Question 7.10: Determine the rotation of joint C of the frame shown in Fig....
Determine the rotation of joint C of the frame shown in Fig. 7.15(a) by the virtual work method.
The real and virtual systems are shown in Fig. 7.15(b) and (c), respectively. The x coordinates used for determining the bending moment equations for the three segments of the frame, AB, BC, and CD, are also shown in these figures. The equations for M and M_v obtained for the three segments are tabulated in Table 7.8. The rotation of joint C of the frame can now be determined by applying the virtual work expression given by Eq. (7.38).
1(θ) = ∑∫\frac{M_vM}{EI}dx (7.38)
1(θ_C) = ∑∫\frac{M_vM}{EI}dx= \frac{1}{EI}\int_{0}^{12}{(\frac{x}{2})(180x – 20\frac{x^2}{2}) dx}
(1 kN-m)θ_C = \frac{4320 kN^2-m^3}{EI}
Therefore,
θ_C =\frac{4320 kN^2-m^3}{EI} = \frac{4320}{200(10^6)1000(10^{-6})} = 0.0216 rad
θ_C = 0.0216 rad 
| TABLE 7.8 | ||||
| Segment | x Coordinate | M (kN-m) | M_v (kN-m) | |
| Origin | Limits (m) | |||
| AB | A | 0-4 | 180x | 0 |
| CB | C | 0-4 | 720 | 0 |
| DC | D | 0-12 | 180x – 20\frac{x^2}{2} | \frac{x}{12} |
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