Chapter Two
Element Technology
Element Technology
Chapter Overview
Training Manual
– The 18x elements include robust element formulations and an
extensive library of constitutive laws.
• For 18x elements, material and element technologies have
been separated. This provides a smaller element library
which can be used as a “toolkit” to handle different situations
and various constitutive models.
– SHELL181 and BEAM188/189 also include advanced pre- and
post-processing tools which are specific to beams and shells.
Advanced Structural Nonlinearities 6.0
• The majority of the discussion in this chapter will focus on
the 18x series of elements. Over the past few releases of
ANSYS, the 18x elements have become the elements of
choice for nonlinear applications.
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Element Technology
... Chapter Overview
Training Manual
– Fully-integrated, conventional displacement-based continuum
elements underpredict displacements under certain situations
• This is called mesh locking
– As a result, we have different element formulations to deal with
these problems, based on:
• Bulk- or bending-dominated problems (Structural behavior)
• Elasticity, Plasticity, or Hyperelasticity (Material behavior)
• Efficiency in nonlinear solutions
– Besides continuum elements, ANSYS has a large library of shell
and beam elements
• A main consideration of the choice of element is based on
‘thin’ or ‘moderately thick’ shells/beams
Advanced Structural Nonlinearities 6.0
• The main points in this chapter are the following:
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Element Technology
... Chapter Overview
A. Conventional Displacement-Based Continuum Elements
B. Shear and Volumetric Locking in Continuum Elements
C. Selective Reduced Integration (B-bar)
D. Uniform Reduced Integration (URI)
E. Enhanced Strain Formulation
F. Mixed U-P Formulation
G. General Recommendations for Continuum Elements
H. Shell Elements
I.
Beam Elements
Advanced Structural Nonlinearities 6.0
• In this chapter, we will cover the following topics:
Training Manual
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Element Technology
A. Conventional Displ. Formulation
Training Manual
– SOLID45 with KEYOPT(1)=1 and PLANE42 with KEYOPT(2)=1 are
examples of lower-order, fully-integrated conventional
displacement formulations.
– SOLID95 with KEYOPT(11)=0 is an example of a higher-order,
fully-integrated conventional displacement formulation.
• This is actually a 14pt integration point rule rather than a
3x3x3 integration scheme, as will be discussed later. This
14pt integration point rule is considered a more efficient full
integration scheme.
Advanced Structural Nonlinearities 6.0
• Fully integrated lower- and higher-order elements with no
additional DOF are examples of conventional displacementbased elements.
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Element Technology
... Conventional Displ. Formulation
Training Manual
– For any element, DOF solution {Du} is solved at nodes
– Stresses and strains are calculated at
integration points. They are derived from
DOF. For example, we can determine
strains from displacements via:
s, e
De   B Du
[B] is called the strain-displacement matrix
u
– When we post-process results, stress/strain values at
integration points are extrapolated or copied to nodal locations
• The image on the right shows a 4-node quad element with
2x2 integration, integration points shown in red.
Advanced Structural Nonlinearities 6.0
• Recall a few important details about integration points:
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Element Technology
... Conventional Displ. Formulation
Training Manual
Element Type Full Integration Order
4 Node Quad
2x2
8 Node Quad
3x3
8 Node Hex
2x2x2
20 Node Hex
3x3x3
• In other words, full integration means that the numerical
integration rule is accurate for all components of strain
energy for geometrically undistorted elements.
Advanced Structural Nonlinearities 6.0
• Integration points for conventional displacement-based
elements follow Gauss quadrature rules and are the same
order as the element. This is called full integration.
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Element Technology
... Conventional Displ. Formulation
Training Manual
• Fully integrated, higher-order conventional displacement
elements are also prone to volumetric locking.
Advanced Structural Nonlinearities 6.0
• Fully integrated, lower-order conventional displacement
elements are susceptible to shear and volumetric locking, so
they are rarely, if ever, used.
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Element Technology
B. Shear and Volumetric Locking
Training Manual
– Shear Locking results in bending behavior being too stiff
(parasitic shear stresses). This is a property of the geometry,
when thin members are subject to bending.
– Volumetric Locking results in overly stiff response. This is a
property of the material, when the Poisson’s ratio is near or
equal to 0.5.
• The majority of this chapter will focus on ways of resolving
these two issues with different element formulations. Our
discussion will be focused on continuum (solid) elements.
• Some of these element formulations also provide more
efficient ways of solving nonlinear problems since nonlinear
analysis can be computationally expensive.
Advanced Structural Nonlinearities 6.0
• There are two problems with conventional displacementbased elements: shear locking and volumetric locking:
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Element Technology
... Shear Locking
Training Manual
y
M
M
Pure bending deformation for
a differential volume, plane
sections remain plane, top
and bottom edges become
arcs, gxy = 0.
x
M
M
Fully integrated lower order
element deformation, top and
bottom edges remain straight,
right angles are not preserved,
gxy is non zero.
Advanced Structural Nonlinearities 6.0
• Fully integrated lower order elements exhibit “overstiffness”
in bending problems. This formulation includes shear strains
in bending which do not physically exist, called parasitic
shear. (From beam theory in pure bending the shear strain,
recall that gxy = 0.)
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Element Technology
... Shear Locking Example
Training Manual
– The resulting displacements will be underpredicted because of
spurious shear strains/stresses.
– An example of a beam in bending below. Shear stresses should
be near zero for this case, but, as shown in the contour plot of
SXY, shear locking occurs.
Advanced Structural Nonlinearities 6.0
• As the length:thickness ratio increases, a model may become
more susceptible to shear locking.
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Element Technology
... Shear Locking Example
Element 182 (B-Bar),
plane strain with nearlyincompressible MooneyRivlin hyperelastic material
Advanced Structural Nonlinearities 6.0
• Does this model exhibit shear locking?
Training Manual
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Answer: Surprisingly, no. This model has hyperelastic material properties, run with
both B-Bar and Enhanced Strain, producing very similar results.
Element Technology
... Volumetric Locking
Training Manual
– The incompressibility can occur from a hyperelastic material or
plastic flow (discussed later).
– Spurious pressure stresses develop in the element, which cause
the element to have an “overstiffness” for deformations that
should not cause any volume change.
– Volumetric locking may also cause convergence problems.
• Volumetric locking can occur for various stress states,
including plane strain, axisymmetric, and 3-D stress.
– For plane stress problems, volumetric locking does not occur
because out-of-plane strains are used to satisfy
incompressibility condition.
Advanced Structural Nonlinearities 6.0
• Volumetric locking occurs in fully integrated elements when
the material behavior is nearly or fully incompressible
(Poisson’s ratio approaches or equals 0.5).
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Element Technology
... Volumetric Locking
Training Manual
s ij  pij  sij
• The hydrostatic stress (p) is defined as the product of the
bulk modulus (k) and volumetric strain (ev):
p  k ev
1
 s x  s y  s z 
3
E
k
31  2 
ev  e x  e y  e z

1  2  
E
s x s y s z 
Advanced Structural Nonlinearities 6.0
• We can separate stress into hydrostatic (p) and deviatoric (s)
components:
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Element Technology
... Volumetric Locking
Training Manual
– Bulk modulus k will be very large or infinite
– Volumetric strain ev will be near or equal to zero
– This is called nearly or fully incompressible material behavior
• Nearly or fully incompressible materials present numerical
difficulties, and they also exhibit overly stiff behavior.
– This is most clearly seen in bulk deformation problems
– From a computational standpoint, nearly incompressible and
fully incompressible problems are treated differently.
Advanced Structural Nonlinearities 6.0
• From the equations on the previous slide, if Poisson’s ratio is
near or equal to 0.5, we can see that:
• Volumetric locking results in an alternating (checkerboard)
pattern of the hydrostatic pressure (p) which can be postprocessed as NL,HPRES for elements when nonlinear
materials are present.
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Element Technology
... Volumetric Locking Example
Postprocessing
hydrostatic pressure
(NL,HPRES) using
element solution
(PLESOL) will allow a
user to verify whether
or not volumetric
locking is a problem.
Advanced Structural Nonlinearities 6.0
Contours of NL,HPRES
are shown on the right.
This output quantity is
available whenever
nonlinear materials are
present.
Training Manual
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Element Technology
... Workshop Exercise
• Workshop 1: Shear Locking
Advanced Structural Nonlinearities 6.0
Please refer to your Workshop Supplement:
Training Manual
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Chapter Two, Sections C-G
Continuum Elements
Element Technology
Continuum Element Formulations
Training Manual
C. Selective Reduced Integration (B-bar)
D. Uniform Reduced Integration (URI)
E. Enhanced Strain Formulation
F. Mixed U-P Formulation
Advanced Structural Nonlinearities 6.0
• Some general guidelines and recommendations will be
discussed next. However, the following subsections will
provide details on these element technologies used to
overcome shear and volumetric locking.
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Element Technology
... Continuum Element Formulations
Training Manual
• We use different element formulations to address this by
relaxing the constraint or introducing additional equations to
solve these constraints.
Relax
Constraints
Element
(Reduced
Formulation Integration)
B-Bar
x
URI
x
ES
Mixed U-P
x
•
Additional
Equations
(Add Extra
DOF)
x
Unfortunately, there is no element formulation which acts as a silver
bullet to solve locking problems in the most efficient manner.
Hence, we will discuss the pros and cons of each formulation in the
next sections.
Advanced Structural Nonlinearities 6.0
• As a simplified explanation, shear and volumetric locking are
due to overconstraints on our system.
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Element Technology
... Continuum Element Formulations
Training Manual
Y
Y
Y
Y
Y
Y
FullyIncompressible
(Hyperelasticity)
Shear Locking
(Bending)
NearlyIncompressible
(Plasticity,
Hyperelasticity)
Higher-Order
Elements
Lower-Order
Elements
Element
Technology
B-Bar
Enhanced Strain
URI
Mixed U-P
N
Y
Y
N
Y
Y
Y
Y
N
N
N
Y
– Higher-order 18x elements (PLANE183, SOLID186-187) always use URI.
– Lower-order 18x elements (PLANE182, SOLID185) use B-Bar by default.
– B-Bar and Enhanced Strain are not applicable to higher-order elements.
– Mixed U-P technology is independent of the others, so it can be used in
conjunction with B-Bar, Enhanced Strain, or URI.
Advanced Structural Nonlinearities 6.0
Present in the 18x elements are four different element
technologies: B-Bar, URI, Enhanced Strain, and Mixed U-P.
These are used to address shear and volumetric locking:
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Element Technology
... Continuum Element Formulations
Training Manual
– Main Menu > Preprocessor > Element Type > Add/Edit/Delete…
“Options” button in dialog box
Example for SOLID185:
“Full Integration” is B-Bar
“Reduced integr” is URI
Enhanced Strain is 3rd option
“Pure displacemnt” is default
“Mixed U/P” can also be
selected
– If using commands,
• KEYOPT(1) is used for PLANE182 for B-bar, URI, Enhanced Strain
• KEYOPT(2) is used for SOLID185 for B-bar, URI, Enhanced Strain
• KEYOPT(6) is used for Mixed U-P for all solid/planar 18x elements.
Advanced Structural Nonlinearities 6.0
• The element keyoption allows the user to choose the
appropriate element formulation.
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Element Technology
C. Selective Reduced Integration
Training Manual
– Recall that the stress state can be separated in hydrostatic (p)
and deviatoric (s) terms.
s  ps
p  ke v
s  2Ge d
s  ke v  2Ge d
In the above equation, ev is volumetric strain and ed is deviatoric
strain. k is the bulk modulus and G is the shear modulus.
Advanced Structural Nonlinearities 6.0
• Selective Reduced Integration (a.k.a. B-bar method, constant
dilatational elements) uses an integration rule one order
lower for volumetric terms.
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Element Technology
... Selective Reduced Integration
Training Manual
B   Bv   Bd 
Bv dV

B  
v
V
B  Bv  Bd 
De  B Du
– When evaluating [B], however, we will use two different
integration orders for volumetric and deviatoric components.
[Bv] is evaluated with one integration point
(reduced integration)
On the other hand, [Bd] is evaluated with 2x2
integration points (full integration)
Advanced Structural Nonlinearities 6.0
– Strains are related to displacements via the following:
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Element Technology
... Selective Reduced Integration
Training Manual
De  B  Du
• The fact that the volumetric term [Bv] has reduced integration
allows it to be ‘softer’ since it is not fully integrated. This
allows for solution of nearly incompressible behavior and
overcomes volumetric locking.
• However, because the deviatoric term [Bd] remains the same,
parasitic shear strains still exist, so this formulation is still
susceptible to shear locking.
Advanced Structural Nonlinearities 6.0
• As shown on the previous slide, the volumetric and
deviatoric components of [B] are not evaluated at the same
order of integration. Only the volumetric component [Bv] has
reduced integration. That is why this method is called
selective reduced integration. It is also known as the B-bar
method because [B] is averaged on the volumetric term.
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Element Technology
... Selective Reduced Integration
Training Manual
– B-Bar method alone does not work for fully incompressible
problems but can be used in conjunction with Mixed U-P
elements (discussed later) for fully incompressible materials.
– B-Bar method should not be used in bending-dominated models
• Certain elements support selective reduced integration:
– Available for plane strain, axisymmetric, and 3D stress states.
Recall that volumetric locking is not a problem for plane stress,
so B-Bar method is not needed in those situations.
– PLANE182 and SOLID185 use B-Bar method by default
(KEYOPT(1)=0). Can be used for various constitutive models.
Advanced Structural Nonlinearities 6.0
• In summary, selective reduced integration is useful for nearly
incompressible material behavior (e.g., plasticity,
hyperelasticity) in problems dominated by bulk deformation.
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Element Technology
D. Uniform Reduced Integration
Training Manual
Element Type
4 Node Quad
8 Node Quad
8 Node Hex
20 Node Hex
Full Integration Reduced Integration
Order
Order
2x2
1x1
3x3
2x2
2x2x2
1x1x1
3x3x3
2x2x2
• This is similar to selective reduced integration, but both
volumetric and deviatoric terms have reduced integration.
• This formulation leads to a more flexible formulation which
helps eliminate shear and volumetric locking.
– Reduced integration of volumetric terms allows solution of
nearly incompressible problems.
Advanced Structural Nonlinearities 6.0
• Uniform Reduced Integration (URI) uses an integration rule
one order lower than needed for numerically exact integration
– Reduced integration of deviatoric terms prevents shear locking
in bending problems.
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Element Technology
... Hourglass Modes
Training Manual
• In the lower order element with one integration point shown
below, two modes of deformation are illustrated where the
single integration point does not capture any strain energy in
the element.
Advanced Structural Nonlinearities 6.0
• Unfortunately, the reduced integration of deviatoric terms
causes modes of deformation which have zero strain energy,
called zero energy or hourglass modes. By themselves,
these are uncontrollable modes of deformation which lead to
physically unrealistic behavior.
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Element Technology
... Hourglass Modes
Training Manual
• To control hourglass modes in lower-order URI elements, an
hourglass stiffness is added. This provides a stiffness to
resist the zero energy modes.
– Although the default hourglass stiffness value is usually
sufficient, the user can override this through a real constant.
– This hourglass stiffness does not have physical significance, so
specifying too large of a value is not recommended. Artificial
energy (element table AENE) can be retrieved -- this is the
energy due to this hourglass stiffness. Artificial energy should
not be more than 5% of total energy (e.g., strain energy). This
can be done via dividing element tables AENE and SENE.
Advanced Structural Nonlinearities 6.0
• Hourglass modes are usually only a problem in lower-order
URI elements. Higher-order URI elements’ zero energy
modes will not propagate as long as there is more than one
element in each direction.
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Element Technology
... Hourglass Modes
Training Manual
– Do not apply any point loads or single-point constraints since
these tend to excite hourglass modes
– Refining the mesh usually helps prevent hourglass modes from
propagating
– Change to other element formulations to prevent hourglassing
On right, lower order URI
elements with a point load
applied at corner is shown. The
hourglass modes are clearly
shown to be propagating in the
mesh. (Displacements are
scaled to exaggerate effect)
Advanced Structural Nonlinearities 6.0
• Besides the hourglass stiffness, there are ways in which the
user can prevent hourglassing:
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Element Technology
... Uniform Reduced Integration
Training Manual
– Can be used in nearly incompressible problems to overcome
volumetric locking
– Can be used in bending problems without worrying about shear
locking
– No additional DOF are required, and, in fact, less CPU time is
required for element calculations. File sizes (e.g., *.esav) are
reduced. This provides efficient solutions, especially for
nonlinear problems.
– Elements have same formulation and are compatible with
ANSYS/LS-DYNA explicit-dynamics elements
– Higher-order URI elements have no hourglass modes, as long as
there is more than one element in any direction (e.g., thickness).
Advanced Structural Nonlinearities 6.0
• URI elements have many nice benefits:
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Element Technology
... Uniform Reduced Integration
Training Manual
– Lower-order URI elements are susceptible to hourglassing, and
this needs to be checked.
– Lower-order URI elements may be too flexible, especially in
bending-dominated problems, so a finer mesh may be required
such that displacements are not over-predicted
– Both lower- and higher-order URI elements have an integration
rule which is one order lower than full integration. This means
stresses are evaluated at 1 point for lower-order elements and
2x2 or 2x2x2 for higher-order elements. Hence, more elements
may be required to capture stress gradients.
– URI cannot be used in fully incompressible analyses.
Advanced Structural Nonlinearities 6.0
• On the other hand, a user needs to consider a few things
when using URI:
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Element Technology
... Uniform Reduced Integration
Training Manual
– SOLID95 uses a modified 14-point integration rule but will use
URI if KEYOPT(11)=1
• Most lower-order elements, by default, do not use URI. This
can be activated with KEYOPT(2)=1 for SOLID45 and
SOLID185 or KEYOPT(1)=1 for PLANE182
– URI is not available with PLANE42. It is recommended to use
PLANE182 instead, which supports URI.
– Unless required for special reasons (e.g., compatibility with LSDYNA elements), the user is encouraged to use B-bar or
Enhanced Strain for lower-order elements instead of URI.
Advanced Structural Nonlinearities 6.0
• Most ANSYS higher-order structural elements (PLANE82,
PLANE183, SOLID186) use URI by default. This is because
higher-order elements are not prone to hourglassing but have
many benefits, so these are very attractive to use.
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Element Technology
E. Enhanced Strain Formulation
Training Manual
– Enhanced Strain elements are useful when shear or volumetric
locking are encountered (e.g., bending dominated problems or
nearly incompressible material behavior).
Advanced Structural Nonlinearities 6.0
• Enhanced Strain Formulation (a.k.a. Incompatible Modes,
Assumed Strain) adds internal degrees of freedom to lowerorder quad/hex elements. The displacement gradient tensor
is modified with these extra ‘enhanced’ terms, hence the
name “Enhanced Strain”.
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Element Technology
... Enhanced Strain Formulation
Training Manual
– Element performs best when nearly rectangular; on the other
hand, they do not perform well when trapezoidal. This is a
limitation of the Enhanced Strain technology.
– Lower-order tri or tet elements (constant strain elements) cannot
incorporate the enhanced strain formulation.
– Higher-order elements do not suffer from shear locking.
Axial Mode: 1st Natural Frequency Ratio
Shape
PLANE182 PLANE183 SOLID185 SOLID187 SOLID186
Rectangular
1.004
1.001
1.005
1.000
1.002
Trapezoid (15°)
1.004
1.001
1.005
1.000
1.002
Trapezoid (30°)
1.004
1.001
1.005
1.000
1.002
Trapezoid (45°)
1.005
1.001
1.006
1.000
1.002
Parallelogram (15°)
1.004
1.001
1.005
1.000
1.002
Parallelogram (30°)
1.004
1.001
1.005
1.000
1.002
Parallelogram (45°)
1.004
1.001
1.005
1.000
1.002
Bending Mode: 1st Natural Frequency Ratio
Shape
PLANE182 PLANE183 SOLID185 SOLID187 SOLID186
Rectangular
1.010
0.999
1.010
1.004
0.999
Trapezoid (15°)
1.567
1.000
1.596
1.005
1.000
Trapezoid (30°)
1.973
1.003
2.009
1.008
1.003
Trapezoid (45°)
2.207
1.012
2.245
1.020
1.012
Parallelogram (15°)
1.040
0.999
1.042
1.005
0.999
Parallelogram (30°)
1.091
0.999
1.097
1.009
0.999
Parallelogram (45°)
1.119
0.999
1.126
1.020
0.999
Advanced Structural Nonlinearities 6.0
• This formulation is only applicable for lower-order elements
in quad or hex shape only.
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Element Technology
... Enhanced Strain Formulation
Training Manual
182, Plane Stress
182, Plane Strain
182, Axisymmetric
185
Displ.-Based Mixed U-P
Additional
Additional
Internal DOF Internal DOF
4
N/A
4+1=5
4
4+1=5
4
9+4=13
9
– Recall that plane stress elements do not suffer from volumetric
locking, so that is why there are only 4 extra DOF (bending
terms) for PLANE182 when used in plane stress applications.
Advanced Structural Nonlinearities 6.0
• There are two sets of terms for Enhanced Strain in 2D and 3D
– one set to treat shear locking (4 and 9 internal DOF,
respectively) and another for volumetric locking (1 and 4
internal DOF, respectively).
– When Enhanced Strain is used with Mixed U-P formulation
(discussed later), only bending terms (4 and 9) are used since
Mixed U-P handles the volumetric terms.
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Element Technology
... Enhanced Strain Formulation
Training Manual
• They are also known as “incompatible modes” because they
lead to gaps and overlaps in the mesh.
F
F
2F
2F
F
F
Enhanced Strain
F
F
2F
2F
F
F
No Enhanced Strain
Advanced Structural Nonlinearities 6.0
• A simplified explanation is that the additional DOF augment
the element’s shape function to allow curvature. Another
term helps to treat volumetric locking for nearly
incompressible materials
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Element Technology
... Enhanced Strain Formulation
Training Manual
– PLANE182 when KEYOPT(1)=2
– SOLID185 when KEYOPT(2)=2
• Older elements support a subset of Enhanced Strain, called
“Extra Displacement Shapes” or “Bubble Functions”.
– Most PLANE elements (e.g., PLANE42)
– Most SOLID elements (e.g., SOLID45)
– Most SHELL elements (e.g., SHELL63, 181)
Advanced Structural Nonlinearities 6.0
• There are two elements which can use Enhanced Strain,
when in quad or hex shape:
– These elements have 4 internal DOF (2D) and 9 internal DOF
(3D), respectively. These terms help overcome shear locking,
but are meant for small strain applications only. Use PLANE182
and SOLID185 instead for large strain applications. The “extra
displacement shape” formulation of these elements are not
covered in this chapter.
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Element Technology
... Example of Enhanced Strain
– A Mooney-Rivilin beam (20X1)
– SOLID182 with Enhanced Strain and Mixed U-P (fully
incompressible)
– HYPER56 (nearly incompressible, nu=0.4999)
– Plane strain, NLGEOM,ON, Pressure loading
Advanced Structural Nonlinearities 6.0
Example of Shear Locking in Bending of Rubber Beam
Training Manual
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Element Technology
... Example of Enhanced Strain
Training Manual
Erroneous results with
Hyper56 elements
Correct results with
enhanced 182 elements
Advanced Structural Nonlinearities 6.0
Example of Shear Locking in Bending of Rubber Beam
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Element Technology
... Example of Enhanced Strain
Training Manual
– Ri=3,Ro=9
– SOLID185 with enhanced strain
– SOLID45 with extra shape
– Pure elastic material (E=1000)
– Different Poisson’s ratios (nu=0.0, 0.25, 0.3, 0.49, 0.499,0.4999)
– Linear analysis
Advanced Structural Nonlinearities 6.0
Example of Volumetric Locking in Thick-Walled Cylinder
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Element Technology
... Example of Enhanced Strain
Results from Element 45
%18 Error in
displacement calculation
Results from Element 185
%1.6 Error in
displacement calculation
Advanced Structural Nonlinearities 6.0
Example of Volumetric Locking in Thick-Walled Cylinder
Training Manual
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Element Technology
... Enhanced Strain Summary
Training Manual
– Enhanced Strain cannot be used for fully incompressible
analyses, but it can be used with Mixed U-P formulation for
PLANE182 and SOLID185, discussed in the next section.
• Enhanced Strain has the above advantages, but it may be
more computationally expensive
– The extra internal DOF mentioned on the previous slides are
condensed at the element level, but there is still extra
computational time (and larger *.esav file) associated with it.
• Only lower-order quad PLANE182 and hex SOLID185 support
Enhanced Strain.
– The Enhanced Strain terms will have little benefit in bending if
the element is distorted, especially if trapezoidal.
Advanced Structural Nonlinearities 6.0
• Enhanced Strain Formulation was designed for bending and
nearly incompressible applications in mind
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Element Technology
F. Mixed U-P Formulation
Training Manual
– Separate interpolation functions are used for displacement and
hydrostatic pressure DOF.
• There are three different mixed u-p formulations in ANSYS,
used for cases of either nearly or fully incompressible
analyses
– We will first present the basic idea behind mixed u-p elements,
then discuss the three different methods of implementing this
technology
Advanced Structural Nonlinearities 6.0
• Mixed U-P elements (a.k.a. Hybrid or Herrmann Elements) are
used to treat volumetric locking by interpolating (and solving)
hydrostatic pressure as an additional DOF.
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Element Technology
... Mixed U-P Formulation
Training Manual
s  ps
p  k ev
E
31  2 
1  2  
ev 
s x s y s z 
E
k
• As Poisson’s ratio approaches 0.5, bulk modulus becomes
infinite, and volumetric strains approach zero.
Advanced Structural Nonlinearities 6.0
• Recall that, for volumetric locking, Poisson’s ratio near or
equal to 0.5 causes numerical difficulty:
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Element Technology
... Mixed U-P Formulation
Training Manual
• As a result, we can solve for pressure (p) as an independent
DOF. That way, we do not have to worry about high bulk
modulus or very low volumetric strain terms.
s  ps
s  2Ge d
s  p  2Ge d
Advanced Structural Nonlinearities 6.0
• Since volumetric strains are calculated from derivatives of
displacements, these values are not as accurate as
displacements. Any small error in volumetric strain will
appear as large error in hydrostatic pressure (and,
consequently, stresses). This, in turn, will affect
displacement calculations (the mesh will ‘lock’)
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Element Technology
... Mixed U-P Formulation
Training Manual
 K uu
K
 pu
K up  Du  DF 
  

K pp  Dp   0 
• Now, since pressure may be solved for independently, the
accuracy of hydrostatic pressure is independent of
volumetric strain, bulk modulus, or Poisson’s ratio.
• We have two methods of implementing mixed u-p in ANSYS
– Penalty-based Mixed U-P formulation for nearly incompressible
– Lagrange Multiplier for nearly and fully incompressible
Advanced Structural Nonlinearities 6.0
• Solving displacements {u} and hydrostatic pressure {p} as
unknowns leads to the name “mixed u-p” formulation
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... Penalty-based Mixed U-P
Training Manual
– This formulation is used for HYPER56, 58, 74, and 158 with
hyperelastic material (Mooney-Rivlin)
– Also used for VISCO106-108, which supports rate-dependent
and rate-independent plasticity (Anand, Isotropic Hardening)
• This formulation can be used for nearly incompressible
analyses.
– Note that, depending on whether hyperelasticity or plasticity is
used, the user must select the appropriate HYPER or VISCO
element type.
Advanced Structural Nonlinearities 6.0
• The basic idea behind penalty-based Mixed U-P is that we
condense out the hydrostatic pressure (p) DOF in the element
level through volumetric constraint equations. In this
manner, our stiffness matrix will still be displacement-based,
and we do not worry about extra DOF.
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Element Technology
... Penalty-based Mixed U-P
Training Manual
Element Displacement
Type
DOF
HYPER56
8 or 121
HYPER58
24
HYPER74
16 or 241
HYPER158
30
VISCO106
8 or 121
VISCO107
24
VISCO108
16 or 241
1
Pressure
DOF
1
1
3
1
1
1
3
Depends on 2D plane strain or 2D axisymmetric KEYOPT(3). If axisymmetric,
more DOF due to torsion (UZ) DOF.
Advanced Structural Nonlinearities 6.0
• As noted earlier, pressure and displacement DOFs are
interpolated with separate functions.
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Element Technology
... Lagrange Multiplier U-P
Training Manual
• As noted earlier, the assembled equations for Mixed U-P are:
 K uu
K
 pu
K up  Du  DF 
  

K pp  Dp   0 
Advanced Structural Nonlinearities 6.0
• For nearly and fully incompressible analyses using 18x
elements, a special element formulation is used, which we
will refer to as the Lagrange Multiplier method.
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Element Technology
... Lagrange Multiplier U-P
Training Manual
– This hydrostatic pressure DOF is associated with an ‘internal
node’ which is automatically generated by ANSYS and is
inaccessible/transparent to the user.
– This formulation is used for 18x series of elements with
KEYOPT(6)>0 (PLANE182-183, SOLID185-187)
– The next few slides will discuss formulation of nearly vs. fully
incompressible materials using the Lagrange Multiplier method.
Note that ANSYS will automatically use the appropriate
formulation, depending on the material, so this is transparent to
the user. The subsequent information is provided for better
understanding of this element technology.
Advanced Structural Nonlinearities 6.0
• Unlike the Penalty-based Mixed U-P formulation, the
Lagrange Multiplier method keeps P as an independent DOF
which is solved for.
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Element Technology
... Lagrange Multiplier U-P
Training Manual
PP
k
0
1
P  s ii
[elasto - plastic]
3
2
P   J  1
[hyperelastic]
d
P  Pressure DOF
k  Bulk Modulus
d2
k
J  dV
dVo
Advanced Structural Nonlinearities 6.0
• Let us review the nearly incompressible case first. We can
rewrite the volumetric compatibility equation as follows:
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Element Technology
... Lagrange Multiplier U-P
Training Manual
PP
V k dV
 Vtol
V
where V refers to volume of the element.
– This volumetric constraint is incorporated into the final
equations as an extra condition which must be satisfied. The
Output Window/File will report the number of elements, if any,
which do not satisfy this constraint.
Advanced Structural Nonlinearities 6.0
• Because of numerical precision, we solve the volumetric
compatibility equation to a given tolerance (default 1e-5),
specified in the Vtol argument of SOLCONTROL command.
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Element Technology
... Lagrange Multiplier U-P
Training Manual
V  Vo
 Vtol
Vo
where V and Vo are the updated and original volumes of the
element, respectively.
– Similar to the nearly incompressible case, Vtol is specified via
SOLCONTROL command (default value is 1e-5).
Advanced Structural Nonlinearities 6.0
• For fully incompressible analyses of hyperelastic materials, a
different equation is used to enforce volumetric constraint.
Unlike other materials, we cannot obtain hydrostatic pressure
from material constitutive law (for example, we cannot
determine P from 1/3sii). Instead, we focus on the volumetric
constraint to ensure that the volume is constant, which is
true for fully incompressible materials:
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Element Technology
... Lagrange Multiplier U-P
Training Manual
 K uu
K
 pu
K up  Du  DF 
  

0  Dp   0 
Advanced Structural Nonlinearities 6.0
• For the case where the material is fully incompressible, it is
worthwhile to note that [Kpp]=0 for this formulation. Hence,
our stiffness matrices have some zero diagonals.
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Element Technology
... Lagrange Multiplier U-P
Training Manual
– SOLID187 with KEYOPT(6)=2 has 4 Pressure DOF, also
consistent with volumetric strain interpolation function.
Element Displacement B-bar or URI Enhanced Strain
Type
DOF
Pressure DOF Pressure DOF
PLANE182
8
1
3
PLANE183
16
3
N/A
SOLID185
24
1
4
SOLID186
60
4
N/A
SOLID187
30
1 or 4
N/A
Advanced Structural Nonlinearities 6.0
• Pressure and displacement DOFs are interpolated with
separate functions. Note that, because of URI or B-bar
technology in PLANE182 and SOLID185, volumetric strain is
constant in each element, consistent with constant P (1 DOF)
interpolation function.
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Element Technology
... Lagrange Multiplier U-P
Training Manual
• When the number of pressure DOF (Np) is greater than the
number of active (unconstrained) displacement DOF (Nd), this
is an over-constrained model, which results in locking.
Ideally, the ratio of Nd/Np should be 2/1 for 2D problems or 3/1
for 3D problems. Over-constrained models can be overcome
by mesh refinement, especially in areas without displacement
b.c.
Advanced Structural Nonlinearities 6.0
• Because the Lagrange Multipliers (internal DOF P) are kept in
the assembled stiffness matrix, direct solvers must be used
with this formulation. Iterative solvers such as PCG cannot
handle zero diagonals. It is recommended to use the frontal
solver over the sparse solver due to robustness reasons.
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... Lagrange Multiplier U-P
Training Manual
Advanced Structural Nonlinearities 6.0
• For a fully incompressible problem, no unique solution may
exist if all boundary nodes have prescribed displacements.
This is due to the fact that hydrostatic pressure (internal
DOF) is independent of deformation. Hydrostatic pressure
needs to be determined by a force/pressure boundary
condition. Without this, the hydrostatic pressure cannot be
calculated -- i.e., there is no unique solution. For these
situations where this occurs, having at least one node
without applied boundary condition will remedy this situation.
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Element Technology
... Mixed U-P Formulation
Training Manual
– For nearly-incompressible hyperelastic materials, use HYPER56,
58, 74, 158 or Mixed U-P 18x family of elements.
– For nearly-incompressible elasto-plastic materials, use Mixed UP formulation of 18x family or VISCO106-108 elements.
– For fully-incompressible hyperelastic materials, use Mixed U-P
formulation of 18x elements.
• Mixed U-P Formulation in 18x elements can be combined with
other element formulations discussed in earlier sections.
– Mixed U-P, by itself, addresses the issue of volumetric locking
– For 18x elements, one can combine Mixed U-P (KEYOPT(6)>0)
with B-bar, URI, or Enhanced Strain Formulations.
Advanced Structural Nonlinearities 6.0
• In summary, ANSYS provides an extensive library of element
technology using Mixed U-P formulation for nearly and fully
incompressible materials.
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Element Technology
G. Solid Element Recommendations
Training Manual
– Choice of element technology often depends on model,
including bending/bulk deformation and material behavior.
• Whenever possible, it is recommended to use 18x elements
for nonlinear problems because of the following:
– Newest element technology incorporated into 18x elements,
including B-bar, URI, Enhanced Strain, and Mixed U-P.
– 18x family has separated element technology from material
technology. These elements have wealth of constitutive models
which will be discussed later in this seminar. This also makes
for fewer elements to choose from.
Advanced Structural Nonlinearities 6.0
• Recall that conventional elements are susceptible to shear
and volumetric locking. There are many types of element
technology in ANSYS to address these two problems.
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... Solid Element Recommendations
Training Manual
– Some guidelines on lower-order element selection below:
18x Element Pros
Formulation
B-Bar
Efficient for nearlyincompressible, bulkdeformation problems
B-Bar with
Efficient for fullyMixed U-P
incompressible, bulkdeformation problems
Enhanced
Will handle bendingStrain
dominated, nearlyincompressible problems.
Enhanced
Will handle bendingStrain with
dominated, fullyMixed U-P
incompressible problems.
URI
Will handle shear locking
and nearly incompressible
problems.
URI with
Will handle shear locking
Mixed U-P
and fully incompressible
problems.
Cons
When to Use
May be susceptible to shear
locking
Recommended for
most analyses,
especially bulk deformation problems.
Same considerations as above,
plus direct solvers required due
to Mixed U-P
Extra CPU time required for
Enhanced Strain terms
(condensed out at element level)
Same considerations as above,
plus direct solvers required due
to Mixed U-P
Because of hourglass modes,
not recommended choice
Same considerations as above,
plus direct solvers required due
to Mixed U-P
Recommended if
problem is bendingdominated. If problem
has some bending, use
when accuracy is an
issue.
Recommended choice
only for compatibility
with LS-DYNA in
implicit-to-explicit or
explicit-to-implicit
simulations.
Advanced Structural Nonlinearities 6.0
– For higher-order elements, URI is used by default. The only
choice users need to consider is that, if the material is fully
incompressible, Mixed U-P should be used.
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... Solid Element Recommendations
Training Manual
• Any first-order quad/hex element with extra displacement
shapes (default for PLANE42, SOLID45 in non-degenerate
form). These elements are useful for both shear locking and
nearly incompressible material behavior.
• Any second-order element, especially SOLID92 (or SOLID187)
for CAD geometries requiring tet meshes. Higher-order
quad/hex elements such as PLANE183 or SOLID186 use URI,
which is also useful to overcome shear locking and nearly
incompressible behavior.
Advanced Structural Nonlinearities 6.0
Linear Analyses and Small-Strain Nonlinear Analyses
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... Solid Element Recommendations
Training Manual
• For large strain applications, use of lower-order quad/hex
elements are preferred (less chance of midside nodes getting
inverted). Use B-Bar method first; if shear locking becomes
an issue, the user can switch to Enhanced Strain.
• Higher-order elements (which, by default, use URI) are also
acceptable.
• For 18x elements, one can use Mixed U-P KEYOPT(6) with
other technology for nearly or fully incompressible analyses.
• For large strains, one may need to grade the mesh and
anticipate large strain areas to ensure that good element
quality persists throughout solution.
Advanced Structural Nonlinearities 6.0
Finite-Strain Nonlinear Analyses
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... Workshop Exercise
• Workshop 2: Volumetric Locking
Advanced Structural Nonlinearities 6.0
Please refer to your Workshop Supplement:
Training Manual
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Chapter Two, Sections H-I
Shell and Beam Elements
Element Technology
H. Shell Elements
Training Manual
– The physical dimensions are usually based on the distance
between supports/constraints or the wavelength of the
modeshapes of interest.
– A ratio of 20:1 of length to thickness can be used as a general
guideline to determine the applicability of shell elements.
– As a result of this assumption, stresses through the thickness of
the shell are assumed to be negligible
Advanced Structural Nonlinearities 6.0
• Shell elements are used to model structures where one
dimension (the thickness) is much smaller than the other
dimensions, and the dimensions are based on the physical
structure, not the element size.
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Element Technology
... Shell Elements
Training Manual
SHELL63
Kirchhoff
Neglected
Shell Theory
Transverse
Shear
Element Order Lower-order
Kinematics
Small strain
Materials
Linear;
Homogenous
Section
Definition
(SECxxx
commands)
Bending
Response
Thru-plane
Integration
Points
No, input via real
constants only
In-plane
Behavior
Drilling DOF
Extra displacement
shapes
Artificial spring,
Allman-type DOF
Cubic
3 integration points
SHELL91, 93
Reissner/Mindlin
Included (constant
transverse shear)
Higher-order
Finite strain
SHELL181
Reissner/Mindlin
Included (constant
transverse shear)
Lower-order
Finite strain; includes
change of thickness
effects
Plasticity;
Plasticity, Creep;
Composite for SHELL91 Hyperelasticity;
Composite
No, input via real
Yes (can also use
constants only
real constants for
homogenous
material)
Quadratic
Linear
SHELL93: 2 for linear, 5
for nonlinear materials;
SHELL91: 3 for each
layer
URI
Artificial spring
User-definable
number of integration
points (default: 5).
URI (default); Extra
displacement shapes
Penalty method
Advanced Structural Nonlinearities 6.0
• The table below summarizes some of the differences in shell
elements available in ANSYS:
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Element Technology
... Shell Elements
Training Manual
– Plane stress behavior describes the membrane behavior. As
noted in the previous slide, we assume that through-thickness
stress (element z-stress) is zero.
• In ANSYS, there are mainly two distinctions of shell elements:
thin and thick shells
– Thin shells neglect transverse shear deformation, whereas thick
shells include these effects as a first-order approximation
(transverse shear strain is constant through thickness).
– Thin shells are also known in literature as Kirchhoff elements.
– Thick shells are known as Mindlin/Reissner elements.
Advanced Structural Nonlinearities 6.0
• Basic shell theory is a superposition of membrane and platebending theories
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... Shell Elements
– The cross-sectional plane initially
normal to the midsurface of the shell is
assumed to remain straight and normal
to the neutral axis during loading. This
assumption excludes shear
deformations.
Moderately-Thick Shells (Mindlin/Reissner)
– The cross-sectional plane initially
normal to the midsurface of the shell is
assumed to remain straight but not
remain normal to the neutral axis during
loading. The shear strain, as a result, is
constant across the section.
Advanced Structural Nonlinearities 6.0
Thin Shells (Love/Kirchhoff)
Training Manual
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Element Technology
... Shell Elements
Training Manual
• For in-plane behavior, since we are assuming a state of plane
stress, volumetric locking is not an issue. (SHELL181
supports fully incompressible hyperelastic behavior.)
• For in-plane behavior, shear locking is still a problem. Most
shells use extra displacement shapes for in-plane response
(SHELL43, 63, 143). SHELL181 also supports URI.
• The sixth DOF, in-plane rotation, is a fictitious DOF since
translations fully describe in-plane behavior. As a result, this
“drilling DOF” is usually controlled by a small stiffness.
• SHELL43 and SHELL63 (KEYOPT(1)=1) are membrane-only
elements. This formulation neglects bending and transverse
shear effects.
Advanced Structural Nonlinearities 6.0
In-plane (Membrane) Shell Behavior:
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... Shell Elements
Training Manual
• Most ANSYS shell elements assume that the element and
nodal locations describe the midsurface of the element. For
lower-order shells, use SHELL181 with SECOFFSET (top,
bottom, or user-defined offset). For higher-order shells, use
SHELL91 and 99 (composite shells) to model shell offset to
top or bottom surfaces.
• For nonlinear analyses, various shell elements have different
number of integration points in-plane as well as throughplane. For example, SHELL181 with URI has 1 integration
point in-plane and 5 through-thickness. Check Theory
Manual Ch. 14 for details on specific elements. Note that
SHELL181 can have user-defined integration points through
the thickness.
Advanced Structural Nonlinearities 6.0
Additional Considerations for Shell Elements:
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... Key Features of SHELL181
Training Manual
• Element Technology
– Moderately ‘thick’ shell which accounts for transverse shear
– URI (default) or Incompatible Modes for in-plane behavior
– Employs concept of sections for shell cross-section definition
– Supports layer definition (composite)
• Constitutive Models
– Compared with other shells, SHELL181 supports the most
nonlinear constitutive models including rate-independent
plasticity, viscoplasticity/creep, and hyperelasticity
– For composites, user can combine linear and/or nonlinear
material properties
Advanced Structural Nonlinearities 6.0
• SHELL181 is part of the 18x family of elements which are the
recommended choice for nonlinear applications because of
its wealth of features.
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... SHELL181 Section Definition
Training Manual
– Still supports real constant definition if no shell section defined.
However shell sections are more powerful, flexible, and easier to
use, so this is the recommended input method.
– Section definition allows for input of layers (composite) with any
type of linear or nonlinear material supported by 18x elements.
– Layer orientation and through-thickness integration points
defined through section
– Allows for definition of nodes to be top or bottom surface or
user-defined position
– Allows for easier tapered shell definition through SECFUN
command (does not require use of RTHICK)
– Philosophy of shell section consistent with beam cross-section
definition of BEAM188/189 (discussed in next subsection)
Advanced Structural Nonlinearities 6.0
• SHELL181 has employed a “section” definition
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... SHELL181 Section Definition
Training Manual
Main Menu > Preprocessor > Sections > -Shell- Add/Edit …
Advanced Structural Nonlinearities 6.0
• Define the SHELL181 element type and all applicable material
properties first. Then, activate the Shell Section GUI via:
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... SHELL181 Section Definition
Training Manual
Shell Section GUI > Layup Tab
– Input shell section name (up to 8 chars) and unique ID number.
– Add or delete various layers through simple button.
Advanced Structural Nonlinearities 6.0
• In this example, a sample composite section will be defined.
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... SHELL181 Section Definition
Training Manual
– Thickness, material ID, orientation, and number of integration
points can easily be defined.
– Section offset allows the specification of mid-, top, bottom, or
user-defined surface.
Advanced Structural Nonlinearities 6.0
• The cross-section definition is conveniently laid out
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... SHELL181 Section Definition
Training Manual
Shell Section GUI > Section Controls Tab
– Most of the time, the default values will suffice, but the user can
define hourglass control, transverse shear stiffness, etc.
Advanced Structural Nonlinearities 6.0
• Additional parameters may be specified in the “Controls” tab.
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... SHELL181 Section Definition
Training Manual
Shell Section GUI > Summary Tab
Main Menu > Preprocessor > Sections > -Shell- Plot Section …
Advanced Structural Nonlinearities 6.0
• The summary tab provides a text-based summary, and the
user can also plot the section to visually confirm the layup.
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... SHELL181 Section Definition
Training Manual
–
Define shell section and name
SECTYPE, SECID, Shell, Subtype, Name, REFINEKEY
–
Define each shell layer property
SECDATA, A, Iyy, Iyz, Izz, Iw, J, CGy, CGz, SHy, SHz
–
Define shell offsets
SECOFFSET, Location
–
Define additional shell controls
SECCONTROLS, --, TXZ, --, TXY, ADDMAS
–
Define shell thickness variation
SECFUN, %table%
• Additional information can be found in the following
references
– Structural Analysis Guide, “Shell Analysis and Cross Sections”
– Commands Manual for /PREP7 Section Commands
Advanced Structural Nonlinearities 6.0
• The following commands provide equivalent functionality as
present in the Shell Section GUI:
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... SHELL181 Section Definition
Training Manual
Command: SECNUM,value
Main Menu > Preprocessor > MeshTool
Meshtool > -Element Attributes- Global
Advanced Structural Nonlinearities 6.0
• After specifying the element type and shell section definition,
the default attributes can be set to mesh areas with the
specified shell section.
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Element Technology
... SHELL181 Section Definition
Command: /ESHAPE,1
Utility Menu > PlotCtrls > Style > Size and Shape
Advanced Structural Nonlinearities 6.0
• After meshing, the shell cross-sections can also be
visualized on the mesh.
Training Manual
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... Shell Element Recommendations
Training Manual
• For thin shells, SHELL63 should be used. SHELL63 is also
fine in triangular shape for bending-dominated problems.
SHELL63 supports small-strain, large rotation effects.
• For thick shells (include transverse shear effects), use of
quad-shaped SHELL181 is recommended for lower-order
shells. SHELL93 can be used for higher-order shells (when
curvature is important).
• SHELL181, SHELL91, and 99 can be used for composite
shells.
Advanced Structural Nonlinearities 6.0
Linear Analyses and Large Rotation Analyses:
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... Shell Element Recommendations
Training Manual
• In general, use SHELL181 for homogenous materials or
composites. SHELL181 supports isotropic and kinematic
hardening, hyperelasticity, as well as a host of other
constitutive models. SHELL181 supports thickness change
effects and user-defined number of integration points.
• SHELL93 and 91 can be used for higher-order homogenous
and composite shells, respectively. However they do not
support nearly as many materials as SHELL181, such as
various hyperelasticity or Chaboche plasticity models.
Advanced Structural Nonlinearities 6.0
Nonlinear Material and Finite Strain Analyses:
• With the robust Q-morph quad meshing algorithm in ANSYS,
it shouldn’t be difficult to obtain a sufficient, quad-dominant
lower-order mesh for use with SHELL181 in most analyses.
Consider using SHELL181 for most linear or nonlinear, finiteSeptember 30, 2001
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Element Technology
I. Beam Elements
Training Manual
– The physical dimensions are usually based on the distance
between supports/constraints or the wavelength of the
modeshapes of interest.
– A ratio of 20:1 or 30:1 of length to cross-sectional dimensions
can be used as a general guideline to determine the applicability
of beam elements.
– As a result of this assumption, stresses through the thickness of
the cross-section (element y- and z-axes) are assumed to be
negligible
Advanced Structural Nonlinearities 6.0
• Beam elements are used to model structures which are
longer in one dimension than they are in the other two
dimensions, and the dimensions are based on the physical
structure, not the element size
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Element Technology
... Beam Elements
Training Manual
BEAM4 and others
Beam Theory Euler-Bernoulli
Transverse
Neglected (but can
Shear
add as shear factor)
Torsion
Approximate
Warping
Unrestrained
Kinematics
Materials
Bending
Response
Section
Definition
(BeamTool)
Tapered
Beams
Small strain
Linear;
Homogenous
Cubic
BEAM44 only
BEAM44, 54
BEAM188/189
Timoshenko
Included
Poisson's Equation
Unrestrained or
Restrained
(KEYOPT(1))
Finite strain
Plasticity, Creep;
Composite
Linear or Quadratic
Yes, and results
calculated at userdefinable section
integration points
Not currently
available
Advanced Structural Nonlinearities 6.0
• In ANSYS, there are mainly two distinctions of beam
elements:
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... Beam Elements
– The cross-sectional plane initially
normal to the neutral axis of the beam is
assumed to remain straight and normal
to the neutral axis during loading. This
assumption excludes shear
deformations.
Moderately Thick Beams (Timoshenko)
– The cross-sectional plane initially
normal to the neutral axis of the beam is
assumed to remain straight but not
remain normal to the neutral axis during
loading. The shear strain, as a result, is
constant across the section.
Advanced Structural Nonlinearities 6.0
Thin Beams (Euler-Bernoulli)
Training Manual
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... Beam Elements
Training Manual
– All cross-sections, except for solid circular sections, can warp.
– Thin walled open sections exhibit substantial warping. Torsional
stiffness of such sections is negligible, and restraining of
warping provides resistance against twist.
Advanced Structural Nonlinearities 6.0
Warping (Unrestrained vs. Restrained)
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... Key Features of BEAM188/189
Training Manual
• Element Technology
– Moderately ‘thick’ beam which accounts for transverse shear
– Unrestrained or restrained warping
– Finite strain capability
– Employs concept of sections for beam cross-section definition
– Supports multi-material cross-section definition (composite)
Advanced Structural Nonlinearities 6.0
• BEAM188 and 189 are part of the 18x family of elements
which are the recommended choice for nonlinear
applications because of its wealth of features.
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... Key Features of BEAM188/189
Training Manual
– Compared with other beams, BEAM188/189 support the most
nonlinear constitutive models including rate-independent
plasticity, and viscoplasticity/creep.
– Although ANSYS 6.0 calculates equivalent strains correctly for
SOLID, PLANE, and SHELL elements, the user must still set
effective Poisson’s ratio (AVPRIN,,effnu) for BEAM elements
(and other line elements).
• Be sure to select BEAMs only
via select logic
• Because BEAMs are assumed
to be incompressible, the user
can set effnu to 0.5 when postprocessing BEAMs only.
Main Menu > General Postproc > Plot Results > -Contour Plot-
Advanced Structural Nonlinearities 6.0
• Constitutive Models
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... BEAM188/189 Features
Training Manual
– KEYOPT(1) allows for unrestrained or restrained warping.
– KEYOPT(2) allows for change in cross-section as function of
axial stretch to preserve volume.
– KEYOPT(4) allows for specification of output. Shear stresses
caused by flexural/transverse
and torsional loads. Either or
both can be output.
– KEYOPT(6-9) specify printout
controls (text output)
Advanced Structural Nonlinearities 6.0
• The various element options (KEYOPT) are shown below.
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... BEAM188/189 Section Definition
Training Manual
– Real constants supply extra parameters, but main definition of
beam cross-section is done through sections.
– Section definition allows for easier input from library of common
cross-sections. User can also define his/her own cross-section.
– Each cross-section is made of a number of ‘cells’ with section
integration points. For nonlinear material applications, more
integration points allow for accurate calculations.
– Allows for definition of nodes to be at centroid, shear center, or
user-defined location/offset.
– Philosophy of beam section consistent with shell cross-section
definition of SHELL181 (discussed in previous subsection)
Advanced Structural Nonlinearities 6.0
• BEAM188/189 utilize a “section” definition
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... BEAM188/189 Section Definition
Training Manual
Main Menu > Preprocessor > Sections > -Beam- Common Sectns …
Specify a unique Section ID number
Specify a 8-character name
Selection of common beam crosssections. User can also define his/her own
cross-section via an arbitrary mesh.
Advanced Structural Nonlinearities 6.0
• Define the BEAM188 or 189 element type and all applicable
material properties first. Then, activate the BeamTool via:
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Specify node location
(centroid, shear center, user-defined)
An pictorial representation of the beam
Parameters to define the selected cross-section
Specify the density of the section mesh
Preview the cross-section and/or section mesh
Advanced Structural Nonlinearities 6.0
• All of the beam cross-section parameters
can be defined through the BeamTool
Training Manual
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... BEAM188/189 Section Definition
Training Manual
Note that crosssectional properties
are automatically
calculated and
shown on the right.
(They are displayed
for summary
information only.)
The centroid and
shear centers are
also marked with
symbols.
Advanced Structural Nonlinearities 6.0
• Sample cross-section shown (“Preview” and “Meshview”)
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... BEAM188/189 Section Definition
Training Manual
Main Menu > Preprocessor > MeshTool
Meshtool > -Element Attributes- Lines
Advanced Structural Nonlinearities 6.0
• When assigning attributes to lines, note that sections must
be assigned. Moreover, an orientation keypoint needs to be
defined.
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... BEAM188/189 Section Definition
Command: /ESHAPE,1
Utility Menu > PlotCtrls > Style > Size and Shape
Advanced Structural Nonlinearities 6.0
• After meshing, the beam cross-sections can also be
visualized on the mesh.
Training Manual
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... Notes on BEAM188/189
Training Manual
– No warping is considered due to direct shear stresses
• Applicability to non-homogeneous cross section is limited by
the approximations of beam theory
– Validate by solid/shell models first
I
• Note that if restrained warping is
considered, ensure that nodes at
intersection do not share WARP DOF.

J
M

L

K
J  L  0
cp,,ux,j,l
cp,,uy,j,l
cp,,uz,j,l
cp,,rotx,j,l
cp,,roty,j,l
cp,,rotz,j,l
Advanced Structural Nonlinearities 6.0
• Beam elements 188/189 are based on a first order shear
deformable theory
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... Beam Element Recommendations
Training Manual
• For ‘thin’ beams (neglect transverse shear), BEAM4, 188, 189
can be used. Note that BEAM4 uses Hermitian polynomials
for shape functions, resulting in cubic response in bending.
BEAM188/189 have linear and quadratic responses in
bending, respectively, so a finer mesh may be required.
• For ‘moderately thick’ beams, use of BEAM188/189 is
recommended since it includes transverse shear
deformation.
Advanced Structural Nonlinearities 6.0
Linear Analyses and Large Rotation Analyses:
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... Beam Element Recommendations
Training Manual
• BEAM188/189 is recommended for finite strain applications
since it supports many plasticity and creep models,
composite definition, and finite strain capabilities.
BEAM188/189 can also serve as shell stiffeners with an
appropriate section offset.
• In general, the enhanced pre- and post-processing
capabilities of BEAM188/189 make it the preferred choice for
linear or nonlinear applications.
Advanced Structural Nonlinearities 6.0
Nonlinear Material and Finite Strain Analyses:
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References for Further Reading
Training Manual
1. Non-Linear Finite Element Analysis of Solids and Structures
Vol.1 and 2, M.A. Crisfield, John Wiley & Sons, 1996 & 1997.
2. Nonlinear Continuum Mechanics for Finite Element Analysis,
Bonet and Wood, Cambridge University Press, 1997.
3. Introduction to the Mechanics of a Continuous Medium, Malvern,
Prentice-Hall, 1969.
Advanced Structural Nonlinearities 6.0
Some useful references on numerical theory:
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