Questions: Statically Determinate Systems Analysis

Question 1

A bridge beam is supported by a pin at the left end, a roller at the middle, and a roller at the right end. How many unknown reaction forces does this create, and what does that imply about the beam?

• A. 3 unknowns (pin = 2, two rollers = 1 each = 2, total = 4) — wait, that's 4: statically indeterminate
• B. 4 unknowns (pin = 2, roller = 1, roller = 1) — statically indeterminate to the first degree
• C. 3 unknowns (pin = 1, two rollers = 1 each) — statically determinate
• D. 5 unknowns — requires material stiffness properties to solve

Answer: (B)

A pin provides two unknown reactions (horizontal and vertical force components); each roller provides one (perpendicular force only). So: 2 + 1 + 1 = 4 unknowns, but a 2D structure has only 3 equilibrium equations (ΣFx = 0, ΣFy = 0, ΣM = 0). With 4 unknowns and 3 equations, the beam is statically indeterminate to the first degree — you cannot solve for all reactions from equilibrium alone. Option C is the most common error: students undercount the pin's contribution, assigning it only one unknown instead of two.

Question 2

A structural engineer calculates member forces in a statically determinate truss and then switches to using steel that is twice as stiff as originally planned. How does this affect the calculated member forces?

• A. The member forces double, because stiffer members attract more load through the structure
• B. The member forces are unchanged — in a statically determinate structure, forces depend only on geometry and loading, not on material stiffness
• C. The member forces halve, because the stiffer structure deflects less and therefore absorbs less energy
• D. The engineer must redo the analysis using the stiffness matrix method to account for the material change

Answer: (B)

This is the defining practical property of static determinacy: all reactions and internal forces follow from equilibrium equations alone. Equilibrium only involves geometry (member directions, loading positions) and applied loads — material stiffness does not appear in ΣF = 0 or ΣM = 0. Option A describes the behavior of a statically indeterminate structure, where stiffer members do attract more load because the load distribution depends on relative stiffnesses. Choosing between determinate and indeterminate analysis is therefore not just a math question — it's a question about what information you actually need.

Question 3

Converting a statically determinate beam from a pin-and-roller support to a pin-and-pin support makes it statically indeterminate, requiring the beam's flexural stiffness to find all support reactions.

• T. True
• F. False

Answer: True

Pin-and-roller gives 2 + 1 = 3 unknowns, matching the 3 equilibrium equations exactly — determinate. Replacing the roller with a pin adds one more unknown (now 2 + 2 = 4), exceeding the 3 available equations. The extra reaction force cannot be found from statics alone; it depends on how the beam flexes under load, which depends on the material's stiffness EI. This is the transition from statics (forces from equilibrium) to mechanics of materials (forces from compatibility of deformations).

Question 4

A statically determinate structure is typically stronger and safer than a statically indeterminate structure with the same loading, because its forces can be uniquely solved from equilibrium.

• T. True
• F. False

Answer: False

Determinacy refers to analytical tractability, not structural performance. Statically indeterminate structures have multiple load paths — if one member fails, others can redistribute the load. Determinate structures lack redundancy: if one critical member or support fails, the entire structure can collapse. Indeterminate structures are typically stronger and more robust under overload or partial failure. Determinacy is valuable for analysis simplicity and for understanding how forces flow, not as a measure of structural quality.

Question 5

Explain why statically determinate structures can be analyzed without knowing material properties, and what additional information is needed to analyze a statically indeterminate structure.

Model answer: Equilibrium equations (ΣF = 0, ΣM = 0) express balance of forces and moments — they contain only geometric quantities (distances, angles) and force magnitudes. Material properties like Young's modulus or cross-sectional moment of inertia do not appear in the equilibrium equations. A statically determinate structure has exactly as many unknowns as equations, so the system is solvable from equilibrium alone. A statically indeterminate structure has more unknowns than equilibrium equations, so some unknowns cannot be resolved by statics. The additional equations needed come from compatibility conditions — constraints on how the structure deforms — which depend on material stiffness (EI for beams, AE for axial members). You must solve equilibrium and compatibility simultaneously.

This is why engineering curricula introduce statics before mechanics of materials: determinate structures can be fully analyzed with Newton's laws alone, providing a clean foundation. Indeterminate analysis adds deformation compatibility, requiring the material constitutive law (stress = E × strain) to close the system of equations.