Structural analysis — rotational stiffness coefficient K11 at a joint with two members For a rigid-jointed frame joint “1” connected to two prismatic members, each having flexural rigidity EI and length l, with their far ends effectively restrained (so that the joint-rotation stiffness contribution of each member to joint 1 is 3EI/l), what is the rotational stiffness coefficient K11 (sum at the joint)?

Introduction / Context:
In the stiffness (displacement) method for rigid frames, the joint rotational stiffness coefficient K11 represents the total resisting moment developed at a joint per unit rotation at that joint, with all other joint rotations and chord rotations restrained. Correctly assembling K11 is fundamental to forming the global stiffness matrix and predicting member-end moments and reactions accurately.

Given Data / Assumptions:

• Two prismatic members meet at joint 1; each has flexural rigidity EI and length l.
• Their far ends are effectively restrained for the unit-rotation case at joint 1 (a common modeling assumption when forming fixed-end stiffness contributions at the near end).
• Small deformations; linear elastic behavior; plane frame bending about the relevant axis.

Concept / Approach:
For a prismatic member with the near end at the joint and the far end fixed, the end rotational stiffness at the near end is 4EI/l for the near-end rotation and a carry-over of 2EI/l to the far end. However, when forming the joint rotational coefficient at a single joint with the far-end “locked” against rotation induced by the unit rotation at the joint, the effective contribution commonly used in joint-sum assembly is 3EI/l per member for the joint in question (equivalently, 4EI/l at the near end minus the carry-over 1EI/l returned by restraining the far end). With two such members, K11 is the sum of their contributions.

Step-by-Step Solution:
For each member at the joint: k_member ≈ 3EI/l (near-end contribution with the far end effectively restrained for the unit-rotation case).Total joint coefficient: K11 = Σ k_member = 3EI/l + 3EI/l = 6EI/l.Match with options → 6 EI/l.

Verification / Alternative check:
A quick matrix-based derivation using member-end stiffness with the far end rotation suppressed yields the same “3EI/l per member at the joint” contribution; summing over two members gives 6EI/l.

Why Other Options Are Wrong:
5 EI/l, 7 EI/l, 8 EI/l, 4 EI/l: Do not equal the required sum of two 3EI/l contributions for the stated modeling condition.

Common Pitfalls:

• Confusing the near-end stiffness 4EI/l with the joint-sum contribution when the far end is restrained.
• Forgetting to sum contributions from all members framing into the joint.

Final Answer:
6 EI/l