & Constraints
Rows
Values
- No
- Both members are zero-force.
- Rare in practice; if it occurs, both are useless for that load case.
Property
Exactly 2 members, non-collinear (e.g., L-shaped)
Values
- No
- The perpendicular (non-collinear) member is zero-force; the collinear pair carries equal and opposite forces.
- Common in web diagonals; vertical member at a joint with two horizontal members carries no load.
Property
Exactly 3 members: 2 collinear, 1 perpendicular (e.g., T-shaped or cross)
Values
- Yes
- No zero-force members (load must be carried by the members).
- Each member contributes; cannot drop any.
Property
Any number of members, with external load at joint
Values
- No
- Must analyze by method of joints; cannot deduce from simple rules.
- Use equilibrium equations; do not assume zero-force without proof.
Property
4+ members, no external load, complex arrangement
Columns
- Joint Configuration
- External Load at Joint?
- Zero-Force Member(s)
- Example Sketch / Note
Table Title
Zero-Force Member Rules: Quick Recognition
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