Structural Analysis Calculator (Simple Truss)

What is a Structural Analysis Calculator (Simple Truss)?

A structural analysis calculator helps engineers determine forces within members of a structure, and reaction forces at its supports. This specific calculator focuses on a simple 2D triangular truss, a fundamental structural form made of interconnected straight members.

This tool analyzes a common configuration: a symmetrical triangular truss with a pinned support at one end, a roller support at the other, and a vertical point load applied at its apex. It uses the "Method of Joints" to calculate the axial forces (tension or compression) in each truss member and the support reactions.

Formulas for a Simple Triangular Truss Analysis

Consider a symmetrical triangular truss ABC, with base AC of length L, and height H from base to apex B. Support A is pinned (R_Ax, R_Ay), support C is a roller (R_Cy, R_Cx=0). A vertical load P acts downwards at apex B.

1. Support Reactions:

From overall equilibrium of the truss:

• ΣF_x = 0 => R_Ax = 0 (since no horizontal external loads)
• ΣM_A = 0 => P * (L/2) - R_Cy * L = 0 => R_Cy = P / 2
• ΣF_y = 0 => R_Ay + R_Cy - P = 0 => R_Ay + P/2 - P = 0 => R_Ay = P / 2

2. Member Forces (Method of Joints):

Let θ be the angle between the base member AC and the diagonal members AB and BC. tan(θ) = H / (L/2) = 2H / L.

Joint A:

• ΣF_y = 0 => R_Ay + F_AB * sin(θ) = 0 => P/2 + F_AB * sin(θ) = 0 => F_AB = - (P / 2) / sin(θ) (Compression)
• ΣF_x = 0 => R_Ax + F_AC + F_AB * cos(θ) = 0 => 0 + F_AC + F_AB * cos(θ) = 0 => F_AC = - F_AB * cos(θ) (Tension)

Joint C (similar by symmetry):

• F_BC = F_AB (Compression, by symmetry)

Alternatively, at Joint B:

• ΣF_y = 0 => -P - F_AB*sin(θ) - F_BC*sin(θ) = 0. If F_AB = F_BC => -P - 2*F_AB*sin(θ) = 0 => F_AB = -P / (2*sin(θ))

Note: Negative force indicates compression, positive indicates tension.

Helper for calculations:

• Length of diagonal members (L_diag) = √(H² + (L/2)² )
• sin(θ) = H / L_diag
• cos(θ) = (L/2) / L_diag

How to Analyze a Simple Truss: Example

Consider a triangular truss with:

• Base Length (L) = 6 m
• Height (H) = 2 m
• Vertical Load at Apex (P) = 10 kN

1. Calculate Support Reactions:

   R_Ax = 0 kN

   R_Cy = P / 2 = 10 kN / 2 = 5 kN (upwards)

   R_Ay = P / 2 = 10 kN / 2 = 5 kN (upwards)

2. Calculate Angle θ and its trigonometric functions:

   Half base length = L/2 = 6m / 2 = 3m

   Length of diagonal member AB (L_diag) = √(H² + (L/2)²) = √(2² + 3²) = √(4 + 9) = √13 ≈ 3.606 m

   sin(θ) = H / L_diag = 2 / 3.606 ≈ 0.5547

   cos(θ) = (L/2) / L_diag = 3 / 3.606 ≈ 0.8321

3. Calculate Force in Member AB (F_AB):

   F_AB = - (P / 2) / sin(θ) = - (10 kN / 2) / 0.5547 = -5 kN / 0.5547 ≈ -9.014 kN (Compression)

4. Calculate Force in Member AC (F_AC):

   F_AC = - F_AB * cos(θ) = - (-9.014 kN) * 0.8321 ≈ 7.500 kN (Tension)

5. Force in Member BC (F_BC):

   F_BC = F_AB ≈ -9.014 kN (Compression, by symmetry)

Summary of forces: F_AB ≈ 9.01 kN (C), F_BC ≈ 9.01 kN (C), F_AC ≈ 7.50 kN (T).

Common Applications

• Roof Trusses: Analyzing forces in common roof structures to ensure stability.
• Bridge Design: Determining member forces in truss bridges under various load conditions.
• Construction Scaffolding: Ensuring the safety and load-bearing capacity of temporary structures.
• Cranes and Lifting Devices: Analyzing forces in the structural members of cranes.
• Educational Purposes: Teaching fundamental principles of statics and structural mechanics.