Let's break down the solution into **two main steps** as you requested. --- ## **Step 1: Draw the Free Body Diagram (FBD) of Joint E** ### **Explanation:** We need to analyze the forces acting at joint **E**. The struts converging at joint E are: - EA - EB - EF - EG - ED - EH For each strut, we must represent its force as acting **away from the joint** (assuming tension; if negative, it's in compression). ### **Coordinates (for direction vectors)** From the diagram, we assign coordinates: - \( A = (, , ) \) - \( B = (4, , ) \) - \( C = (4, 3, ) \) - \( D = (, 3, ) \) - \( E = (, 3, 3) \) - \( F = (4, 3, 3) \) - \( G = (4, 3, 6) \) - \( H = (, 3, 6) \) **E is at (, 3, 3).** #### **Strut Direction Vectors (from E):** - **EA:** A(,,) → E(,3,3): vector = (-, -3, -3) = (, -3, -3) - **EB:** B(4,,) → E(,3,3): vector = (4-, -3, -3) = (4, -3, -3) - **EF:** F(4,3,3) → E(,3,3): vector = (4-, 3-3, 3-3) = (4, , ) - **EG:** G(4,3,6) → E(,3,3): vector = (4-, 3-3, 6-3) = (4, , 3) - **ED:** D(,3,) → E(,3,3): vector = (-, 3-3, -3) = (, , -3) - **EH:** H(,3,6) → E(,3,3): vector = (-, 3-3, 6-3) = (, , 3) --- ## **Step 2: Write Equilibrium Equations for Joint E** ### **Explanation:** Sum of forces at joint E must be zero in all directions (x, y, z): \[ \sum F_x = , \quad \sum F_y = , \quad \sum F_z = \] Each force in a strut can be written as: \[ \vec{F} = F \cdot \hat{u} \] Where \( \hat{u} \) is the unit vector along the strut. #### **Find Unit Vectors:** For each strut, calculate its length and unit vector: - **EA:** Length = \( \sqrt{^2 + 3^2 + 3^2} = \sqrt{18} = 4.24 \) - Unit vector: \( (, -3/4.24, -3/4.24) = (, -.707, -.707) \) - **EB:** Length = \( \sqrt{4^2 + 3^2 + 3^2} = \sqrt{34} = 5.83 \) - Unit vector: \( (4/5.83, -3/5.83, -3/5.83) = (.686, -.514, -.514) \) - **EF:** Length = 4, Unit vector: (1, , ) - **EG:** Length = \( \sqrt{4^2 + ^2 + 3^2} = 5 \), Unit vector: (.8, , .6) - **ED:** Length = 3, Unit vector: (, , -1) - **EH:** Length = 3, Unit vector: (, , 1) #### **Write Equilibrium Equations:** Let: - \( F_{EA}, F_{EB}, ... \) be the force magnitudes in each strut (positive = tension). Write equations (sum of all components in x, y, z must be zero): **X-direction:** \[ = F_{EB} \cdot .686 + F_{EF} \cdot 1 + F_{EG} \cdot .8 \] **Y-direction:** \[ = F_{EA} \cdot (-.707) + F_{EB} \cdot (-.514) \] **Z-direction:** \[ = F_{EA} \cdot (-.707) + F_{EB} \cdot (-.514) + F_{EG} \cdot .6 + F_{ED} \cdot (-1) + F_{EH} \cdot 1 \] (Notice: Since no external force is directly applied at E, all equations sum to zero.) --- ## **Final Answer Format** You now have: - The **FBD for joint E** (step 1) - The **equilibrium equations** relating the six unknown strut forces (step 2). **Next steps (not requested):** To fully solve, you would need additional equations from other joints or use a matrix method, since you have more unknowns than equations at this joint alone. --- ### **Final Summary** - **Step 1:** Drew the Free Body Diagram for joint E and listed the direction vectors for all converging struts. - **Step 2:** Wrote the force equilibrium equations at joint E using the unit vectors for each strut. **To determine the actual forces and their nature (tension/compression), solve these equations (and possibly others from adjacent joints).**