Structural idealization and modeling simplify complex structures for analysis. Engineers use assumptions and simplified models to focus on critical aspects, balancing accuracy and efficiency. This process involves identifying key components, load paths, and choosing appropriate model complexity.

Fundamental assumptions like linear elasticity and small deformations form the basis of structural analysis. Engineers compare 2D and 3D modeling approaches, selecting the most suitable representation based on geometry, loading conditions, and required accuracy.

Model Simplification

Simplifying Complex Structures

• Simplified models reduce complex structures to manageable representations for analysis
• Assumptions form the basis of structural idealization, allowing engineers to focus on critical aspects
• 2D models represent structures in two dimensions, suitable for planar analysis (beams, trusses)
• 3D models capture spatial behavior, necessary for complex geometries (space frames, shells)
• Simplification process involves identifying key structural components and load paths
• Engineers balance accuracy and computational efficiency when choosing model complexity

Fundamental Assumptions in Structural Analysis

• Linear elastic behavior assumes materials return to original shape after load removal
• Small deformations theory applies when displacements are small compared to overall structure size
• Static equilibrium presumes structure remains at rest under applied loads
• Principle of superposition allows combining effects of multiple loads linearly
• Homogeneous material properties assume uniform characteristics throughout structural elements
• Plane sections remain plane after deformation, crucial for beam theory

Comparing 2D and 3D Modeling Approaches

• 2D models offer simplicity and faster analysis for planar structures (building frames, bridges)
• 3D models provide comprehensive behavior analysis, capturing torsional effects and out-of-plane loading
• 2D idealizations include plane stress, plane strain, and axisymmetric models
• 3D representations encompass solid elements, shell elements, and space frame formulations
• Model dimensionality selection depends on structure geometry, loading conditions, and required accuracy
• Hybrid approaches combine 2D and 3D elements to optimize analysis efficiency (floor diaphragms with 3D columns)

Boundary Conditions

Types of Structural Supports

• Boundary conditions define constraints on structural movement and rotation
• Pinned supports allow rotation but prevent translation (bridge bearings)
• Fixed supports restrict both rotation and translation (column bases in moment frames)
• Roller supports permit translation in one direction while preventing movement in others (expansion joints)
• Elastic supports provide partial restraint, modeled with springs (soil-structure interaction)
• Internal hinges allow relative rotation between connected elements (pin-connected trusses)
• Support conditions significantly influence structural behavior and load distribution

Modeling Connections in Structural Systems

• Connections transfer forces between structural elements
• Rigid connections maintain constant angles between connected members (welded joints)
• Pinned connections allow relative rotation between members (bolted truss connections)
• Semi-rigid connections exhibit partial moment transfer (partially restrained beam-column joints)
• Connection stiffness affects overall structural behavior and load distribution
• Proper modeling of connections ensures accurate representation of force transfer mechanisms
• Engineers must consider connection flexibility in analysis for realistic structural response prediction

Implementing Boundary Conditions in Analysis

• Boundary conditions translate to mathematical constraints in structural equations
• Displacement method enforces known displacements at supports ($u = 0$ for fixed support)
• Force method applies reaction forces at constrained degrees of freedom
• Penalty method introduces large stiffness values to approximate rigid constraints
• Lagrange multipliers enforce exact constraint conditions without numerical ill-conditioning
• Proper boundary condition implementation ensures solution uniqueness and stability
• Sensitivity analysis assesses the impact of boundary condition variations on structural response

Element Idealization

Idealized Structural Elements

• Truss elements carry axial loads only, assuming pinned connections (roof trusses, bridge members)
• Beam elements resist bending, shear, and axial forces (floor joists, cantilever beams)
• Frame elements combine beam and truss behavior for space frame analysis
• Plate elements model thin, flat structures subjected to out-of-plane loading (concrete slabs)
• Shell elements represent curved surfaces, combining membrane and bending behavior (domes, pressure vessels)
• Solid elements model three-dimensional stress states in thick structures (dams, machine parts)
• Element selection depends on structural behavior, geometry, and analysis requirements

Degrees of Freedom in Structural Analysis

• Degrees of freedom (DOF) represent independent displacements and rotations at nodes
• Truss elements typically have 3 DOF per node in 3D space (translations only)
• Beam elements in 2D have 3 DOF per node (2 translations, 1 rotation)
• Frame elements in 3D space have 6 DOF per node (3 translations, 3 rotations)
• Plate and shell elements may have 5 or 6 DOF per node, depending on formulation
• Solid elements usually have 3 translational DOF per node
• Number of DOF affects computational complexity and solution time
• Reduced DOF formulations (e.g., Mindlin plate theory) balance accuracy and efficiency

Element Formulation and Behavior

• Shape functions describe displacement field within elements
• Isoparametric formulation uses same functions for geometry and displacements
• Strain-displacement relations link nodal displacements to internal strains
• Constitutive equations relate strains to stresses based on material properties
• Element stiffness matrices derived from energy principles or virtual work
• Higher-order elements provide improved accuracy at the cost of increased complexity
• Special elements address specific structural behaviors (crack tip elements, contact elements)
• Element technology advancements continually improve analysis capabilities and efficiency