{ "openapi": "3.1.1", "info": { "title": "Section Properties Calculator: Area, Moments of Inertia, Torsion Constant API", "version": "1.0.0", "summary": "A, Iy, Iz, J and section moduli of rectangles, round bars, tubes, box and I-sections.", "description": "Every endpoint performs the same calculation as the human page and returns inputs, result, units, assumptions, sources, warnings and computedAt. MCP endpoint: /mcp." }, "servers": [ { "url": "https://prooflinesectionproperties.usefulweb.app" } ], "paths": { "/api/v1/calculate": { "post": { "operationId": "calculate", "summary": "A, Iy, Iz, J and section moduli of rectangles, round bars, tubes, box and I-sections.", "description": "Enter the shape and dimensions of a beam cross-section — solid rectangle, solid circle, circular tube, rectangular hollow section or doubly symmetric I-section — and get the area, the second moments of area about both axes, the St. Venant torsion constant, the elastic section moduli, the torsion modulus, the radii of gyration and, with a typical steel, stainless, aluminium or timber material, the mass per metre and the bending and torsional stiffness. A drawing shows the section with its local axes, and a ready line feeds the 3-D frame model of Proofline’s truss and frame solver. Part of the Proofline math notebook.\n\nMethodology:\n- Local axes as in Eurocode 3: y across the width b, z along the depth h, so Iy is the strong axis of an upright rectangle or I-section. A, Iy and Iz from the exact formulas of each shape (I-sections and box sections without root or corner radii).\n- Torsion: circles and tubes J = πd⁴/32 (difference for the tube), τ = T·r/J; solid rectangles by Roark’s formulas, J = a·c³·[1/3 − 0.21·(c/a)·(1 − c⁴/(12a⁴))] and τmax = 3T/(a·c²)·[1 + 0.6095r + 0.8865r² − 1.8023r³ + 0.9100r⁴]; box sections by Bredt’s thin-wall formula J = 4Am²t/p; I-sections as thin open rectangles J = (2·b·tf³ + (h − 2tf)·tw³)/3, warping ignored.\n- Section moduli W = I/(distance to the extreme fibre), torsion modulus Wt = T/τmax, radii of gyration i = √(I/A); with a material the mass per metre ρ·A and the stiffnesses E·I and G·J.\n- Verified against closed forms (b·h³/12, πd⁴/64), the square-bar torsion constant 0.1406·a⁴ and τmax = T/(0.208·a³), Bredt’s formula for a thin square tube and steel-table values of IPE 200 (Iz 142 cm⁴; A and Iy within 6 % because of the fillets).\n\nNominal geometry without fillets, corner radii or tolerances; open sections ignore warping torsion. Material values are typical, not those of a certified grade. Use the manufacturer’s section tables for design.", "tags": [ "apps" ], "x-mcp-tool": "proofline_section_properties", "x-owner-app": "proofline", "requestBody": { "required": true, "content": { "application/json": { "schema": { "$ref": "#/components/schemas/ProoflineSectionPropertiesInput" }, "examples": { "example1": { "summary": "I-section like an IPE 200 (100 × 200, tf 8.5, tw 5.6)", "value": { "shape": "i", "b": 100, "h": 200, "tf": 8.5, "tw": 5.6, "material": "steel", "name": "IPE200" } }, "example2": { "summary": "Timber joist 100 × 200 mm, C24", "value": { "shape": "rect", "b": 100, "h": 200, "material": "timber", "name": "C24" } }, "example3": { "summary": "Steel tube 88.9 × 4 mm", "value": { "shape": "tube", "d": 88.9, "t": 4, "material": "steel", "name": "CHS89" } }, "example4": { "summary": "Aluminium box section 100 × 50 × 4 mm", "value": { "shape": "rhs", "b": 50, "h": 100, "t": 4, "material": "aluminium", "name": "RHS" } }, "example5": { "summary": "Solid round bar 30 mm, stainless", "value": { "shape": "circle", "d": 30, "material": "stainless", "name": "BAR30" } } } } } }, "responses": { "200": { "description": "Calculation result envelope.", "content": { "application/json": { "schema": { "$ref": "#/components/schemas/ProoflineSectionPropertiesResult" } } } }, "400": { "description": "Malformed JSON body.", "content": { "application/problem+json": { "schema": { "$ref": "#/components/schemas/Problem" } } } }, "422": { "description": "Invalid input (see errors[]).", "content": { "application/problem+json": { "schema": { "$ref": "#/components/schemas/Problem" } } } }, "429": { "description": "Rate limit exceeded (see Retry-After).", "content": { "application/problem+json": { "schema": { "$ref": "#/components/schemas/Problem" } } } } } } }, "/api/v1/tool": { "get": { "operationId": "getToolDefinition", "summary": "Inputs, output schema and examples for this tool", "responses": { "200": { "description": "Tool definition.", "content": { "application/json": { "schema": { "type": "object" } } } } } } } }, "components": { "schemas": { "Problem": { "type": "object", "description": "RFC 9457 problem details. 422 responses include per-field `errors`.", "required": [ "type", "title", "status", "detail" ], "properties": { "type": { "type": "string" }, "title": { "type": "string" }, "status": { "type": "integer" }, "detail": { "type": "string" }, "code": { "type": "string" }, "errors": { "type": "array", "items": { "type": "object", "required": [ "field", "message" ], "properties": { "field": { "type": "string" }, "message": { "type": "string" } } } } } }, "ProoflineSectionPropertiesInput": { "type": "object", "additionalProperties": false, "properties": { "shape": { "title": "Shape", "type": "string", "enum": [ "rect", "circle", "tube", "rhs", "i" ], "default": "rect" }, "b": { "title": "Width b", "description": "Rectangle, hollow section and I-section: the width along the local y axis (flange width of an I-section).", "type": "number", "minimum": 0, "maximum": 100000, "x-unit": "mm" }, "h": { "title": "Depth h", "description": "Rectangle, hollow section and I-section: the depth along the local z axis.", "type": "number", "minimum": 0, "maximum": 100000, "x-unit": "mm" }, "d": { "title": "Diameter d", "description": "Circle and tube: the outside diameter.", "type": "number", "minimum": 0, "maximum": 100000, "x-unit": "mm" }, "t": { "title": "Wall thickness t", "description": "Tube and hollow section.", "type": "number", "minimum": 0, "maximum": 100000, "x-unit": "mm" }, "tf": { "title": "Flange thickness tf", "description": "I-section.", "type": "number", "minimum": 0, "maximum": 100000, "x-unit": "mm" }, "tw": { "title": "Web thickness tw", "description": "I-section.", "type": "number", "minimum": 0, "maximum": 100000, "x-unit": "mm" }, "material": { "title": "Material", "description": "Typical values for the mass per metre and the stiffnesses EI and GJ.", "type": "string", "enum": [ "steel", "s355", "stainless", "aluminium", "timber", "none" ], "default": "steel" }, "name": { "title": "Section name for the frame model", "type": "string", "maxLength": 24, "default": "S1" } }, "required": [] }, "ProoflineSectionPropertiesResult": { "type": "object", "additionalProperties": false, "required": [ "schemaVersion", "tool", "computedAt", "inputs", "summary", "result", "units", "assumptions", "sources", "warnings", "presentation", "links" ], "properties": { "schemaVersion": { "type": "string", "const": "1.0" }, "tool": { "type": "object", "additionalProperties": false, "required": [ "id", "name", "version" ], "properties": { "id": { "type": "string" }, "name": { "type": "string" }, "version": { "type": "string" } } }, "computedAt": { "type": "string", "format": "date-time" }, "inputs": { "type": "object", "description": "Normalised inputs actually used, including applied defaults." }, "summary": { "type": "string" }, "result": { "type": "object", "additionalProperties": false, "required": [ "shape", "dims", "A", "Iy", "Iz", "J", "Wy", "Wz", "Wt", "iy", "iz", "zMax", "yMax", "material", "massPerMetre", "EIy", "EIz", "GJ", "modelLine" ], "properties": { "shape": { "type": "string", "enum": [ "rect", "circle", "tube", "rhs", "i" ] }, "dims": { "type": "object", "additionalProperties": { "type": "number" }, "description": "Dimensions used, mm." }, "A": { "type": "number", "description": "Area, cm²." }, "Iy": { "type": "number", "description": "Second moment of area about the local y axis (horizontal), cm⁴." }, "Iz": { "type": "number", "description": "Second moment about the local z axis, cm⁴." }, "J": { "type": "number", "description": "St. Venant torsion constant, cm⁴." }, "Wy": { "type": "number", "description": "Elastic section modulus Iy/zmax, cm³." }, "Wz": { "type": "number", "description": "Elastic section modulus Iz/ymax, cm³." }, "Wt": { "type": "number", "description": "Torsion modulus T/τmax, cm³." }, "iy": { "type": "number", "description": "Radius of gyration, mm." }, "iz": { "type": "number" }, "zMax": { "type": "number" }, "yMax": { "type": "number" }, "material": { "type": [ "object", "null" ], "additionalProperties": false, "required": [ "id", "name", "E", "G", "fy", "rho" ], "properties": { "id": { "type": "string" }, "name": { "type": "string" }, "E": { "type": "number" }, "G": { "type": "number" }, "fy": { "type": "number" }, "rho": { "type": "number" } } }, "massPerMetre": { "type": [ "number", "null" ], "description": "kg/m." }, "EIy": { "type": [ "number", "null" ], "description": "kN·m²." }, "EIz": { "type": [ "number", "null" ] }, "GJ": { "type": [ "number", "null" ] }, "modelLine": { "type": "string", "description": "Section line for the 3-D frame model of proofline-truss-frame-solver." } } }, "units": { "type": "object", "additionalProperties": { "type": "string" } }, "assumptions": { "type": "array", "items": { "type": "string" } }, "sources": { "type": "array", "items": { "type": "object", "additionalProperties": false, "required": [ "title" ], "properties": { "title": { "type": "string" }, "url": { "type": "string" }, "publisher": { "type": "string" }, "retrievedAt": { "type": "string" }, "note": { "type": "string" } } } }, "warnings": { "type": "array", "items": { "type": "string" } }, "presentation": { "type": "array", "items": { "type": "object" }, "description": "Display blocks used by the human page (same data)." }, "links": { "type": "object", "additionalProperties": false, "required": [ "human", "api", "openapi", "methodology" ], "properties": { "human": { "type": "string", "format": "uri" }, "api": { "type": "string", "format": "uri" }, "openapi": { "type": "string", "format": "uri" }, "methodology": { "type": "string", "format": "uri" } } } } } } } }