Structure analysis—slope deflection: The number of simultaneous equations that must be solved in the slope-deflection method equals which measure of the structure?

Introduction / Context:
The slope-deflection method expresses member end moments as functions of joint rotations and joint translations (sways) plus fixed-end moments from loads. Solving the structure reduces to writing joint equilibrium equations in terms of these unknown joint displacements and then solving them simultaneously.

Given Data / Assumptions:

• Members obey linear elastic behavior.
• Unknowns are joint rotations and joint translations (if sway is permitted).
• End releases and boundary conditions are known.

Concept / Approach:

Kinematic indeterminacy is the number of independent displacement unknowns required to define the deformed shape (excluding known boundary displacements). Each unknown displacement leads to one equilibrium equation (sum of moments at a joint or overall sway equilibrium), hence the number of simultaneous equations equals the degree of kinematic indeterminacy (not the static indeterminacy).

Step-by-Step Solution:

Verification / Alternative check:

Matrix stiffness method generalizes the same concept: system size equals DOFs (kinematic indeterminacy). Slope-deflection is a precursor with explicit end-moment formulations.

Why Other Options Are Wrong:

• Statical indeterminacy counts redundant forces, not displacement unknowns.
• Number of joints/members is not a direct measure of unknown DOFs.

Common Pitfalls:

• Forgetting to include frame sway DOFs when lateral loads act.

Final Answer:

the degree of kinematic indeterminacy