# virtual work and energy method

\begin{align*} W_{v,e} &= W_{v,i} \\ \text{External Virtual Work} &= \text{Internal Virtual Work} \end{align*} \begin{equation} \boxed{ \sum \left( \begin{array}{l} \text{Virtual Ext. This service is more advanced with JavaScript available, Structural Theory and Analysis Use it at your own risk. Since the force $P$ is constant the entire time that force $Q$ is applied to the bar, the additional external work caused by force $P$ is equal to: \begin{align*} W_{P,e} = P \Delta_Q \end{align*}. The same is true for the moment. • Need to repeat analysis for different points. Part (b) of Figure 5.17 shows the same beam structure, but this time the two point loads are applied separately and a full analysis is done for each, including determination of the displaced shape. +Now we have the external virtual work, we also need the internal virtual work. The information on this website, including all content, images, code, or example problems may not be copied or reproduced in any form, except those permitted by fair use or fair dealing, without the permission of the author (except where it is stated explicitly). A simple beam structure is shown in part (a) of the figure. where $L$ is the length of the bar, $E$ is the Young's Modulus of the bar material, and $A$ is the cross-sectional area of the bar. The total strain energy in a simple bar that is deformed axially by a force that gradually increases from zero up to a maximum force $p$ is equal to: \begin{equation} U = \frac{p\delta}{2} \tag{6} \end{equation}. In this figure, first a force $P$ is gradually added to the bar. The force is just enough to overcome the surface friction between the body and the ground. Virtual work is perhaps the most useful and widely applicable of the methods for calculating deflections that are described in this chapter. There are two different principles of virtual work, one for rigid bodies, and one for deformable bodies. These keywords were added by machine and not by the authors. These equations do not have the factor of $\frac{1}{2}$ since the areas are rectangles as shown in the darkly shaded portions of the bottom left and bottom right plots (the force remaining constant). We know from equation \eqref{eq:virtual-work} that: \begin{align*} W_{v,e} = W_{v,i} \end{align*}. Like the previous example, this structure has two loads. Similar to the situation described in previous Figure 5.16, in this structure, there is a single bar with a pin at one end and a roller at the other. Input Work OutputWork e For simple machines with SDOF & which operates in uniform manner, mechanical efficiency may be determined using the method of Virtual Work >>When you're done reading this section, check your understanding with the interactive quiz at the bottom of the page. The deflections are what we would like to find. Download preview PDF. Since the behaviour of the virtual system does not affect the behaviour of the real system, then we can choose any arbitrary set of virtual external loads that we want (position, magnitude and direction). This strain energy ($U$) can also be called the internal work (here written as $W_i$). Although the internal and external virtual work both include real and virtual components, we can find the individual component using the real system and virtual system in isolation (again because of superposition). The principle of virtual work for rigid bodies is akin to equilibrium, but is beyond the scope of this text. Forces} \; \times \text{Real Int. © 2020 Springer Nature Switzerland AG. This will be the force that we will consider to be our `virtual' force. The definition of the external virtual work is what makes the principle of virtual work useful for finding structural deflections. You may recall that the expression for work ($W$) is equal to: \begin{equation} \boxed{W = P\Delta} \label{eq:work} \tag{1} \end{equation}. Therefore since the total must be equal, then the internal virtual work must equal the external virtual work: \begin{align*} W_e &= W_i \\ W_{0,e} + W_{v,e} + W_{r,e} &= W_{0,i} + W_{v,i} + W_{r,i} \\ \text{but} \; W_{0,e} &= W_{0,i} \\ \text{and} \; W_{r,e} &= W_{r,i} \end{align*}, \begin{equation*} W_{v,e} = W_{v,i} \end{equation*}. Since energy must be conserved in the system, all of the work that is applied to from the outside, must be balanced by the work done on the inside (through strain energy), so: \begin{equation} \boxed{W_e = W_i} \label{eq:consv-energy} \tag{9} \end{equation}. it can only resist axial loads). This process is experimental and the keywords may be updated as the learning algorithm improves. The bar force is higher than the external force (because of the angle), and the deformation of the bar will be less than the displacement (also because of the angle, try out the trigonometry for yourself); however, the external work must still be equal to the internal work to maintain the conservation of energy: \begin{align*} W_{0,e} &= W_{0,i} \\ \frac{P_v \Delta_v}{2} &= \frac{p_v \delta_v}{2} \end{align*}. The external work done by the addition of force $Q$, which gradually increases from zero up to its maximum value, is similar to the original work that was done by force P when it was applied: \begin{equation*} W_{Q,e}=\frac{Q \Delta_Q}{2} \end{equation*}. where $M$ is a moment acting on a body or structure and $\theta$ is the rotation of that body or structure. This is the dark shaded triangular area in the bottom right plot of Figure 5.16. Over 10 million scientific documents at your fingertips. If there is only one bar in the system, then the total internal work (strain energy) in the system is equal to: \begin{align*} W_i = U = \frac{p^2L}{2EA} \end{align*}. where $\delta$ is the axial deformation of the bar. For beam bending problems like this, the axial and shear deformations are typically insignificant in comparison to the deformations caused by bending stresses and strains. A frame element, unlike the truss bar element that we discussed previously, has multiple different ways to store strain energy. Part of Springer Nature. This is a preview of subscription content, log in … This work is shown as the dark shaded regions in the plots on the right side of Figure 5.18 part (c). One $P_v$ which is added first, and another one $P_r$ that is added after the full application of $P_v$. The structure with the real forces may be called the real system and the structure with only the virtual force (without the real forces) may be called the virtual system. Reciprocal theorems and theorems of plastic analysis for plane frames are also discussed. virtual work and energy concepts, calculation of deflections, and analysis of indeterminate structures using the compatibility and equilibrium methods. While the force $P_r$ is doing work, displacing the end and deforming the bar, force $P_v$ is still present, and therefore also still does work (even though it is not causing the new displacement $\Delta_r$). The information on this website is provided without warantee or guarantee of the accuracy of the contents. If we can find these, and if we select an appropriate virtual external load, then we can use virtual work to solve for the real external displacement at a specific point. Notice that this is similar to the general work equation from equation \eqref{eq:work}, but is divided by two because the area under the plot is a triangle instead of a rectangle. The chapter also provides an introduction to a later chapter on stiffness and flexibility methods, which are fundamental to a modern approach in structural analysis and can in fact be applied to all types of structures, including framed structures, plates, and shells. Finally, an example of what each of these components actually look like for a beam is shown in Figure 5.19. or the force $P$ multiplied by the deformation caused by $Q$. What does any of this mean, anyway? , and one for deformable bodies required to move mass around in space of course to solve for displacement. To understand before we can use the virtual force is a force does n't matter the... Structural parts } \tag { 15 } \end { array } \right ) = \sum \left ( {... Work must equal the area underneath the force does not have to.! Forces and the keywords may be updated as the external virtual work is applicable when the force P! 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