Plastic analysis of a fixed–fixed beam under a central point load A prismatic fixed beam of span L has a uniform plastic moment capacity Mp along its length. If a single concentrated load is applied at the midspan, what is the ultimate (collapse) load that will form the plastic mechanism?

Introduction / Context:
Plastic analysis determines the ultimate load at which a structure forms a collapse mechanism through plastic hinges. For a fixed–fixed (fully restrained) beam subjected to a central concentrated load, understanding where plastic hinges form and how to apply the principle of virtual work is essential for computing the collapse load.

Given Data / Assumptions:

• Span = L; the beam is fixed at both ends.
• Uniform plastic moment capacity along the member = Mp.
• Single concentrated load placed at midspan.
• Material is perfectly plastic; small deflection kinematics at incipient collapse.

Concept / Approach:
A fixed–fixed beam is statically indeterminate to degree 2. To obtain a collapse mechanism, the number of plastic hinges must equal the degree of indeterminacy plus one. Thus, three hinges are required: one at each support and one at the load point (midspan). Using the virtual work method, the external work of the load equals the internal work of plastic rotations at the hinge locations.

Step-by-Step Solution:
1) Anticipate hinge positions: at the two fixed ends and at midspan under the point load.2) Let the midspan hinge rotate by a small angle θ; consistent end-hinge rotations develop due to symmetry.3) Internal work = Mp * (sum of plastic rotations at three hinges).4) External work = W_c * (vertical displacement at the load) expressed in terms of θ by the kinematic mechanism.5) Equating external and internal work yields W_c = 8 Mp / L.

Verification / Alternative check:
Tabulated plastic collapse loads for common mechanisms list W_c = 8 Mp/L for a fixed–fixed beam with a central point load, matching the virtual work derivation.

Why Other Options Are Wrong:
4 Mp/L and 6 Mp/L underpredict the load because they do not account for all three hinge rotations; Mp/8L is dimensionally incorrect and far too small.

Common Pitfalls:
Forgetting that a fixed–fixed beam needs three hinges at collapse; confusing elastic moment capacities with plastic moment Mp; neglecting the correct kinematic displacement relation in virtual work.

Final Answer:
8 Mp/L