12.3. Kinematic Relations

Whereas in the flexibility method we sought to obtain a displacement in terms of the forces (through virtual work) for an entire structure, our starting point in the stiffness method is to develop a set of relationship for the force in terms of the displacements for a single element.

In structural analysis, there are two main methods: the flexibility method and the stiffness method. In the flexibility method, the goal is to express the displacements of a structure based on applied forces. This approach considers how the entire structure deforms when forces are applied. Conversely, in the stiffness method, we begin with the concept of relating forces to displacements but focus on a single element of the structure. In this framework, we formulate the relationships that will allow us to express how much force is generated in response to a specific displacement at that element.

Imagine bending a simple ruler (representing a beam) with your hands at two points. The ruler flexes and bends, and the extent of bending (displacement) depends on the force you apply (the force-displacement relationship). If we treat each segment of the ruler (as an individual element), we can predict the required force to achieve any desired position of the ruler.

We start from the differential equation of a beam, where we have all positive known displacements, we have from strength of materials d2v/dx2 = M/EI = M(Vx+m(x)) where m(x) is the moment applied due to the applied load only. It is positive when counter-clockwise.

The behavior of a beam under load can be described using a differential equation that relates the beam's deflection (v) to the applied moments and forces. The equation d2v/dx2 gives us the curvature of the beam, which is related to the moments (M) and the properties of the material (EI, where E is Young's modulus and I is the moment of inertia). The term m(x) accounts for the effect of the loads applied to the beam and indicates how these moments change along the length of the beam. Significantly, we note that a moment is considered positive when it causes a counter-clockwise rotation.

Consider a diving board as a beam. If a diver jumps on one end, the board bends down at the point of impact. The relationship defining how much it bends depends on various factors, such as the diver's weight and where they land. Just like the mathematical description of the forces and bending moments, we can predict the board's shape under different loads.

Integrating twice leads to the equations: EIv = M x V x^2 + f(x) + C1, EIv = M x^2 + V x^3 + g(x) + C2, where f(x) = m(x)dx, and g(x) = f(x)dx.

To find the deflection of a beam from the differential equation, we integrate the equation twice. The first integration gives us a relationship that includes the slope (related to the first derivative of the deflection). The second integration provides the deflection itself, consisting of terms accounting for applied moments and boundary conditions, denoted by integration constants C1 and C2. The functions f(x) and g(x) represent the cumulative effect of the applied moments and loads along the length of the beam, respectively.

Think of a piece of pasta (representing a beam) that you bend. The initial shape as you start bending it (the slope) affects how much the entire pasta will bend (the deflection). Integrating our equations is like keeping track of how each small bend accumulates to form the overall curvature of the pasta.

Applying the boundary conditions at x = 0 gives us: v = C1 = EItheta0, v = v1 implies C2 = EIv1.

Boundary conditions are crucial in structural analysis as they help define specific scenarios at particular points on the beam, typically where it is supported or loads are applied. By substituting values into our earlier equations, we can determine the constants C1 and C2, which set the initial conditions for the beam's deflection and slope. This ensures that our solutions reflect the exact conditions of the structure, enabling accurate predictions of behavior under load.

Consider a seesaw (the beam) resting on a pivot point (a support). When you sit down on one side (the applied load), you can think of the seesaw's reactions as boundary conditions. By knowing how high each end of the seesaw sits at rest, we can predict how it will behave when force is applied.

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