Technical Notes on Virtual Work Theory

Energy methods such as Virtual Work method have traditionally been used to calculate deflections in determinate and indeterminate truss structures. The principle of Virtual Work states that the external work done in a physical domain should be equal to the internal work done.

This principle is further used in such theories as the Castigliano’s theorem that states that the work done by virtual loads going through real displacements is the same as the internal work done by real loads causing virtual displacements. This can be expressed as follows:

{D P F}_{i} = \int {(σ_{i}^{v})}^{T} ε_{i}^{r} ⅆ V

Equation 3-1

where

DPF_i	=	the displacement participation factor for member i
dV	=	the volume of the member i
$σ_{i}^{v}$	=	the stresses in member i due to a virtual load
$ε_{i}^{r}$	=	the strains member i due to real loads

The virtual stresses and real strains are computed during analysis in RAM Frame due to real and virtual load cases, considering element properties and structural configurations.

In its most general form, the contributions to displacements could be expanded through its component stresses and strains as:

F_{v i r} Δ_{r e a l} = \int_{Ω}^{} \int_{- h / 2}^{h / 2} (σ_{x x}^{v} ε_{x x}^{r} + σ_{y y}^{v} ε_{y y}^{r} + σ_{z z}^{v} ε_{z z}^{r} + σ_{x z}^{v} ε_{x z}^{r} + σ_{y z}^{v} ε_{y z}^{r} + σ_{x y}^{v} ε_{x y}^{r}) ⅆ z ⅆ A

Equation 3-2

As can be seen from the above equation, the contribution to the component participation is made of strain energy due to in-plane, transverse shear and transverse normal stresses and strains.

This breakdown of contribution helps to identify which behavior is dominant and what sectional or material property needs to be modified to arrive at acceptable and desired response.

The same principles are extended here to a case of civil engineering structures to get a useful quantitative assessment of contribution of member flexibility to structural responses such as roof displacement (or drift) and fundamental periods. Furthermore, the methodology also helps evaluate the contribution of each energy component, i.e., shear, flexure, axial and joint deformation to the structure’s response under consideration thereby indicating the member properties that need to be modified for an optimized design.

By multiplying the contribution of both a "virtual" load case and a "real" load to each of the energy components (i.e., shear, flexure, axial and torsion), the contribution of each member in the structure to drift is evaluated. The elemental contribution (also called DPF - displacement participation factor) to drift (or frequency, as the case maybe) is further broken down to each of the components such as shear, flexure, axial, joint and torsion displacements.

This breakdown of contribution helps to identify which behavior is dominant and what sectional or material property needs to be modified to arrive at acceptable and desired responses. For instance, to reduce drift or frequency, members with large participation factors should be made stiffer and contributing members with very small participation factors could be made smaller. Furthermore, an important piece of information is the per volume contribution (or participation) of each structural element, which is also referred to as Sensitivity Index (SI). It is obtained by dividing the element participation factor by its volume. From a weight optimization point of view, the Sensitivity Index provide valuable information as summarized below:
1. "When adding material to a structure to reduce displacement, the material should be added to the member (s) with the largest sensitivity index."
2. "When removing material from a structure to improve economy, the material should be removed from member (s) with the smallest SI values". [1]

The reader is referred to References [1] - [3] for further reading on Virtual Work optimization.