Structural Mechanics

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This section covers internal force resultants in beams — axial force, shear force, and bending moment — the effect of different load types, superposition of loading effects, and how to construct and interpret AF/SF/BM diagrams.

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Internal Forces in Beams Internal Force Definition Axial force (AF) The internal force acting along the longitudinal axis of the member, tensile (positive) or compressive (negative) Shear force (SF) The internal force acting transverse (perpendicular) to the member's axis at a section, tending to slide one part of the section relative to the other Bending moment (BM) The internal moment at a section that tends to bend/curve the member, causing fibres on one side to stretch (tension) and the other to shorten (compression) Types of Loads Point (concentrated) load: acts at a single point along the span.

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Uniformly distributed load (UDL): constant intensity (force/length) spread over a length of the member.

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Uniformly varying load (UVL): intensity varies linearly from zero (or some value) to a maximum over a length (triangular/trapezoidal loading).

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Applied (concentrated) moment/couple: a pure moment applied at a point, causing a discontinuity (jump) in the BM diagram at that point with no effect on the SF diagram.

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Superposition: for a linear elastic system, the combined SF/BM/deflection due to several loads acting together equals the algebraic sum of the effects of each load acting separately — a key simplification for analysing beams under multiple loads.

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Load–Shear–Moment Relationships Relationship Meaning dV/dx = −w(x) The rate of change of shear force along the span equals the negative of the distributed load intensity w(x); a concentrated load causes a sudden (step) change in V equal to the load dM/dx = V(x) The rate of change of bending moment along the span equals the shear force; the slope of the BM diagram at any section equals the SF value there Area rule The change in SF between two sections equals the area under the load diagram between them; the change in BM between two sections equals the area under the SF diagram between them BM is maximum/minimum Where SF = 0 (or changes sign), since dM/dx = V = 0 there; concentrated moments produce a discontinuous jump in the BM diagram equal to the moment's magnitude Diagram Interpretation A cantilever under a tip point load has a constant SF diagram and a BM diagram varying linearly from zero at the free end to maximum (hogging) at the fixed support.

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A simply supported beam under a central point load has SF constant (of opposite sign) on each half, with BM varying linearly to a peak (sagging) at mid-span.

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A simply supported beam under a UDL has SF varying linearly (zero at mid-span) and BM varying parabolically, maximum (sagging) at mid-span where SF = 0.

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An overhanging beam can show both sagging (positive) and hogging (negative) bending moment regions along its length, with a point of contraflexure (BM = 0) where the curvature reverses.

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This section covers normal and shear stresses at a point, principal stresses and planes, maximum shear stress, the stress-strain behaviour of common materials, and torsion of circular shafts.

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Normal and Shear Stress Normal stress (σ) acts perpendicular to a section (tensile or compressive); shear stress (τ) acts tangential (parallel) to the section's plane.

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At any point in a stressed body, the state of stress on an arbitrarily oriented plane can be expressed in terms of the normal and shear stresses on a reference (e.g. x-y) set of planes, related by the transformation equations of plane stress.

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Principal Stresses and Principal Planes Principal planes are the particular orientation of planes at a point on which the shear stress is zero and the normal stress is either maximum or minimum; the normal stresses on these planes are the principal stresses (σ1, σ2).

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Maximum shear stress at a point occurs on planes oriented at 45° to the principal planes, with magnitude τmax = (σ1 − σ2)/2.

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Mohr's circle is a graphical method for determining principal stresses, maximum shear stress, and stresses on any inclined plane, by plotting normal stress (horizontal axis) against shear stress (vertical axis) for all possible plane orientations at a point.

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Stress–Strain Curves Region/Point Significance Proportional limit Stress up to which stress is directly proportional to strain (Hooke's law applies); slope of this linear region is the modulus of elasticity, E Elastic limit Maximum stress beyond which the material no longer returns to its original shape upon unloading (permanent/plastic deformation begins) Yield point Stress at which the material begins to deform plastically with little or no increase in stress (a distinct yield plateau is characteristic of ductile materials like mild steel) Ultimate tensile strength The maximum stress reached on the stress-strain curve, corresponding to the onset of necking in a ductile specimen Fracture (rupture) point The stress/strain at which the specimen actually breaks Ductile vs. brittle behaviour A ductile material (e.g. mild steel) shows large plastic strain before fracture with a clear yield point; a brittle material (e.g. cast iron, concrete) fractures with little or no plastic deformation and no distinct yield point Torsion For a circular shaft in pure torsion, the torsion formula relates applied torque T, polar moment of inertia J, shear stress τ at radius r, shear modulus G, angle of twist θ, and length L:

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Shear stress in torsion varies linearly from zero at the shaft's centre to a maximum at the outer surface; a hollow circular shaft is more efficient in torsion than a solid shaft of the same cross-sectional area, since material near the centre contributes little to torsional resistance.

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This section covers the theory of pure bending, the flexure formula, the elastic curve and beam deflection, and the buckling behaviour of long columns under axial compression via Euler's formula.

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Pure Bending and the Flexure Formula Pure (co-planar) bending occurs when a beam segment is subjected only to a constant bending moment (zero shear force), so plane cross-sections remain plane and perpendicular to the beam's longitudinal axis after bending (Euler-Bernoulli assumption).

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The flexure formula relates bending moment M, second moment of area I, bending stress σ at distance y from the neutral axis, modulus of elasticity E, and radius of curvature R:

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Bending stress varies linearly across the depth of the section, from maximum compressive stress at the extreme fibre on one side, through zero at the neutral axis (which passes through the section's centroid for elastic bending), to maximum tensile stress at the extreme fibre on the other side.

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Elastic Curve, Curvature & Deflection The elastic curve is the deflected shape of a beam's longitudinal axis under load; its curvature (1/R) at any section is related to the bending moment by 1/R = M/(EI).

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The angle of rotation (slope) and deflection of the elastic curve are obtained by successive integration of the curvature-moment relationship (the double integration method);

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Macaulay's method extends this using singularity functions to handle beams with discontinuous loading conveniently in a single expression.

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Flexural stiffness (EI) is the product of the modulus of elasticity and the second moment of area, representing a member's resistance to bending deformation — a higher EI produces smaller curvature (and deflection) for a given bending moment.

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Columns & Euler's Buckling Formula A long (slender) column under axial compressive load can fail by buckling (sudden lateral instability) at a stress well below the material's compressive yield/crushing strength, rather than by direct material crushing as in a short column.

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Euler's critical (buckling) load for a long column:

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Pcr = π²EI/Le², where Le is the effective length, depending on end support conditions.

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Effective length: both ends pinned, Le=L; one end fixed, other free, Le=2L; both ends fixed, Le=L/2; one end fixed, other pinned, Le≈0.7L (L/√2).

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Slenderness ratio λ = Le/r, where r = √(I/A) is the radius of gyration of the cross-section; a higher slenderness ratio increases susceptibility to buckling and lowers the critical buckling stress.

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This section covers the degree of static determinacy of structures, energy methods and the virtual work (unit load) method for computing deflections, and their application to beams and portal frames.

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Degree of Static Determinacy Structure Type Determinacy Condition Plane truss Ds = m + r − 2j (m = number of members, r = number of reaction components, j = number of joints);

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Ds=0 statically determinate, Ds>0 indeterminate, Ds<0 unstable mechanism Plane frame/beam Ds = 3m + r − 3j − c (c = additional condition equations from internal hinges/rollers); same sign convention as for trusses Energy Methods Strain energy (U) is the energy stored in an elastic member due to deformation under load; for axial, bending, shear, and torsional effects, U can be expressed as integrals of the respective internal force/moment squared divided by the corresponding stiffness (e.g.

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U = ∫M²dx/(2EI) for bending).

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Castigliano's theorem: the partial derivative of the total strain energy of a linear elastic structure with respect to an applied load (or moment) gives the displacement (or rotation) at the point of application of that load, in its direction: δ = ∂U/∂P.

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Virtual Work (Unit Load) Method The virtual (unit) load method computes the deflection or rotation at a point by applying a virtual unit load (or unit moment) at that point in the direction of the desired displacement, then evaluating δ = ∫(mM/EI)dx, where M is the bending moment due to the real loading and m is the bending moment due to the virtual unit load.

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This method is widely used for deflection of beams (at any point, not just standard cases) and for portal frames, where deflections/rotations at the frame joints or along the members are found by integrating the product of real and virtual moment diagrams over each member and summing.

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This section covers influence lines for determinate structures under moving point loads and UDLs, and the analysis of two-hinged arches.

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Influence Lines An influence line diagram (ILD) for a given response function (a reaction, shear force, or bending moment at a specific section) plots the value of that response as a unit load moves across the structure — distinct from a SF/BM diagram, which shows the response along the structure for a fixed load position.

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For a simply supported beam, the ILD for a support reaction is a straight line, maximum (=1) at that support and zero at the other; the ILD for shear force at a section is bilinear (jumping by 1 as the unit load crosses the section); the ILD for bending moment at a section is triangular, with maximum ordinate ab/L at the section (a, b = distances from the section to each support, L = span).

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Point loads: the maximum response due to a series of moving point loads is found by positioning the loads to maximize the relevant ILD ordinate(s), often via trial positions or the ILD's peak.

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UDL: the response due to a uniformly distributed load equals the load intensity multiplied by the net area under the ILD over the loaded length; the UDL is positioned over the portion(s) of the ILD with the same sign to maximize the response.

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Two-Hinged Arches A two-hinged arch has hinges (pins) at both supports only, with the arch rib continuous (no internal hinge at the crown); this makes it statically indeterminate to the first degree — the horizontal thrust H at the supports is the redundant, found using strain energy/Castigliano's theorem (minimizing strain energy with respect to H, ∂U/∂H = 0), since equilibrium equations alone are insufficient.

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For a two-hinged parabolic arch of span L and rise h under a UDL w over the full span, the horizontal thrust works out to H = wL²/(8h) — identical in form to the thrust of the corresponding three-hinged parabolic arch under the same UDL.

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Once H is known, the bending moment at any section of the arch is found as the simple-beam bending moment at that section minus H multiplied by the rise of the arch axis above the springing line at that section.

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This section covers classical methods for analysing statically indeterminate structures — the flexibility (force) method, slope-deflection method, moment distribution method, and stiffness (matrix) method — along with influence lines for continuous beams and elementary plastic analysis.

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Flexibility (Force) Method The flexibility method treats one or more redundant reactions/forces as unknowns, removes them to obtain a statically determinate primary (released) structure, and applies compatibility equations (the actual displacement at each redundant's location, considering both the real loads and the redundants, must match the known support condition, typically zero) to solve for the redundants.

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For a two-hinged parabolic arch, the flexibility method with H as the redundant (as in section 4.5) is a specific application of this general approach.

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Slope-Deflection Method The slope-deflection method is a displacement (stiffness-based) method expressing the end moments of each member in terms of the (unknown) joint rotations, chord rotation (due to sidesway), and fixed-end moments (FEM), via the slope-deflection equations; these are combined with joint/storey equilibrium equations to solve for the unknown displacements, then back-substituted for member end moments.

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Moment Distribution Method The moment distribution method (Hardy Cross method) is an iterative displacement method: members are initially treated as fixed at all joints (computing fixed-end moments), then joints are successively “released” and balanced, distributing the unbalanced moment at each joint to connected members in proportion to their distribution factors (DF = relative member stiffness k / Σk at that joint), with a portion (carry-over factor = 1/2 for a far end that is fixed, 0 for a far end that is pinned) carried over to the far end of each member; the process repeats until moments converge.

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Stiffness (Matrix) Method The stiffness (matrix displacement) method systematizes the displacement-based approach in matrix form: it assembles a global stiffness matrix relating all joint displacements (unknowns) to the applied joint loads, solves the resulting system of equations for the displacements, then recovers member end forces/moments — the basis of modern computer structural analysis software.

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Influence Lines for Continuous Beams For statically indeterminate (continuous) beams, influence lines are typically obtained using the Müller- Breslau principle: the influence line for a response function (reaction, shear, or moment) has the same shape as the deflected shape of the structure obtained by releasing the restraint corresponding to that response and applying a unit displacement/rotation there.

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Elementary Plastic Analysis A plastic hinge forms at a section once the bending moment reaches the section's plastic moment capacity, Mp = σyZ (σy = yield stress, Z = plastic section modulus); beyond this, the section rotates freely at essentially constant moment, like a mechanical hinge.

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The shape factor, S = Zp/Ze (ratio of plastic to elastic section modulus), measures the reserve strength beyond first yield due to plastic stress redistribution across the section (S = 1.5 for a rectangular section, ≈1.7 for a solid circular section).

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Collapse (mechanism) analysis determines the ultimate (collapse) load of a structure by identifying the combination and locations of plastic hinges that convert the structure (or part of it) into a mechanism, then applying the principle of virtual work to equate external work done by the collapse load to internal work absorbed by the plastic hinges.