實驗力學研究室 1 Common Model and Element Types. 實驗力學研究室 2 Common Modeling...
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Transcript of 實驗力學研究室 1 Common Model and Element Types. 實驗力學研究室 2 Common Modeling...
實驗力學研究室
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Common Model and Element Common Model and Element TypesTypes
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Common Modeling Types Planar simulations
• Plane stress
• Plane strain
• Axisymmetric
3Dsimulation and modeling
• Beam simulation
• Symmetry or anti-symmetry
• Plate or shell models
• Solid models
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Plane Stress Modeling
The definition of plane stress requires that the behavior of interest
occurs in such a manner that there is no stress component normal to
the plane of action. This means that one of the three principal
stresses is zero.
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Identifying Plane Stress Models
A practical method for identifying plane stress opportunities is to
look for geometry or systems that are essentially extrusions of a
group of planar cross section. The loads and constrains must be
defined such that all resulting defections allow the planar cross
sections to remain in their initial plane.
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Geometry in a Plane Stress Model
If plane stress is valid, it can be assumed that any cross section or
slice parallel to the generating cross sections would have the same
stress distribution as any other. Consequently, the geometry for the
model can be generated by cutting a solid model or assembly with
an appropriately oriented plane.
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Plane Strain Modeling
While the depth of the plane stress model is usually small compared
to cross-sectional size, the depth of a plane strain model must be
large in comparison to the section. In fact, it is common to assume
an infinite depth so that any effects from end conditions are so far
removed from the modeled cross section they can be ignored.
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Axisymmetric Modeling
In an axisymmetric model, the geometry and
boundary conditions are or can be assumed to
be revolved 360° about an axis.axisymmetry
problems are planar models in which the solv
er understands that the modeled half of the cr
oss section is revolved 360°.
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Identifying an Axisymmetric Model
When attempting to identify axisymmetric problems, the most obvio
us means of qualification is that the base feature of the part should b
e a solid of revolution. In many cases, axisymmetry is still valid if th
ere are small features that breaks up the revolution.
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Loading
Loading is applied to an axisymmetric model differently in different
solvers. Some codes allow you to apply the actual, total load to the
model, while others forces you to divide it by 2π, or 6.28, before app
lying the load to the model.
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Constraints
Axisymmetric models only require constrains parallel to the axis of r
evolution. Being a planar approximation, no out-of-plane translation
or rotation is permitted by definition. In addition, the model is constr
ained automatically by its axis of revolution.
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Symmetry
Reflective Symmetry
Symmetry conditions require that the geometry and boundary
conditions are , or can be approximated as being, equal across one,
two, or three planes.
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Near-symmetry
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Loading
The total load applied to a symmetric model should be divided by
the number of symmetry planes used.
Constraints
The constraints on a reflective symmetry model define the symmetry
to the solver. The constraints on a solid model must prevent
translation through the plane of symmetry on the entire cut face and
the constrains on beams and shell models must also prevent rotation
in the components parallel to the cut planes.
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Cyclic Symmetry
Cyclic symmetry is a more specialized condition where features that
are repeated about an axis can be modeled by a single instance of
that feature.
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Boundary Conditions
In cyclic symmetry, each instance of the feature must see the same
boundary conditions in its respective frame of reference. Acceptable
loading might be centrifugal forces, radial displacement due to a
press fit, or uniform wind or fluid resistance due to spinning.
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Using Reflective Symmetry to Approximate Cyclic Symmetry
The restrictions on using planar symmetry to model cyclic symmetry
are that the geometry conforms to the definition of cyclic symmetry
and that the only loading and resultant displacement are radial or
coaxial.
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Beam Models
Two Basic Types of Beam Elements
Most beams can be categorized as able to transmit moments or not
able to transmit moments.
Rod Elements
Common names for beam elements which cannot carry moments are
rod, bar, or truss elements.
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Beam Elements
Beam elements are defined by the geometric position of the end poin
ts, a material, a cross-sectional area, an orientation vector, the area
moments of inertia, and torsional stiffness. A restriction on beam ele
ments is that the cross sections specified remain planar and perpendi
cular to the axis throughout the solution.
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Beam Coordinate System
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Stress Recovery in Beams
Tensile and compressive stress is calculated for the entire beam, but
reported bending stress and stress from torsion will depend on your
choice of stress recovery points.
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Beams in Torsion
Beams in torsion also require the specification of a torsional constan
t. For circular cross sections, the torsional constant, K, equals the pol
ar moment of inertia, J.
Section Orientation
While two I-beams rotated 90° about their axes from each other hav
e the same cross-sectional properties, they obviously will not suppor
t the same load in the orientations depicted in Fig. 4.31.
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Plate and Shell Modeling
The terms plate and shell are often used interchangeably and refer to
surface-like elements used to represent thin-walled structures.
板結構特性分析薄板( thin plate )係基 kirchhoff 假設:
1. 板之材料是彈性的、均質的( homogeneous )、且為等向性的( isotropic )。
2. 板之最初形狀是平的( flat )。
3. 板之厚度須小於板之其他尺寸,以最短邊之長度大於厚度的10 倍以上。
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4. 板厚度方向之側向位移須小於板之厚度,以最大側向位移與厚度比在 1/10~1/5 為小變形範圍。
5. 板內垂直於中性面( neutral surface )之法線,於變形後依然為中性面之法線。
6. 板之中性面之斜率變形遠小於 1 。
7. 板之側向位移以中性面之側向位移表示,此側向位移並垂直於中性面。
8. 垂直於中性面之應力,亦即厚度方向之應力可忽略。
9. 平行於中性面之外力所產生之剪應變,通常遠小於彎曲應變,也可以忽略。
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在厚板理論分析,係基於 Mindlin 假設考慮了厚度方向之剪變形( shear deformation )效應:
薄板:
厚板: yzyxzx x
w
y
w
,
x
w
y
wyx
,
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殼結構特性分析傳統小變形之薄殼理論假設:
1. 殼厚度比其他尺寸小。
2. 應變與應力相當小。
3. 側方向之正向應力( σz )比其他方向正向應力相較較為小可忽略,故令 σz = 0 。
4. 垂直於中性面之法線變形後,仍然為直線且垂直於中性面。
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殼元素之特性分析板結構分析可概分為:
1. 剛板( stiff plate ):具有側向剛性( flexural rigidity )可承受彎曲、扭曲及側向力,其特性如同一為維樑( beam )結構,僅擴充為二維板。
2. 薄模板( membrane ):不具側向剛性,只可承受軸向及沿中性面剪力,即為薄膜效應。
3. 撓曲板( flexible plate ):係剛板及薄膜兩種效應之組合。
4. 厚板( thick plate ):則加上厚度方向剪應變效應。
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殼結構包含:
1. 剛殼( stiffshelll )
2. 薄膜殼( membrane shell )
3. 撓曲殼( flexible shell )
4. 厚殼( thick shell )
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殼元素節點之自由度,以線性四邊形殼元素為例,每個節點自由度為 6 個,三個方向位移( u,v,w )及三方向旋轉角度( θx ,θ
y , θz ),而其對應之節點外力,分別為三個方向力( fx , fy ,fz )及三方向力矩( Mx , My , Mz )。 殼元素之分析設定如下:
1. 僅彎曲效應:對應於具剛板或剛殼特性之薄殼結構,亦即進可使板殼彎曲或扭曲之現象。
2. 僅薄膜效應:對應於具薄膜特性之薄殼結構,只可承受延中性面之軸向或剪力,沒有彎曲力矩、扭曲力或側向力。
3. 包含彎曲及薄膜效應
4. 剪應變效應:適用於殼結構,即 , 在 ~ 之間,加入 γyz 、 γxz 之效應。
xR
t
yR
t
5
110
1
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Orientation
The default interpretation of a shell mesh is that it is centered at the
mid-surface of the modeled geometry. Many codes offer the ability
to numerically offset the mid-surface of selected elements to better
represent the geometry.
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Shell meshes are usually a constant thickness but some codes also
provide the option for tapering a shell mesh.
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Shell elements have an orientation similar to beam elements. The x
and y coordinate axes are oriented in the plane of the shell, and z
axis is normal to it. The element is defined with a top and bottom
side. The two primary requirements for understanding the
orientation of a shell mesh involve using pressure loads and
evaluating bending stress. Pressure loads on shells are typically
oriented from the bottom to the top.
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Stress Recovery
Interpolates the stress in the outer fibers of a shell. Sell elements ass
ume a linear stress distribution across the defined thickness. It is imp
ortant to know which side of the element has the stresses of interest
and to adjust the stress display accordingly. It is equally important th
at the shell normals be consistently aligned so that there are no anno
ying discontinuities in the contour results due to flipped shell normal
s.
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Identifying Shell Model Candidates
Shell modeling is appropriate is that the wall thickness of the part or
assembly is small compared to the overall size or surface area of
system 10:1.
One rule of thumb: if the part would be understandable when
modeled with zero thickness surfaces to someone unfamiliar with its
actual form, a shell model is a likely candidate.
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Two shell elements that are continuous but not co-planar will be
interpreted with some “virtual” overlap of their thickness. This can
be ignored for gross displacement models but the stress at a corner
will contain some error resulting from such overlap.
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Another good indicator of the validity of a shell element is spatially
located at the mid-surface of the geometry it is simulating. Fig 4.38
shows the steps required to turn a simple I-beam into a shell model.
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The cast cross brace shown in Fig. 4.40 has relatively thick
transitions at the ends which would not intuitively suggest shells.
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However, under the loading which resulted in the deformed shape in
Fig. 4.41, the behavior of interest is far from questionable geometry
and a solid model would not have significantly improved the
accuracy.
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Accuracy
Shell models, where applicable, may be significantly more accurate
than solid models in bending with reasonable solution times.
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Additional Benefits
First, it is a simple matter to delete the mesh on only one or two
surfaces that might warrant topology changes. After adjusting the
geometry, these surfaces can be meshed and merged with the
previous mesh in a matter of minutes. Second, making a change to
wall thickness is as simple as typing in a different number. Finally,
achieving convergence by locally refining an existing shell mesh is
much more straightforward than in a solid model.
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Solid Element Modeling
Identifying Solid Model Candidates
The ideal solid model is a bulky, low aspect ratio part.
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Solid Element Basics
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Special Elements
Spring Elements
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Damper Elements
Damper elements provide dashpot type damping for dynamic
models only. With units of force per velocity, they are rarely used in
a static analysis.
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Mass Elements
The primary use of a mass element is to idealize the mass of a
component that provides a contribution to the loading of the part
being studied, but is much more rigid and/or too complex to include
as a mesh. Mass elements are used to represent engines in cars or
motorcycles, display tubes in televisions or monitors, and pumps
and motors on models of machinery. Mass elements are typically
single node elements with no geometric properties.
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Rigid Elements
Rigid elements may go by the terms rigid links, links, or multi-point
constraints (MPCs). They connect the degrees of freedom of one no
de or entity to the degrees of freedom of one or more other nodes or
entities. A node tied to X translational degrees of freedom of another
node is mathematically constrained to translate an equivalent X dista
nce for any X displacement of the independent node. Another way to
look at it is that their relative position in the X direction is fixed. A s
imilar link between rotations or any combination of translations and
rotations can be made.
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Uses of Rigid Elements
Rigid elements are extremely valuable in assembly modeling
because you can easily tie together meshes that do not touch or lack
aligned nodes.
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Rigid elements are also critical in transitional meshing due to the inh
erent incompatibility of different element types. These elements can
tie the rotational degrees of freedom of a shell element to the translat
ional degrees of freedom of an adjacent solid.
While rigid elements are versatile and handy, care must be taken to a
void overstiffening a model due to overuse of rigid connectivity.
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Contact Modeling
1. 間隙元素( gap element )。
2. 介面元素( interface element )。
彈簧間隙元素模型如下,節點 1,2 分別隸屬兩個物體, Δu=u2-u
1 代表間隙,當 Δ u>0 ,代表兩物體未接觸;當 Δ u<0 ,代表兩物體接觸,則所得到的緩衝力須代入接觸接觸物體 u2 , u1 使Δ u = 0 不形成干涉( Interface )或是嵌入( penetration )現象。
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節點對節點間隙型接觸元素,模擬方式:
1. 模擬被接觸物體之支撐勁度( supporting stiffness )。
2. 模擬兩物體間之力傳遞( force transfer )。
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接觸元素形式如下:
1. 節點對節點( node-to-node )接觸元素。
2. 節點對線( node-to-line )接觸元素。
3. 節點對面( node-to-surface )接觸元素。
4. 介面型接觸元素。
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Slide Line Elements
Slide line elements are essentially contact curves that will allow
significant relative sliding between the contacting parts. A slide line
is created via the connectivity of two selected sets of curves or
nodes that define the curves.
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General Contact Elements
General contact elements are conceptually the simplest, bur the most
computationally intensive. The user will typically define a contact
pair consisting of two surface, curves, surface meshes, or edge
meshes.
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Crack Tip Elements
Elements have been developed in some codes to capture the
singularity at a crack tip. They go by the names of quarter point,
crack tip, or singularity elements.
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Part versus Assembly Modeling
Component Contribution Analysis
Isolating each part or rigidly connected subassembly for a separate
initial analysis. The interaction of the mating components should be
accounted for using boundary conditions, or in extreme cases,
simplified representations of the other parts and, possibly, contact.
Turning the One Disadvantage into an Advantage
Disadvantage it takes longer to examine each part individually
before building the assembly model.
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Considering interaction in terms of Boundary Conditions
It forces you to dedicates some serious thought to the interactions of
the external loads with each of the parts in the assembly and
prepares you to be able to evaluate the end results with confidence.
Isolating the Performance of Each Part
Consider each component on its own merits and allows you early
identification of the weak link or links in the assembly.
Keep It Simple…
Look for the simplest means of tying parts together to minimize the
error introduced.
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Transitional Meshing
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Use Test Models to Debug Idealizations