Unit dummy force method

The Unit dummy force method provides a convenient means for computing displacements in structural systems. It is applicable for both linear and non-linear material behaviours as well as for systems subject to environmental effects, and hence more general than the Castigliano's second theorem.

Discrete systems

Consider a discrete system such as trusses, beams or frames having members interconnected at the nodes. Let the consistent set of members' deformations be given by $\mathbf {q} _{M\times 1}$. These member deformations give rise to the nodal displacements $\mathbf {r} _{N\times 1}$, which we want to determine.

We start by applying N virtual nodal forces $\mathbf {R} _{N\times 1}^{*}$, one for each wanted r, and find the virtual member forces $\mathbf {Q} _{M\times 1}^{*}$ that are in equilibrium with $\mathbf {R} _{N\times 1}^{*}$:

$\mathbf {Q} _{M\times 1}^{*}=\mathbf {B} _{M\times N}\mathbf {R} _{N\times 1}^{*}\qquad \qquad \qquad \mathrm {(1)}$

In the case of a statically indeterminate system, matrix B is not unique because the set of $\mathbf {Q} _{M\times 1}^{*}$ that satisfies nodal equilibrium is infinite.

Let the sets of internal and external virtual forces undergo, respectively, the real deformations and displacements, the external and internal virtual work can be written as:

• External virtual work: $\mathbf {R} ^{*T}\mathbf {r}$
• Internal virtual work: $\mathbf {Q} ^{*T}\mathbf {q}$

According to the virtual work principle, the two work expressions are equal:

$\mathbf {R} ^{*T}\mathbf {r} =\mathbf {Q} ^{*T}\mathbf {q}$

Substitution of (1) gives

$\mathbf {R} ^{*T}\mathbf {r} =\mathbf {R} ^{*T}\mathbf {B} ^{T}\mathbf {q}$

Since $\mathbf {R} ^{*}$ contains arbitrary virtual forces, the above equation gives

$\mathbf {r} =\mathbf {B} ^{T}\mathbf {q} \qquad \qquad \qquad \mathrm {(2)}$

It is remarkable that the computation in (2) does not involve any integration regardless of the complexity of the systems. It is thus far more convenient and general than the classical form of the dummy unit load method, which varies with the type of system as well as with the imposed external effects. On the other hand, it is important to note that Eq.(2) is for computing displacements or rotations of the nodes only. This is not a restriction because we can make any point into a node when desired.

Finally, the name unit load arises from the interepretation that the coefficients $B_{i,j}$ in matrix B are the member forces in equilibrium with the unit nodal force $R_{j}^{*}=1$, by virtue of Eq.(1).

General systems

File:DummyUnitForce-01.jpg

Virtual work by internal stresses.

For a general system, the unit dummy force method also comes directly from the virtual work principle. Fig.(a) is the system with actual deformations ${\boldsymbol {\epsilon }}$. Theses deformations, supposedly consistent, give rise to displacements throughout the system. For example, a point A has moved to A', and we want to compute the displacement r of A in the direction shown. For this particular purpose, we choose the virtual forces system in Fig.(b) which shows:

• The unit force R* is chosen to be at A and in the direction of r so that the external virtual work is