ECIV 720 A Advanced Structural Mechanics and Analysis Non-Linear Problems in Solid and Structural Mechanics Special Topics

Introduction Nonlinear Behavior: Response is not directly proportional to the action that produces it. P 

Introduction Recall Assumptions Small Deformations Linear Elastic Behavior

Introduction Linear Behavior

Introduction A. Small Displacements Inegrations over undeformed volume Strain-displacement matrix does not depend on

Introduction B. Linear Elastic Material Matrix [E] does not depend on

Introduction C. Boundary Conditions do not change (Implied Assumption) Constraints do not depend on

Introduction If any of the assumptions is NOT satisfied NONLINEARITIES Material Assumption B not satisfied Geometric Assumption A & or C not satisfied

Classification of Nonlinear Analysis Small Displacements, small rotations Nonlinear stress-strain relation

Classification of Nonlinear Analysis Large Displacements, large rotations and small strains – Linear or nonlinear material behavior

Classification of Nonlinear Analysis Large Displacements, large rotations and large strains – Linear or nonlinear material behavior

Classification of Nonlinear Analysis Change in Boundary Condition

Classification of Nonlinear Analysis

Nonlinear Analysis Cannot immediately solve for {d} Iterative Process Required to obtain {d} so that equilibrium is satisfied

Solution Methods

Newton-Raphson

Newton Raphson With initial conditions

Modified Newton-Raphson

SPECIAL TOPICS Boundary Conditions Elimination Approach Penalty Approach Special Type Elements

Boundary Conditions – Elimination Approach Consider B C u1u1 u2u2 u3u3 u4u4 uiui u i+1 u n-1 unun P4P4 PiPi PnPn u 1 =a Singular, No BC Applied

Boundary Conditions – Elimination Approach Boundary Conditions u 1 =a

Boundary Conditions – Elimination Approach Consequently, Equilibrium requires that Since u 1 =a known, DOF 1 is eliminated from

Boundary Conditions – Elimination Approach ……… K ff u f =P f + K fs u s

Boundary Conditions – Elimination Approach k ii k ij k ik k il k im uiui ujuj ukuk ulul k ji k jj k jk k jl k jm k ki k kj k kk k kl k km k li k lj k lk k ll k lm k li k lj k lk k ll k lm umum = PiPi PjPj PkPk PlPl PmPm K ff K fs K sf K ss uf uf Pf Pf usus PsPs

Boundary Conditions – Elimination Approach k ii k ij k ik k il k im uiui ujuj ukuk ulul k ji k jj k jk k jl k jm k ki k kj k kk k kl k km k li k lj k lk k ll k lm k li k lj k lk k ll k lm umum = PiPi PjPj PkPk PlPl PmPm K ff K fs uf uf Pf Pf usus K ff u f + K fs u s =P f K sf K ss PsPs K sf u f + K ss u s =P s u f = K ff (P f + K fs u s )

Boundary Conditions Penalty Approach u1u1 u2u2 u3u3 u4u4 uiui u i+1 u n-1 unun P4P4 PiPi PnPn u1u1 u2u2 u3u3 u4u4 uiui u i+1 u n-1 unun P4P4 PiPi PnPn k=C large stiffness Boundary Conditions u 1 =a

Boundary Conditions Penalty Approach u1u1 u2u2 u3u3 u4u4 uiui u i+1 u n-1 unun P4P4 PiPi PnPn k=C large stiffness Consequently, for Equilibrium Contributes to 

Boundary Conditions Penalty Approach The only modifications Support Reaction is the force in the spring

Choice of C Rule of Thumb Error is always introduced and it depends on C Penalty approach is easy to implement

Changing Directions of Restraints x,u y,v 

Changing Directions of Restraints e.g. for truss

Changing Directions of Restraints

Introduce Transformation In stiffness matrix…

Connecting Dissimilar Elements Simple Cases

a b L 

Hinge Beam

Connecting Dissimilar Elements Simple Cases Beam Stresses are not accurately computed

Connecting Dissimilar Elements Eccentric Stiffeners

Use Eccentric Stiffeners Connecting Dissimilar Elements Eccentric Stiffeners Slave Master

Connecting Dissimilar Elements Eccentric Stiffeners bb

bb 3,4 Slave 1,2 Master

Connecting Dissimilar Elements Eccentric Stiffeners bb The assembly displays the correct stiffness in states of pure stretching and pure bending The assembly is too flexible when curvature varies – Use finer mesh

Connecting Dissimilar Elements Rigid Elements Rigid element is of any shape and size Generalization of Eccentric Stiffeners – Multipoint Constraints Use it to enforce a relation among two or more dof

Connecting Dissimilar Elements Rigid Elements e.g a b Perfectly Rigid Rigid Body Motion described by u 1, v 1, u 2

Connecting Dissimilar Elements Rigid Elements

Elastic Foundations Strain Energy RECALL

Elastic Foundations RECALL

Elastic Foundations Additional stiffness Due to Elastic Support RECALL

Elastic Foundations + RECALL

Elastic Foundations – General Cases Soil x y z Foundation Plate/Shell/Solid of any size/shape/order Winkler Foundation

Elastic Foundations – General Cases Winkler Foundation Stiffness Matrix s is the foundation modulus H are the Shape functions of the “attached element”

Winkler Foundations Resists displacements normal to surface only Deflects only where load is applied Adequate for many problems

Other Foundations Resists displacements normal to surface only They entire foundation surface deflects More complicated by far than Winkler Yields full matrices

Elastic Foundations – General Cases Soil x y z Infinite

Infinite Elements

Use Shape Functions that force the field variable to approach the far-field value at infinity but retain finite size of element Use conventional Shape Functions for field variable Use shape functions for geometry that place one boundary at infinity or

Shape functions for infinite geometry Element in Physical Space Mapped Element Reasonable approximations

Shape functions for infinite geometry Node 3 need not be explicitly present