Matrix Structural Analysis (Introduction) - Examples & Applications

Assemble the global stiffness matrix for a simple 1D system composed of two sequential springs connected in series. Node 1 is the left support (fixed). Node 2 connects Spring 1 (stiffness $k_1$) and Spring 2 (stiffness $k_2$). Node 3 is the right end (free). There are 3 total nodes, meaning the initial global matrix will be 3x3 before applying boundary conditions.

Calculate the element stiffness matrix $[k]$ in global coordinates for a 2D truss element. The element has length $L=5\text{ m}$, cross-sectional area $A=2\times 10^{-3}\text{ m}^2$, and modulus of elasticity $E=200\text{ GPa}$. It connects Node 1 at $(0,0)$ to Node 2 at $(3,4)$.

Calculate the element stiffness matrix $[k]$ for a 2D horizontal beam element. The beam has length $L=4\text{ m}$, moment of inertia $I=300\times 10^6\text{ mm}^4$, and modulus of elasticity $E=200\text{ GPa}$. Assume axial deformation is neglected, leaving 2 DOFs per node (vertical displacement $v$ and rotation $\theta$).