Question

In the force method of analysis of indeterminate trusses, if the truss is indeterminate to degree one, the change in length of redundant, member due to unit force is found by using the formula where A is cross-sectional area

I – Moment of Inertia

n – force in the member due to unit load application

N – force in the member due to actual load

E – Modulus of Elasticity

1. \(\sum \frac{{nNL}}{{EI}}\)
2. \(n\sum \frac{NL}{AE}\)
3. \(\sum \frac{nNL}{AE}\)
4. \(\sum \frac{NL}{AE}\)

Answer 1

Correct Answer - Option 3 : \(\sum \frac{nNL}{AE}\)

Explanation:

From the virtual work principle, we know that

External virtual work = Internal virtual work 

When the external virtual loads are multiplied by the real displacement, we get the external virtual work.

If a unit load virtual load produces internal loading equal to n_i in various members and the real displacements of the member is dl_i,

then the internal virtual work done is w.

w = ∑ n_i × dl_i.

Thus, if the virtual load at any point is 1 and the displacement at that point due to external effects is Δ then,

Virtual load × real displacement = Virtual load × Real displacement

1 × Δ  = ∑ n_i × dl_i

∴ dl_i = \(\frac{{{N_i}{L_i}}}{{{A_i}{E_i}}}\) = Change in length of any member due to external load

Hence, from the virtual work principle, we have

Δ = \(\mathop \sum \limits_{i = 1}^n {n_i} \times \frac{{{N_i}{L_i}}}{{{A_i}{E_I}}}\)