13.2. Stiffness Matrices

From strength of materials, the force/displacement relation in axial members is

$\sigma = E\epsilon$ $\sigma = \frac{A}{L} P$ Hence, for a unit displacement, the applied force should be equal to $AE$. From statics, the force at the other end must be equal and opposite.

In structural engineering, truss elements are foundational components that experience axial forces. This means they only take tension or compression along their length. The relationship between force and displacement in these axial members follows Hooke's Law, which states that stress ($\sigma$) is proportional to strain ($\epsilon$). The equations provided express this relationship using area (A), modulus of elasticity (E), and length (L) of the truss element. When a truss undergoes a unit displacement, the force generated due to that displacement is dictated by the product of the cross-sectional area and the modulus of elasticity.

Think of a rubber band. When you stretch it (applying a displacement), it resists that stretch by getting tighter (creating a force). Similarly, in a truss element, the material properties determine how much force it can handle when stretched or compressed.

The truss element (whether in 2D or 3D) has only one degree of freedom associated with each node. Hence, from Eq. 13.1, we have $\mathbf{K_t} = \frac{AE}{L} \begin{bmatrix} 1 & -1 \\ -1 & 1 \end{bmatrix}$

The stiffness matrix ($\mathbf{K_t}$) for a truss is crucial for analyzing its behavior under loads. Each node within a truss element has a single degree of freedom, which typically represents the axial displacement. The stiffness matrix captures how forces at the two ends of the truss relate to the displacements at those ends. In the matrix, the values represent how a unit displacement of one node affects the force at the other node. The coefficients indicate equal but opposite forces due to the axial nature of the truss.

Imagine a see-saw; when one end goes down, the other end goes up. In the truss model, pushing down one side (applying a force) causes a predictable upward reaction on the other side, reflecting the essential balance of forces modeled in the stiffness matrix.

Using Equations 12.10, 12.10, 12.12 and 12.12 we can determine the forces associated with each unit displacement. The stiffness matrix of the beam element (neglecting shear and axial deformation) will thus be $k_b = \begin{bmatrix} 12EI/L^3 & 6EI/L^2 \\ 6EI/L^2 & 4EI/L \end{bmatrix}$

For beam elements, the stiffness matrix is generated from fundamental principles of mechanics, considering bending behavior under loads. This matrix accounts for moments and deflections, providing a relationship between the applied forces and the resulting displacements. It reflects how a beam will resist bending when forces are applied at different points, and the coefficients relate to the flexibility and stiffness of the beam based on its elastic modulus (E) and moment of inertia (I).

Picture a diving board. When a person jumps on one end, it bends down while the other end rises. The degree to which the board bends (its displacement) for a certain weight (force) is comparable to how the stiffness matrix calculates the relationship between load and deflection for a beam element.

The global matrix can be rewritten and stored in the same global matrix. The structure stiffness matrix is assembled, combining the stiffness relationships of all individual elements to build the overall response of the structure.

In practice, the global stiffness matrix represents the entire structure composed of multiple elements (like trusses or beams). Each element’s stiffness contributes to this larger matrix, allowing engineers to evaluate how the entire system will react under various loads. This assembly process is essential to understanding structural integrity and flexibility. Correctly assembling these matrices enables us to predict displacements and internal forces mathematically and ensures the safety and usability of buildings and structures.

Consider assembling a Lego structure. Each Lego piece (element) has a specific strength and flexibility. When you connect them all (assemble the global matrix), the combined structure's ability to withstand bending and forces is enhanced. The total stability relies on the individual contributions of each piece, just like the overall stiffness of a structure depends on its individual elements.

--