PROBLEM 1.44
The 250 N luminaire is supported by three steel rods interconnected by an A-ring. Determine the orientation angle θ of AC so that the mean normal stress at rod AC is twice the mean normal stress at rod AD. What is the amount of tension on each rod? The diameter of each rod is given in the figure.
00:00 Making the Free Body Diagram (d.c.l.)
✅ The first step in solving the problem is to make the free-body diagram (d.c.l.), that is, to make a schematic drawing of all the forces involved in the system.
02:00 Decomposition of Forces
✅ After doing the d.c.l., we must find the components of the forces acting on the rods, decomposing them in the x and y axes.
03:00 Applying Equilibrium Equations
✅ Having found the components of the forces on the respective axes, we will now apply the equilibrium equations in order to determine the strengths of the forces on the rods and the angle ϴ.
05:15 Calculation of Normal Pull Force on Each Rod and Angle ϴ
✅ Note that an equation is missing to solve the system, because when we apply the equilibrium equations (ΣFx = 0 and ΣFy = 0), we get two equations and three unknowns to be determined, which are: FAC, FAD and ϴ.
This equation is contained in the problem statement, we were given that:
σAC = 2σAD, breaking down the equation, we have:
FAC/AAC = 2FAD/AAD
FAC = 2FAD.AAC/AAD
FAC = 2FAD.dAC²/dAD²
Now, we get the third equation that was missing to solve the system.
11:00 Calculation of Average Normal Stress in each Rod
✅ Very well, we have already found all the forces developed in the rods and the angle ϴ, now it is easy to determine the average normal stress in each rod:
σAB = 4FAB/πdAB²
σAC = 4FAC/πdAC²
σAD = 4FAD/πdAD²
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📚 Source: Mechanics of Materials 7th Ed. (R.C. Hibbeler)
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