Basic Geometric Nonlinearities

Basic Geometric Nonlinearities
Chapter Five Basic Geometric Nonlinearities

5. Basic geometric nonlinearities
What is geometrically nonlinear behavior? A structure’s overall stiffness depends on the orientation and stiffness of its individual component parts (elements). As an element’s nodes displace, the element’s contribution to overall stiffness can change in a number of ways. Stiffness changes due to geometric deformations are categorized as geometric nonlinearities. ANSYS features three different kinds of geometric nonlinearities: Large strain. Large deflection (large rotation). Stress stiffening. October 15, 2001 Inventory #001565 5-2

… Basic geometric nonlinearities
In this chapter, we will introduce you to the basics of geometric nonlinearities via the following topics: A. Overview B. Three kinds of geometric nonlinearities C. Consistent tangent matrix D. Building the model E. Obtaining the solution F. Postprocessing The purpose is to give you an understanding of how to account for geometric nonlinear effects in your analysis. October 15, 2001 Inventory #001565 5-3

Basic geometric nonlinearities A. Overview
Consider three phenomena associated with geometric nonlinearities: If an element’s shape changes (area, thickness, etc.), its individual element stiffness will change. If an element’s orientation changes (rotation), the transformation of its local stiffness into global components will change. X Y October 15, 2001 Inventory #001565 5-4

Basic geometric nonlinearities … Overview
If an element’s strains produce a significant in-plane stress state (membrane stresses), the out-of-plane stiffness can be significantly affected. F Y F X uy As the vertical deflection increases (UY), significant membrane stresses (SX) lead to a stiffening response. October 15, 2001 Inventory #001565 5-5

Large strain behavior encompasses all three of these phenomena. Large deflection behavior encompasses only the last two. Stress stiffening behavior encompasses only the third phenomenon. Thus, stress stiffening theory is a subset of large deflection theory, which is in turn a subset of large strain theory. Large strain Large rotation Stress stiffening October 15, 2001 Inventory #001565 5-6

Geometric nonlinearities will be included in your analysis if: You specify a Large Displacement analysis, and The element types in your model support geometric nonlinear effects. You will find this information in the Special Features list in the element description. For example, note that SHELL63 supports stress stiffening and large deflection, but not large strain. October 15, 2001 Inventory #001565 5-7

By contrast, SHELL181 supports all three types of geometric nonlinearities: stress stiffening, large deflection, and large strain. Be sure to choose an element type that supports the necessary nonlinear geometric behavior! October 15, 2001 Inventory #001565 5-8

Large strain analyses benefit from an improved mathematical definition of strain. In a physical sense, strains are always defined as a normalized measure of the deformations of a body. However, there are many possible mathematical definitions of strain. Although a mathematical definition of strain can be somewhat arbitrary, it must meet certain requirements: Strain should be zero when there is no deformation (i.e., pure rigid-body motion, including rotations). Strain should be non-zero when there is deformation. Strain should relate to stress through a material stress-strain relationship. October 15, 2001 Inventory #001565 5-9

In a nonlinear large strain analysis the stress measures employed must be conjugate to the strain measures. Conjugacy means that the strain energy (a scalar quantity which is a function of the product of stress and strain) is independent of the stress and strain measures selected. Stress Strain energy value must be the same for any conjugate measure of stress and strain Strain October 15, 2001 Inventory #001565 5-10

The ANSYS program uses three measures of strain and stress: Engineering strain and engineering stress. Logarithmic strain and true stress. Green-Lagrange strain and 2nd Piola-Kirchoff stress. The program automatically chooses which measure to use, depending on the type of analysis and element used. We will examine these various stress and strain definitions through a simple one-dimensional example. F October 15, 2001 Inventory #001565 5-11

Engineering strain is a small strain measure, which is computed using the original geometry: The engineering strain measure is a linear measure since it depends on the original geometry, (i.e. length) which is known beforehand. It is limited to small rotations of the material because a moderate rigid body rotation will result in non-zero strains. ANSYS uses it in small-displacement analyses. October 15, 2001 Inventory #001565 5-12

Engineering stress (s), is the conjugate stress measure to engineering strain (e). It uses the current force F and the original area A0 in its computation. October 15, 2001 Inventory #001565 5-13

In large-displacement analyses of elements that support large deflection but not large strain, the program uses a corotational approach, which extracts rigid-body rotations from the total displacement. This eliminates non-zero strains due to large rotations, leaving only the small-strain deformational component. Thus, large-deflection, small-strain analyses also use engineering strain (e) and engineering stress (s). October 15, 2001 Inventory #001565 5-14

Logarithmic strain is a large strain measure, which is computed as This measure is a nonlinear strain measure since it is a nonlinear function of the unknown final length l . It is also referred to as the log strain. The 3-D equivalent of the log strain is the Hencky strain. ANSYS uses it in large-displacement analyses, for most elements that support large strain. October 15, 2001 Inventory #001565 5-15

True stress (t) is the conjugate 1-D stress measure to the log strain (el ), which is computed by dividing the force F by the current (or deformed) area A: This measure is also commonly referred to as the Cauchy stress. October 15, 2001 Inventory #001565 5-16

Green-Lagrange strain is another large strain measure, which is computed in 1-D as This measure is nonlinear because it depends on the square of the updated length l , which is an unknown. A computational advantage of this strain measure, over the log or Hencky strain, is that it automatically accommodates arbitrarily large rotations in large strain problems. ANSYS uses it in large displacement analyses for some elements that support large strain. October 15, 2001 Inventory #001565 5-17

The conjugate stress measure for the Green-Lagrange strain (eG ), is the Second Piola-Kirchoff stress (S). It is computed in 1-D by It should be noted that this stress has little physical meaning. For output purposes, ANSYS always reports stresses for options that use this measure as Cauchy or True stresses (t). October 15, 2001 Inventory #001565 5-18

Which strain and stress measure does ANSYS use? You must be aware of which measure the ANSYS program uses for input and output, in order to be able to enter data and interpret results correctly. For a given element type and analysis option (large or small displacement), the program chooses the strain measure. Other than your choice of element type and analysis option, you have no control over what strain measure the program uses. October 15, 2001 Inventory #001565 5-19

In general: ANSYS uses engineering stress and strain for small displacement analyses, or for large displacement with elements that support only large deflection. ANSYS uses log strain and true stress for large deflection with most elements that support large strain. Exceptions arise for Mooney-Rivlin hyperelasticity, as noted below. October 15, 2001 Inventory #001565 5-20

You might have to convert data from one measure to another. To use correct measure for input data. To compare ANSYS results to known response data, using a consistent measure. For uniaxial stress-strain data, engineering stress versus engineering strain can be converted to true stress versus log strain by el = ln (1 + e) t = s (1 + e) Note that this stress conversion assumes that the material undergoing the large strains is incompressible or nearly incompressible. This assumption is valid for large plastic strains or hyperelastic materials. October 15, 2001 Inventory #001565 5-21

To convert from true to engineering, use the inverse relationships: To convert from engineering strain to Green-Lagrange: The inverse relationship is October 15, 2001 Inventory #001565 5-22

Basic geometric nonlinearities A. Overview ...Workshop
Please refer to your Workshop Supplement for instructions on: W9. Basic Geometric Nonlinearities - Strain Measure Study October 15, 2001 Inventory #001565 5-23

Basic geometric nonlinearities B
Basic geometric nonlinearities B. Three kinds of geometric nonlinearities Large strain: When the strains in a material become “large” (typically more than a few percent), the changing geometry due to this deformation can no longer be neglected. “Large” is problem-dependent. Large strain analyses assume that strain is no longer infinitesimal, but rather it is finite or large. Large strain theory accounts for shape changes (such as thickness, area, etc.) and any large rotations. It also inherently accounts for stress-stiffening effects. Large strain Large rotation Stress stiffening Large strain behavior includes all three phenomena associated with geometric nonlinearities. October 15, 2001 Inventory #001565 5-24

Basic geometric nonlinearities ... Three kinds of geometric nonlinearities
Large deflection: When the rotation of an element becomes “large” (typically more than one or two degrees), the transformation of the element’s local stiffness into global components will change significantly. “Large” is problem-dependent. The terms large deflection and large rotation are used interchangeably in ANSYS. Large deflection theory accounts for large rotations, but it assumes that strains are small. It also inherently accounts for stress-stiffening effects. Large strain Large deflection theory is a subset of large strain theory. Large rotation Stress stiffening October 15, 2001 Inventory #001565 5-25

The stress state in a part can affect the stiffness of the part.
Basic geometric nonlinearities ... Three kinds of geometric nonlinearities Stress stiffening: The stress state in a part can affect the stiffness of the part. Lateral stiffness of a cable increases with increasing tension. Lateral stiffness of a column decreases with increasing compression (leading eventually to complete loss of stiffness – i.e., buckling). When stress stiffening is activated, the program calculates a stress stiffening matrix and adds it to the original stiffness matrix to include this effect. Stress stiffness matrix is a function only of stress and geometry. Stress stiffness matrix helps the tangent stiffness matrix to be more consistent (generally improves convergence). Stress stiffening theory is a subset of large deflection theory. Large strain Large rotation Stress stiffening October 15, 2001 Inventory #001565 5-26

Basic geometric nonlinearities … Three kinds of geometric nonlinearities
Thus, an element’s available special features determine its behavior in a large displacement analysis: Large strain elements account for shape changes and large rotations, and also inherently include stress-stiffening effects. You cannot separate large-strain effects from large-rotation effects in such elements. Large deflection elements account for large rotations, and also inherently include stress-stiffening effects. Many of the older spar, beam, and shell elements have large deflection capability, but not large strain (BEAM4, SHELL63, etc.). Elements that list stress stiffening as a special feature will by default include a stress stiffening term in the stiffness matrix. This generally helps improve the rate of convergence. October 15, 2001 Inventory #001565 5-27

Basic geometric nonlinearities C. Consistent tangent stiffness matrix
The program can use almost any arbitrary form of the stiffness matrix, and still obtain an accurate solution when converged. Initial stiffness Secant stiffness Tangent stiffness F u Accuracy is determined by the convergence tolerance, not by the form of [K]. F u F u October 15, 2001 Inventory #001565 5-28

Basic geometric nonlinearities … Consistent tangent stiffness matrix
Although accuracy is not affected by the form of the stiffness matrix, the rate of convergence can be strongly affected. The fully consistent tangent stiffness matrix usually gives the best rate of convergence. A fully consistent tangent stiffness matrix [Kenl] is made up of four components: [Kenl] = [Keinc] + [Kes] + [Keu] + [Kea] [Keinc] is the main tangent matrix. [Kes] is the stress-stiffening matrix. [Keu] is the initial displacement-rotation matrix, which includes the effect of changing geometry in the stiffness relation. [Kea] is the pressure load stiffness matrix, which includes the effect of changing pressure load orientation in the stiffness relation. October 15, 2001 Inventory #001565 5-29

Basic geometric nonlinearities … Consistent tangent stiffness matrix
The first three components are included automatically by default for large displacement analyses for most elements. Use Solution Controls to control how the fourth component (pressure load stiffness) is included: The default (“Program Chosen”) includes the pressure load stiffness for elements SURF153, SURF154, SHELL181, PLANE182, PLANE183, SOLID185, SOLID186, SOLID187, BEAM188, and BEAM189 Use the non-default settings only if you encounter convergence difficulties. For elements that don’t directly support the pressure load stiffness you can include the calculations by using SURF154 on pressure-loaded surfaces. October 15, 2001 Inventory #001565 5-30

Basic geometric nonlinearities D. Building the model
In the remainder of this chapter, we will describe various procedural tips that are often useful when building, running, and postprocessing a large-displacement model. Let’s first examine some tips for building the model. Element selection. Meshing. Coupling and constraint equations. October 15, 2001 Inventory #001565 5-31

Basic geometric nonlinearities … Building the model
Use appropriate element types. Not all elements support geometric nonlinearities! Some have no geometric nonlinear capability. For example, CONTAC52 and PRETS179. Others have only limited geometric nonlinear capability. For example, VISCO88 does not support large strain or large rotation, and SHELL63 does not support large strain. You must check the Special Features list in the element description for each element type that you plan to use. October 15, 2001 Inventory #001565 5-32

Basic geometric nonlinearities … Building the Model
… use appropriate element types: In addition, for models that experience plasticity, creep, or hyperelasticity, the material becomes nearly incompressible at large strains. Incompressibility can cause convergence difficulties due to shear locking or volumetric locking. You can overcome these difficulties through careful choice of element type and element option. This topic is covered in detail in the Advanced Structural Nonlinearities training course. October 15, 2001 Inventory #001565 5-33

Anticipate mesh distortion. ANSYS shape-checking examines the quality of the mesh prior to the first iteration. In a large strain analysis, the mesh can become significantly distorted after the first iteration. Poor element shapes are undesirable in every iteration. Prevent poor shapes from developing by modifying the original mesh. October 15, 2001 Inventory #001565 5-34

… anticipate mesh distortion: For instance, grade the initial mesh in anticipation of subsequent distortion in the neck-down region of this tensile specimen. Maintains reasonable aspect ratios in the deformed mesh. October 15, 2001 Inventory #001565 5-35

… anticipate mesh distortion: Replace quad elements with two triangles to prevent 180° corner angles. Undeformed Mesh Deformed Mesh Large interior angle develops. Corner elements maintain better shapes as triangles. October 15, 2001 Inventory #001565 5-36

Use an adequate mesh density. Of course, mesh density must be adequate to prevent mesh- discretization error. (As evidenced by element contour plot discontinuities.) In addition, shell and beam element meshes must be adequate to capture the bending response. No one element should experience more than 30° of flexure. 30° max. October 15, 2001 Inventory #001565 5-37

Generally avoid coupling and constraint equations in large displacement analyses. Nodal coordinate systems are not updated to account for large rotations. Coupling and constraint equations always act in the original directions. For example, in a linear analysis, a pinned joint is often modeled using coupling. However, in a large deformation analysis, the axis of rotation’s orientation needs to be updated. To model a 3D pinned joint, use the nonlinear COMBIN7 element instead of coupling in a nonlinear analysis. October 15, 2001 Inventory #001565 5-38

… generally avoid coupling and constraint equations: Constraint equations that link rotational and displacement DOFs are based on linear, small-deflection theory. The constraint equation that would transfer action between ROTZ at node 2 and UY at nodes 1 and 3 has this form: 0 = UY3 - UY1 - 10*ROTZ2 Obviously, this equation is valid only for small rotations. October 15, 2001 Inventory #001565 5-39

… generally avoid coupling and constraint equations: However, recognize that there are some situations in which coupling or constraint equations can be valid in a nonlinear analysis. For example: You can couple together constrained DOFs at a rigid boundary. You can model 2D pinned joints with coupling. Constraint equations can be valid for large strain, small rotation response. But … think carefully before using coupling or constraint equations! October 15, 2001 Inventory #001565 5-40

Basic geometric nonlinearities E. Obtaining the solution
Let’s next examine some tips for obtaining the solution. When to use large displacements. Loads and boundary conditions. Step size and convergence. October 15, 2001 Inventory #001565 5-41

Basic geometric nonlinearities … Obtaining the solution
When should you choose Large Displacement? Large displacement effects improve the accuracy of your solution, but at the expense of running an iterative nonlinear solution. If you are 100% certain that large displacement effects are insignificant, then choose a small displacement analysis for best solution efficiency. However, recall from your first workshop that large displacement effects can be surprisingly significant! If you have any question about it, always use large displacement. If you have other nonlinearities in your model, they will require an iterative solution anyway. The added expense of large displacement in such cases is minimal. When in doubt, always use Large Displacement! October 15, 2001 Inventory #001565 5-42

Loads and BCs. Consider what happens to the loads as a structure experiences large deflections: In many instances, loads will maintain constant direction. In other cases, loads will change direction, “following” the elements as they undergo large rotations. ANSYS can model both situations, depending on the type of load applied. Additionally, in large strain analyses, pressures are applied to updated areas. Therefore, the total load exerted by a pressure will change as the pressure surface stretches or shrinks. October 15, 2001 Inventory #001565 5-43

… loads and BCs: Direction Before Deflection Direction After Deflection Load Acceleration (constant direction) Nodal Force (constant direction) Element Pressure (follower; always normal to surface) October 15, 2001 Inventory #001565 5-44

… loads and BCs: Be sure to specify accurate boundary conditions. Avoid over-constraining the deformation at the boundaries: Generally avoid single-node constraints and forces in large-strain analyses: F F October 15, 2001 Inventory #001565 5-45

… loads and BCs: Some analysts become confused when trying to specify nonzero rotations to a solid model. Some analysts believe that by rotating nodal coordinate systems into a cylindrical system, they can specify rotations by specifying Y () -direction displacements. However, nodal coordinate systems are always Cartesian systems. A nodal system that has been rotated into a cylindrical system is still a Cartesian system. It has simply been reoriented, such that nodal X is radial, and nodal Y is tangential (not circumferential). Nodal coordinate systems do not update in a large-displacement analysis. October 15, 2001 Inventory #001565 5-46

… loads and BCs: A modeling trick, such as a spiderweb of beams, is usually required to apply nonzero rotations accurately. No-separation contact, using surface-to-surface contact elements, can also be useful. A specified nonzero rotation can be applied using an artificial spiderweb of beam elements. October 15, 2001 Inventory #001565 5-47

Stepsize and convergence. The time step size should be small enough that no element experiences more than 10° rotation in any one substep. You can easily check for this by animating the displaced shape. PlotCtrls > Animate > Over Results Using too large a time step size can sometimes lead to elements turning inside out. If this happens, reduce the time step size. ANSYS performs an automatic Jacobian check that will cause the solution to bisect and restart for most such cases. October 15, 2001 Inventory #001565 5-48

… stepsize and convergence: If, after repeated bisection, your model fails to converge at full load, the cause may be true physical instability (buckling or full plastic section). Plot the load-deflection curve to find out if the tangent stiffness is approaching zero. Probable instability indicated by zero slope October 15, 2001 Inventory #001565 5-49

Basic geometric nonlinearities F. Postprocessing
When postprocessing, realize that: Calculated nodal displacements are reported in the original directions, because nodal coordinate system orientations are not updated for large deflections. Stress and strain components rotate with most elements, because most element coordinate systems follow the element. Exceptions – hyperelastic (HYPER56, 58, 74, 84, 86, 158) maintain original element coordinate system orientations. SX UX UX SX October 15, 2001 Inventory #001565 5-50

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