(a) Show that, if a vertical force 𝐏 is applied at point A of the section shown, the equation of

(a) Show that, if a vertical force 𝐏 is applied at point A...

(a) Show that, if a vertical force $\mathbf{P}$ is applied at point $A$ of the section shown, the equation of the neutral axis $B D$ is \[\left(\frac{x_{A}}{r_{z}^{2}}\right) x+\left(\frac{z_{4}}{r_{x}^{2}}\right) z=-1\] where $r_{z}$ and $r_{x}$ denote the radius of gyration of the cross section with respect to the $z$ axis and the $x$ axis, respectively. ( $b$ ) Further show that, if a vertical force $\mathbf{Q}$ is applied at any point located on line $B D,$ the stress at point $A$ will be zero.

(a) Show that, if a vertical force $\mathbf{P}$ is applied at point $A$ of the section shown, the equation of the neutral axis $B D$ is
\[\left(\frac{x_{A}}{r_{z}^{2}}\right) x+\left(\frac{z_{4}}{r_{x}^{2}}\right) z=-1\]
where $r_{z}$ and $r_{x}$ denote the radius of gyration of the cross section with respect to the $z$ axis and the $x$ axis, respectively. ( $b$ ) Further show that, if a vertical force $\mathbf{Q}$ is applied at any point located on line $B D,$ the stress at point $A$ will be zero.

Key Concepts

Bending Moment and Stress Distribution

In the context of beam bending, the applied vertical force produces a bending moment that generates normal stresses throughout the cross?section according to the bending stress formula. The stress at any point is proportional to the moment and the distance from the neutral axis, and inversely proportional to the moment of inertia. This basic concept underpins the derivation of conditions under which the stress at a specific point becomes zero, reflecting the transition from compressive to tensile stress.

Neutral Axis

The neutral axis is the line within the cross section of a beam where bending stresses vanish under an applied moment. The location and orientation of the neutral axis result from the distribution of the cross-sectional area and the applied force, and its equation can be derived by setting the moment-induced stress to zero. Understanding the neutral axis is essential in determining the bending behavior and designing sections to manage stresses efficiently.

Moment of Inertia and Radius of Gyration

The moment of inertia quantifies a cross section's resistance to bending about a given axis, while the radius of gyration provides a measure by relating the moment of inertia to the area. These properties are critical in establishing how the beam's geometry influences its response under bending loads. They appear prominently in the derivation of the neutral axis equation, linking geometric characteristics with the stress distribution.

Static Equilibrium and Moment Balance

The analysis of bending under a vertical force relies on the principles of static equilibrium, where sums of forces and moments are balanced. By considering the moments generated by applied forces about different axes, one can derive relationships and equations—such as the equation for the neutral axis—that ensure internal equilibrium. This concept is a core element in structural analysis and mechanics, ensuring that the deformation response is consistent with the applied loads.