Prestressed concrete beam with bent tendon — for an external load W and prestress P at inclination θ (two symmetric bends), the net downward load at midspan equals:

Introduction / Context:
In prestressed beams, deviated (bent) tendons produce vertical components of prestressing that counteract gravity loads. Understanding this vertical component is key to load balancing design and to estimating net design load on the concrete section at midspan or at deviator points.

Given Data / Assumptions:

• Two symmetric tendon legs inclined at angle θ to the beam axis at midspan region.
• Prestressing force magnitude in each leg ≈ P (neglecting minor losses/curvature friction).
• External downward load W acting at midspan.

Concept / Approach:

Each inclined tendon leg contributes an upward vertical component equal to P sin θ. With two symmetric legs, the total upward component is 2 P sin θ. The load-balancing concept subtracts this from the external downward load to obtain the net downward action on the concrete section.

Step-by-Step Solution:

Verification / Alternative check (if short method exists):

Draw a free-body diagram at the deviated zone; equilibrium in vertical direction confirms V_up = 2 P sin θ. This is standard in load balancing methods.

Why Other Options Are Wrong:

Terms with cos θ are horizontal components; single P sin θ neglects the symmetric second leg; adding 2P sin θ would reverse the intended balancing effect.

Common Pitfalls (misconceptions, mistakes):

Using cos instead of sin; forgetting there are two inclined tendon legs contributing vertical components.

Final Answer:

W - 2P sin θ