Question 13.12: Determine the reactions and the force in each member of the ...

Degree of Indeterminacy i = (m + r) – 2j = (14 + 4) – 2(8) = 2.

Primary Truss The vertical reaction D_y at the roller support D and the axial force F_{BG} in the diagonal member BG are selected as the redundants. The roller support D and member BG are then removed from the given indeterminate truss to obtain the primary truss shown in Fig. 13.17(b). The primary truss is subjected separately to the external loading (Fig. 13.17(b)), a unit value of the redundant D_y (Fig. 13.17(c)), and a unit tensile force in the redundant member BG (Fig. 13.17(d)).

Compatibility Equations The compatibility equations can be expressed as

Δ_{DO} + f_{DD}D_y + f_{D,BG}F_{BG} = 0                (1)

Δ_{BGO} + f_{BG,D}D_y + f_{BG,BG}F_{BG} = 0                  (2)

in which Δ_{DO} = vertical deflection at joint D of the primary truss due to the external loading; Δ_{BGO} = relative displacement between joints B and G due to the external loading; f_{DD} = vertical deflection at joint D due to a unit load at joint D; f_{BG,D} = relative displacement between joints B and G due to a unit load at joint D; f_{BG,BG} = relative displacement between joints B and G due to a unit tensile force in member BG; and f_{D,BG} = vertical deflection at joint D due to a unit tensile force in member BG.

Deflections of Primary Truss The virtual work expressions for the preceding deflections are

Δ_{DO} = ∑\frac{F_Ou_DL}{AE}                   Δ_{BGO} = ∑\frac{F_Ou_{BG}L}{AE}

f_{DD} = ∑\frac{u^2_DL}{AE}                    f_{BG,BG} = ∑\frac{u^2_{BG}L}{AE}

in which F_O, u_D, and u_{BG} represent the member forces due to the external loading, a unit load at joint D, and a unit tensile force in member BG, respectively. The numerical values of the member forces, as computed by the method of joints (Fig. 13.17(b) through (d)), are tabulated in Table 13.4. Note that since the axial rigidity EA is the same for all the members, only the numerators of the virtual work expressions are evaluated in Table 13.4. Thus

Δ_{DO} = -\frac{4,472.642 kN-m}{AE}                 Δ_{BGO} = -\frac{992.819 kN-m}{AE}

f_{DD} = \frac{48.736 m}{AE}                    f_{BG,BG} = \frac{48.284 m}{AE}

Magnitudes of the Redundants By substituting these deflections and flexibility coefficients into the compatibility equations (Eqs. (1) and (2)), we write

-4,472.642 + 48.736D_y – 6.773F_{BG} = 0                     (1a)

-992.819 – 6.773D_y + 48.284F_{BG} = 0                          (2a)

Solving Eqs. (1a) and (2a) simultaneously for D_y and F_{BG}, we obtain

D_y = 96.507 kN ↑                F_{BG} = 34.1 kN (T)

Reactions The remaining reactions of the indeterminate truss can now be determined by superposition of reactions of the primary truss due to the external loading and due to each of the redundants. The reactions thus obtained are shown in Fig. 13.17(e).

Member Axial Forces The forces in the remaining members of the indeterminate truss can be determined by using the superposition relationship:

The member forces thus obtained are shown in Table 13.4 and Fig. 13.17(e).