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

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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