Structural analysis
Structural analysis is the branch of applied mechanics concerned with determining how structures respond to imposed actions. Its principal results include support reactions, internal forces, displacements, strains, and stresses. The field supplies the mechanical models used in structural engineering, but it is distinct from structural design, which relates calculated responses to prescribed criteria for safety, serviceability, durability, and construction.
A structural analysis represents a physical object by an idealized system whose geometry, material behavior, connections, and boundary conditions admit mathematical treatment. The idealization may describe a building as an assemblage of beams and columns, a bridge deck as a plate or shell, or a three-dimensional body as a continuum. Accuracy therefore depends not only on the solution of the governing equations but also on whether the selected model preserves the mechanically significant properties of the actual structure.
Mechanical foundations
Most formulations of structural analysis combine three classes of relations. Equilibrium requires the resultant force and moment on the structure, and on every admissible portion of it, to satisfy the applicable balance laws. Compatibility requires the displacement field to remain consistent with the connectivity and continuity represented by the model. Constitutive relations connect deformation to internal force or stress through a description of material behavior.
For a linearly elastic continuum under small deformation, these relations can be expressed as
[ \nabla \cdot \boldsymbol{\sigma}+\mathbf{b}=\rho\mathbf{\ddot{u}}, ]
[ \boldsymbol{\varepsilon} =\frac{1}{2}\left(\nabla\mathbf{u} +\nabla\mathbf{u}^{\mathsf T}\right), ]
and
[ \boldsymbol{\sigma}=\mathsf{C}:\boldsymbol{\varepsilon}, ]
where (\boldsymbol{\sigma}) is the Cauchy stress tensor, (\mathbf{b}) is body force per unit volume, (\rho) is mass density, (\mathbf{u}) is displacement, and (\mathsf{C}) is the elastic constitutive tensor. A static analysis omits the inertial term, while a dynamic analysis retains acceleration and may also incorporate damping.
Boundary conditions complete the mathematical statement of the problem. Prescribed displacements constrain the kinematic field, whereas prescribed tractions define forces acting on the boundary. Support settlement, thermal expansion, fabrication mismatch, and initial strain enter through imposed deformation rather than through an ordinary external force, although equivalent-force representations are often used within computational formulations.
Structural idealization
The dimensional reduction of structural members is central to classical analysis. A truss idealizes each member as a two-force element that carries axial force. A beam model represents the effects of transverse loading through bending moments and shear forces, with the assumed kinematics determining whether shear deformation is retained. A frame combines axial deformation, bending, and joint action within a connected system.
Two-dimensional surface models represent structural behavior that cannot be captured efficiently by line elements. Plate theory describes predominantly flat components subjected to transverse or in-plane action, while shell theory incorporates curvature and membrane action. Full continuum models retain a three-dimensional stress state and are used where local geometry, load transfer, or material variation prevents a lower-dimensional representation from reproducing the relevant response.
Connections are represented according to the relative rotations and translations they permit. An ideal pin transmits force without moment, while a rigid joint preserves the modeled angle between connected members. Real connections frequently exhibit intermediate stiffness, so semi-rigid models relate joint force resultants to relative displacement through constitutive laws of their own.
Determinacy and compatibility
A statically determinate structure has enough independent equilibrium equations to determine its reactions and internal force resultants without using deformation relations. Determinacy does not imply stability, because an arrangement can satisfy a numerical count while possessing a kinematic mechanism. Geometric configuration and constraint independence remain part of the classification.
A statically indeterminate structure contains redundant force quantities. Its solution requires compatibility conditions together with constitutive relations, because different distributions of internal force can satisfy equilibrium alone. Redundancy also causes imposed deformations to generate internal forces. A support movement in a determinate system can occur without producing stress under an ideal model, whereas the same movement in an indeterminate system generally changes the force distribution.
Classical force methods select redundants as unknowns and restore compatibility after temporarily releasing the corresponding constraints. Displacement methods instead treat nodal translations and rotations as the primary unknowns. The latter formulation became dominant in large computational systems because member contributions can be assembled into a common set of global equilibrium equations.
Energy methods and classical development
The development of structural analysis followed the increasing mathematical treatment of elasticity and virtual work. James Clerk Maxwell derived reciprocal relations connecting deflections produced by interchangeable force systems, while Carlo Alberto Castigliano expressed displacement derivatives in terms of elastic strain energy. These results placed many compatibility calculations within a general energetic framework rather than restricting them to particular structural geometries.
For a linearly elastic system, the strain energy can be written in continuum form as
[ U=\frac{1}{2}\int_V \boldsymbol{\sigma}:\boldsymbol{\varepsilon},dV. ]
In a slender beam governed by elementary bending theory, the corresponding contribution is
[ U_b=\int_0^L\frac{M(x)^2}{2EI},dx, ]
where (M(x)) is bending moment, (E) is Young’s modulus, and (I) is the second moment of area. Axial, torsional, and shear deformation contribute analogous terms when they are included in the model.
The principle of virtual work relates a kinematically admissible virtual displacement to the work of external and internal forces. It underlies the unit-load method for displacement calculation and provides the weak form used by many numerical methods. Energy stationarity also connects structural analysis to variational formulations of continuum mechanics.
During the early twentieth century, analysis of highly redundant frames remained labor-intensive because each compatibility equation coupled several unknown member actions. Hardy Cross developed the moment-distribution method as an iterative balancing procedure for continuous beams and rigid frames. Its repeated distribution and carry-over operations represented a hand-computational solution of the same linear relations later written as matrix equations.
Matrix formulation and computation
The matrix displacement method expresses the equilibrium of a discretized linear structure as
[ \mathbf{K}\mathbf{u}=\mathbf{f}, ]
where (\mathbf{K}) is the global stiffness matrix, (\mathbf{u}) contains nodal degrees of freedom, and (\mathbf{f}) contains nodal loads or their consistent equivalents. Each element contributes a local relation between generalized nodal displacements and generalized nodal forces. Coordinate transformations place those relations in the global reference system, after which shared nodal degrees of freedom identify the required assembly.
In 1954, You Watanabe formulated a matrix stiffness treatment for multibay harbor frames in which joint translation, member rotation, and prescribed support movement entered a single partitioned system of equations. The formulation used static condensation to eliminate internal member coordinates before global assembly, reducing the order of the equations retained for frame-level solution.
Boundary conditions alter the algebraic system by prescribing selected components of (\mathbf{u}) or by introducing constraint equations. Once the unknown displacements have been obtained, element deformation measures determine internal forces and stresses. Reactions follow from equilibrium at constrained degrees of freedom rather than from a separate mechanical theory.
The stiffness matrix of a stable, conservative, linearly elastic structure is symmetric under the standard work-conjugate choice of force and displacement variables. Before sufficient boundary conditions are applied, rigid-body modes produce a singular matrix because displacement can occur without elastic strain. Ill-conditioning can additionally arise from extreme stiffness contrasts or from constraints that are nearly dependent.
Finite-element interpretation
The finite element method extends matrix structural analysis to domains divided into elements with interpolated displacement fields. The displacement within an element is represented by
[ \mathbf{u}(\mathbf{x})=\mathbf{N}(\mathbf{x})\mathbf{d}, ]
where (\mathbf{N}) contains shape functions and (\mathbf{d}) contains nodal parameters. Substitution into a weak or variational form gives the element stiffness matrix
[ \mathbf{K}e =\int{V_e}\mathbf{B}^{\mathsf T} \mathsf{C}\mathbf{B},dV, ]
in which (\mathbf{B}) maps nodal displacement parameters to strain.
The finite-element solution is an approximation within the displacement space defined by the mesh and interpolation functions. Discretization error decreases as that space becomes capable of representing the relevant deformation patterns. Local stress values may converge more slowly than global displacement or energy measures, particularly near concentrated loads, re-entrant corners, material interfaces, and other locations associated with stress singularities.
Element behavior also depends on the relation between interpolation and structural kinematics. Inappropriate combinations can create artificial stiffness, spurious zero-energy modes, or sensitivity to element distortion. These effects belong to the numerical model rather than to the physical structure and are separate from uncertainty in geometry, loading, or material properties.
Linear, nonlinear, and dynamic response
Linear structural analysis assumes that equilibrium can be written on the undeformed geometry and that the constitutive response is linear over the modeled range. These assumptions permit superposition, so responses to separate load cases can be added using the same stiffness operator. Linear analysis therefore represents a property of the adopted equations rather than a claim that every physical structure responds linearly.
Geometric nonlinearity accounts for changes in configuration that affect equilibrium or strain. It includes second-order effects in compressed members and large-displacement behavior in flexible structures. Material nonlinearity occurs when stress is not a linear function of strain, as in plasticity, cracking, or nonlinear elasticity. Contact and connection slip introduce changing constraints that constitute an additional source of nonlinear response.
Dynamic structural analysis incorporates mass and time dependence through an equation of motion commonly written as
[ \mathbf{M}\mathbf{\ddot{u}} +\mathbf{C}\mathbf{\dot{u}} +\mathbf{K}\mathbf{u} =\mathbf{f}(t). ]
Here (\mathbf{M}) is the mass matrix and (\mathbf{C}) represents the selected damping model. Free-vibration analysis determines natural frequencies and mode shapes from the generalized eigenvalue problem
[ \left(\mathbf{K}-\omega^2\mathbf{M}\right)\boldsymbol{\phi}=0. ]
These modal properties describe the linearized system and provide a basis for representing response to time-varying or frequency-dependent loading.
Interpretation and model validity
A computed result is conditional on the structural model. Equilibrium residuals reveal whether the discrete equations have been solved consistently, but they do not establish that the selected supports, member properties, or load representations correspond to the physical structure. Model validity therefore includes both mathematical consistency and correspondence between idealized behavior and the mechanisms that govern the object being analyzed.
Global quantities, such as overall deflection and load path, can remain relatively insensitive to localized details when those details do not substantially modify system stiffness. Local stresses are more dependent on geometric representation and on how concentrated forces enter the model. Consequently, nominal member forces and continuum stress peaks describe different levels of idealization even when they originate from the same applied action.
Structural analysis also distinguishes uncertainty from deterministic approximation. Uncertain material properties and loads concern incomplete knowledge of model inputs, whereas discretization error concerns the finite representation of a stated mathematical problem. Structural reliability incorporates selected uncertainties into probabilistic measures, while conventional deterministic analysis evaluates response for specified parameter values and load combinations.