Non-linear & Buckling analysis with GSA

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Non-linear analysis with GSA

Non-linear Basics

Linear analysis Based on the assumptions of Results Implication Displacement - Strain relationship is linear (valid if displacements are small) Stress - strain relationship is linear (valid if strains are small) Equilibrium conditions are established at the original (non-deformed) geometry Results Equilibrium equations are linear Load – displacement relationships are linear Load – internal forces relationships are linear Implication Stresses, internal forces etc are obtained based on non-deformed geometry. If deformation or strain are large, results are not accurate

Non-linear analysis If either of the following conditions applies Displacement – strain relationship is not linear (large displacement problem - geometrical non-linearity) Stress – strain relationship is not linear Material is elastic-plastic (material non-linear problem) Material is still elastic, but strain is large (large strain problem, such as rubber) Results Equilibrium equations are no longer linear Load – displacement relationships are not linear Load – internal forces relationships are not linear Implication Equilibrium condition is established on deformed configuration Structural responses may be path dependent Analysis results are not combinable

Geometric non-linear

Geometric non-linear analysis Ignore deformation when establishing equilibrium P Q h M F V So: F = P V = Q M = Qh

Geometric non-linear analysis Equilibrium is established at the deformed configuration P Q d h M F V q So: F = Pcosq – Qsinq V = Qcosq + Psinq M = Qh + Pd

Example Geometric non-linear

Non-linear elements

Non-linear Elements Strut Tie Cable Fabric Compression-only 1D Tension-only 1D Cable Tension-only chain 1D Fabric Tension-only 2D

Example Non-linear elements

Material non-linear

Material non-linear Engineering stress (e.g. bar in tension) s = F/A0 where A0 – the original section area e s

Material non-linear Simplified s – e curves in GSA e s

Example Material non-linear

Buckling

Buckling analysis Linear buckling analysis Eigen value or Modal buckling Elastic non-linear buckling analysis Include effects of nodal displacement Elasto-plastic non-linear buckling analysis Add material plasticity

Linear buckling analyses (2 step analysis) Results: The structure is assumed in equilibrium at any deformed positions given by the mode shape Search for load factor l that factors the loads to make the element stiffness K = 0 (Eigen value extraction) P P lP lP V lV V lV lP lV K(p) < Ke Ke Step 1 Step 2 Note: 1. Shear force V has little effect on buckling load capacity 2. Linear buckling analysis does not give actual deflection of the structure for the given loads

Elastic non-Linear buckling analysis (multi-step analysis) d 1 d i d n aoP a1P aiP aoV anP a1V Vr1 aiV Vri anV Vrn Pr1 Pri Fo = aoP Ko = f(EI,Fo) < Ke Prn F1 = f(a1P, a1V, d) K1 = f(EI,F1) Fi = f(aiP, aiV, d) Ki = f(EI,Fi) Prn  anP Vrn  anV . Note: 1. d is the actual deflection of the column under the imposed loads 2. If material plasticity is taken into account, this is also the elastic-plastic non-linear buckling analysis

Load - deflection curves from the three buckling analyses Displacement (d) Linear buckling analysis (no unique relationship between P and d ) Elastic non-linear buckling analysis (may approach the linear buckling load capacity) Elastic-plastic non-linear buckling analysis

Examples Buckling

Non-linear analysis cases and combinations

Cases and Combinations Load Cases Loads are assigned to a Load Case Referenced as L1, etc Analysis Cases Analyse single or multiple load cases for results Load factors can be applied: 1.35L1+1.5L2 Referenced as A1, etc Combination cases Combine multiple linear analysis cases Load factors can be applied: 1.35A1+1.5A2 Envelope Analysis and Combination cases

Cases and Combinations Linear Analysis Load cases Analysis Cases Combination Cases L1 (e.g. Dead Load) A1 = L1 C1 = 1.35A1 + 1.5A2 L2 (e.g. Live Load) A2 = L2 C2 = 1.35A1 + 1.5A3 L3 (e.g. Wind Load) A3 = L3 C3 = 1.35A1 + 1.5A2 + 0.75A3 C4 = C1 or C2 or C3 Non-Linear Analysis Load cases Analysis Cases Combination Cases L1 (e.g. Dead Load) A1 = 1.35L1 + 1.5L2 C1 = A1 or A2 or A3 L2 (e.g. Live Load) A2 = 1.35L1 + 1.5L3 L3 (e.g. Wind Load) A3 = 1.35L1 + 1.5L2 + 0.75L3

Dynamic Relaxation

Dynamic relaxation solution method Basic idea of dynamic relaxation Most unbalanced position: zero speed, acceleration at peak, zero kinetic energy Equilibrium position: maximum speed, zero acceleration, Kinetic energy at peak vibration

Damping used in GsRelax analysis 1 Viscous Damping Apply resistance against the vibration, the resistance is proportional to the velocity of the node 2 Kinetic Damping Reposition nodes once the system kinetic energy exceeds peak value (pass equilibrium position) Equilibrium position: maximum speed peak kinetic energy zero acceleration Most unbalanced position: zero speed zero kinetic energy peak acceleration

Effect of viscous damping 24/04/2017 Effect of viscous damping no damping Deflection Time equilibrium deflection damping just smaller than critical damping larger than critical

Example projects

'Water Cube' bubbles A recent job: the Beijing Olympics Water Cube

MOMA – Liquid Skies This is a GSA job from the New York Museum of Modern Art’s summer exhibition. it might not be a standard fabric structure, but you can see all the elements Struts Edge cables Double curved surfaces

Khalifa Stadium Note the tension mesh and the slender compression arch, which also needs to be part of the form finding The arch is also prevented from buckling by the cable net It looks so simple, but this is a very complex example of form finding

Melbourne Cricket Ground

Singapore Flyer

United States Air Force Memorial

National Assembly for Wales Awards RIBA Awards - Wales Royal Institute of British Architects RIBA Winner Structural Steel Design Award Sustainable Regeneration Award - Winner Building Exchange BEX Awards Innovative Regeneration Award - Runner-up Building Exchange BEX Awards

Royal Ontario Museum

Kurilpa Bridge

Infinity Footbridge In 2009, Expedition won the Institution of Structural Engineer’s Supreme award for the Infinity Footbridge in Stockton-on-Tees With its primary span of 120m and secondary of 60m, it is the biggest footbridge in the country Awards IStructE Supreme Award for Structural Engineering Excellence 2009 IStructE Award for Pedestrian Bridges 2009 The judges praised it as "a bold and daring design which demonstrates what can be done with artistic flair, technical excellence, complex analysis and excellent engineering teamwork. The bridge will be one of those iconic designs to be talked about for years." ICE Robert Stephenson Award 2009 The judges described it as "imaginatively engineered" acting as "a catalyst for regeneration and a landmark much appreciated by the community." North East Constructing Excellence Awards 'Project of the Year' 2009 Concrete Society Civil Engineering Award 2009 International Green Apple Gold Award for Civic Pride: New Build 2009

Any Questions? GSA basic concepts Now does anyone have any questions? Just write them in the question box in your right hand toolbar. Let me know if there’s some part of the software that you’d like me to go over. Ok well lets see what questions we have. Sorry we don’t have time to answer all your questions right now but we will be answer all your questions by email very soon. And if you didn’t get a chance to put your questions in I’ll send out a survey which you can put any additional questions in.

Future Webinars Soil Movement and Slope Stability 14th March What’s New in GSA 8.6 21st March