Truss joint H: temperature and long vertical members

Given data and assumed geometry

Top chord joints C(0,4), D(3,4), E(6,4), F(9,4), G(12,4); bottom joints A(3,0), H(6,0), B(9,0). Members: top chord CD, DE, EF, FG; bottom chord AH, HB; verticals DA, EH, FB; diagonals CA, DH, FH, GB. A hinge, B roller. 20 kN down at E. $A = 1500$ mm$^2 = 1.5\times10^{-3}$ m$^2$, $E = 200$ kN/mm$^2 = 2\times10^8$ kN/m$^2$, $\alpha = 12\times10^{-6}$/$^\circ$C. Unit vertical load (down) at H.

Member forces (tension +)

(i) Loading

$$\Delta_1 = \frac{\sum kFL}{AE} = \frac{135.000}{1.5\times10^{-3}\times2\times10^8} = 0.450\ \text{mm}\ (\downarrow)$$

(ii) Top chord temperature rise 20$^\circ$C

$$\Delta_2 = \sum k\alpha\Delta T L = 2\times(-0.375)(12\times10^{-6})(20)(3000) = -0.540\ \text{mm}$$

(DE and EF have $k = -0.375$; CD and FG have $k = 0$.)

(iii) All vertical members 10 mm too long

$$\Delta_3 = (-0.5)(10) + (0)(10) + (-0.5)(10) = -10.000\ \text{mm}$$

Total

$$\Delta_H = 0.450 - 0.540 - 10.000 = -10.090\ \text{mm}$$

Answer: vertical deflection of joint H = 10.09 mm upward (net).