Analisa Struktur Truss Metode Matriks part 2

Introduction

This tutorial provides a step-by-step guide on analyzing truss structures using the matrix method, as demonstrated in the second part of Muhamad Rusli Ahyar's video. Understanding this method is crucial for structural engineering and design, allowing for the efficient calculation of forces within truss members.

Step 1: Understand Truss Structure Basics

Familiarize yourself with key components of truss structures

• Nodes: Points where members connect.
• Members: Straight elements that form the truss.
• Supports: Points that provide stability, typically at the ends of the truss.

Recognize the types of loads that can act on a truss, including

• Dead loads (permanent/static)
• Live loads (temporary/dynamic)

Step 2: Set Up the Coordinate System

Choose a coordinate system that simplifies the analysis

• Typically use a Cartesian coordinate system (X, Y).

Assign coordinates to each node in the truss

• For example, Node A at (0,0), Node B at (L,0), etc.

Step 3: Establish the Global Stiffness Matrix

• Prepare to create the global stiffness matrix for the truss:

  • Each member can be represented with its stiffness matrix, derived from its material and geometric properties.

• The stiffness matrix ( k ) for a member can be calculated 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

Step 4: Assemble the Global Stiffness Matrix

Combine the individual stiffness matrices into a global stiffness matrix

• Ensure that the orientation of each member is considered.
• Use systematic indexing to keep track of all degrees of freedom for each node.

Step 5: Apply Boundary Conditions

Identify and apply boundary conditions to the global stiffness matrix

• Remove rows and columns corresponding to fixed supports or constraints.
• This helps in simplifying the calculations and focusing on the active degrees of freedom.

Step 6: Formulate the Load Vector

Create a load vector that represents external forces applied to the nodes

• Each entry corresponds to a force at a specific node in the direction of the coordinate system.
• Ensure that your load vector aligns with the dimensions of the modified global stiffness matrix.

Step 7: Solve the System of Equations

• Use numerical methods or software to solve the system of equations derived from:

  K * u = F
  

  Where:

  • ( K ) is the global stiffness matrix
  • ( u ) is the displacement vector
  • ( F ) is the load vector

• Determine the displacements at each node.

Step 8: Calculate Member Forces

• With displacements known, calculate forces in each member:

  • Use the relationship between member forces and node displacements.

• For member ( i ), the force can be calculated as:

  F_i = k_i * u_i
  

Conclusion

This tutorial covered the essential steps for analyzing truss structures using the matrix method. Key takeaways include setting up a proper coordinate system, assembling the global stiffness matrix, applying boundary conditions, and solving for member forces. For further study, consider practicing with different truss configurations and loads to deepen your understanding of structural analysis.