Two Theorems for Computation of Projections of Virtual Displacements and Its Application in Structural Analysis

A theorem for planar case and its generalization for spatial case are proposed to determine the projection of a virtual displacement to the orientation under the case of knowing the projections of a virtual displacement to the given two or three orientations for object systems subject to holonomic and scleronomic constraints. Some lemmas corresponding to the two theorems for special cases are given. Applications to structural static analysis are investigated using the two theorems in this paper. Result reveals that the two theorems and corresponding lemmas are easy to be used, shorten the distance between the principle of virtual displacement and its application, and the relating problems can be solved quickly with them.


Introduction
It is well known that the principle of virtual displacement (or virtual work) is a main part in analytical statics, and the important basis for analytical dynamics and structural analysis.This principle provides an excellent tool for people to investigate the equilibrium laws of object systems, and plays a supporting role in classical mechanics.
Generally speaking, some current college textbooks [1][2][3][4][5][6][7] for engineering mechanics or structural analysis course often propose two main methods for building relations among virtual displacements of different points for the system with holonomic and scleronomic constraints, ie, analytical method and geometrical method.Using analytical method, one can determine the virtual displacements of interest points (usually points of forces) by taking differentials of their position coordinates, and geometrical methods by introducing the concept of the instant center of rotation of virtual displacements.These methods are very useful in application of the principle of virtual displacement.However, in actual experience, we realize that these methods are sometimes not sufficient for analyzing complex structures.For some difficult problems, such as the examples given in this paper, it is impossible or not sufficient to solve them out if only directly using the results given in these current textbooks.Then, how to shorten the distance between the principle of virtual displacement and its application, and give some feasible approaches to realize, is very important and great significance.
In this study, we will gain an insight into the application of the principle of virtual displacement and present two new theorems on how to build up the relations among the projections to different orientations of the virtual displacement of a point.Several examples are given to illustrate the application of two theorems and corresponding lemmas.The article is organized as follows.In Section 2, a theorem is given to find a projection under knowing the two projections to different orientations of a point for planar systems, and two lemmas for special cases are proposed.In Section 3, a theorem for spatial systems as the planar extending situation is given to find a projection under knowing the three projections to different orientations of a point, and three lemmas for special cases are proposed.

The first theorem for plane cases
For a planar object system, if the projections of a virtual displacement dr of a certain dot to orientations n 1 and n 2 are dr 1 and dr 2 separately, shown in Fig. 1, then the projection of the virtual displacement to orientation n 3 is where, angles θ and φ are angles between n 1 and n 2 , n 1 and n 3 separately.Hence, one can get  Solution.To calculate the force in rod CD, it is isolated from the system in Fig. 2 (b).Then, the system in Fig. 2 (b) is a mechanism.Obviously, equilateral triangles AED and DBG can be regarded as rigid plates, and DAED can be assumed to rotate about A point with δθ .Then, δr E = δr D = aδθ .Considering the orientation of δr D and character of support B, point B is the virtual displacement center of DDBG, therefore δr G = δr D = aδθ .Based on the principle of virtual displacement, one reads Based on the theorem of projection of virtual displacement (i.e. the projections of the virtual displacements of the points from a straight line belonging to a body, on that line, are equal.),the projections of δr C to the orientations E→C and G→C are zeros.Then, by employing the lemma 1 of the above theorem, δr C must be zero.Therefore, based on the above analysis and (6), one can get Because of θ ≠ 0 , then (bar CD in compression).
Example 2. Determine the force in member DG of the truss, shown in Fig. 3. Assume all members are pin connected. ( (5) Solution.To calculate the force in rod DG, it is isolated from the system in Fig. 3  For the point G, the projections δr GH and δr CG are known.By employing the formula (1) of the above theorem, the projection of δr G to the orientation G→D is By employing the principle of virtual displacement, one can get Therefore, based on the above analysis and ( 7), one can get .
where e ix , e iz , e iz are projections of unit vector e i of spatial orientation n i to the Cartesian coordinate axes x, y, z, separately.(i = 1, 2, 3, 4) Proof.Assuming that i, j, k are the standard unit orthogonal vectors of the spatial coordinate system, based on the given conditions, one reads x x y z Because the three orientations e 1 , e 2 , e 3 are non-coplanar, the coefficient matrix is reversible.Hence, one can get Thus, the projection of the virtual displacement δr to orientation n 4

Fig. 1 A
Fig.1A virtual displacement and its projections

4
(b).Obviously, triangles ACE and DBE can be regarded as rigid plates, and DACE can be assumed to rotate about A point with δθ .Then, Considering that the orientation of δr B is horizontal, the virtual displacement δr D must be horizontal, and For the bar BH, it is easy to get that Then, for the bar GH, the projection of δr G to the orientation G→H is For the bar CG, the projection of δr G to the orientation C→G is

3
The second theorem for spatial cases If the projections of the virtual displacement δr of a dot to three non-coplanar orientations n 1 , n 2 , n 3 are δr 1 , δr 2 , δr 3 , then the projection δr 4 of the virtual displacement to the orientation n

Lemma 3 .Lemma 4 . 5 .Example 3 .
If the unit vectors e i (i = 1, 2, 3, 4) are normal orthogonal unit vectors, the projection δr 4 of the virtual displacement to the orientation n 4 is If the unit vectors e i of orientations n i (i = 1, 2, 3, 4) are non-planar vectors, and the projections of the virtual displacement of a dot to n i are all zeros, then the virtual displacement must be zero vector, and projection of the virtual displacement to n 4 (every orientation) must be zero.Lemma If the unit vectors e i (i = 1, 2, 3, 4) are standard unit orthogonal vectors of the Cartesian coordinate axes x, y, z, separately, the projection of the virtual displacement to e i is δr i , then the projection of the virtual displacement to the orientation n Determine the force in member AC of the spatial truss, shown in Fig.4 (a).Assume all members are pin connected.F 1 =5kN, F 2 =4kN, F 3 =2kN, F 4 =1.5kN,F 5 =1.5kN.Solution.To calculate the force in rod AC, it is isolated from the system in Fig.4 (b).Obviously, hinge points E, D, C and B are fixed.Therefore, the projection of point A to the orientations B→A and D→A are zeros.Assuming that the projection along the F 4 direction is dr , considering that the unit vectors of the direction B→A, A→C, D→A and F 4 are as follows the second theorem for spatial case, the projection along the A→C direction is Because δθ ≠ 0 , then one can get