Question 6.14: Review again Example Analyses 11 to 13. Let τmax be the maxi......
Review again Example Analyses 11 to 13. Let τ_{max} be the maximum allowable shear stress in the rivets. Determine relations between the weight load W and τ_{max}. Let the dimensions and distances be the same as in the foregoing example analyses and as shown again in Figure a.
This problem is statically indeterminate: If we envision a FBD of the bracket we immediately see that the downward directed shear forces on the rivets, from the bracket, are resisted by equal upward directed forces on the bracket, by the rivets. If we call these shear forces V_{A} and V_{B,} on rivets A and B, respectively, the vertical force balance yields:
W=V_{A}+V_{B} (a)
Observe from an envisioned FBD that neither a balance of horizontal forces nor a balance of moments will involve either V_{A} and V_{B}. Therefore, Eq. (a) is insufficient in itself for determining V_{A} and V_{B}.
The usual approach with statically indeterminate systems is to 1) make a force balance; 2) examine the deformation; and 3) relate the forces and deformation via Hooke’s law. Steps 2) and 3) then hopefully produce additional equations, enabling the determination of all unknown forces and deformations.
In the current example, however, this procedure requires some additional assumptions before we can determine the unknown forces: If we assume ideal geometry, and if we also assume relatively rigid bracket material, we find by a symmetry argument that V_{A} and V_{B} are equal: Then in view of Eq. (a) we have:
V_{A}=V_{B}=\frac{W}{2} (b)
Alternatively, if we assume ideal geometry, but if more realistically, we assume the bracket material is deformable, then with rivet A being closer to the applied load W than rivet B, the deformation of the bracket material in contact with rivet A will relieve the force on rivet B. Therefore, in the interest of safe design, in the extreme, we can assume that rivet A absorbs all the load. Then from Eq. (a) we have:
V_{A}=W and V_{B}=0 (c)
Finally, the shear stresses on the rivets are simply:
τ_{A}=\frac{V_{A}}{\pi r^{2}} and τ_{B}=\frac{V_{B}}{\pi r^{2}} (d)
Then for safe design, in view of Eq. (c), we obtain the requested relations as:
τ_{max}=\frac{W}{\pi r^{2}} and W=\pi r^{2}τ_{max} (e)
Comment
Observe the simplicity of the results, and specifically, observe that aside from the rivet radius r, none of the other dimensions nor distances affect the result.