12.3.1. Force-Displacement 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 primary methods: the flexibility method and the stiffness method. While the flexibility method focuses on how much an entire structure moves when subjected to forces, the stiffness method approaches the problem differently. Instead of looking at the whole structure, it zeroes in on individual elements and establishes a relationship between the forces acting on these elements and their respective displacements. This foundational difference represents a shift in perspective from connecting forces to relate them with movement.

Think of a rubber band. If you pull on a rubber band (force), you can measure how much it stretches (displacement). The stiffness method is like studying a single rubber band to understand how pulling it affects its stretch, while the flexibility method considers how the entire rubber band system behaves when you pull different parts.

We start from the differential equation of a beam, Fig. 12.4 in which we have all positive known displacements, we have from strength of materials d²v/M = EI = M/Vx + m(x).

The behavior of beams under load can be described mathematically using differential equations. In this equation, 'M' represents the bending moment at a certain point along the beam, 'EI' is the flexural rigidity (a product of the modulus of elasticity 'E' and the moment of inertia 'I' of the beam's cross-section), and 'm(x)' denotes the applied moment due to external loads. The equation establishes how the curvature of the beam (represented by 'v') changes based on the forces acting on it, capturing how external forces influence internal stresses.

Imagine bending a ruler. When you apply a force at the center, that force creates a bending moment that causes the ruler to curve. The differential equation helps us quantify how much the ruler will bend under different loads, guiding engineers in designing safe structures.

Integrating twice: EIv = Mx + Vx²/2 + f(x) + C1; EIv = Mx² + Vx³/3 + g(x) + C2.

To solve the differential equation describing beam behavior, engineers integrate the equation twice. The first integration provides the relationship between bending and displacement. The second integration leads to additional terms that account for constants of integration (C1 and C2) and specific functions that represent physical effects like loading. These integrated equations help establish the relationship between internal forces and displacements in the structure.

Consider making a sculpture out of clay. If you push down on the clay (applying a force), the shape of your sculpture changes. First, you understand that pushing changes the surface (first integration), but then you realize how the base might also adjust to maintain stability, and you account for that second adjustment (second integration).

Applying boundary conditions at x = 0: v = 0; C1 = EIθ.

Applying boundary conditions means enforcing specific known values at certain points on the beam, such as supports or loads. For example, if the displacement (v) is zero at one end (x=0), we can use this information to solve for the constants (C1). Boundary conditions are critical for ensuring that the mathematical model accurately reflects the physical support conditions of the structure.

Imagine you are building a bridge. You know the bridge starts at the ground (x=0) and cannot bend down there (v=0). By applying this condition, you can determine how the rest of the bridge will behave, ensuring safety and structural integrity.

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