Question 13.13: Determine the reactions and draw the shear and bending momen...
Determine the reactions and draw the shear and bending moment diagrams for the frame shown in Fig. 13.18(a) by the method of consistent deformations.
Degree of Indeterminacy i = 2.
Primary Frame The reactions D_X and D_Y at the hinged support D are selected as the redundants. The hinged support D is then removed to obtain the primary frame shown in Fig. 13.18(b). Next, the primary frame is subjected separately to the external loading and the unit values of the redundants D_X and D_Y , as shown in Fig. 13.18(b), (c), and (d), respectively.
Compatibility Equations Noting that the horizontal and vertical deflections of the actual indeterminate frame at the hinged support D are zero, we write the compatibility equations:
| TABLE 13.5 | |||||
| Member | x coordinate | M_O (kN-m) | m_{DX} (kN-m/kN) | m_{DY} (kN-m/kN) | |
| Origin | Limits (m) | ||||
| AB | A | 0-5 | -1,750 + 50x | -x | 10 |
| CB | C | 0-10 | -15x² | -5 | x |
| DC | D | 0-5 | 0 | x | 0 |
Δ_{DXO} + f_{DX,DX}D_X + f_{DX,DY}D_Y = 0 (1)
Δ_{DYO} + f_{DY,DX}D_X + f_{DY,DY}D_Y = 0 (2)
Deflections of Primary Frame The equations for bending moments for the members of the frame due to the external loading and unit values of the redundants are tabulated in Table 13.5. By applying the virtual work method, we obtain
Δ_{DXO} =∑∫\frac{M_Om_{DX}}{EI} dx = \frac{44791.7 kN-m^3}{EI}Δ_{DYO} =∑∫\frac{M_Om_{DY}}{EI} dx = \frac{83593.75 kN-m^3}{EI}
f_{DX,DX} = ∑∫ \frac{m^2_{DX}}{EI} dx = \frac{333.33 m^3}{EI}
f_{DY,DY} = ∑∫ \frac{m^2_{DY}}{EI} dx = \frac{833.33 m^3}{EI}
f_{DX,DY} = f_{DY,DX} =∑∫\frac{m_{DX}m_{DY}}{EI}dx = -\frac{375 m^3}{EI}
Magnitudes of the Redundants By substituting these deflections and flexibility coefficients into the compatibility equations, we write
44791.7 + 333.33D_X – 375D_Y = 0 (1a)
– 83593.75 – 375D_X + 833.33D_Y = 0 (2a)
Solving Eqs. (1a) and (2a) simultaneously for D_X and D_Y , we obtain
D_X = 52.52 kN← D_Y = 166.13 kN ↑
Reactions The remaining reactions and the member end forces of the indeterminate frame can now be determined by applying the equations of equilibrium. The reactions and member and forces thus obtained are shown in Fig. 13.18(e).
Shear and Bending Moment Diagrams See Fig. 13.18(f ).
