Beginner — Fundamentals of Statics
Start here if you are new to structural analysis.
Free-Body Diagrams and Equilibrium
Every structural analysis begins with a correct free-body diagram (FBD). The FBD isolates a structural element from its supports and connections, replacing them with reaction forces. The three equations of equilibrium — sum of horizontal forces, sum of vertical forces, and sum of moments — must all equal zero for a structure at rest. Mastering FBDs is the single most important skill for structural analysis, as errors here propagate through every subsequent calculation.
Support types determine the reaction forces available: pinned supports provide two reaction components (horizontal and vertical), roller supports provide one (vertical), and fixed supports provide three (horizontal, vertical, and moment). Understanding how each support constrains movement is essential before attempting any analysis. Practice drawing FBDs for simply supported beams, cantilevers, and overhanging beams with various load configurations.
Determinacy and Stability
A structure is externally determinate if the number of unknown reaction components equals the number of equilibrium equations available. For plane structures, three equations are available, so three unknowns indicate a determinate system. If unknowns exceed equations, the structure is indeterminate and requires compatibility conditions to solve. An unstable structure has fewer constraints than required for equilibrium and will collapse under load.
Internal determinacy for trusses is checked using the equation m = 2j - 3, where m is the number of members and j is the number of joints. A truss satisfying this equation is statically determinate internally. For frames, the degree of indeterminacy equals 3m + r - 3j (for plane frames), where m is members, r is reactions, and j is joints. These checks must be performed before selecting an analysis method.
Shear Force and Bending Moment Diagrams
Shear force diagrams (SFD) and bending moment diagrams (BMD) graphically represent the internal forces along a beam. The shear force at any section is the algebraic sum of all vertical forces to one side of the section. The bending moment is the algebraic sum of moments about that section. The relationship between load, shear, and moment is fundamental: the slope of the shear diagram equals the negative of the distributed load intensity, and the slope of the moment diagram equals the shear force.
Construct SFDs and BMDs by first calculating support reactions, then cutting the beam at critical sections (supports, concentrated loads, and points of zero shear). The maximum moment occurs where the shear force crosses zero, which is critical for design. Uniformly distributed loads produce parabolic moment diagrams, while concentrated loads produce linear segments. Practice with the Bending Moment Calculator to verify your manual calculations.
Intermediate — Trusses, Frames, and Influence Lines
Build on fundamentals with classical analysis methods.
Truss Analysis: Method of Joints and Method of Sections
Trusses are assemblies of slender members connected at pinned joints, carrying only axial forces (tension or compression). The method of joints applies equilibrium at each joint sequentially, solving for the two unknown member forces using the two force equilibrium equations. Start at a joint with at most two unknowns, and work through the truss systematically. Members in tension pull on the joint, while members in compression push.
The method of sections is more efficient when only specific member forces are needed. A cutting line is passed through the truss, dividing it into two parts. The internal forces in the cut members become external forces on the free-body diagram of either part. Taking moments about the intersection point of two of the cut members directly solves for the force in the third member. Use the Truss Analysis Calculator to check complex truss configurations.
Analysis of Determinate Frames
Frames differ from trusses in that their members resist bending moments, shear forces, and axial forces. Rigid (moment-resisting) connections transfer moments between members, creating a more complex internal force distribution. Portal frames, gable frames, and multi-bay frames are common in building structures. Analysis follows the same equilibrium principles but must account for all three internal force components at every section.
For determinate frames, the axial force, shear force, and bending moment diagrams are constructed by analyzing each member individually, carrying over the internal forces from connections. The moment diagram for frames is typically drawn on the tension side of each member. Portal frames subjected to lateral loads develop characteristic moment patterns with points of contraflexure where the moment changes sign — useful for simplified analysis methods like the portal method.
Influence Lines for Determinate Structures
Influence lines show how a particular response quantity (reaction, shear, moment) varies as a unit point load moves across the structure. They are essential for determining critical load positions for moving loads on bridges, crane runways, and floor systems. For determinate structures, influence lines consist of straight-line segments constructed using the Müller-Breslau principle: the influence line for a response quantity is proportional to the deflected shape when the corresponding constraint is released.
Once the influence line is drawn, the maximum effect from a series of concentrated loads is found by positioning the loads where the influence line ordinates are largest. For uniform loads, the maximum occurs when the load covers the areas where the influence line has the same sign. Influence lines for indeterminate structures are curved but follow the same principle. Mastering influence lines is critical for bridge design and moving-load analysis.
Advanced — Indeterminate Structures and Matrix Methods
For senior students and practicing engineers.
Moment Distribution Method (Hardy Cross)
The moment distribution method, developed by Professor Hardy Cross in 1930, is a powerful iterative technique for analyzing continuous beams and rigid frames without solving simultaneous equations. The method distributes unbalanced moments at joints to adjacent members based on their relative stiffness (k = EI/L). Each member's carry-over factor (typically 1/2 for prismatic members) transfers half the distributed moment to the far end.
The procedure involves: calculating fixed-end moments for each span, computing distribution factors at each joint (DF = k/sum(k) for members meeting at the joint), then iteratively balancing and carrying over until the unbalanced moments become negligible. The method converges rapidly — typically 3-5 cycles for frame analysis. Modified stiffness factors can account for far-end pinned or fixed conditions, reducing the number of cycles required.
Slope-Deflection Method
The slope-deflection method relates the end moments of a member to its end rotations and displacements. The fundamental equation for a prismatic beam: M_AB = (2EI/L)(2θ_A + θ_B - 3Δ/L) + FEM_AB. This method provides exact solutions for indeterminate structures by writing moment equilibrium equations at each joint and solving the resulting system of linear equations. It is particularly suitable for structures with a small degree of indeterminacy.
Sidesway (lateral displacement) in frames introduces additional terms in the slope-deflection equations. Non-sway frames are analyzed first by setting Δ = 0, then sway corrections are applied by considering the horizontal equilibrium of the entire frame. The slope-deflection method bridges the gap between classical hand methods and modern matrix analysis, making the transition to computational methods more intuitive.
Matrix Structural Analysis (Direct Stiffness Method)
The direct stiffness method is the foundation of modern structural analysis software (SAP2000, ETABS, STAAD.Pro). The structure is modeled as an assembly of finite elements connected at nodes. Each element has a stiffness matrix [k] relating its nodal forces to nodal displacements. The global stiffness matrix [K] is assembled by summing individual element contributions at shared nodes. The system equation [K]{D} = {F} is solved for unknown displacements {D}, then element forces are recovered.
For 2D frame elements, the element stiffness matrix is 6x6 (three degrees of freedom per node: horizontal displacement, vertical displacement, and rotation). Transformation matrices convert element stiffness from local coordinates to global coordinates before assembly. Boundary conditions are applied by modifying the global stiffness matrix — removing rows and columns corresponding to restrained degrees of freedom. Post-processing computes member end forces, reactions, and internal force diagrams.
Practice Exercises
Exercise 1: Simply Supported Beam
A simply supported beam of span 8 m carries a uniformly distributed load of 12 kN/m over its entire length and a concentrated load of 30 kN at midspan. Calculate the support reactions, draw the shear force diagram, and determine the maximum bending moment. Verify your answer with the Bending Moment Calculator.
Exercise 2: Truss Analysis
A Warren truss with 6 panels, each 3 m wide and 3 m deep, supports point loads of 20 kN at each top chord joint. Using the method of sections, determine the forces in members L0-U1, U1-U2, and L1-U1. Identify whether each member is in tension or compression. Use the Truss Analysis Calculator to verify your results.
Exercise 3: Continuous Beam by Moment Distribution
A two-span continuous beam has spans of 6 m and 8 m, with a uniformly distributed load of 15 kN/m on both spans. All members have constant EI. Determine the support moments using the moment distribution method (Hardy Cross). Calculate the final bending moments and draw the bending moment diagram.
Exercise 4: Deflection by Virtual Work
A cantilever beam of length 5 m with EI = 200,000 kN·m² carries a concentrated load of 40 kN at its free end and a uniformly distributed load of 8 kN/m over its entire length. Using the principle of virtual work (unit load method), calculate the deflection at the free end. Compare with the Cantilever Beam Calculator.
Related Calculators
Bending Moment Calculator
Compute shear forces and bending moments for beams under various loading conditions.
Moment of Inertia Calculator
Calculate section properties including area, centroid, and moments of inertia.
Truss Analysis Calculator
Analyze determinate trusses using the method of joints and method of sections.
Cantilever Beam Calculator
Analyze cantilever beams for reactions, shear, moment, and deflection.
Euler Buckling Calculator
Calculate critical buckling loads for columns based on Euler's formula.
Shear Force Diagram Calculator
Draw shear force and bending moment diagrams for determinate beams.
References
- Hibbeler, R.C. Structural Analysis. 10th ed., Pearson, 2019.
- Kassimali, A. Structural Analysis: SI Edition. 6th ed., Cengage Learning, 2020.
- Leet, K.M., Uang, C.M., and Gilbert, A.M. Fundamentals of Structural Analysis. 5th ed., McGraw-Hill, 2018.
- McCormac, J.C. and Brown, R.H. Structural Analysis: A Unified Approach. 9th ed., Wiley, 2020.
- ASCE 7-22. Minimum Design Loads and Associated Criteria for Buildings and Other Structures. American Society of Civil Engineers, 2022.
- Civil Engineering Handbook — Structural Analysis chapter with detailed design guidance.
- Engineering Formula Library — Beam bending, section properties, and buckling formulas.
- Engineering Standards Reference — ACI 318, IS 456, Eurocode 2 provisions.
- Engineering Glossary — Definitions of structural analysis terms.