Method of tension coefficients – definition of the coefficient In the method of tension coefficients for analyzing pin-jointed frames, the “tension coefficient” of a member is defined as:

Introduction / Context:
The method of tension coefficients is a convenient approach for solving pin-jointed frames by resolving joint equilibrium in terms of member “tension coefficients,” avoiding repeated trigonometric calculations for direction cosines.

Given Data / Assumptions:

• Pin-jointed truss with two-force members.
• Linear elasticity is not required; only static equilibrium and geometry are used.
• Members are massless for analysis purposes.

Concept / Approach:
The tension coefficient t_ij of member ij is defined as the member tension T_ij divided by its length L_ij: t_ij = T_ij / L_ij Joint equilibrium can then be written as the sum of t_ij times the respective coordinate differences, simplifying hand calculations.

Step-by-Step Solution:
Definition: T_ij acts along the member.Unit vector from i to j has components proportional to (Δx_ij / L_ij, Δy_ij / L_ij).Joint equilibrium at node i becomes Σ(t_ij * Δx_ij) = −Σ external_x and Σ(t_ij * Δy_ij) = −Σ external_y.Thus, the fundamental quantity is T_ij / L_ij, i.e., the tension coefficient.

Verification / Alternative check:
Compare to direction cosines method: replacing T_ij * (Δx_ij / L_ij) with t_ij * Δx_ij reduces arithmetic complexity.

Why Other Options Are Wrong:

• (a) is stress, not a tension coefficient.
• (c) is a component of force, not normalized by length.
• (d) is just the force magnitude.
• (e) is strain (dimensionless), unrelated to this definition.

Common Pitfalls:
Mixing up stress (force/area) with tension coefficient (force/length). Keep track of geometry clearly.

Final Answer:
Tension in the member divided by its length