# Structural Analysis Fundamentals ## Table of Contents - [Introduction](#introduction) - [Statics and Equilibrium](#statics-and-equilibrium) - [Conditions of Equilibrium](#conditions-of-equilibrium) - [Free Body Diagrams](#free-body-diagrams) - [Support Types and Reactions](#support-types-and-reactions) - [Load Types](#load-types) - [Dead Loads](#dead-loads) - [Live Loads](#live-loads) - [Wind Loads](#wind-loads) - [Seismic Loads](#seismic-loads) - [Snow Loads](#snow-loads) - [Other Load Types](#other-load-types) - [Load Combinations](#load-combinations) - [Eurocode Combinations](#eurocode-combinations) - [IBC and ASCE 7 Combinations](#ibc-and-asce-7-combinations) - [Factor of Safety](#factor-of-safety) - [Design Philosophies](#design-philosophies) - [Working Stress Design](#working-stress-design) - [Limit State Design](#limit-state-design) - [Comparison of Methods](#comparison-of-methods) - [Internal Forces and Diagrams](#internal-forces-and-diagrams) - [Practical Notes for Architects](#practical-notes-for-architects) - [Related Topics](#related-topics) - [References](#references) --- ## Introduction Structural analysis is the determination of the effects of loads on physical structures and their components. It is the foundational discipline upon which all structural design is built. For the practicing architect, a working knowledge of structural analysis fundamentals enables effective collaboration with structural engineers, informed early-stage design decisions, and the ability to critically evaluate structural proposals. This article covers the core principles that every chartered architect should understand. ## Statics and Equilibrium ### Conditions of Equilibrium A structure in static equilibrium satisfies three fundamental conditions in two-dimensional analysis: - **Sum of horizontal forces equals zero:** ΣFx = 0 - **Sum of vertical forces equals zero:** ΣFy = 0 - **Sum of moments about any point equals zero:** ΣM = 0 For three-dimensional structures, six equations govern equilibrium (three force equations and three moment equations about the three axes). All building structures must satisfy these conditions under every load case considered in design. ### Free Body Diagrams A free body diagram (FBD) isolates a structural element or portion of a structure and represents all external forces, reactions, and internal forces acting upon it. Constructing an accurate FBD is the essential first step in any structural analysis. **Steps for constructing an FBD:** 1. Isolate the element or joint of interest 2. Replace all supports with their corresponding reaction forces 3. Show all applied loads (point loads, distributed loads, moments) 4. Show all internal forces at cut sections 5. Indicate dimensions and angles relevant to the analysis ### Support Types and Reactions | Support Type | Reactions Provided | Degrees of Freedom Restrained | |---|---|---| | Roller | 1 vertical force | 1 (vertical translation) | | Pin / Hinge | 1 vertical + 1 horizontal force | 2 (both translations) | | Fixed | 1 vertical + 1 horizontal force + 1 moment | 3 (all) | | Spring | Force proportional to displacement | 1 (with flexibility) | A statically determinate structure has exactly as many unknown reactions as equilibrium equations. When unknowns exceed equilibrium equations, the structure is **statically indeterminate** and requires additional compatibility equations for solution. ## Load Types ### Dead Loads Dead loads (permanent actions in Eurocode terminology, denoted **Gk**) include the self-weight of all permanent construction: structural frame, floors, roofing, finishes, fixed partitions, cladding, and permanently installed services. Typical values include: - Reinforced concrete: 24-25 kN/m³ - Structural steel: 78.5 kN/m³ - Masonry blockwork: 12-20 kN/m³ - Timber: 4-7 kN/m³ (species dependent) - Plasterboard ceiling: 0.15-0.25 kN/m² - Screed (50mm): ~1.15 kN/m² ### Live Loads Live loads (variable actions, **Qk**) arise from occupancy and use. They are specified by codes based on building use: | Occupancy | Eurocode (kN/m²) | IBC/ASCE 7 (kN/m²) | |---|---|---| | Residential | 1.5 - 2.0 | 1.92 (40 psf) | | Office | 2.5 - 3.0 | 2.40 (50 psf) | | Retail | 4.0 - 5.0 | 4.79 (100 psf) | | Assembly (fixed seats) | 4.0 | 2.87 (60 psf) | | Storage / Archives | 7.5+ | 5.99+ (125 psf) | ### Wind Loads Wind loads are dynamic in nature but are typically treated as equivalent static pressures for most buildings. The basic wind pressure is derived from the fundamental formula **q = 0.5 * rho * v²**, where rho is air density (approximately 1.25 kg/m³) and v is the design wind velocity. See [[Wind Engineering for Buildings]] for detailed treatment. ### Seismic Loads Seismic loads are inertial forces generated by ground acceleration during earthquakes. The equivalent lateral force procedure calculates base shear as **V = Cs * W**, where Cs is the seismic response coefficient and W is the effective seismic weight. Refer to [[Seismic Design Principles]] for comprehensive coverage. ### Snow Loads Snow loads depend on geographic location, roof geometry, exposure, and thermal conditions. The ground snow load is modified by shape, exposure, and thermal coefficients. Drifting snow on lower roofs adjacent to taller structures requires special attention. Eurocode uses **s = μi * Ce * Ct * sk** where sk is the characteristic ground snow load. ### Other Load Types - **Construction loads:** Temporary loads during erection - **Thermal loads:** Forces from restrained thermal expansion/contraction - **Settlement loads:** Forces from differential foundation settlement - **Impact / accidental loads:** Vehicle impact, explosion, progressive collapse scenarios - **Hydrostatic and earth pressure:** On basement walls and retaining structures ## Load Combinations Load combinations represent the simultaneous occurrence of multiple load types with appropriate factors. The principle is that the maximum of all loads acting simultaneously is statistically unlikely. ### Eurocode Combinations **Ultimate Limit State (ULS) — Fundamental combination (EN 1990 Eq. 6.10):** `1.35 * Gk + 1.5 * Qk,1 + 1.5 * Σ(ψ0,i * Qk,i)` Where ψ0 is the combination factor (e.g., 0.7 for office live load, 0.5 for snow, 0.6 for wind). **Serviceability Limit State (SLS) — Characteristic combination:** `Gk + Qk,1 + Σ(ψ0,i * Qk,i)` ### IBC and ASCE 7 Combinations Key LRFD combinations from ASCE 7: - 1.4D - 1.2D + 1.6L + 0.5(Lr or S or R) - 1.2D + 1.6(Lr or S or R) + (L or 0.5W) - 1.2D + 1.0W + L + 0.5(Lr or S or R) - 1.2D + 1.0E + L + 0.2S - 0.9D + 1.0W - 0.9D + 1.0E See [[Load Path and Load Combinations]] for detailed application. ## Factor of Safety The factor of safety (FoS) represents the ratio of the capacity of a system to the expected load. In traditional **working stress design**, a single global factor of safety is applied: **FoS = Ultimate capacity / Working load** Typical global factors of safety range from 1.5 to 3.0 depending on material, loading conditions, and consequences of failure. For foundations, FoS of 2.5-3.0 is common. ## Design Philosophies ### Working Stress Design Also known as Allowable Stress Design (ASD) or permissible stress method. The principle is that stresses under service loads must not exceed a permissible value (material strength divided by a factor of safety). This was the predominant method before the 1970s and is still used in some contexts (e.g., timber design, geotechnical engineering). **Permissible stress = Material strength / Factor of safety** ### Limit State Design Limit State Design (LSD) is the modern approach adopted by Eurocodes, BS EN standards, and most international codes. It applies partial safety factors separately to loads (gamma_f) and materials (gamma_m), recognizing that uncertainty in loading differs from uncertainty in material strength. **Design requirement:** Factored resistance >= Factored load effect (Rd >= Ed) Two limit states are checked: - **Ultimate Limit State (ULS):** Safety against collapse, instability, or failure - **Serviceability Limit State (SLS):** Deflection, vibration, cracking under service conditions Typical partial factors: - Permanent loads (unfavourable): γG = 1.35 - Variable loads: γQ = 1.50 - Concrete (γc): 1.50 - Reinforcing steel (γs): 1.15 - Structural steel (γM0): 1.00, (γM1): 1.00, (γM2): 1.25 ### Comparison of Methods | Aspect | Working Stress | Limit State | |---|---|---| | Safety factor | Single global FoS | Partial factors on loads and materials | | Load combinations | Service loads | Factored loads for ULS, service for SLS | | Economy | Generally conservative | More efficient use of materials | | Code adoption | Legacy codes, some geotechnical | Eurocodes, modern standards | | Serviceability | Implicitly addressed | Explicitly checked | ## Internal Forces and Diagrams Structural analysis determines four types of internal forces at any section: - **Axial force (N):** Tension or compression along the member axis - **Shear force (V):** Transverse force perpendicular to the member axis - **Bending moment (M):** Moment causing flexure of the member - **Torsion (T):** Twisting moment about the member axis Shear force and bending moment diagrams are essential tools. The relationship between load intensity (w), shear (V), and moment (M) is: `dV/dx = -w` and `dM/dx = V` This means shear is the derivative of moment, and load intensity is the derivative of shear — a relationship that architects should understand for intuitive structural reasoning. ## Practical Notes for Architects 1. **Early design stages:** Use approximate methods (span/depth ratios, load takedowns) to estimate member sizes before detailed analysis 2. **Structural grid:** Align structural grids with architectural planning grids to minimise transfer structures 3. **Load path clarity:** The simplest, most direct load path is usually the most efficient and economical — see [[Structural Systems Overview]] 4. **Redundancy:** Statically indeterminate structures provide inherent redundancy and are preferred for robustness 5. **Communication:** Understanding FBDs and load diagrams enables architects to discuss structural intent meaningfully with engineers 6. **Software literacy:** Modern analysis uses FEM software (SAP2000, ETABS, Robot, STAAD), but architects benefit from hand-check capability ## Related Topics - [[Load Path and Load Combinations]] - [[Structural Systems Overview]] - [[Seismic Design Principles]] - [[Wind Engineering for Buildings]] - [[Reinforced Concrete Design]] - [[Structural Steel Design]] ## References - EN 1990: Eurocode 0 — Basis of Structural Design - EN 1991-1-1: Eurocode 1 — Actions on Structures (General Actions) - ASCE 7-22: Minimum Design Loads and Associated Criteria for Buildings - IS 875 (Parts 1-5): Code of Practice for Design Loads - Hibbeler, R.C. *Structural Analysis*, Pearson Education --- #engineering #structural #analysis #statics #loads #design-philosophy