# Calculate the residual stresses in the bars and the residual displacement of the $C$ point when the $P$ force is removed

Consider the rigid bar $$ABC$$ linked to two bars, $$AD$$ and $$BF$$, as shown in the figure. All bars are made of mild steel which is admitted to be an elastoplastic material ($$E$$ = 210 GPa, $$\sigma_Y$$ = 250 MPa) but the bar $$AD$$ has a uniform and rectangular cross section of 20 mm $$\times$$ 6 mm while the $$BF$$ bar consists of two sections, $$BE$$ and $$EF$$, with section area respectively equal to $$A_ {BE} = 1200$$ mm $$^ 2$$ and $$A_ {EF} = 2400$$ mm $$^ 2$$. The strength of the $$P$$ force applied in $$C$$ is gradually increased until the displacement of the $$C$$ point reaches 2.5 mm.

Calculate the residual stresses in the bars and the residual displacement of the $$C$$ point when the $$P$$ force is removed.

The resolution says:

Now consider the extreme case where the $$C$$ point offset is increased to 2.5 mm. What will happen to the displacements and forces at the other points of interest on the rigid bar $$ABC$$?

Since the maximum force $$P$$ does not change during the path $$\delta_C in [1.206 \ , ~ 2.5]$$ mm the forces on the bars $$AD$$ and $$BF$$ do not change:

$$$$\begin{cases} F_{BF} &= P + F_{AD} = \left(\displaystyle\frac{L_{BC}}{L_{AB}} + 1\right) P \\ F_{AD} &= \left(\displaystyle\frac{L_{BC}}{L_{AB}}\right) P \end{cases}$$$$

In this case, the displacement of the point $$B$$ remains fixed ($$F_ {BF}$$ constant) and the geometric rotation of the mechanism occurs around $$B$$. At constant force, there will be a continuous increment of variation in the length of the bar $$AD$$ being the total displacement value in $$\delta_A ^ t$$ (for $$\delta_ {C_p} =$$ 2.5 mm) given by:

\begin{aligned} \frac{\delta_A^t+\delta_B}{L_{AB}} = \frac{\delta_A^t+\delta_{C_p}}{L_{AC}} \end{aligned} \quad\Leftrightarrow\quad \begin{aligned} \delta_A^t = \frac{\delta_{C_p}L_{AB} - \delta_BL_{AC} }{L_{AC}-L_{AB}} \end{aligned}

deltaA tot = 1.6983 mm

I can't understand how they calculated deltaA tot ? I also tried to use the relationship $$\frac{\delta}{L}$$ but i'm not getting it.

Could someone explain it to me?