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# Non-Linear Behaviour

## material non-linear, general:

**general set of benchmarks:**

- a review of numerous potential benchmark problems, not worked up into formal benchmarks

*Benchmark P15 .*

## creep;

**2D single element tests subject to secondary creep (Norton law):**

- plane stress with constant uniaxial load
- plane stress with constant applied uniaxial displacement
- plane stress with constant biaxial load (3 different ratios)
- plane stress with constant applied biaxial displacement (3 different ratios)
- plane strain with constant unequi-biaxial load
- plane strain with constant applied unequi-biaxial displacement

**3D single element tests subject to secondary creep (Norton law):**

- with constant unequi-triaxial load
- with constant applied unequi-triaxial displacement

**axisymmetric multi-element test subject to secondary creep (Norton law):**

- internally pressurised thick hollow cylinder

**2D single element tests subject to primary creep (Norton-Bailey law):**

- plane stress with constant uniaxial load
- plane stress with constant uniaxial stepped load
- plane stress with constant applied uniaxial displacement
- plane stress with constant biaxial load (2 different ratios)
- plane stress with constant biaxial stepped load (2 different ratios)
- plane stress with constant applied biaxial displacement (2 different ratios)

**3D single element test subject to primary creep (Norton-Bailey law):**

- with constant unequi-triaxial load

**2D single element tests subject to combined primary and secondary creep:**

- plane stress with constant uniaxial load
- plane stress with constant uniaxial stepped load
- plane stress with constant applied uniaxial displacement

*Benchmarks R0027 , R0049 (subset of above), R0080 (subset of above).*

**3D multi-element test subject to secondary Norton creep:**

- torsional creep of long prismatic circular shaft under uniform twist held constant with time
- torsional creep of long prismatic circular shaft under uniform twist steadily increasing with time

**3D multi-element test subject to primary creep (Norton-Bailey law) with time hardening:**

- torsional creep of a square shaft under constant twist
- torsional creep of a square shaft under steady twist rate

**2D plane strain thick cylinder under temperature transient followed by fluctuating internal pressure:**

- both elastic-plastic and creep behaviour over 4 stages; non-proportional loading

**Axisymmetric multi-element test subject to temperature-dependent secondary creep (Norton law):**

- thick hollow cylinder with temperature gradients and thermally dependent creep laws

*Benchmarks R0026 , R0030 , R0049 (subset of above), R0080 (subset of above).*

## plasticity:

**2D single element tests:**

- plane strain with perfect plasticity
- plane strain with isotropic hardening
- plane strain comparing kinematic and isotropic hardening
- plane stress with perfect plasticity
- plane stress with isotropic hardening

**3D single element tests:**

- with perfect plasticity
- with isotropic hardening

Axisymmetric test with perfect plasticity

Axisymmetric test with isotropic hardening

Plane strain cylinder incompressibility test

*Benchmarks P06 , R0049 (subset of above), R0072 (subset of above), R0080 (subset of above).*

**2D two-bar assembly under constant axial load and cyclic temperatures of differing values in each bar:**

- with kinematic hardening and thermal load in one bar only, to test elastic shakedown
- with kinematic hardening and larger thermal load amplitude, to test alternating plasticity
- with kinematic hardening and thermal load in both bars, to test different elastic shakedown in each bar
- with perfect plasticity and thermal load in one bar only, to test thermal ratcheting

**2D beam with constant end load and cyclic temperature gradient:**

- with kinematic hardening, to test elastic/plastic shakedown and cyclic plastic strain accumulation

**2D plane strain rigid punch test under prescribed displacements, with many elements:**

- with perfect plasticity, testing the solution towards the limit load
- with isotropic bilinear hardening

**Thin square plate with uniform load and perfect plasticity:**

- with simply supported edges, testing the solution towards the limit load
- with clamped edges, testing the solution towards the limit load

- with both elastic-plastic and creep behaviour over 4 stages; non-proportional loading

**2D plane stress half of an eccentric tube under constant internal pressure and cyclically varying through-thickness temperature gradient:**

- with perfect plasticity, testing ratcheting and alternating plasticity

*Benchmarks R0026 , R0030 , R0049 (subset of above), R0072 (subset of above), R0080 (subset of above).*

## geometric non-linearity:

**Large displacements in truss elements:**

- the unrotated element (1)
- limit point behaviour with one variable (2)
- hardening with one variable (3)
- bifurcation problem (4)
- limit point with two variables (5)
- hardening with two variables (6)
- snap back (7)

*Benchmarks P03 , R0065 (subset of above), R0072 (subset of above),* *R0080 (subset of above).*

**Large displacements and rotations in thin 2D beams and axisymmetric shells:**

- rigid body rotation (NLGB1)
- straight cantilever with end moment (NLGB2)
- curved cantilever with end moment (NLGB3)
- straight cantilever with transverse free end point load (NLGB4)
- straight cantilever with axial free end point load (NLGB5)
- symmetric shallow pinned arch buckling (NLGB6)
- unsymmetric deep arch buckling (NLGB7)
- Lee’s frame buckling problem (NLGB8)
- built-in circular plate with uniformly distributed load (NLGS1)
- annular plate with inside edge load (NLGS2)
- truncated cone with outside edge load (NLGS3)
- built-in spherical cap with central point load (NLGS4)

*Benchmarks P10 , R0010, R0065 (subset of above), R0072 (subset of above),* *R0080 (subset of above).*

**Assembly tests for 3D beams and shells:**

- elastic large deflection response of a Z-shaped cantilever beam under end load (3DNLG-1)
- elastic large deflection response of a pear shaped cylinder under end shortening (3DNLG-2)
- elastic lateral buckling of a right angle frame under in-plane end moments (3DNLG-3)
- lateral-torsional buckling of an elastic cantilever subjected to a transverse follower force (3DNLG-4)
- large deflection of a curved elastic cantilever under transverse end load (3DNLG-5)
- buckling of a flat plate with an initial imperfection when subjected to in-plane shear (3DNLG-6)
- large deflection elastic response of a hinged spherical shell with uniform pressure loading (3DNLG-7)
- collapse of a straight pipe under pure bending (3DNLG-8)
- large elastic deflection of a pinched hemispherical shell (3DNLG-8)
- elasto-plastic behaviour of a stiffened cylindrical shell panel under compressive end load (3DNLG-10)

*Benchmarks R0024 , R0029 , R0065 (subset of above), R0080 (subset of above).*

**A survey of possible benchmarks for 3D beams:**

- elastic lateral-torsional buckling of a cantilever (B1)
- elastic lateral buckling of a right angle frame under in-plane end moments (B2)
- lateral buckling of an I beam cantilever subjected to transverse force (B3)
- elastic circular ring subjected to non-uniform pressure (B4)
- elastic large deflection of a cantilever under follower transverse force (B5)
- elasto-plastic large deflection of a cantilever under transverse end load (B6)
- elastic large deflection of pinned deep arch subjected to central load (B7)
- elastic large deflection of cantilever 45
^{0}bend subject to end load (B8) - elastic large deflection analysis of a prestressed cable (B9)
- elastic large deflection behaviour of a cable under gravity load (B10)
- torsional buckling of end loaded cantilever (B11)
- elastic large deflection behaviour of Lee’s frame (B12)
- elastic large deflection of curved beam under non-conservative in-plane load (B13)
- clamped-hinged deep circular arch subjected to a point load (B14)
- large deflection of a diamond shaped frame (B15)
- planar square frame loaded at the midpoints on opposite sides (B16)
- cantilever under end moment (B17)
- large deflection of a cantilever right angle frame under end load (B18)
- elasto-plastic collapse of a 90
^{0}pipe bend (B19) - elasto-plastic collapse of a thin walled elbow under in-plane loading and internal pressure (B20)
- uniform collapse of a curved pipe under pure bending and internal pressure (B21)
- elasto-plastic large deflection response of a Z-shaped cantilever beam (B22)
- elasto-plastic analysis of I beams in torsion (B23)
- finite rigid body rotation of a curved arch (B24)
- collapse analysis of elasto-perfectly plastic triangular truss structure (B25)
- elasto-plastic large deformation of a clamped beam under point load (B26)
- plastic collapse of a two bay two story frame (B27)

*Benchmark R0009 .*

**A survey of possible benchmarks for 3D shells:**

- elastic large deflection response of a centrally loaded spherical shell (S1)
- large deflection elasto-plastic analysis of a cylindrical shell roof (S2)
- thermal buckling of a simply supported rectangular plate (S3)
- elastic post-buckling behaviour of a square plate subjected to shear (S4)
- elastic large deflection of a cantilever plate with end moment (S5)
- elasto-plastic buckling of a clamped imperfect spherical cap subjected to uniform pressure (S6)
- elastic large deflection analysis of a pinched hemispherical shell (S7)
- elastic large deflection of a hyperbolic paraboloid under concentrated moment at corners (S8)
- elastic large deflection of a cylindrical shell under point load (S9)
- elastic large deflection of a curved beam (S10)
- finite rigid body rotation of a singly curved shell (S11)
- elastic large deflection response of a pear shaped cylinder under end shortening (S12)
- buckling and post-buckling response of axially compressed perfect and imperfect elliptical cylinders (S13)
- elastic large deflection response of a cylindrical shell with two diametrically opposite cutouts (S14)
- elastic large deflection response of a clamped square plate under uniform pressure (S15)
- post buckling analysis of cylindrical shell under axial compression (S16)
- post buckling behaviour of cylindrical panel under central point load (S17)
- expansion bellows under axial end load (S18)
- elasto-plastic response of a spherical cap under a central load (S19)
- large deflection of a clamped cylindrical panel subjected to uniform pressure (S20)
- snap through of hinged cylindrical shell (S21)
- post buckling of a simply supported circular plate loaded in radial edge compression (S22)
- elastic lateral buckling and large deflection response of a narrow cantilever beam (S23)
- elasto-plastic lateral buckling of a narrow cantilever beam (S24)
- large deflection elasto-plastic response of a circular plate (S25)

*Benchmark R0009 .*

**A survey of possible benchmarks for 3D branched shells:**

- elasto-plastic behaviour of a stiffened cylindrical shell panel under compressive end load (BS1)
- post buckling of stiffened panel under axial compression (BS2)
- elastic buckling of T-ring stiffened cylinders under external pressure (BS3)
- buckling of spherical shells ring supported at the edges (BS4)
- elastic buckling of I-ring stiffened cylinders under external pressure (BS5)
- 3D finite rigid body rotation of stiffened shell connected to a hinged bar (BS6)
- non-symmetric elasto-plastic bifurcation buckling of torispherical head with axisymmetric nozzle (BS7)
- elasto-plastic large deflection response of a multi-mitre pipe bend (BS8)

*Benchmark R0009 .*

## contact analysis:

**2D multi-element plane strain tests:**

- contact patch test
- rigid punch on deformable foundation
- Hertzian contact
- sliding wedge with linear springs
- bending of a plate over a stiff cylinder
- sliding and rolling of a ring on a rigid surface

**2D multi-element plane stress tests:**

- cantilevered beam loaded against a rigid curvilinear surface

**2D solid multi-element tests:**

- two contacting rings
- buckling of a curved column with self-contact

**2D axisymmetric multi-element tests:**

- interference between two cylinders

*Benchmark R0081 .*

**Multi-element 2D/3D tests of a range of physical features:**

- 2D contact of cylindrical roller
- 3D punch with rounded edges
- 3D sheet metal forming
- 3D loaded pin
- 3D steel roller on rubber

*Benchmark R0094 .*