3. Alternative way to prove superposition

Consider an arbitrary body clamped at two locations. We apply a force F at point 1 and measure vertical component of displacement at point 3 (see Figure 3). The displacement will be δ = k F.

In this part, we introduce a scenario where we have a body that is fixed at two points. When we apply a force (F) at a specific point (point 1), the body will deform, leading to a measurable displacement (δ) at another point (point 3). The relationship between the displacement at point 3 and the applied force is linear, indicated by the equation δ = kF, where k is an influence coefficient that characterizes how much displacement occurs for a given force.

Imagine a trampoline. When someone jumps on it (applying a force F), the fabric stretches, and the point where the jumper lands goes down (displacement δ). The trampoline's stiffness determines how much it stretches for a given force, much like the influence coefficient k determines how much a solid body displaces for a given force.

Due to the application of force, the system must have deformed and changed its shape and size (see blue dotted lines in Figure 4). In the next step, we apply force F at point 2’ (originally at 2) to the deformed body.

Once the initial force is applied, the shape and dimensions of the body change. We then continue to apply an additional force (second force) at a point that has moved due to this deformation. This means that the reaction and influence of forces within the body are contingent upon its current state after deformation, affecting the influence coefficient.

Think of a piece of clay; once you push into it (the first force), the clay changes shape. If you then push down on a different part that has moved as a result of your first push (the second force), the amount it moves will change because the clay is no longer in its original state.

The total vertical displacement at point 3 will be expressed as δ = k F (1) + k F (2). This means that if both forces act together, the resulting displacement is simply the sum of the displacements caused by each force acting separately.

We derive that when multiple forces act on the body simultaneously, the total displacement at point 3 is the sum of the individual displacements caused by each applied force. This holds true under the superposition principle, which is valid as long as the governing equations are linear.

Consider a seesaw with two kids sitting at different ends. If one child pushes down (applying force F1), the seesaw tilts one way. If both kids push down together (applying an additional force F2), the seesaw tilts even more. The total tilt (displacement) is the sum of each child’s push on the seesaw.

Notice that we have used the original influence coefficient in the above equation as if only force F2 were acting right from the beginning, because the displacement in an elastic body only depends on the final state of applied loads and does not depend on how those loads were applied.

The final displacement depends solely on the last state of applied loads rather than the sequence of application. In other words, it doesn't matter if we applied force F1 first or F2; the resulting displacement will be the same as long as the final load configuration is the same. This illustrates an important property of elastic systems and the nature of superposition.

Imagine climbing a staircase. Whether you step on the first stair first and then the second or vice versa, you'll still end up at the same height. In both cases, your final position (displacement) depends only on the total height of the stairs you climbed, not the order in which you stepped on them.

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