Question 6.11: Figure a shows a sketch of a bracket attached to a vertical ......

Let \rho be the point at the lower inside corner of the bracket, adjacent to the wall. With the bracket assumed to be rigid, and as the weight is placed on the shelf portion of the bracket, the rivets will be elongated as the bracket pivots about P.

With the bracket portion against the wall being rigid and thus remaining straight during the loading, the deformations of rivets A and B will be directly proportional to their distances from the pivot point P. Therefore, if δ_{A} and δ_{B} are the respective rivet elongations, we can express δ_{A} and δ_{B} as:

δ_{A}=k\left(d+e\right)    and     δ_{B}=ke   (a)

where k is the constant of proportionality.

From the principles of elementary strength of materials we know that the respective rivet forces, F_{A} and F_{B}, are proportional to the rivet deformations δ_{A} and δ_{B}. Specifically,

F_{A}=\left\lgroup\frac{AE}{\ell } \right\rgroup \delta _{A}     and    F_{B}=\left\lgroup\frac{AE}{\ell } \right\rgroup \delta _{B}  (b)

where A is the rivet shaft cross-section area, \ell is rivet length, and E is the modulus of elasticity of the rivet material.

By substituting from Eqs. (a) into (b) the rivet forces become:

F_{A}=k\left\lgroup\frac{AE}{\ell } \right\rgroup\left(d+e\right) =k\left(d+e\right)    (c)

and

F_{B}=k\left\lgroup\frac{AE}{\ell } \right\rgroup e =ke     (d)

where K is a constant which by inspection is defined as:

K\triangleq \frac{AEk}{\ell }     (e)

Next, envision a FBD of the bracket as in Figure b, where R_{X} and R_{Y} are the horizontal and vertical components of the reaction force exerted by the wall on the bracket at Let \rho. By balancing moments about Let \rho we obtain

Wb=F_{A}\left(d+e\right) + F_{B}\left(e\right)    (f)

By substituting for F_{A} and F_{B} from Eqs. (c) and (d) Eq. (f) becomes:

Wb=K\left(d+e\right) ^{2}+Ke^{2}=K\left[\left(d+e\right)^{2}+e^{2} \right]   (g)

Then K is seen to be:

K=\frac{Wb}{\left[\left(d+e\right)^{2}+e^{2} \right] } (h)

Finally, by back substituting from Eq. (h) into Eqs. (c) and (d) the requested rivet forces are:

F{A}=\frac{Wb\left(d+e\right)}{\left[\left(d+e\right)^{2}+e^{2} \right] }  (i)

and

F{B}=\frac{Wb e}{\left[\left(d+e\right)^{2}+e^{2} \right] }   (j)

The rivet stresses σ_{A} and σ_{B} are then

\sigma _{A}=\frac{Wb \left(d+e\right)}{\left(\pi r^{2} \right) \left[\left(d+e\right)^{2}+e^{2} \right] }   (k)

and

\sigma _{B}=Wbe/\left(\pi r^{2} \right)\left[\left(d+e\right)^{2}+e^{2} \right]  (Ρ)

Comment

With the use of rivets, as opposed to bolts, we assume no preload tension. If, however, both had been used with a preload of say F_{O,} then F_{O} would need to be added to the values for F_{A} and F_{B} in Eqs. (i) and (j).