Structural Load Decomposition Solution

In civil and structural engineering, complex loads acting on bridge trusses and framework joints must be resolved along individual steel members. By projecting external force vectors onto member axes, engineers determine whether structural beams are under tension or compression.

A structural node on a pedestrian bridge truss experiences a downward gravitational load vector of $\vec{F}_{\text{load}} = \langle 0, -50 \rangle\text{ kN}$. One of the diagonal steel support beams framing into this junction is oriented along the vector direction $\vec{a} = \langle 4, 3 \rangle$.

1. Find the unit vector $\hat{u}_a$ pointing in the direction of the support beam $\vec{a}$.
2. Compute the scalar component ($\text{comp}_{\vec{a}}\vec{F}_{\text{load}}$) of the load directed along the axis of this support beam.
3. Find the vector projection ($\text{proj}_{\vec{a}}\vec{F}_{\text{load}}$) of the load onto the support beam.

Step-by-Step Solution:

1. Find the Unit Vector $\hat{u}_a$: First, compute the magnitude of $\vec{a}$:

   $$|\vec{a}| = \sqrt{4^2 + 3^2} = \sqrt{16 + 9} = \sqrt{25} = 5$$
   Divide vector $\vec{a}$ by its magnitude:
   $$\hat{u}_a = \frac{\vec{a}}{|\vec{a}|} = \left\langle \frac{4}{5}, \frac{3}{5} \right\rangle = \langle 0.8, 0.6 \rangle$$

2. Compute the Scalar Component ($\text{comp}_{\vec{a}}\vec{F}_{\text{load}}$): Take the dot product of the load vector and the unit vector:

   $$\text{comp}_{\vec{a}}\vec{F}_{\text{load}} = \vec{F}_{\text{load}} \cdot \hat{u}_a = (0)(0.8) + (-50)(0.6) = 0 - 30 = -30\text{ kN}$$

3. Compute the Vector Projection ($\text{proj}_{\vec{a}}\vec{F}_{\text{load}}$): Multiply the scalar projection by the unit vector:

   $$\text{proj}_{\vec{a}}\vec{F}_{\text{load}} = (\text{comp}_{\vec{a}}\vec{F}_{\text{load}}) \hat{u}_a = -30 \langle 0.8, 0.6 \rangle = \langle -24, -18 \rangle\text{ kN}$$

Meaning of the Answer & Real-Life Application:

The scalar component value of $-30\text{ kN}$ indicates that exactly $30\text{ kN}$ of the $50\text{ kN}$ downward load is transmitted directly along the axis of the support beam. The negative sign signifies that the axial force acts in the direction opposite to $\vec{a}$, putting the steel member into compression. Civil engineers use this projection calculation to size the beam cross-section and select steel grade to prevent buckling under compressive loads.