Over View of Beam Elements

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Date October 29, 2000 Memo Number STI69:001029 Subject CSI Tip of the Week: Structural Beam ElementsCSI Tip of the Week: Structural Beam ElementsCSI Tip of the Week: Structural Beam ElementsCSI Tip of the Week: Structural Beam Elements Keywords Structural: BEAM 1.1.1.1. Introduction:Introduction:Introduction:Introduction:

Due to the number of beam elements available in ANSYS, it may be difficult to distinguish between the various formulations/features. This memo will cover some background information on beam elements in hopes of aiding the user in proper element selection.

Some comparisons to ANSYS shell elements will be made, so please refer to CSI’s Tip of the Week, issue #8 “Structural Shell Elements”, for further information. 2.2.2.2. Background Discussion:Background Discussion:Background Discussion:Background Discussion:

Beams are geometrically 1-D but spatially 2-D or 3-D elements used to approximate structures which are longer in one dimension than they are in the other two dimensions. In other words, for thin, column-like structures, if the dimensions of the cross-section are smaller than the length of the beam, then BEAM elements can be used. (This is similar to shell elements that are thinner in one direction than they are in the other two) Generally speaking, the cross-sectional dimensions should be less than 1/20 or 1/30 of the length of the member, where the distance between supports of the physical system defines the length of the member. The beam elements themselves can be of any mesh size and does not dictate the applicability of beam theory; instead, as noted above, the physical dimensions and characteristics should determine whether beam elements can be used.

There are basically two types of beam theory: Timoshenko (shear-deformable) and Euler-

Bernoulli theory. All of the older beam elements are based on Euler-Bernoulli beam theory; these include:1

• 2D/3D elastic BEAM3/4 • 2D plastic BEAM23 • 2D/3D offset tapered, unsymmetric BEAM54/44 • 3D thin-walled, plastic BEAM24

The basic features and restrictions of Euler-Bernoulli beams are as follows: • Uses Hermitian polynomials for shape functions, so has cubic response in bending • Bending moment can vary linearly • Transverse shear is not taken into account • Unrestrained section warping is assumed (no warping under torsion) • Elastic behavior only is assumed or limited nonlinear material capabilities (BEAM23/24) • Very limited pre- and post-processing capabilities (except for BEAM44) There are two beam elements introduced at ANSYS 5.5: 3D 2-node BEAM188 and 3D 3-node

BEAM189.2 These are based on Timoshenko theory, which includes the following assumptions: • Exhibits linear (188) or quadratic (189) response under bending • Transverse shear stress is constant through thickness (first-order shear-deformable beams) • Unrestrained or restrained warping can be modeled at ANSYS 5.6 • Rich set of complex constitutive models available (plasticity and creep) • Finite strain applications • Enhanced pre- and post-processing capabilities

BEAM188/189 are part of the 18x family of elements which are being developed as a “next generation” of nonlinear elements. These set of elements are continually being enhanced at every new revision of ANSYS, including version 5.7 which supports many more nonlinear material models.

1 Also included but not listed here are the PIPE family of elements which assume a circular cross-section rather than a rectangular cross-section as BEAM4 does. 2 Although the author refers to BEAM188 and BEAM189 and 2-node and 3-node elements, respectively, an additional ‘orientation’ node is present, so, technically speaking, they are 3-node and 4-node elements.





3.3.3.3. Enhanced PreEnhanced PreEnhanced PreEnhanced Pre---- and Post and Post and Post and Post----Processing for BEAM44/188/189:Processing for BEAM44/188/189:Processing for BEAM44/188/189:Processing for BEAM44/188/189: For the older ANSYS beam elements, the cross-sectional properties must be input via REAL

constants. This usually involves calculating the area, area moment of inertia, torsional moment of inertia, and shear deflection constants by hand.

However, at ANSYS 5.5, the BeamTool and SECxxxx family of commands have been introduced for BEAM44/188/189. These introduce the concept of a “section” property as an additional element attribute. The shape of the cross-section is defined (I-beam L-channel, etc.), and the cross-sectional properties are automatically calculated by ANSYS.

Above is a screenshot of the BeamTool (right) and the calculated cross-section properties (left). There are many pre-defined beam cross-sections which can be selected from the “Sub-type” menu. A user-defined cross-section may also be input.3 The offset (where the nodes are located) can be the centroid or shear center (as calculated by ANSYS) or even a user-defined offset. This is useful when modeling beam elements as stiffeners for shell elements.4

The beam elements with cross-sections may be shown graphically with /ESHAPE,1. For most beam elements, a simple rectangular represen-tation will be shown using the effective thickness-es input via REAL constants. For BEAM188/189, however, the actual cross-sections can be shown on the right. In fact, BEAM188/189 has ‘section integration points’ where stresses are evaluated after deflections are calculated. As a result, besides deflections, accurate stresses on the flanges of the I-beam, as shown on the right, can be determined.

3 Please refer to the online help documentation for more details since, depending on the revision of ANSYS being used, the user-defined cross-sections may have limitations compared with other versions. 4 Note that, as of ANSYS 5.7, none of the shell elements in ANSYS support a user-specified offset. The nodes of the shell elements represent the midsurface, except for the composite SHELL91/99 which can be offset to the top or bottom surfaces, respectively.





4.4.4.4. Transverse Shear (Thin/Thick BeamsTransverse Shear (Thin/Thick BeamsTransverse Shear (Thin/Thick BeamsTransverse Shear (Thin/Thick Beams):):):): A mathematical discussion of Timoshenko and Euler-Bernoulli beam theories is beyond the scope

of the present discussion, but some basics will be covered. For a more detailed treatment of the subject matter, the reader is referred to the references at the end of this memo.

Euler-Bernoulli beam theory is often covered in basic mechanics texts. Transverse shear is not

accounted for, although a flexibility factor (shear deflection constant) can be included as a REAL constant in the ANSYS elements. This shear deflection constant changes the effective shear area in calculations, such that the shear strain energy can be accounted for correctly. The reason why transverse shear is not accounted for is because the beam in bending is assumed to behave in such a way that the cross-section normal to the neutral axis remains normal to the neutral axis after bending.

For thin beams, transverse shear is negligible, so this assumption is quite valid in those cases. As a result, the elements which are based on Euler-Bernoulli theory – e.g., BEAM3/4 – are appropriate for thin beams. For moderately thick beams, the shear deflection constant can be included as a real constant for these classes of elements. [When comparing with shell element formulations, Euler-Bernoulli beams are analogous to Kirchhoff plate theory.]

Timoshenko beam theory considers the effects of transverse shear. This is because the manner

in which the beam acts in bending is as follows: the cross-section initially normal to the neutral axis remains plane but does not remain normal to the neutral axis. This is taken into account by BEAM188/189. [Timoshenko beam theory is similar to Mindlin shell theory as both consider transverse shear.]

Unlike the other ANSYS beams,5 the shear constants need not be calculated separately since the BeamTool (SECxxxx commands) calculates these based on the type of cross-section the user chooses.

Timoshenko beam theory is good for thin to moderately-thick beams. If the physical members are too thick/stubby, the first-order transverse shear approximation is no longer accurate for the beam elements since the actual variation of transverse shear stress would not be constant anymore. In this case, modeling the physical members with SOLID elements may be more appropriate, so a non-constant transverse shear stress can be captured. As noted in the ANSYS online help, the following guideline may be used:

302

>EI

GAL

where

inertia of moment area

modulus elastic

member oflength

area sectional-cross

modulus shear

=====

I

E

L

A

G

5 BEAM44 does support the BeamTool (SECxxxx family of commands) functionality.





5.5.5.5. Torsion and Restrained Warping (Warping DOF):Torsion and Restrained Warping (Warping DOF):Torsion and Restrained Warping (Warping DOF):Torsion and Restrained Warping (Warping DOF): Torsion is usually due to torsional moments or a shear force which does not act upon the shear

center of a beam. These loads twists the beam about its axis and may produce warping, where cross-sections of the beam do not remain planar. Warping is important since it introduces axial strains/stresses as well as affects the torsional resistance.

Except for solid circular sections, all beams warp under torsional loads. However, the significance of warping is usually negligible for most solid or even closed cross-sections, but it is most pronounced for open, thin-walled sections since these have very little torsional stiffness. Moreover, the shear center and centroid often do not coincide for open sections, so any off-center shear forces or pressures will result in torsional loads.

Only BEAM188/189 support unrestrained or restrained warping starting from ANSYS 5.6. All other BEAM/PIPE elements in ANSYS assume unrestrained warping; i.e., warping is assumed to be negligible. For BEAM188/189, KEYOPT(1) determines whether warping is considered by adding an extra degree of freedom, WARP.

To understand this better, two input files “ibeam_188.inp” and “ibeam_181.inp” are provided which demonstrate a lateral torsional buckling of an I-beam. The effects of unrestrained (default) or restrained warping can be shown by running these input files. For the first input file, “ibeam_188.inp”, BEAM188 is used. Restrained warping is assumed (“keyopt,1,1,1”) but this line can be commented out to run with unrestrained warping. The results are compared with an I-beam comprised of SHELL181 elements, which presents a more detailed model of the I-beam:

Table Table Table Table 1111

Critical LoadBEAM188 (Unrestrained Warping) 387.3BEAM188 (Restrained Warping) 569.6SHELL181 564.1

As shown above, unrestrained warping (as is present with all ANSYS beam elements) greatly

underpredicts the buckling load. However, when warping is considered (restrained warping), the critical lateral torsional buckling load is calculated accurately and is in agreement with the more detailed shell model (SHELL181).

Please note that when using

BEAM188/189 with all seven degrees of freedom, the WARP DOF should NOT be the same at beam intersections. When connecting beams perpendicular to each other, care should be taken to ensure that the nodes are coincident at the intersection, but only translational and rotational degrees of freedom are coupled (CP) together. If not, the WARP DOF (warping magnitude) will be the same at that location for the perpendicular beams, which will result in physically inaccurate behavior.

On the right is an example of this where, at the intersection of two sets of beams, the coincident nodes are coupled in U and ROT DOF only.





6.6.6.6. Geometric and Material Nonlinearities:Geometric and Material Nonlinearities:Geometric and Material Nonlinearities:Geometric and Material Nonlinearities: The majority of the older beams (e.g., BEAM4) support large rotation only. These beams do not

allow for finite-strains, but they can be used in large deflection/rotation analyses. However, the new BEAM188/189 support both large rotation and finite strain applications. In fact, with KEYOPT(2), the cross-section can be assumed to be rigid or can be scaled as a function of axial stretch (to preserve volume in finite strain cases).

While some of the older beams such as BEAM23/24 support plasticity and explicit creep, most of

the commonly-used beams such as BEAM4 are elastic only in nature (recall that these older elements do not support finite-strain, either). The newer BEAM188/189 support implicit creep as well as many new constitutive models for plasticity. At 5.7, these have been extended to include Hill’s anisotropic plasticity and the Perzyna and Peirce viscoplasticity models. Although BEAM188/189 are recommended in most nonlinear applications, please refer to the online help for more information since not all constitutive models are supported by BEAM188/189, such as MELAS, or even element birth-and-death.

7.7.7.7. Dynamics and Mesh Density:Dynamics and Mesh Density:Dynamics and Mesh Density:Dynamics and Mesh Density:

In static and dynamic applications, mesh density is important to consider. Because the elements based on Euler-Bernoulli beam theory (e.g., BEAM4) exhibit a cubic response in bending, usually a few elements along a member suffice for dynamic applications (and even 1 element along a member may give satisfactory answers in static analyses).

On the other hand, BEAM188/189 use either linear or quadratic shape functions. Consequently, more elements should be present along a member length when using these elements in order to capture the dynamic response (mode shape) accurately. BEAM189, because of its quadratic shape function, is strongly recommended for curved beams.

Another important note is that, for modal analyses with BEAM188/189, if the user wants to “see”

the results on the actual cross-sections with /ESHAPE,1, the element stress calculations must be activated with MXPAND.6 Also, OUTRES,MISC must be issued as well (save miscellaneous data) to store this information.

BEAM188/189 have section integration points at each node, as mentioned earlier. This means that for each node of BEAM188/189, many element calculations are performed after displacements are solved for. For example, if each cross-section of a BEAM188 model has 20 “cells” where each cell has 4 integration points, there are 80 integration points/section nodes where stresses, etc. are to be calculated. This will result in a much larger results file. Also, for computers with slower disk I/O, this may noticeably increase solution time as well. (The older beam elements do not have this problem as they do not have section integration points)

To circumvent this, the user is strongly advised to use OUTRES commands, starting with OUTRES,ALL,NONE to wipe out current specifications. OUTRES allows the user to save specific data for specific groups of nodes/elements.

An alternative is to use the older BEAM elements in dynamic applications, if stresses at non-rectangular cross-sections are not required. The older beam elements do not take longer for stress calculations, so they are more efficient in this respect. To simplify the use of older beam elements, the BeamTool can be used to set up a fictitious “section” – note the values of the cross-sectional properties reported by ANSYS when defining cross-sections. These values can then be directly input as REAL constants for the older beam elements.

6 Even if only displacements are desired with /ESHAPE,1 element “stress” calculations must be activated with MXPAND. These are actually not limited to stress calculations but all element calculations.





8.8.8.8. Conclusion:Conclusion:Conclusion:Conclusion: This memo hoped to cover some of the basics of beam theory as well as the differences in the

older and newer (BEAM188/189) beam elements available in ANSYS. For most applications, the use of BEAM188/189 is recommended because of easier pre-

processing, useful post-processing features, and much more sophisticated element technology including finite strain, nonlinear constitutive models, restrained warping, and transverse shear capabilities. At 5.7, these capabilities have been extended to include:

• Composite beam definition • “True” combined axial and torsional state of stress for plasticity and creep (at 5.5 and 5.6,

axial/bending and shear effects were uncoupled, so sometimes, the actual stress might have exceeded yield strength of the material, yet no yielding was reported with PRSSOL)

There are many other features such as pressure-load stiffness matrix, orientation nodes/keypoints, listing section results, and the other KEYOPTs available with BEAM188/189 which have not been discussed. The author will refer the reader to the online documentation for further details on BEAM188/189, especially Chapter 15 “Beam Analysis and Cross Sections” in the ANSYS 5.6 Structural Analysis Guide and the appropriate sections in the Elements Manual.

Lastly, the author would like to point out a few Class3 Errors which may affect users running

models with beam elements: • Class3 2000-31 notes that if the BeamTool is used with BEAM44, the maximum distance of

the fibers to the centroid are not calculated properly. • Class3 2000-21 is related to using BEAM44/188/189 in a modal cyclic analysis • Class3 2000-25 deals with BEAM188/189 with kinematic hardening and plotting/listing

equivalent plastic strain Please contact CSI for further details on any of the error reports as noted above.

9.9.9.9. References:References:References:References:

• Cook, R.D., Malkus, D.S., Plesha, M.E., “Concepts and Applications of Finite Element Analysis”, 3rd ed, John Wiley and Sons, 1989

• Bathe, K.J., “Finite Element Procedures”, Prentice Hall, 1996 • ANSYS Theory Manual, Revision 5.6 • ABAQUS Theory Manual, Revision 5.8 • Young, W.C., “Roark’s Formulas for Stress & Strain”, 6th ed., McGraw-Hill, 1989

__________________________ Sheldon Imaoka Collaborative Solutions, Inc. (LA Office) Engineering Consultant





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Over View of Beam Elements