Question 7.11: Use the virtual work method to determine the vertical deflec...
Use the virtual work method to determine the vertical deflection at joint C of the frame shown in Fig. 7.16(a).
The real and virtual systems are shown in Figs. 7.16(b) and (c), respectively. The x coordinates used for determining the bending moment equations for the two members of the frame, AB and BC, are also shown in the figures. The equations for M and M_v obtained for the two members are tabulated in Table 7.9. The vertical deflection at joint C of the frame can now be calculated by applying the virtual work expression given by Eq. (7.37):
1(Δ) = ∑∫\frac{M_vM}{EI} dx (7.37)
1(Δ_C) = ∑∫\frac{M_vM}{EI} dx1(Δ_C) = \frac{1}{EI}[\frac{1}{2}\int_{0}^{5}{(-4)(76x – 530) dx }+ \int_{0}^{5}{(-\frac{4}{5}x)(-6x^2) dx}]
(1 kN)Δ_C = \frac{4,150 kN^2-m^3}{EI}
Therefore,
Δ_C = \frac{4,150 kN-m^3}{EI} = \frac{4,150}{70(554)} = 0.107 m = 107 mmΔ_C = 107 mm ↓
| TABLE 7.9 | |||||
| Segment | x Coordinate | EI (I = 554 – 10^6 mm^4) |
M (kN-m) |
M_v (kN-m) |
|
| Origin | Limits (m) | ||||
| AB | A | 0-5 | 2EI | 76x – 530 | -4 |
| CB | C | 0-5 | EI | -12\frac{x^2}{2} | -\frac{4}{5}x |
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