## 48 Unit load method

For a stationary value of C

from which as before as before

4.8 Unit load method 101

If instead of the arbitrary dummy load Pf we had placed a unit load at C, then the load in the z'th linearly elastic member would be

Therefore, the term dFt/9Pf in Eq. (4.25) is equal to the load in the i'th member due to a unit load at C, and Eq. (4.25) may be written where Fl 0 is the force in the i'th member due to the actual loading and F^ is the force in the i'th member due to a unit load placed at the position and in the direction of the required deflection. Thus, in Example 4.2 columns (4) and (6) in Table 4.1 would be eliminated, leaving column © as FB1 and column (7) as FD1. Obviously column (3) is F0.

Similar expressions for deflection due to bending and torsion of linear structures follow from the well-known relationships between bending and rotation and torsion and rotation. Hence, for a member of length L and flexural and torsional rigidities EI and GJ respectively where M0 is the bending moment at any section produced by the actual loading and Mi is the bending moment at any section due to a unit load applied at the position and in the direction of the required deflection. Similarly for torsion.

Generally, shear deflections of slender beams are ignored but may be calculated when required for particular cases. Of greater interest in aircraft structures is the calculation of the deflections produced by the large shear stresses experienced by thin-walled sections. This problem is discussed in Chapter 9.

### Example 4.8

A steel rod of uniform circular cross-section is bent as shown in Fig. 4.23, AB and BC being horizontal and CD vertical. The arms AB, BC and CD are of equal length. The rod is encastré at A and the other end D is free. A uniformly distributed load covers the length BC. Find the components of the displacement of the free end D in terms of EI and GJ.

Since the cross-sectional area A and modulus of elasticity E are not given we shall assume that displacements due to axial distortion are to be ignored. We place, in turn, unit loads in the assumed positive directions of the axes xyz.

First, consider the displacement in the direction parallel to the x axis. From Eqs

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