Where did I go wrong conceptually when attempting to calculate the maximum force on a truss at a given joint?

In the problem statement below, the question states to find the maximum load $\vec{P}$ that the truss can support. My method of approach was:

1. Draw a FBD for the entire structure.
2. Generally I would identify all external forces, however for this problem I felt it was unnecessary because I felt they weren't needed if I knew already the forces in each member.
3. So I then dove straight into joint $D$, and assumed each member connected to $D$ was in tension and experienced the maximum tensile force $\vec{T} = 1500$ lb.

However, this led to the wrong answer when solving for $\vec{P}$. The solution manual instead found each external force in terms of $\vec{P}$ and started at joint $A$ to find $\overrightarrow{AD}$ and $\overrightarrow{AB}$ in terms of $\vec{P}$. In addition, in order to get numerical values, the solution manual assumed member $\overrightarrow{AB}$ was experiencing the maximum compression force of 660 lb. However, when I assume that member $\overrightarrow{AD}$ is experiencing the maximum tensile force, it doesn't work out the same.

My question is, conceptually, why must I find each each member's force in terms of $\vec{P}$, and why must one assume $\overrightarrow{AB}$ is experiencing maximum compression (but not $\overrightarrow{AD}$ in maximum tension)?

EDIT: I want to note that I do not need help with solving this problem, nor the math required. I simply am just looking for a conceptual answer as to why my approach did not work (i.e. only analyzing joint $D$). • Finally, a homework question that follows our guidelines! Mar 23 '16 at 12:50

The reason is that you assumed that the elements around node $$\text{D}$$ will be the first to fail. That is not the case. Indeed, it is the elements under compression ($$\text{AB}$$ and $$\text{BC}$$) that will fail first. Also, you assumed that all the members around $$\text{D}$$ will present the same axial force, which is untrue.

To see this, here's your structure with a unitary load (the units are irrelevant): And here are the axial forces in each member under this unitary force (positive in tension): Since these results are linearly proportional to the force applied, we can now find the concentrated load $$P$$ at node $$\text{D}$$ that each member can withstand:

\begin{align} P_{\text{AB}} = P_{\text{BC}} &= \dfrac{660}{0.943} = 700 \\ P_{\text{AD}} = P_{\text{CD}} &= \dfrac{1500}{0.687} = 2183 \\ P_{\text{BD}} &= \dfrac{1500}{1.333} = 1125 \\ \end{align}

Therefore, the maximum load supported by the structure is the minimum of these values, which is 700 lb.

To show the validity of these results, let's now apply each of these forces and check the results in the relevant members:   All results obtained with Ftool, a free 2D frame analysis program.

• Do members $AB$ and $BC$ fail first because their maximum force is less than that of members $AD, CD,$ and $BD$ (i.e. 800 lb < 1500 lb)? Or because of something intrinsic about compression forces? Mar 22 '16 at 23:59
• @Hunter, because their maximum force is the lesser of the members. The only way to know the maximum load of a structure is to check the maximum load each element of the structure can support and getting the minimum of that set of values. And "first" is not meant chronologically: if you apply 10000 lb instantaneously to the structure, all the elements will fail at the same time.
– Wasabi
Mar 23 '16 at 0:06
• @Hunter It's just that if you apply 701 lb, $AB$ and $BC$ will fail while the other elements of the structure won't (well, once they fail, the other elements will fail as well since the structure becomes hipostatic, but that's irrelevant). So it doesn't matter if the other elements can still withstand more load, because these have already failed. All that means is that the structure could possibly be more efficient by using weaker (and therefore cheaper) sections for $AD$, $BD$ and $CD$ which have maximum loads in this structure closer to 700 lb as well.
– Wasabi
Mar 23 '16 at 0:09
• Thank you, this is extremely informative and helped tremendously. Also, one slight thing, it might help to edit the post and make the math more precise. Although I knew exactly what it meant, the values represent $P$ and not the respective forces Mar 23 '16 at 0:12