How to calculate von Mises equivalent stress yield

Calculate von Mises equivalent stress sigma_VM = sqrt(0.5*((sx-sy)^2+(sy-sz)^2+(sz-sx)^2)+3*(txy^2+tyz^2+tzx^2)) distortion energy yield criterion. Steel S235 235 MPa S355 355 MPa Al6061 240 Ti grade 5 880 MPa.

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How to calculate the von Mises equivalent stress (VM stress): equivalent uniaxial stress producing the same distortion energy as a generic 3D stress state, plasticity criterion ductile materials. General 3D formula: sigma_VM = sqrt(0.5 * ((sigma_x - sigma_y)^2 + (sigma_y - sigma_z)^2 + (sigma_z - sigma_x)^2) + 3 * (tau_xy^2 + tau_yz^2 + tau_zx^2)). In principal stresses: sigma_VM = sqrt(0.5 * ((sigma_1 - sigma_2)^2 + (sigma_2 - sigma_3)^2 + (sigma_3 - sigma_1)^2)). Plane stress 2D (sigma_z=0): sigma_VM = sqrt(sigma_x^2 + sigma_y^2 - sigma_x*sigma_y + 3*tau_xy^2). Pure shear: sigma_VM = sqrt(3)*tau.

Yield criterion: sigma_VM >= sigma_yield material yields plastic deformation. Safety factor SF = sigma_yield/sigma_VM. History: Richard von Mises (Austria-USA 1883-1953) Berlin TH 1913 paper Göttingen Nachrichten distortion energy theory ductile materials. Maxwell-Huber-Hencky-von Mises criterion. Heinrich Hencky 1924 thermodynamic interpretation.

Essential tool for: structural engineering bridge building steel design Eurocode 3 EN 1993-1-1; mechanical engineering pressure vessels ASME BPVC SF=4; aerospace fatigue USA Boeing Lockheed SpaceX Northrop Grumman; automotive USA Detroit MI Ford GM Chrysler Tesla; petrochemical ExxonMobil Chevron; nuclear Westinghouse pressure vessels.

USA structural engineering organizations: ASME American Society Mechanical Engineers NYC 1880 Boiler Pressure Vessel Code BPVC + B31 piping codes 130000 members; ASCE American Society Civil Engineers Reston VA 1852 150000 members ASCE 7 minimum design loads; AISC American Institute Steel Construction Chicago IL 1921 AISC 360 Steel Construction Manual specifications LRFD ASD; AISI American Iron Steel Institute Washington DC 1855 cold-formed steel; AWS American Welding Society Miami FL 1919 structural welding D1.1; ACI American Concrete Institute Farmington Hills MI 1904 318 Building Code Reinforced Concrete; ASTM American Society Testing Materials West Conshohocken PA 1898 A36 A572 A992 steel standards; NSPE National Society Professional Engineers Alexandria VA 1934 PE license.

USA universities engineering: MIT Cambridge MA 1861 mechanical aerospace; Stanford Mechanical Palo Alto CA 1925; Caltech Pasadena CA 1891 structures; UC Berkeley CA 1868; Princeton NJ 1746; Cornell NY 1865; Carnegie Mellon Pittsburgh PA 1900; Georgia Tech Atlanta GA 1885; Purdue West Lafayette IN 1869; Illinois Urbana-Champaign 1867; Texas A&M College Station 1876; Michigan Ann Arbor 1817 mechanical engineering; Northwestern Evanston IL 1851; Penn State University Park 1855; Virginia Tech Blacksburg 1872.

USA software FEA: ANSYS Canonsburg PA 1970 founded John Swanson + Fluent + Mechanical 5500 employees market leader; Abaqus Dassault Systèmes Vélizy France 1978 acquired by 3DS 2005; NASTRAN MSC Software Newport Beach CA 1965 NASA originally + Hexagon AB acquisition 2017; LS-DYNA Livermore Software Technology LSTC Livermore CA 1976 crash simulation acquired by Ansys 2019; COMSOL Stockholm Sweden 1986 multiphysics; Hyperworks Altair Engineering Troy MI 1985; SolidWorks Simulation Dassault Waltham MA 1995; Autodesk Inventor San Rafael CA 1999; Siemens Simcenter Plano TX.

USA steel industry: Nucor Charlotte NC 1955 founded 31000 employees largest US steel producer 27 milioni tonnellate/anno; US Steel Pittsburgh PA 1901 founded JP Morgan Andrew Carnegie 23000 employees; Cleveland-Cliffs Cleveland OH 1847 21000 employees acquired AK Steel 2020; Steel Dynamics Fort Wayne IN 1993 11000 employees; Commercial Metals Irving TX 1915.

USA engineering firms: Bechtel San Francisco CA 1898 founded Warren Bechtel 55000 employees engineering construction; Fluor Irving TX 1912 41000 employees; Jacobs Engineering Dallas TX 1947 60000 employees; AECOM Los Angeles CA 1980 51000 employees; Parsons Corporation Centreville VA 1944; Black & Veatch Overland Park KS 1915; CH2M HILL Englewood CO 1946 (acquired by Jacobs 2017). Steel materials USA: ASTM A36 yield 250 MPa structural; ASTM A572 Grade 50 yield 345 MPa; ASTM A992 yield 345 MPa wide-flange shapes; ASTM A500 yield 290-345 MPa hollow structural sections HSS; AISI 304 stainless yield 207 MPa; AISI 316 stainless yield 200 MPa; Al 6061-T6 yield 240 MPa; Al 7075-T6 yield 503 MPa aerospace; Ti-6Al-4V Grade 5 yield 880 MPa aerospace; CFRP carbon fiber yield 1000+ MPa.

von Mises formula

3D fundamental equation: sigma_VM = sqrt(0.5 * ((sx-sy)^2 + (sy-sz)^2 + (sz-sx)^2) + 3 * (txy^2 + tyz^2 + tzx^2)). Equivalent uniaxial stress same distortion energy as 3D state. Richard von Mises 1913 Berlin TH paper.

Principal stresses form: sigma_VM = sqrt(0.5 * ((s1-s2)^2 + (s2-s3)^2 + (s3-s1)^2)). s1>=s2>=s3 principal stresses (eigenvalues stress tensor). Uniaxial test: s2=s3=0, sigma_VM = s1 = sigma_yield (validation).

Plane stress 2D: sigma_z = tau_yz = tau_zx = 0. Reduces to sigma_VM = sqrt(sx^2 + sy^2 - sx*sy + 3*txy^2). Pure shear (sx = sy = 0, txy = tau): sigma_VM = sqrt(3) * tau ≈ 1.732 * tau. Yield shear: tau_y = sigma_y / sqrt(3).

Yield criterion: sigma_VM yield plastic flow. SF = sigma_yield / sigma_VM. Eurocode 3 SF=1.0 partial factor; ASME BPVC SF=4.0 conservative.

Tresca comparison: Henri Tresca (France 1814-1885) maximum shear: tau_max = (s1-s3)/2 < tau_y. Tresca conservative von Mises (inscribed). Plane stress: Tresca hexagonal yield surface; von Mises elliptical inscribed Tresca polygon.

History and context

Richard von Mises (Austria-USA 1883-1953) Lemberg L'viv Ukraine born Austro-Hungarian Empire, Vienna TH mathematics 1907, Brünn Brno 1909, Strasbourg 1909-1918, Frankfurt 1918-1919, Berlin TH 1919-1933 director Institut für Angewandte Mathematik. Emigration 1933 Nazi: Istanbul University 1933-1939 Atatürk reforms Turkish exile; Harvard University Cambridge MA USA 1939-1953. 1913 paper Nachrichten der Gesellschaft der Wissenschaften zu Göttingen 1:582 'Mechanik der festen Körper im plastisch deformablen Zustand' distortion energy yield criterion. Maxwell-Huber-Hencky-von Mises: Maxwell 1856 letter Kelvin first idea; Tytus Huber (Poland 1872-1950) Lemberg L'viv 1903 paper Lwów; Heinrich Hencky (Germany 1885-1951) 1924 thermodynamic distortion energy interpretation. Plasticity history: Tresca 1864 maximum shear; Coulomb 1773; St-Venant 1864 elastic-plastic; Drucker (USA 1918-2001) Brown 1951 postulate; Prager (Germany-USA 1903-1980) Brown kinematic hardening. USA structural engineering history: Galileo 1638 Two New Sciences first systematic; Hooke 1660 sigma=E*epsilon; Euler 1744 buckling; Cauchy 1822 stress tensor; Saint-Venant 1855; Castigliano 1873 strain energy theorem. Modern: Stephen Timoshenko (Russia-USA 1878-1972) Stanford 'Theory of Elasticity' 1934 textbook classic; James Goodier (UK-USA 1905-1969) Stanford coauthor; Daniel Drucker Brown plastic limit theorems; William Prager Brown kinematic hardening 1955;

Egor Popov (Russia-USA 1913-2001) Berkeley earthquake engineering; Karl Terzaghi (Austria-USA 1883-1963) MIT Harvard soil mechanics 1925; Ralph Peck (USA 1912-2008) Illinois soil mechanics. USA universities: MIT Cambridge MA 1861 founded; Stanford 1925; Caltech 1891; Berkeley 1868; Princeton 1746; Cornell 1865; Carnegie Mellon Pittsburgh PA 1900; Georgia Tech 1885; Purdue 1869; Illinois Urbana 1867; Texas A&M 1876; Michigan Ann Arbor 1817; Northwestern 1851; Virginia Tech 1872; Lehigh University Bethlehem PA 1865 Fritz Engineering Lab seismic. USA professional organizations: ASME 1880 Boiler Pressure Vessel Code; ASCE 1852 ASCE 7 loads; AISC 1921 Steel Construction Manual; AISI 1855 cold-formed; AWS 1919 D1.1 welding; ACI 1904 318 Code; ASTM 1898 standards. USA steel industry: Nucor Charlotte NC 1955 founded F. Kenneth Iverson 31000 employees largest US producer 27 million tons/year electric arc furnace; US Steel Pittsburgh PA 1901 founded JP Morgan Andrew Carnegie 23000 employees; Cleveland-Cliffs Cleveland OH 1847 21000 employees; Steel Dynamics Fort Wayne IN 1993 11000 employees; Commercial Metals Irving TX 1915; ArcelorMittal USA Burns Harbor Indiana subsidiary; AK Steel acquired by Cleveland-Cliffs 2020. USA engineering firms: Bechtel San Francisco CA 1898 55000 employees; Fluor Irving TX 1912 41000 employees; Jacobs Engineering Dallas TX 1947 60000 employees; AECOM Los Angeles CA 1980 51000; Parsons Centreville VA 1944; Black & Veatch Overland Park KS 1915;

Stanley Consultants Muscatine IA 1913; HDR Omaha NE 1917; Burns & McDonnell Kansas City MO 1898; CDM Smith Boston MA 1947. USA software FEA: ANSYS Canonsburg PA 1970 founded John Swanson finite element + Fluent + Mechanical 5500 employees largest market leader; Abaqus Dassault Systèmes 1978; NASTRAN MSC Software Newport Beach CA 1965 NASA + Hexagon AB acquisition 2017; LS-DYNA LSTC Livermore CA 1976 crash + Ansys 2019; COMSOL Stockholm 1986 multiphysics; Hyperworks Altair Troy MI 1985; SolidWorks Simulation Dassault Waltham MA 1995; Autodesk Inventor San Rafael CA 1999; Siemens Simcenter Plano TX. USA aerospace structural: NASA Washington DC 1958 17000 employees; Boeing Chicago IL 1916 100000 employees 737 747 777 787; Lockheed Martin Bethesda MD 1995 116000 employees F-35 F-22 F-16; SpaceX Hawthorne CA 2002 Falcon Dragon Starship; Northrop Grumman Falls Church VA 1994; Raytheon Technologies Arlington VA 2020 RTX; Pratt & Whitney East Hartford CT 1925 jet engines.

von Mises values table

  • 3D formula: sqrt(0.5*((sx-sy)^2+(sy-sz)^2+(sz-sx)^2)+3*(txy^2+tyz^2+tzx^2))
  • 2D plane stress: sqrt(sx^2+sy^2-sx*sy+3*txy^2)
  • Pure shear yield: tau_y = sigma_y/sqrt(3)
  • Tresca: tau_max = (s1-s3)/2 < tau_y
  • ASTM A36 yield: 250 MPa structural standard
  • ASTM A572 Grade 50: 345 MPa
  • ASTM A992: 345 MPa wide-flange shapes
  • ASTM A500 HSS: 290-345 MPa hollow sections
  • AISI 304 stainless: 207 MPa
  • AISI 316 stainless: 200 MPa
  • Al 6061-T6: 240 MPa
  • Al 7075-T6: 503 MPa aerospace
  • Ti-6Al-4V Grade 5: 880 MPa aerospace
  • Concrete C30 fck: 30 MPa compression
  • Concrete C40 fck: 40 MPa
  • CFRP carbon fiber: 1000+ MPa
  • Kevlar: 3000 MPa
  • HDPE polyethylene: 25 MPa
  • PVC plastic: 50 MPa
  • Glass: 30-90 MPa
  • SF Eurocode 3: 1.0 partial factor
  • SF ASME BPVC: 4.0 pressure vessels
  • SF aerospace: 1.5
  • SF buildings: 1.5-2.0 ASCE 7
  • SF crane lifting: 4-10 OSHA
  • von Mises 1913 paper: Göttingen Nachrichten
  • Hencky 1924: distortion energy
  • USA Nucor 1955: 31000 employees largest steel
  • USA US Steel 1901: Pittsburgh PA Andrew Carnegie
  • USA ASME 1880: BPVC pressure vessel code
  • USA ASCE 1852: ASCE 7 loads code
  • USA AISC 1921: Steel Construction Manual
  • USA ANSYS 1970: FEA market leader 5500
  • USA NASA 1958: 17000 aerospace
  • USA Boeing 1916: 100000 aircraft Chicago
  • USA Lockheed Martin 1995: F-35 Bethesda MD

Numerical examples

Example 1 - Uniaxial tension steel A36. Tie steel A36 yield 250 MPa. sigma_x=200, others=0. sigma_VM=200 MPa. SF=1.25 adequate margin. USA: ASTM A36 structural standard; AISC Chicago 1921 LRFD specifications; Nucor Charlotte NC 1955 31000 employees.

Example 2 - Biaxial + shear A572. Plate A572 Grade 50 yield 345 MPa. sigma_x=150, sigma_y=100, tau_xy=50. sigma_VM=158 MPa. SF=2.18 good. USA: US Steel Pittsburgh PA 1901; Cleveland-Cliffs OH 1847; Steel Dynamics Fort Wayne IN 1993.

Example 3 - Pure torsion shaft. Pure shear tau=100 MPa. sigma_VM=sqrt(3)*100=173.2 MPa. Steel yield 250 MPa: SF=1.44. USA automotive: Ford Dearborn MI 1903; GM Detroit MI 1908; Tesla Austin TX 2003.

Example 4 - 3D complex state. yield 500 MPa. sigma_x=100, sigma_y=50, sigma_z=-30, tau_xy=40, tau_yz=20, tau_zx=15. sigma_VM=140 MPa. SF=3.57 high. USA aerospace: NASA 1958; Boeing Chicago 1916; Lockheed Martin Bethesda MD 1995 F-35.

Example 5 - Pressure vessel cylindrical. p=10 MPa, R=200 mm, t=10 mm. sigma_hoop=200, sigma_long=100. sigma_VM=173.2 MPa. A36 yield 250 MPa: SF=1.44. ASME BPVC SF=4 conservative. USA: Bechtel San Francisco CA 1898; Fluor Irving TX 1912; Jacobs Dallas TX 1947.

Limitations

  1. Ductile materials only: VM suitable metals (steel, aluminum, copper). Brittle (cast iron, concrete, glass) require Mohr-Coulomb, Drucker-Prager, Rankine.
  2. Isotropy required: assumes isotropic. Anisotropic (CFRP composites, wood, laminates) require Tsai-Hill, Tsai-Wu, Hill anisotropic.
  3. Hydrostatic independent: VM ignores hydrostatic stress. Pressure-dependent (rocks, concrete) require Drucker-Prager.
  4. Yield onset only: predicts plasticity start, not post-yield flow (hardening law required: isotropic kinematic mixed).
  5. Static monotonic: applicable static loads. Fatigue cyclic requires S-N or E-N curves Coffin-Manson.
  6. Ambient temperature: yield varies with T. High T (>0.4 T_melt) creep; low T (<0°C) brittleness.
  7. Strain rate: rate sensitivity metals. Static yield differs from impact dynamic (>1 1/s).
  8. Confusing sigma_VM and sigma_max: VM equivalent measure single value; sigma_1 sigma_2 sigma_3 principal singles.
  9. Coordinate frame: stresses depend coordinates. Invariants I1 I2 I3 independent orientation, sigma_VM invariant.
  10. Linear elasticity only: VM yield criterion onset plasticity. Post-yield finite plasticity J2 flow theory.

Common errors

Confusing sigma and tau: sigma normal stress (positive tension, negative compression); tau shear stress. Convention: sigma_x sigma_y sigma_z normal; tau_xy tau_yz tau_zx shear.

Plane stress vs plane strain: plane stress sigma_z=0 (thin plates); plane strain epsilon_z=0 (thick components). Different formulas, sigma_VM differs.

Principal stress ordering: sigma_1 >= sigma_2 >= sigma_3 standard convention (algebraic). Tresca tau_max = (sigma_1 - sigma_3)/2 always with this ordering.

Units MPa vs ksi: 1 MPa = 1 N/mm² = 10^6 Pa. USA ksi = 6.895 MPa. ASTM A36 yield 36 ksi = 248 MPa.

Yield vs ultimate: sigma_yield onset plastic; sigma_ultimate UTS rupture. von Mises uses yield, not ultimate. SF typically vs yield not ultimate.

Practical applications

  1. Structural engineering: bridges buildings steel ASCE 7 + AISC 360. USA Bechtel Fluor Jacobs.
  2. Mechanical engineering: ASME BPVC SF=4 pressure vessels.
  3. Aerospace fatigue: NASA Washington DC 1958; Boeing Chicago 1916 100000; Lockheed Martin Bethesda MD F-35; SpaceX Hawthorne 2002.
  4. Automotive: Ford Dearborn MI 1903; GM Detroit MI 1908; Chrysler Auburn Hills MI 1925; Tesla Austin TX 2003 EV.
  5. Petrochemical pipelines: ExxonMobil Irving TX; Chevron San Ramon CA; ConocoPhillips Houston TX.
  6. Nuclear: Westinghouse Pittsburgh PA 1886 nuclear reactor pressure vessels.
  7. FEA software: ANSYS Abaqus NASTRAN LS-DYNA Hyperworks COMSOL.
  8. Pipeline gas: API 5L X70 yield 485 MPa; ASME B31.
  9. Civil seismic: ASCE 7 seismic + IBC International Building Code.
  10. Steel industry: Nucor Charlotte NC 1955; US Steel Pittsburgh PA 1901; Cleveland-Cliffs OH 1847.

Updated July 2026

Frequently Asked Questions

What is von Mises equivalent stress?

Equivalent uniaxial stress producing same distortion energy as 3D state, plasticity criterion ductile materials. Formula: sigma_VM = sqrt(0.5*((sx-sy)^2+(sy-sz)^2+(sz-sx)^2)+3*(txy^2+tyz^2+tzx^2)). Plane stress 2D: sqrt(sx^2+sy^2-sx*sy+3*txy^2). Pure shear: sigma_VM = sqrt(3)*tau. Yield criterion: sigma_VM >= sigma_yield material yields. Richard von Mises 1913 Berlin TH paper Göttingen distortion energy theory; Hencky 1924 thermodynamic. USA: ASME 1880 BPVC; ASCE 1852; AISC Chicago 1921 LRFD; ASTM 1898; ANSYS Canonsburg PA 1970 FEA.

How to calculate sigma_VM 2D?

Plane stress 2D (sigma_z = 0, tau_yz = tau_zx = 0): sigma_VM = sqrt(sigma_x^2 + sigma_y^2 - sigma_x*sigma_y + 3*tau_xy^2). Example: sigma_x=150, sigma_y=100, tau_xy=50 MPa. sigma_VM = sqrt(22500+10000-15000+7500) = sqrt(25000) = 158 MPa. Steel A572 Grade 50 yield 345 MPa: SF = 345/158 = 2.18 good. USA: ASTM A572 standard; AISC Chicago 1921 Steel Construction Manual; Nucor Charlotte NC 1955 31000 employees largest US steel; US Steel Pittsburgh PA 1901.

Who derived von Mises criterion?

Richard von Mises (Austria-USA 1883-1953) Berlin TH 1919-1933 1913 paper Nachrichten Göttingen 1:582 'Mechanik der festen Körper im plastisch deformablen Zustand' distortion energy yield criterion. Maxwell 1856 letter Kelvin first idea; Tytus Huber Poland 1903 paper; Heinrich Hencky Germany 1924 thermodynamic interpretation. Von Mises emigration 1933 Nazi Istanbul 1933-1939 Atatürk reforms then Harvard 1939-1953. USA: Stephen Timoshenko Stanford 'Theory Elasticity' 1934 textbook; Daniel Drucker Brown 1951 postulate; William Prager Brown kinematic hardening 1955.

Difference von Mises vs Tresca?

von Mises (distortion energy): sigma_VM = sqrt(0.5*((s1-s2)^2+(s2-s3)^2+(s3-s1)^2)). Elliptical yield surface. Tresca (max shear): tau_max = (sigma_1-sigma_3)/2 < tau_y. Hexagonal yield surface. Tresca conservative (inscribed in von Mises ellipse). For uniaxial and biaxial coincide along edges. Henri Tresca France 1814-1885 Conservatoire Arts Métiers Paris 1864 paper extrusion lead experiments. USA: ASME BPVC uses Tresca conservative; AISC Chicago 1921 von Mises ductile steel.

What is safety factor SF?

SF = sigma_yield / sigma_VM. SF > 1 design safe; SF = 1 onset yield critical; SF < 1 plastic FAIL. Typical SF: 1.0 Eurocode 3 partial factor; 1.5 aerospace lightweight; 1.5-2.0 steel buildings ASCE 7; 4.0 ASME BPVC pressure vessels conservative + boilers; 4-10 crane lifting OSHA; 10+ elevators safety. USA: ASCE 1852 + AISC 1921 + ASME 1880 + AWS 1919 + ACI 1904 + ASTM 1898 standards organizations. SF integrated via partial factors loads + materials + structural class.

Which materials for von Mises?

Ductile materials: steels (A36, A572, A992), aluminum (6061-T6, 7075-T6), titanium (Ti-6Al-4V Grade 5), copper, brass, nickel, metallic alloys. Yield strength: A36=250 MPa, A572=345, A992=345 wide-flange, A500 HSS=290-345; AISI 304 stainless 207 MPa, 316=200; Al6061=240, Al7075=503; Ti Grade 5=880 aerospace; CFRP>1000. NOT suitable brittle: cast iron, concrete, ceramic, glass → Mohr-Coulomb, Drucker-Prager. USA steel: Nucor Charlotte NC 1955 31000 27M tons; US Steel Pittsburgh 1901; Cleveland-Cliffs OH 1847.

When does material yield?

Yield onset when sigma_VM >= sigma_yield. Plastic deformation begins. Ductile metals: yield then strain hardening (work hardening) increases strength + ductility, eventually UTS ultimate tensile strength + necking + rupture. Stress-strain curve: linear elastic (Hooke sigma=E*epsilon) → yield → plastic work hardening → UTS → necking → fracture. Hardening laws: Ramberg-Osgood, Voce, Hollomon. von Mises J2 flow theory + Prandtl-Reuss equations. USA: ASTM 1898 testing standards; ASME 1880 BPVC; AISC Chicago 1921 LRFD specifications.

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