SkyCiv Truss Calculator Tutorial: Solve 2D Truss Structures

The stiffness matrix is a fundamental concept in Finite Element Analysis that relates forces to displacements in discretized structural systems. The FEA procedure begins with discretization, dividing continuous bodies into finite elements connected by nodes. Two primary solution methods exist: the force method (flexibility method) for unknown forces, and the displacement method (stiffness method) for unknown displacements. The stiffness method is preferred for its simpler formulation from physical properties. Matrix representation converts individual element equations into a global system that can be efficiently solved computationally. The linear elastic spring element serves as the foundational model, with stiffness k following Hooke's Law. When multiple elements assemble, each has its own stiffness between different nodes. The fundamental relationship f = k × u applies to single elements, while F = K × U governs assembled systems.

Finite Element Analysis (FEA) is a numerical method that transforms complex structural problems into manageable systems by dividing structures into small interconnected elements (finite elements) connected at nodes, then using shape functions to calculate displacements and stresses at these nodes, and finally assembling a global stiffness matrix to solve for the overall structural behavior; this method, developed in 1943 for aircraft analysis, is now the foundation of all structural analysis software including SAP2000, allowing engineers to analyze complex geometries, multiple materials, and various boundary conditions that would be impossible to solve using traditional analytical methods.

The first step in FEA analysis is discretization, where the structure is divided into smaller elements. For a stepped bar with 300 kN axial load, the bar is divided into three elements with nodes numbered 1, 2, 3, 4. The degrees of freedom equals the number of nodes (4). The element stiffness matrix is calculated using K = (AE/L), where A is cross-sectional area, E is Young's modulus, and L is element length. For element 1: A1=250 mm², E1=200×10³ MPa, L1=150 mm, giving K1 = 3.33×10⁵ N/mm. For element 3: A3=400 mm², E3=200×10³ MPa, L3=300 mm, giving K3 = 2.66×10⁵ N/mm. All stiffness values must use consistent units.

The Finite Element Method (FEM) is an analytical technique for solving complex engineering problems by discretizing continuous structures into simpler elements. The spring element serves as the foundational element in FEM courses due to its simplicity and instructive nature. The stiffness matrix (K) is defined as a matrix that relates nodal displacements (u) to nodal forces (F) through the equation K × u = F. For continuous structures, the global stiffness matrix relates the displacements of the entire structure to the applied forces. The spring element, with its two nodes and single degree of freedom per node, provides the most general and instructive tool for illustrating these basic concepts.

The Finite Element Method (FEM) is a numerical calculation method essential for structural analysis, used to determine stresses and deformations. It is widely implemented in major structural analysis programs such as SAP, ANSYS, and Nastran. A plane frame model can accommodate up to three degrees of freedom per node: two translational (horizontal and vertical) and one rotational. The stiffness matrix for a plane frame element is a 6x6 matrix derived from superposition of truss element stiffness matrices. When a frame element is divided into smaller segments, each segment retains three degrees of freedom per node. The global stiffness matrix for a structure with five nodes becomes a 15x15 matrix. Material properties include concrete area of 50x20 cm², modulus of elasticity calculated as 0.85 × √fck, and moment of inertia calculated as (base × height³)/12. The coordinate transformation matrix relates local element coordinates to global coordinates using direction cosines.

This video demonstrates the complete process of designing steel truss members under tension and compression using Eurocode 3, covering material properties (E=210,000 N/mm², fy=275 N/mm²), section classification (Class 1 for 50×50×5mm RHS with c/t=8 < 13.36), cross-sectional compression resistance verification (Ncrd = 240 kN), and flexural buckling analysis (λ̄ = 86.39, χ = 0.76) to ensure structural adequacy.

Truss member design for compression must consider buckling (pandeo). The procedure involves: (1) calculating the effective length using Leff = α × L, where α depends on end conditions (0.5 for fixed-fixed, 0.7 for fixed-pinned, 2.0 for pinned-pinned), (2) calculating the slenderness ratio λ = Leff/r_min, where r_min is the minimum radius of gyration, (3) determining the buckling coefficient from tables using λ, (4) calculating the Euler buckling stress σ_Euler = π²E/(λ²), (5) verifying that σ_Euler ≥ F/A. The design must satisfy both tensile and compressive requirements.

In bridge trusses, member sizing reflects their respective force conditions. Tension members (diagonals and bottom chords) can be slender rods since tension doesn't cause buckling. Compression members (verticals and top chords) must be chunkier to resist buckling. This is visible in actual bridges where vertical compression members are thicker than diagonal tension rods. The compression members don't need to be solid; they can be constructed with multiple laced members to provide overall stability against buckling.

This section covers truss member design for tension and compression. Truss members in tension are designed based on axial force and allowable stress. The required area is calculated as force divided by allowable stress. For members composed of multiple elements, the total area is divided among elements. The design considers slenderness ratio to prevent buckling. Truss members in compression are designed based on axial force, effective length, and allowable stress. The effective length is determined by distance between points of lateral restraint. The slenderness ratio is calculated as effective length divided by radius of gyration. The allowable stress is determined from slenderness ratio using buckling formulas. The required area is calculated as force divided by allowable stress. The design must consider local buckling of individual elements and overall stability. The effective length of truss members for buckling analysis depends on end conditions and points of lateral restraint. For members with pinned ends, effective length equals distance between supports. For members with fixed ends, effective length is half the distance between supports. The radius of gyration is determined from member section properties. The slenderness ratio is calculated as effective length divided by radius of gyration. The allowable stress is determined from slenderness ratio using appropriate buckling formulas. Truss members, particularly the top chord working in compression, must be designed to prevent lateral buckling. The lateral restraint is provided by web members and the bracing system. The top chord is typically braced at regular intervals (every two girders) to prevent lateral displacement. The bracing resists lateral forces during wind events or erection. The member design accounts for effective length between restraints and ensures slenderness ratio is within acceptable limits. Truss member design involves selecting appropriate profiles from available material catalogs based on required area, slenderness ratio, and local market availability. The selection considers forces in each member, effective length, and allowable stress. Available profiles are limited by local market availability, which may restrict certain sizes or shapes. The design should minimize material costs while meeting structural requirements. The design must also consider fabrication and erection requirements, including weld accessibility and field welding capabilities. Truss member connections must be designed to transfer required forces between members. The connections typically use gusset plates welded to chords and web members. The design must account for forces in each member and provide adequate weld sizes to transfer loads. The connections must also consider fabrication and erection requirements, including accessibility of welds and ability to perform welding in the field. The connections should be designed to minimize number of welds required while ensuring adequate load transfer.

For truss buckling analysis: (1) Identify compression members using method of joints; (2) Only compression members can buckle; (3) Calculate critical load for each compression member; (4) The minimum critical load among all compression members determines the failure load. For a symmetric truss with members of length L and EI, loaded by force P at 45°: member BC is in compression with force P. For member BC with length L/2 and EI = 4EI, P_critical = 16π²EI/L².

This section teaches how to solve indeterminate problems where equilibrium equations alone are insufficient. The key steps are: (1) Establish force equilibrium (ΣF = 0); (2) Apply geometric compatibility conditions (deformations must be consistent with constraints); (3) Solve the resulting system of equations. The instructor demonstrates this with a parallel bar system where two bars share a load, showing how to set up and solve simultaneous equations to find forces in each member.

The statically determinate/indeterminate discriminant formula (M = unknowns - equilibrium equations) fundamentally represents the difference between the number of unknown forces and the number of available equilibrium equations in a structure. When M = 0, the structure is statically determinate (静定), meaning all unknown forces can be solved using only equilibrium equations. When M > 0, the structure is indeterminate (不静定), requiring additional compatibility conditions. When M < 0, the structure is unstable (不安定). The formula's components include: unknowns (反力 + 部材端内力), equilibrium equations (ΣFx=0, ΣFy=0, ΣM=0 for each member), and the number of equilibrium equations depends on joint types (pins allow only force equilibrium, while rigid joints allow moment equilibrium). This formula is derived from the principle that a structure is solvable when the number of unknowns equals the number of independent equilibrium equations.

Static problems can be solved using only equilibrium equations (force and moment balance), while indeterminate problems require additional compatibility conditions (deformation constraints). To solve indeterminate problems, first draw a free body diagram by cutting the structure from the fixed support, identifying all external forces and reaction forces at the supports. For any mechanical system to be in equilibrium, the sum of all forces in each direction must equal zero. The virtual cut method involves making imaginary cuts through the structure to analyze internal forces. Internal force distribution diagrams show how internal forces vary along the structure. Stress (σ) is calculated by dividing internal force by cross-sectional area (σ = F/A). Strain (ε) is calculated using Hooke's Law: ε = σ/E. Total deformation is calculated by integrating the strain distribution along the structure. Indeterminate problems are solved by combining equilibrium equations with compatibility conditions, where the compatibility condition states that total deformation must satisfy boundary constraints (e.g., zero deformation at fixed supports).

This video demonstrates the graphical method (Cremona diagram) for analyzing statically determinate trusses by applying the joint method, where forces at each joint are represented as vectors forming closed polygons to determine member forces. The instructor calculates reaction forces first, then systematically analyzes each joint (A, B, C, D, E) by drawing force polygons to find axial forces in all members, with tension forces shown as positive values and compression forces as negative values. The final Cremona diagram combines all joint force polygons into a single figure, showing that all joints satisfy force equilibrium.

静定トラスの応力を解くには、まず反力を計算し（ΣFx=0, ΣFy=0, ΣM=0）、次に求める部材を含む切断線で3つ以下の部材を切断し、内力を引っ張り方向に仮定して釣り合い条件式を立てることで応力を求める。斜め方向の力は三角比（1:1:√2, 1:2:√3）を使って分解する。