Questions: Space Trusses: Three-Dimensional Analysis

Question 1

A space truss has 12 joints and is supported by 6 reaction components. How many members are required for it to be statically determinate?

• A. 18 members (applying the 2D rule m = 2n − 3)
• B. 30 members (applying the 3D rule m = 3n − 6: 3×12 − 6 = 30)
• C. 24 members (assuming 3D simply doubles the 2D requirement)
• D. 6 members (using only the reaction component count)

Answer: (B)

The determinacy condition for a space truss is m = 3n − 6, where m is the number of members and n is the number of joints. With n = 12: m = 3(12) − 6 = 30. The 3 in the formula comes from having three equilibrium equations per joint in 3D (ΣFx = ΣFy = ΣFz = 0); the −6 accounts for the six reaction components (three force and three moment) provided by the supports in space. Option A applies the 2D rule m = 2n − 3, which is incorrect for 3D. The tetrahedral base case confirms this: 4 joints, 6 members → 6 = 3(4) − 6 ✓.

Question 2

When solving a joint in a space truss where exactly three member forces are unknown, what is the correct procedure?

• A. Write ΣFx = 0 and ΣFy = 0 only; the third unknown must be found from an adjacent joint
• B. Write ΣFx = 0, ΣFy = 0, and ΣFz = 0; these three equations solve directly for the three unknown member forces
• C. Apply the method of sections by cutting through all three unknown members and solving six equilibrium equations
• D. Use energy methods to avoid the unit vector calculations required for 3D equilibrium

Answer: (B)

At any joint with exactly three unknown member forces, the three 3D equilibrium equations form a system of three equations in three unknowns — exactly solvable. This is the 3D analog of the 2D method of joints. The key setup step is expressing each unknown member force as F·û (where û is the unit vector along the member computed from position vectors), then projecting into x, y, and z components. Option A applies 2D logic and leaves the system underdetermined. Method of sections (option C) is useful for finding specific member forces without solving the whole truss, not as the primary joint method.

Question 3

In a space truss, members can carry both axial and bending loads, which is why 3D analysis requires more equilibrium equations than 2D analysis.

• T. True
• F. False

Answer: False

Space truss members carry axial loads only — exactly like 2D truss members. Joints are spherical pins (ball-and-socket), which cannot transmit moments, so members cannot develop bending moments regardless of geometry. The reason 3D analysis requires more equations is simply that equilibrium must be satisfied in three spatial dimensions (ΣFx = ΣFy = ΣFz = 0) rather than two. The physics — members as two-force elements — is identical to 2D; only the dimensionality of the equilibrium system changes.

Question 4

A space truss that satisfies m = 3n − 6 is expected to be rigid and stable under any loading.

• T. True
• F. False

Answer: False

m = 3n − 6 is a necessary but not sufficient condition for determinacy and stability. A truss can satisfy the member count while still being geometrically unstable if members are arranged so that some are redundant in one region while another region is a mechanism. The classic failure mode is adding extra members in one part of the truss while leaving a joint elsewhere under-constrained. Stability requires both the correct member count AND an appropriate geometric arrangement — satisfying the equation is not enough on its own.

Question 5

A space truss analysis requires computing unit vectors for each member before writing equilibrium equations. Why is this step essential?

Model answer: Each member force acts along the member's axis. To apply 3D equilibrium, you must decompose that force into its x, y, and z components. The unit vector û = (rⱼ − rᵢ)/|rⱼ − rᵢ| gives the direction from joint i toward joint j; multiplying the unknown force magnitude F by û yields the three components (Fx, Fy, Fz). Without explicit unit vectors, it is extremely difficult to correctly distribute a 3D member force into components — especially for non-axis-aligned members. Computing unit vectors from a node coordinate table before writing any equations prevents sign errors, missed components, and incorrect angle assumptions.

The disciplined procedure is: list all node coordinates in a table, compute position vectors and unit vectors for each member, then write the three equilibrium equations at each joint by summing force components from all connected members. The unit vector step converts a geometry problem into a systematic algebraic one — skipping it is the most common source of cascading errors in 3D truss analysis.