Пресс-релиз популярных книг

Авторы: 111 А Б В Г Д Е Ж З И Й К Л М Н О П Р С Т У Ф Х Ц Ч Ш Щ Э Ю Я

Книги: 164 А Б В Г Д Е Ж З И Й К Л М Н О П Р С Т У Ф Х Ц Ч Ш Щ Э Ю Я

На сайте 111 авторов, 92 книг, 72 статей, 5913 глав.

9.1 Problem and Element Classification

Back

The first step in any finite element (FE) analysis is to build a model of the physical structure to be

analyzed. This is an important step and normally requires extensive time and interaction with the analyst.

The adequacy of the model, assumptions involved, and types of elements used for a specific structure and

analysis type establish the accuracy level of the FE analysis. Much research effort is being devoted to

automating this process by providing options for automatic mesh generation, automatic mesh

refinement, error estimation, and error bounds. Such research has resulted in a significant reduction in

the time needed for this step but the role of the analyst is still dominant and vital in obtaining a good

mesh and model for the problem.

The modeling process can be divided into steps as follows:

* Build a geometric database for the structure. This includes description of the characteristic

geometric features of the structure, such as boundaries, holes, intersections, curvatures, etc., in the

finite element program database. The level of geometric detail has an important effect on the

accuracy and the size of the model.

* Build a FE model for the geometric model. This may include important aspects such as:

* Establishing the type of analysis to be performed,

* Choosing the appropriate element or elements for building the model,

* Considering aspects of symmetry in the structure, and

* Establishing critical areas for increasing mesh density.

* Apply constraints and loading boundary conditions.

* Establish the material model(s) to be used in the analysis.

* Perform various options of model checks.

* Solve sample load cases and compare the results with hand calculations or experimental results in

order to check the behavior and response of the model.

* Fine tune the model based on the results obtained from sample load cases.

Most of the time, the above steps are rather linked together, and an overall knowledge of the problem is

required to perform a specific step. An important decision that should be made at the beginning of the

analysis is to identify the category of the problem. This will have an impact on the first three steps

mentioned above. In the first part of the present section, we provide a brief discussion on the geometric

modeling aspects of the problem. The rest of the section then provides general guidelines on classifying

the problem into one of the main categories available in typical commercial finite element programs;

namely, truss, beam, two-dimensional, shell, or three-dimensional problems. We also provide simple

examples for each type of problem. The concepts of element choice and problem classification are

summarized in a table at the end of the section.

9.1.1 Geometric Modeling

Geometric modeling simply means transforming a physical problem into a geometric database in a FE

program. The process is very similar to creating an engineering drawing or a model in a CAD program.

Most FE programs have built-in preprocessors that are dedicated to generating the geometry database.

Generally, FE preprocessors have similar capabilities to those available in CAD programs. In many cases

an existing geometric database may be available for the structure in a CAD program and may be

imported into the FE program database. However, there can be translation problems between the two

databases, especially in three-dimensional and shell structures. In most cases, it is faster to regenerate the

geometric database for the problem directly using the FE program.

9-2 Vibration and Shock Handbook

Finite element programs use specific building blocks or entities to build the geometry of a structure

(Kamel, 1991; NISA User’s Manual, 1992; ANSYS User’s Manual, 2003). The theory on which geometry

generation is based is quite simple and basically relies on parametric cubic modeling of curves and

surfaces in space. Figure 9.1 shows the basic entities used by most programs for building a geometric

model. We define these basic geometric entities and briefly discuss their use in practical modeling of

structures.

9.1.1.1 Key Point

A key point is a coordinate location in space. In two-dimensional space, a key point is uniquely defined

by two coordinates (e.g., x; y; r; u). In three-dimensional space, a key point is uniquely defined by three

coordinates (e.g., x; y; z; r; u; z). Key points are normally the starting building blocks in the geometry.

Connection between key points will generate lines, surfaces, or volumes. On the other hand, most

programs will be capable of extracting key points from the end points or corners of lines, surfaces, and

volumes. Key points should not be confused with nodes, and should be considered only as spatial

locations in space. One may place a node at the same location as a key point or leave the location without

node generation.

9.1.1.2 Line or Line Segment

A line or line segment is a portion of a cubic spline curve bounded on both ends by a key point. A line

may be straight or curved. The curvature of a line is limited only by its parametric cubic equation in the

program. If a physical curve in the structure is presented by up to and including a third-order parametric

cubic equation, it may be modeled exactly by a single line. In many practical situations, however, this is

not the case. As an example, the parametric equation of a circle is not cubic and the recommended way to

model a circle accurately is to break it into at least four lines. Breaking a space curve into many lines will

always increase the accuracy of the geometric modeling but entails increasing the complexity of the

model. The analyst should be careful in situations where the order of the line to be modeled is not known.

A common case is the generation of the line of intersection between two surfaces. In such situations, it is

important to break the intersection line into a few separate lines.

Most programs provide extensive methods for line generation. These may include generation by

joining two grid points, cubic spline fitting of four grid points, best fitting a curve between several grid

points, extracting the edges of a surface or a volume, intersection of surfaces, and mirroring and copying

other lines.

9.1.1.3 Area or Patch

An area or a patch is a portion of a bi-cubic surface completely bounded by three line segments (for

triangular areas) or four line segments (for quadrilateral areas). The same limitations discussed above for

using line segments may be extended here by realizing that the area is modeled by parametric cubic

equations in two directions representing the two edges of the area.

Most programs provide extensive methods for generating areas or patches. These include generation

by sweeping the space between two lines, filling in the area between four edge lines, rotating a line about

an axis, extracting the boundary surfaces of a volume, and intersection between volumes.

FIGURE 9.1 Geometric modeling entities.

Finite Element Applications in Dynamics 9-3

9.1.1.4 Volume or Hyperpatch

A volume or a hyperpatch is a portion of tri-cubic solid completely bounded by four areas (for

tetrahedron volumes) or six areas (for brick volumes). The equations used to model the edge lines of a

volume are still parametric cubic equations and the same restrictions discussed for lines can be extended

to a volume in the three directions of the volume.

As for line and area generation, most programs provide a wide variety of methods for generating

volumes. These may include generation by sweeping the volume between two surfaces, filling the volume

between bounding surfaces, rotating an area about an axis, and copying and mirror imaging the existing

volume.

An important addition in the generation of volumes is the ability of many programs to use solid

primitives as building blocks (Kamel, 1991; ANSYS User’s Manual, 2003). These solid primitives may

include tetrahedrons, cubes, cylinders, conical volumes, spheres, torus elements, and other standard

volumes. These may be used as building blocks that may be combined, subtracted, or intersected with

each other. Many programs provide simple Boolean operations to use for such processes.

9.1.2 Discrete Element Types in FE Programs

Most commercial FE programs have extensive element libraries that may be used in static and dynamic

analyses. For dynamic analysis, it may be convenient to classify elements into discrete and continuum

types. Discrete types include concentrated (lumped) mass and inertia, spring, and damper elements,

whereas continuum (distributed) types include all other one-dimensional (1-D), two-dimensional

(2-D), and three-dimensional (3-D) deformable elements. In this section, we briefly discuss the discrete

type of elements whereas the continuum type will be discussed in detail in subsequent sections.

9.1.2.1 Concentrated Mass/Inertia Element

A concentrated mass/inertia element represents a structural mass and moment of inertia concentrated at

one point and has six DoF: three translational and three rotational. The mass and rotary (moment of)

inertia may be assigned different values in the three coordinate directions (see Figure 9.2), even though,

typically, the mass is the same in all three directions (see Chapter 8). The element is rigid with no

geometrical properties and it only contributes to the global mass matrix of the structure. In building up a

model, the element may be attached to a structural node of other deformable or elastic elements or be

positioned in space and attached to structure nodes through rigid elements or elastic spring and/or

damper elements. Most FE programs provide rotary inertia quantities for various components of the

geometric model as part of the standard preprocessing data. These can be used to model parts of the

structure as lumped mass and inertia that may be connected to the structure through elastic elements.

REMARKS

* In most modeling cases, it is faster to regenerate the geometric database of the problem

directly using the FE program, especially in complicated and three-dimensional models.

* The basic building blocks in geometric modeling are key points, lines, areas, and volumes.

* Many programs provide the ability to create solid primitives as building blocks. These solid

primitives may include tetrahedrons, cubes, cylinders, conical volumes, spheres, toruses,

and other standard volumes.

* Boolean operations are normally used to combine, subtract, or intersect various geometric

entities.

9-4 Vibration and Shock Handbook

An example of the use of a concentrated mass

and inertia element is the modeling of a heavy,

rigid machine mounted on an elastic support.

The mass and inertia effects of the machine may

be represented by a concentrated or lumped

mass/inertia element at the center of the machine.

Care must be taken in connecting this element

to the deformable elements of the structure or

the support. In general, a rigid link may be used

to connect the mass/inertia element to the

nearest node of a deformable element rather than

placing the mass/inertia element directly on that

node (Cook et al., 1989). This will generally

account for proper interaction between translational

and rotational DoF of the mass and the

structure.

It should be noted that some computational

algorithms might have problems with zero diagonal

values of the mass matrix. This happens if the

inertia terms of the mass/inertia element are

assigned as zero. This can be avoided by always

assigning an arbitrary and small value to the rotary

inertia terms.

9.1.2.2 Spring and Damper Elements

Most programs provide 1-D spring and damper elements that can be used to model hydraulic cylinders,

discrete dampers, and shock absorbers. The element normally has one spring constant and one damping

coefficient (along the element axis defined by the nodes I and J as shown in Figure 9.3). It is easy to model

3-D stiffness and damping characteristics by replicating the element in the required directions. A mass

may be attached to one or two nodes of the element. The force transmitted through the element nodes is

the sum of the spring and the damping forces along the element axis.

9.1.3 Truss and Beam Problems

Three-dimensional truss and beam elements are shown in Figure 9.4. The main difference between the

two elements is the fact that beams may support bending moments and have rotations as extra DoF at

Mx, My, Mz

Ixx, Iyy, Izz

FIGURE 9.2 Three-dimensional mass/inertia element.

FIGURE 9.3 One-dimensional spring/damper

element.

REMARKS

* A concentrated mass/inertia element may be used to model lumped mass/inertia at specific

points in the structure. Likewise, spring or damper elements may be used to model stiffness

and damping characteristics between two points in the structure.

* A concentrated mass/inertia element represents a structural mass and inertia at a point and

has six DoF: three translational and three rotational. Different mass and inertia values may

be assigned in different directions.

* Three-dimensional stiffness and damping characteristics between two nodes may be

modeled by replicating 1-D spring or damper elements in the required directions.

Finite Element Applications in Dynamics 9-5

each node. This means that a beam element may have loads along the beam axis and not necessary only at

the end points.

The following conditions should be satisfied for a structure to be classified as a truss or a beam:

* Geometry:

* Thin, slender, straight bars or rods with pin joints at both ends. Depending on the application,

the element may be considered as a truss, a link, a cable, a spring, etc.

If the joints are fixed or the bar is curved, the problem will be classified as a beam type. Fixed

joints may be created by completely fixing the beam ends to a wall or support, or to another

member. This is normally realized if the joint is built-in, welded, or fully bolted to the support

or to other members.

* Geometry may be 1-D, 2-D, or 3-D.

* Loading:

* For truss problems, loading may only be by tension or compression of the members. This may

be achieved by having only concentrated loading at the joints (no bending moment) and no

loadings along the member.

For beam problems, loading may be in any direction and may be applied along the member

axis. This will generally create a bending moment and the member should have rotations as DoF.

* Thermal loading may also be applied to the member.

* Body or inertia forces will generally create a bending moment and violate the truss condition and

so may be applied only to beam problems. Such loads may be applied, however, to truss elements

under the simplification of assuming the inertia effects to be applied only at the nodes or the

joints and ignoring the bending moment that will be created on the member.

Degrees of freedom: For 3-D truss elements, the DoF per node are the displacement in the three

coordinate directions: ux ; uy ; and uz : Two-dimensional truss members only have ux and uy as DoF. Threedimensional

beam elements have all six DoF: three translational, ux ; uy ; and uz ; as well as three rotational,

ux ; uy ; and uz : Two-dimensional beam elements have ux ; uy ; and uz as DoF.

Element shapes: Various element shapes commonly available in commercial finite element programs

are shown in Figure 9.5. The cross section of the element may be a solid or hollow prismatic section, e.g.,

a rectangular, circular, trapezoidal, or thin-walled section or a channel, thin-walled tubular, I-section, etc.

FIGURE 9.4 Three-dimensional truss and beam elements. (Nodal rotations for the beam element are only shown

for node-j.)

2-node (linear)

Truss and beam

3-node (quadratic)

Beam only

4-node (cubic)

Beam only

FIGURE 9.5 Various element shapes for truss and beam elements.

9-6 Vibration and Shock Handbook

The analyst will generally be required to identify the orientation of the beam cross section by

specifying local axes or identifying key points for the program to locate the local axes. Most programs are

capable of modeling variable cross section beam elements to avoid excessive subdivision of the beam into

smaller elements.

Example applications: Figure 9.6 shows sample applications for the use of truss and beam elements.

Cable: tension-only truss elements

Heavy I-beam: beam elements

P3 P1 P2

Truss structure

Springs may be modeled as truss

members. The cross member

should be modeled as beam.

P1 P2

Frames should be modeled as

beam structures

Cantilever with distributed load

Beam model

Beam with fixed ends

Beam with moment load

Thin arches (beam model)

FIGURE 9.6 Typical applications for truss and beam elements.

REMARKS

* Truss elements have translational DoF whereas beam elements have both translational and

rotational DoF at the nodes. Three-dimensional truss elements have 3 DoF/node (ux ; uy ;

and uz ), whereas 3-D beam elements have 6 DoF/node (ux ; uy ; uz ; ux ; uy ; and uz ).

* Truss structures may only carry loads at the end of each truss whereas beam structures may

also carry loads along the beam length.

* Curved members should be modeled with beam elements.

* Problems with body and inertia forces should generally be modeled as beam elements. If

lumping of the body forces is assumed and the effect of the bending moment created by

body forces is neglected, the effect may be modeled with truss elements.

Finite Element Applications in Dynamics 9-7

9.1.4 Two-Dimensional Problems

9.1.4.1 Two-Dimensional Plane Stress Problems

Figure 9.7 shows a typical structure, which may be considered as a 2-D plane stress problem.

The following geometry and load conditions should be satisfied for a structure to be categorized as a

2-D plane stress structure (Boresi and Chong, 2000).

Conditions:

* Geometry:

* A flat, thin surface in one plane (e.g., xy-plane), simply connected or multiply connected with a

small thickness in the third direction (e.g., z-direction).

* The boundary of the structure in the xy-plane may be straight or curved.

* Loading:

* Loading is restricted to the plane of the structure (e.g., xy-plane).

* Thermal loading may also be applied to the plane (i.e., T ¼ Tðx; yÞ).

* Body or inertia forces may be due to linear or angular acceleration in the plane of the structure.

Degrees of freedom: For 2-D elements, the DoF per node are the displacements in the two coordinate

directions, ux and uy :

Typical element shapes (see Figure 9.8): Elements may be triangular or quadrilateral. Most programs

provide the option of having 3, 6, or 9 nodes for triangular elements and 4, 8, or 12 nodes for

quadrilateral elements. Increasing the number of nodes will increase the element accuracy at the expense

of having more DoF and requiring more CPU time to solve the problem. This is normally called a

“p-conversion” approach in finite elements. The same goal may be achieved by increasing the number of

elements while fixing the number of nodes per element, which is usually called “h-conversion” (Zeng

et al., 1992; Babuska and Guo, 1996). The choice of method to obtain greater accuracy is rather arbitrary

and mostly depends on the availability of the option in the program. Only a limited number of programs

provide extensive “p-conversion” options.

Example applications: Figure 9.9 shows some typical examples that may be modeled using a plane stress

assumption. It should be noted such models cannot capture any out-of-plane modes of vibration. If such

vibration modes are of concern, the model should be capable of capturing lateral displacement DoF and

out-of-plane rotations. This may be realized by using shell or 3-D solid elements as will be discussed in

the following sections.

δ1

δ2

Free surface

free

FIGURE 9.7 A general 2-D plane stress structure.

9-8 Vibration and Shock Handbook

9.1.4.2 Two-Dimensional Plane Strain Problems

Figure 9.10 shows a typical structure that may be considered as a 2-D plane strain.

The following geometry and load conditions should be satisfied for a structure to be categorized as a

plane strain structure (Boresi and Chong, 2000).

Conditions:

* Geometry:

* A flat surface in a plane (e.g., xy-plane), simply or multiply connected with large thickness in the

third or z-direction (with much larger dimensions than in the xy-plane).

* The boundary of the structure in the xy-plane may be straight or curved.

* Loading:

* Loading is restricted to the plane of the structure (e.g., xy-plane) with possible uniform loading

or constraint in the z-direction. The loading in the z-direction will remain only as a function of

the x and y coordinates; i.e., Pz ¼ Pz ðx; yÞ:

* Thermal loading may also be applied to the plane (i.e., T ¼ Tðx; yÞ).

* Body or inertia forces may be applied to the plane.

Degrees of freedom: For 2-D plane strain elements, the DoF per node are the displacement in the two

coordinate directions ux and uy : The plane strain assumption means that the strain in z-direction will be

zero and that the stress will be nonzero, 1z ¼ 0 and sz – 0:

Thin plate with a central hole Thin bracket

X Y

Thin arch Through thickness crack in a plate

FIGURE 9.9 Typical plane stress examples.

3 or 4-node (linear) 6 or 8-node (quadratic) 9 or 12-node (cubic)

FIGURE 9.8 Typical plane stress elements.

Finite Element Applications in Dynamics 9-9

Typical element shapes: The same element shapes as in the case of plane stress are available in most

commercial FE programs (see Figure 9.8). The comment above on p-conversion and h-conversion

also applies here.

Example applications: Figure 9.11 shows some typical examples that may be modeled using a plane

strain assumption. Once again, it should be noted that such a model cannot caputre any out-of-plane

modes of vibration. If such vibration modes are of concern, the model should be capable of capturing

lateral displacement DoF and out-of-plane rotations. This may be realized by using shell or 3-D solid

elements as will be discussed in the following sections.

9.1.4.3 Two-Dimensional Axisymmetric Problems

Axisymmetric problems are characterized by having an axis of symmetry or axis of revolution for

geometry and loading. Referring to Figure 9.12, any arbitrary plane that passes by the axis of symmetry

will be a plane of symmetry. Symmetry means that the two halves of the structure on each side of the

plane of symmetry are mirror images of each other. Symmetry must be satisfied for all aspects affecting

the response of the structure including geometry, load, constraint, and material properties.

The following conditions summarize the requirements for categorizing a problem as axisymmetric

(Boresi and Chong, 2000).

Conditions:

* Geometry:

* A solid of revolution formed by rotating a flat area (e.g., in the rz-plane) around an axis of

symmetry (e.g., the z-axis).

* The boundary of the flat area in the rz-plane may be straight or curved and the area may be

simply or multiply connected.

* Loading:

* Loading is restricted to the rz-plane with no variation of loading in the u-direction. No loading

in the u-direction.

* Thermal loading may be applied in the rz-plane (i.e., T ¼ Tðr; zÞ).

* Body or inertia forces may be applied in the rz-plane.

Degrees of freedom: For 2-D-axisymmetric elements, the DoF per node are the displacements in the two

coordinate directions ur and uz : Some programs use x and y coordinates to replace the r and z axes,

respectively.

May be free,

loaded or

constrained

δ1

δ2

Surface may be

free, loaded or

constrained

May be free,

loaded or

constrained

FIGURE 9.10 A general 2-D plane stress structure.

9-10 Vibration and Shock Handbook

z- axis of

symmetry

uq = 0

axis of symmetry

symmetry

plane

(arbitrary)

uq = 0

Sectional view

FIGURE 9.12 Axial symmetry conditions.

FIGURE 9.11 Typical plane strain examples.

Finite Element Applications in Dynamics 9-11

Typical element shapes: Figure 9.13 shows typical axisymmetric element shapes that are commonly

available in commercial FE programs. The elements are shaped as a torus with various cross sections as

shown in the figure.

Application examples: Figure 9.14 shows typical axisymmetric examples.

9.1.5 Shell and Plate Problems

Shell and plate problems are quite similar to plane stress problems (see Figure 9.15). We first recall the

conditions for a plane stress problem: (1) that the geometry has to be flat in one plane with a small

thickness, and (2) that the load has to be in the same plane. If either of these two conditions is violated,

the problem becomes a shell problem. For example, if the load is a moment about any direction other

than the z-direction then it has an out-of-plane component, or if the geometry is not flat then the

problem ceases to be a plane stress problem and should be modeled as a shell problem. Normally, shell

structures would still maintain a small thickness in the direction normal to the surface of the shell. This

maintains the condition that the stress normal to the shell surface will be zero, although there may still be

a pressure applied to the surface of the shell.

The following summarizes the conditions for a shell structure.

* Geometry:

* Thin surfaces or plates that have a thickness in the direction normal to the surface. The

midsurface or midplane of the structure may be flat or curved.

* The boundary or edges of the structure may be straight or curved.

* Loading:

* Both in-plane and out-of-plane loadings are permitted.

* Thermal loading may also be applied in all x-; y-; and z-directions.

* Body or inertia forces may be due to linear or angular acceleration in all three directions.

Degrees of freedom: Three-dimensional shell elements have six DoF: three translational, ux ; uy ; and uz ;

and three global rotational, ux ; uy ; and uz :

Cylinder under axisymmetric

Pressure vessel Loading of a conical washer

FIGURE 9.14 Typical axisymmetric examples.

FIGURE 9.13 Typical axisymmetric elements.

9-12 Vibration and Shock Handbook

Element shapes: Figure 9.16 shows some typical shell elements available in most commercial FE

programs. To properly model a curved shell with linear elements that are flat requires using a large

number of elements to capture the curvature. Quadratic and cubic elements, on the other hand, may have

curvature in two directions and a much reduced number of elements will be needed to model a curved

shell. Most programs are capable of modeling variable thickness shell and plate elements.

P Z 3

δ1

δ2

Surface may be

free, loaded or

constrained

FIGURE 9.15 Shell and plate structures.

REMARKS

* Structures with a flat planar surface of a small thickness and with loading only in the plane

(e.g., xy) of the structure (no out-of-plane loading or constraints) are categorized as plane

stress structures. Such structures will have three nonzero stress components: sxx ; syy ; and txy :

* Structures with a flat planar surface (e.g., xy) but a very large dimension in the third direction

and with loading only in the plane of the structure (with possible uniform loading or

constraints in the third direction) are categorized as plane strain structures. Such structures

will have four nonzero stress components: sxx ; syy ; szz ; and txy :

* Axisymmetric problems are characterized by having an axis of symmetry or axis of revolution

(e.g., z-axis) for geometry and loading. Such structures will have four nonzero stress

components: srr ; suu ; szz ; and trz :

* Two-dimensional problems may be modeled by elements having two translational DoF per

node. The elements may be triangular with 3, 6, or 9 nodes or quadrilateral with 4, 8, or 12

nodes. Axisymmetric elements are shaped as a torus with triangular or quadrilateral crosssections.

* Increasing the number of elements while fixing the nodes per element is called

“h-conversion,” whereas increasing the number of nodes per element while fixing the

number of elements is called “p-conversion.”

* Two-dimensional models cannot capture out-of-plane vibration modes. If such vibration

modes are of concern, the model should be capable of capturing lateral displacement DoF

and out-of-plane rotations.

Finite Element Applications in Dynamics 9-13

Some programs offer shell elements that only have

membrane capabilities and others that have both

membrane and bending capabilities.

In shell and plate analyses, it is important to

note that linear elements, i.e., three-node triangles

and four-node quadrilaterals, may behave in an

unrealistically stiff manner in shear deformation

when the element thickness to size ratio is very

small. This phenomenon is normally called “shear

locking” (Bathe and Dvorkin, 1985; Bathe, 1996;

Luo, 1998; Cesar de Sa et al., 2002). The problem

occurs when in-plane displacements are coupled

with section rotations in the governing equation of

the element and when low-order interpolations (linear elements) are adopted. The same phenomenon is

also evident in beam elements. Similarly, when in-plane displacements are coupled with section rotations

in the governing equations and low-order interpolations are used, “membrane locking” will be evident.

Most programs provide various remedies for shear- and membrane-locking problems. These include

selective/reduced integration, enhanced assumed strain method, mixed field method, etc. Depending on

the availability of a particular method in the program, the user should initiate the remedy. The problem

may also be alleviated by switching to higher order elements, such as quadratic or cubic elements.

Example applications: Figure 9.17 shows typical examples of applications for shell problems.

9.1.6 Three-Dimensional Solid Problems

By default, if the problem is not one of those discussed in Section 9.1.3 to Section 9.1.5 then it will be

classified as a 3-D problem. Three-dimensional problems are easily identified by their 3-D geometry and

loading, as shown in Figure 9.18. Any of the categories of problems discussed above may be solved by

using 3-D elements. The critical drawback is the substantial increase in analyst time for modeling and in

CPU time in processing the solution (the time needed may be one order of magnitude larger than for a

corresponding 2-D problem).

The following summarizes the conditions for a 3-D problem.

* Geometry:

* A 3-D object with no apparant thickness or uniformity in any direction.

* The boundary of the object may be straight or curved.

Linear Quadratic Cubic

FIGURE 9.16 Typical shell elements.

REMARKS

* Shell structures are general 3-D surfaces of small thickness. Loading can be in plane and out

of plane. All stress components except those normal to the surface will be nonzero.

* The shell element may be triangular with 3, 6, or 9 nodes or quadrilateral with 4, 8, or 12

nodes.

* Commercial programs offer shell elements with variable thickness and with membrane

and/or bending capabilities.

* Linear elements may experience “shear locking”: nonphysical, high stiffness in shear.

Various remedies are available in most programs.

9-14 Vibration and Shock Handbook

* Loading:

* Three-dimensional loading. An important

note should be added here: 3-D

elements normally have three displacement

DoF and no rotational DoF. This

means that moments cannot be directly

applied to such elements and the moment

effect should be simulated via concentrated

forces or couples.

* Other loading, including thermal loading

and body or inertia forces, may be applied

in any direction.

Degrees of freedom: Three-dimensional solid

elements have three translational DoF: ux ; uy ; and uz : As mentioned above, concentrated moments

should be modeled by an equivalent system of concentrated forces or couples.

Element shapes: Figure 9.19 shows some typical 3-D element shapes available in most commercial FE

programs.

Example applications: Figure 9.20 shows some typical examples of 3-D structures requiring 3-D solid

modeling and elements. It should be noted that if out-of-plane vibration modes of a 2-D structure are of

concern, then a 3-D solid or shell model should be used even if the loading is in one plane.

F2 δ1

δ2

FIGURE 9.18 Three-dimensional problems.

Thin-walled cylinder with 3D loading C-Channel with 3D loading

Thin-walled T-junction

under internal pressure

Cantliver under in-plane and out-of-plane loading

FIGURE 9.17 Typical shell and plate examples.

Remarks

* Structures that are not classified as truss, beam, 2-D, or shell may be modeled with 3-D solid

elements.

* Three-dimensional elements have three translational DoF per node (ux ; uy ; and uz ) and may

have 4 – 32 nodes per element.

Finite Element Applications in Dynamics 9-15

9.1.7 Synopsis of Problem Classification and Element Choice

Table 9.1 summarizes the concepts discussed above (Section 9.1.2 to Section 9.1.6) for element choice

and problem classification. The table also shows displacement and stress variations within each element

as well as standard element output quantities. The displacement variation within the element represents

the basic element assumption in FE analysis and is called the “shape function” or the “approximation

function” assumption. The strain variation within the element follows by differentiating the displacement

variation according to the strain – displacement relations. The number of nodes per element is linked to

the shape function assumption. For example, in 2-D elements three-node triangles have linear shape

functions and in 3-D elements eight-node bricks have linear shape functions. Elements with linear shape

function are sometimes called low-order or linear elements. Such elements may be used quite efficiently

in both linear and nonlinear analyses.

Thumb nail crack in a thick block

Thick T-junction

Shaft with torque, bending

moment and shear forces Welded joint with three

dimensional loading

FIGURE 9.20 Typical 3-D examples.

4-node tetrahedron,

6-node wedge and

8-node cube (linear)

10-node tetrahedron

15-node wedge and

20-node cube (quadratic)

16-node tetrahedron

21-node wedge and

32-node cube (cubic)

FIGURE 9.19 Typical 3-D solid element shapes.

9-16 Vibration and Shock Handbook

TABLE 9.1 Summary of Problem Classification and Corresponding Element Characteristics

Characteristics Element

Truss Beam Two-Dimensional Plane Elements

UX 1

Node-1

Node-2

Node-3

Node-1

Node-2

Node-3

Dimension 1-, 2-, or 3-D 2- or 3-D 2-D 2-D

Applications Trusses — only axial stiffness

(acts as a spring between

two nodes)

Frames with axial and

bending stiffness

2-D plane stress

or plane strain problems

2-D plane stress or plane

strain problems

Degree of freedom UX, UY, UZ UX, UY, UZ, ROTX, ROTY, ROTZ UX, UY UX, UY

Geometry Linear Linear (principal axes found

from third node)

Linear Quadratic

Displacement Linear Axial: linear Linear Quadratic

Bending: cubic

Stress/strain Constant Axial: constant Constant (triangular) Linear (triangular)

Bending: linear Linear/constant (Quad) Quadratic/linear (Quad)

Loading Only at end nodes

(also possible mass)

Line load

Self-weight

Edge pressure

Self-weight

Edge pressure

Self-weight

No distributed load Centrifugal force Thermal Thermal

Constant thermal Thermal Centrifugal force Centrifugal force

Stress output Axial force or stress Axial force 2-D stresses: sxx ; syy ; txy ;

szz (plane strain)

2-D stresses: sxx ; syy ;

txy ; szz (plane strain)

Bending moment Stress intensities and

principal stresses

Stress intensities and

principal stresses

Shear in two perpendicular

axes or alternatively outer

fiber stresses

(continued on next page)

Finite Element Applications in Dynamics 9-17

TABLE 9.1 (continued)

Characteristics Element

Axisymmetric Solid Three-Dimensional Solids

Dimension 2-D 2-D 3-D 3-D

Applications Axisymmetric structures

with possible

nonaxisymmetric loads

Axisymmetric structures with

possible nonaxisymmetric loads

3-D solids

with general loading

3-D solids

with general loading

Degree of freedom UR, UZ UR, UZ UX, UY, UZ UX, UY, UZ

Geometry Linear Quadratic Linear Quadratic

Displacement Linear Quadratic Linear Quadratic

Stress/strain Constant (triangular) Linear (triangular) Constant (tetrahedron) Linear (tetrahedron)

Linear– constant (quad) Quadratic/linear (quad) Linear/constant

(hexahedron)

Quadratic/linear

(hexahedron)

Distributed loads Pressure Pressure Pressure Pressure

Self-weight Self-weight Self-weight Self-weight

Thermal Thermal Thermal Thermal

Centrifugal force Centrifugal force Centrifugal force Centrifugal force

Stress output Axisymmetric stresses:

srr ; szz ; suu ; trz

Axisymmetric stresses:

srr ; szz ; suu ; trz

3-D stresses:

sxx ; syy ; szz ; txy ; tyz ; tzx

3-D stresses: sxx ; syy ;

szz ; txy ; tyz ; tzx

Stress intensities and

principal stresses

Stress intensities and

principal stresses

Stress intensities and

principal stresses

Stress intensities and

principal stresses

9-18 Vibration and Shock Handbook

TABLE 9.1 (continued)

Characteristics Element

Three-Dimensional Shell Structures

Dimension 3-D 3-D

Applications 3-D shell structures with general loading 3-D shell structures with general loading

Degree of freedom UX, UY, UZ, ROTX, ROTY, ROTZ UX, UY, UZ, ROTX, ROTY, ROTZ

Geometry Linear Quadratic

Displacement In shell local coordinates: linear on midsurface

and linear through the thickness

In shell local coordinates: quadratic on midsurface and linear

through the thickness

Stress/strain On the midsurface and in shell local coordinates: On the midsurface and in shell local coordinates:

Triangle: constant Triangle: linear

Quad: linear/constant (linear through thickness) Quad: quadratic/linear (linear through thickness)

Distributed loads Surface and edge pressure Surface and edge pressure

Self-weight Self-weight

Thermal Thermal

Centrifugal force Centrifugal force

Stress output 3-D stresses sxx ; syy ; szz ; txy ; tyz ; tzx (global) 3-D stresses sxx ; syy ; szz ; txy ; tyz ; tzx (global)

Local stresses Local stresses

Stress intensities and principal stresses Stress intensities and principal stresses

Finite Element Applications in Dynamics 9-19

Пресс-релиз популярных книг

9.1 Problem and Element Classification

Популярные книги

Популярные статьи

Навигация