Mohr Circle 3D

Every structural calculation begins with the same three equations in two dimensions — the sum of forces in x, the sum of forces in y, and the sum of moments about any point all equal zero. Statics is not a preliminary step; it is the constraint the whole solution must satisfy at every section of the structure.

A free-body diagram is the tool that makes this concrete. Isolate the member or joint, replace every support with the reaction it can provide (a pin provides two force components, a roller one normal force, a fixed support two forces and a moment), and apply the loads. Then:

1. Reactions first. Take moments about one support to eliminate the other reactions, then sum forces in each direction. A quick check: reactions must balance the applied load resultant, or the arithmetic is already wrong.
2. Internal forces by sectioning. Cut the member where you need the answer and enforce equilibrium on the cut piece: axial force N (along the member), shear force V (transverse), bending moment M (about the section axis). Sign conventions vary between textbooks — pick one and stay with it, because a flipped moment sign in a shear–moment diagram produces a plausible-looking but mirrored picture.
3. Trusses by joints or sections. The method of joints solves each pin separately (two equilibrium equations per joint); the method of sections cuts through the truss and solves three unknowns at once with the same equations. Both rely on the idealisation that truss members carry axial load only — no bending, no eccentricity. Real gusset plates and welded joints violate that assumption slightly, which is why design codes add detailing rules on top of the analysis.

Beam conventions follow directly: the load, shear and moment relations are differential (w = −dV/dx = d²M/dx²), so a uniformly distributed load produces a linear shear diagram and a parabolic moment diagram — a shape check worth performing on every result. Indeterminate structures (continuous beams, frames with redundant supports) cannot be solved by equilibrium alone; methods such as moment distribution, slope–deflection, the conjugate-beam method and matrix stiffness/flexibility analysis add the compatibility conditions needed to close the system. Influence lines then answer a different question entirely: not "what is the moment under this load" but "where should this moving load sit to maximise the moment" — the idea behind every bridge live-loading specification.

Force divided by area gives normal stress, σ = N/A, in pascals (1 Pa = 1 N/m²; engineering practice uses MPa = N/mm² freely — keep the prefix consistent and the conversion is invisible). Shear stress follows the same pattern, τ = V/A for direct shear, though in beams the distribution is parabolic and the average underestimates the peak at the neutral axis.

Bending is where section geometry enters. The flexure formula σ = My/I relates stress to distance y from the neutral axis, where I is the second moment of area (m⁴). Two derived properties do the real design work:

• Section modulus S = I/y_max — bending stress is simply M/S, and it is the quantity steel tables publish.
• Radius of gyration r = √(I/A) — it feeds directly into slenderness checks for buckling.

For a rectangle of width b and depth h: I = bh³/12 and S = bh²/6. Note the cube: doubling a beam's depth multiplies stiffness by eight and bending strength by four — orientation matters far more than a modest change in cross-sectional area. Torsion of circular shafts uses the polar counterpart, τ = T·r/J, with J = 2I for a solid circle; non-circular sections twist out of plane and need warping analysis instead.

Materials respond to stress through the stress–strain curve: linear-elastic with slope E (Young’s modulus, ~200 GPa for steel, ~70 GPa for aluminium, ~30 GPa for concrete in compression) up to the proportional limit, then yielding, then strain hardening to the ultimate strength, then necking. Ductile materials yield gradually and warn; brittle materials (cast iron, glass, concrete in tension, aged welds) fail without warning — the single most important behavioural difference in design.

Combined stresses are resolved, not added. When normal and shear stresses act on the same plane, Mohr’s circle plots them and reads off the principal stresses σ₁, σ₂ (the orientations where shear vanishes) and the maximum shear stress τ_max = (σ₁ − σ₂)/2. Failure theories then compare an *equivalent* stress against allowable strength: von Mises (distortion energy) for ductile metals — σ_von = √(σ₁² − σ₁σ₂ + σ₂²) — and Tresca (maximum shear) as the more conservative cousin. Using the wrong theory for the material class is a genuine engineering error, not a rounding detail.

A column can fail without its stress ever reaching the yield strength — it buckles. Euler's formula gives the critical load:

P_cr = π²EI / (KL)²

where K is the effective-length factor (1 for pinned-pinned, 0.5 for fixed-fixed, 0.7 for fixed-pinned, 2 for fixed-free — end conditions dominate the answer) and KL is the effective length. The practical reading: buckling resistance scales with the *square* of length and with I, not with area. Slender members therefore fail by stability long before strength, and the slenderness ratio KL/r (r = radius of gyration) decides which regime governs. Stocky columns crush; slender columns buckle; the code transition between the two is where most real column design lives. Plates and shells have their own buckling formulae — the famous cylinder factor of ~0.6 on the classical prediction exists because real shells buckle at a fraction of theoretical load under tiny imperfections, so fabrication tolerances are structural, not cosmetic.

Fatigue is failure under repeated stress far below ultimate strength. Cyclic loading initiates and grows cracks at stress concentrations until the remaining section snaps. The S–N curve (stress amplitude vs cycles to failure) shows ferrous metals levelling off at an endurance limit — below it, infinite life is plausible — while aluminium has no such plateau. Mean stress matters, and three classic criteria correct for it:

• Goodman: linear in stress — σ_a/σ_e + σ_m/σ_u = 1 (most common for ductile design).
• Soderberg: linear but uses yield strength — the most conservative.
• Gerber: parabolic — matches ductile data best but is less conservative.

A stress concentration factor K_t multiplies nominal stress at notches, holes and fillets; the fatigue-notch factor K_f applies the discounted version actually seen in fatigue. Fracture mechanics goes deeper: crack growth per cycle follows the Paris law, da/dN = C(ΔK)^m, with ΔK the stress-intensity range, and fracture toughness K_IC sets the critical crack size at which static fracture becomes unstable. Together they explain why a part with acceptable average stress can fail at a sharp corner — and why inspection intervals are calculated, not guessed.

Throughout all of this runs the safety factor: allowable = material property / factor, with factors spanning roughly 1.5 (controlled, understood, redundant load paths) to well above 3 (brittle failure modes, dynamic or poorly characterised loads). The factor is not pessimism — it is the quantified size of everything the model left out. These tools perform the calculations; design approval belongs to a qualified professional engineer working to the applicable code.

The Mohr Circle 3D is a free engineering tool that applies professional design formulas free mohr circle 3d — engineering reference tool for quick, . It follows industry-standard codes and practices to provide accurate results for engineering calculations. The calculator supports multiple unit systems and uses precise constants appropriate for each discipline. Whether you are an electrical, mechanical, civil, or aerospace engineer, this tool helps you quickly verify designs, size components, and analyze systems. Use it for preliminary design, educational purposes, and quick checks of your engineering calculations.

The Mohr Circle 3D supports engineering professionals with precise calculations for structural, electrical, thermal, and fluid mechanics applications.

Using the Mohr Circle 3D is straightforward. Enter your design parameters, material properties, and loading conditions into the input fields. The calculator applies industry-standard engineering formulas and codes to compute results. You can adjust inputs to optimize your design or check different scenarios. Mohr Circle 3D: always verify critical engineering calculations with manual checks or professional consultation.

The formula used by the Mohr Circle 3D is:

The Mohr Circle 3D uses validated formulas that follow industry standards and academic best practices for engineering calculations.

Understanding the formula behind the Mohr Circle 3D helps you interpret the results more effectively. Each variable in the formula represents a specific input that you provide, and the calculator handles the complex arithmetic automatically.

Key references for this formula

• American Institute of Steel Construction — published specification and design guides for structural steel, the reference behind most steel member checks. ↗
• American Society of Civil Engineers — standards and practice guidance covering structural loading and civil engineering design. ↗
• The Engineering ToolBox — engineering reference data for beam properties, materials and unit conversions used throughout design calculations. ↗