To determine the maximum load $P$ that can be applied to the truss, we need to analyze the forces in each member and ensure they do not exceed the given maximum allowable tensile and compressive forces.

Step-by-Step Solution

1. Free Body Diagram and Reaction Forces:

   • We start by analyzing the overall structure to determine the reactions at points $A$ and $D$ due to the loads.
   • Sum of moments about $A$ to find the reaction at $D$: $\sum M_A = 0$
   • Use equilibrium equations to solve for $A_y$ and $D_y$.

2. Method of Joints:

   • Once the reactions are known, use the method of joints starting from joints where only two unknown forces are present. Typically, we start from joint $A$ or $D$.
   • Each joint will be analyzed to find forces in each truss member by setting up equilibrium equations ($\sum F_x = 0$ and $\sum F_y = 0$).

3. Apply Allowable Forces:

   • After finding the force in each truss member as a function of $P$, ensure that none of these forces exceed the allowable tensile force ($Ft_{max} = 48.0 \, \text{kN}$) or compressive force ($Fc_{max} = 22.1 \, \text{kN}$).
   • Identify the member with the critical (limiting) force that restricts the value of $P$.
   • Calculate $P$ such that the force in this critical member is at its maximum allowable value.

Would you like me to go through the calculations in detail for each step and derive the maximum value of $P$?

Follow-up Questions:

1. How are internal forces distributed within different members of a truss structure?
2. What is the difference between tensile and compressive forces in structural members?
3. Why are truss structures analyzed using methods such as the method of joints or sections?
4. What is the significance of reaction forces at supports in truss problems?
5. How does the angle of truss members affect the force distribution?

Tip:

For truss analysis, the method of joints is helpful for simpler structures, while the method of sections is efficient for analyzing forces in specific members directly.