9.3. Virtual Work

A severe limitation of the method of real work is that only de(cid:13)ection along the externally applied load can be determined. A more powerful method is the virtual work method.

The virtual work method addresses the limitation of the real work method, which can only measure deflections in the direction of the applied loads. This means it can't provide information about other possible deformations in a structure under load. The virtual work method expands beyond this limitation and offers a more comprehensive analysis of a structure's behavior under various forces.

Imagine a trampoline. When someone jumps on it (real load), you can see the deformation (deflection) directly beneath them. However, the trampoline also bends and flexes in other areas, which are not directly under the load. The virtual work method allows us to understand the entire trampoline's response, not just where the person jumps.

The principle of Virtual Force (VF) relates force systems which satisfy the requirements of equilibrium, and deformation systems which satisfy the requirement of compatibility.

This principle suggests that when we analyze a structure, we can consider both the actual forces exerted on it and hypothetical forces that keep the structure in equilibrium (a balanced state). Additionally, the deformations resulting from these forces must also be compatible, meaning they should make sense with respect to how the structure is capable of moving or deforming.

Think of a team of people lifting a heavy table. All of them must coordinate their pushes (actual forces) to lift it properly without tipping or causing the table to bend awkwardly. If one person pushes too hard in the wrong direction, it would not work; thus, their forces need to be both balanced (equilibrium) and coordinated for it to work effectively (compatibility).

In any application the force system could either be the actual set of external loads dp or some virtual force system which happens to satisfy the condition of equilibrium (cid:14)p.

When studying structures, we can examine real forces (actual loads applied to a structure) and virtual forces (hypothetical forces that are also balanced). Both types of forces are crucial when analyzing how the structure behaves under load. By using virtual forces, engineers can simplify complex problems and still extract valuable insights about structural performance.

Consider a bridge over a river. Real loads would be cars crossing it (actual forces). If we want to see how the bridge would manage if it were also carrying more weight than it physically can (like a strong wind), we can apply a virtual force in our calculations to see how it affects its structural integrity without actually putting the bridge under that strain.

Similarly, the deformation could consist of either the actual joint de(cid:13)ections du and compatible internal deformations d" of the structure, or some virtual external and internal deformations (cid:14)u and (cid:14)" which satisfy the conditions of compatibility.

Deformation in a structure can be analyzed through both actual observations (real deformations of joints and internal structures) and hypothetical scenarios (virtual deformations). By applying the concept of virtual deformation, one can assess how the structure would behave under different conditions, ensuring that compatibility with the actual setup is maintained.

Imagine a bendy straw used in a drink. When you sip, the straw deforms in a way that allows you to drink smoothly (actual deformation). However, if you were to visualize how it would bend even more under extreme conditions (like squeezing it harder), that would be a virtual deformation. Engineers can use both observations to ensure that the straw is designed well enough for normal and extreme uses.

It is often simplest to assume that the virtual load is a unit load.

To simplify calculations in the virtual work method, engineers often assume the virtual load to be a unit load—essentially setting it to 1. This assumption allows for easier calculations and clearer understanding of the proportional effects of actual loads on the structure without getting bogged down in complicated numbers.

Think of measuring how heavy a backpack feels when it’s full versus when it’s empty. If every time you want to feel the difference between two states, you just hold a single pound (a unit) in one hand while keeping the full backpack in the other. This simplifies understanding how much extra weight it adds instead of trying to calculate every single heavy item in the backpack.

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