METODE MATRIKS KEKAKUAN #2, Metode Matriks Kekakuan, Plane Truss Structures, Struktur Rangka Bidang

Introduction

This tutorial provides a step-by-step guide to understanding the Direct Stiffness Method for analyzing plane truss structures, as presented in the video by Yoyong Arfiadi. This method is essential for civil engineering students and professionals working with structural analysis, allowing for the determination of forces in truss members efficiently.

Step 1: Understand the Basics of Plane Truss Structures

• A plane truss is a structure composed of members connected at joints, primarily used to support loads.
• Key characteristics include:
  • Members are assumed to carry axial loads only (tension or compression).
  • The structure is two-dimensional, meaning all members and loads lie in a single plane.

Step 2: Set Up the Structural Model

• Begin by defining the geometry of the truss:
  • Identify all nodes (joints) and members (structural elements).
  • Draw a clear diagram to visualize connections and loading conditions.
• Label all forces acting on the truss, including external loads and reactions at supports.

Step 3: Establish the Global Stiffness Matrix

• The stiffness matrix is fundamental for the analysis. Follow these sub-steps:
  1. For each member, calculate the local stiffness matrix using the formula: [ k = \frac{EA}{L} \begin{bmatrix} 1 & -1 \ -1 & 1 \end{bmatrix} ] where:
     • E is the modulus of elasticity,
     • A is the cross-sectional area,
     • L is the length of the member.
  2. Transform the local stiffness matrices to the global coordinate system if necessary.
  3. Assemble the global stiffness matrix by summing the contributions from all members.

Step 4: Apply Boundary Conditions

• Modify the global stiffness matrix to account for supports and constraints:
  • Identify fixed or pinned supports and remove the corresponding rows and columns from the stiffness matrix.
  • Adjust the load vector accordingly to reflect applied loads and reactions.

Step 5: Solve for Node Displacements

• Once the global stiffness matrix and load vector are ready, solve the system of equations: [ \mathbf{K} \mathbf{d} = \mathbf{F} ] where:
  • ( \mathbf{K} ) is the global stiffness matrix,
  • ( \mathbf{d} ) is the vector of displacements,
  • ( \mathbf{F} ) is the load vector.
• Use numerical methods (like Gaussian elimination) or computational tools (e.g., MATLAB) to find the displacements at each node.

Step 6: Calculate Member Forces

• With the displacements known, calculate the forces in each truss member:
  • For each member, use the relationship: [ \mathbf{F} = \mathbf{k} \cdot \mathbf{d} ]
  • This will give you the internal forces (tension or compression) in each member based on the calculated displacements.

Conclusion

The Direct Stiffness Method is a powerful tool for analyzing plane truss structures. By following these steps, you can effectively model a truss, apply loads, and determine member forces. For practical applications, consider using software like MATLAB to automate the calculations and visualize results. Explore the provided links for further reading and examples to deepen your understanding of this analysis technique.