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Challenge Problem 5 - The SolutionDynamic Characteristics of a Truss Structure

In the final year of his engineering degree course a studentwas introduced to finite element analysis and conductedan assessment of a simple, planar, pin-jointed trussstructure. This included a modal analysis to establish thenatural frequencies of the structure. Being a cautiousstudent he thought he’d better conduct some verificationtests on the axial element used to model the trussstructure. He found an analytical solution for a bar, fixedat one end and with uniform cross section and materialproperties. He conducted a convergence study on this teststructure in two commercial finite element systems (CS1and CS2) starting with a single element and thenperforming uniform mesh refinement until he reached amesh with 16 elements. In his model he supported allnodes in the vertical direction and for the node at the left-hand end he also supported is against horizontaldisplacement. His results for the fundamental frequency

are shown in figure 1. The frequencies have beennormalised (divided) by the theoretical value which wasobtained from the equation inset into the figure and heused the same elastic and inertial properties as for his trussstructure.

The results worried him since, although he saw that thefinite element programmes provided results that convergewith mesh refinement to the theoretical solution, he alsosaw that a single element could be in excess of 10% inerror and he realised that using different software wouldprovide different approximations with CS1 giving anupper-bound and CS2 a lower-bound to the theoreticalvalue. As his truss structure analysis had used singleelements for each of the members he was concerned thatthe natural frequencies for the structure might be ratherinaccurate.

The ChallengeThe challenge is to work with this student to understand why different finite element systems provide differentapproximations for coarse meshes and to provide some guidance on how he might obtain accurate frequenciesfor his truss structure.

Copyright © Ramsay Maunder Associates Limited (2004 – 2016). All Rights Reserved

Figure 1: Student’s structures and results

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Raison d’être for the ChallengeThis challenge raises an interesting question namely isthe standard bar element, available in most if not allcommercial finite element systems, able accurately tomodel the dynamic behaviour of truss structures whenonly a single element is used to represent each member?The test structure investigated by the student indicatesthat by using only a single element per member one islikely to have some significant error in the finite elementfrequencies. But whilst the test structure could be easilyrefined and supports added to prevent rigid-body hingingat the nodes (the reason for the vertical supports in figure1), such simple constraints are not appropriate for thetruss structure where each member might undergo arigid-body rotation.

Thus, whilst possible, mesh refinement of the trussstructure will require some thought on how to prevent theintermediate nodes of each member from moving off themember centre line. A different approach would be tomodel the members with beam elements with themoment released at the three joints of the structure.This method is also considered in this response and indoing so demonstrates firstly, in general, that the modesachieved using bar elements are not representative of thereal modes in the structure and secondly that thefundamental frequency of the structure may well besignificantly less than that obtained using bar elements.

Of course, the accurate recovery of frequencies andcorresponding modes shapes is essential information tothe practising engineer particularly if these naturalmodes are likely to be excited by time varying excitationforces in service. For the structure considered suchexcitation forces might come from a seismic event orfrom a hoist positioned above the unsupported joint.

Further Consideration of the Test StructureThe inaccuracy of the frequency for the test structure offigure 1 is likely to be due to the inability of the barelement to model the actual mode shape. This is seen tobe the case in figure 2 where the theoretical shapes formodes 1 and 2 have been plotted.

The theoretical mode shapes express the axialdisplacement, Δx, as a function of the axial position, X, inthe bar and these are simple sinusoidal functions withone quarter and three quarter waves, respectively, formodes 1 and 2. The figure also shows the convergence ofthe (normalised) frequencies with uniform meshrefinement – the result for mode 1 being identical to thatshow in figure 1.

A number of points may be drawn from figure 2. Firstly,with the single element model having only a single freedegree of freedom, only a single mode of vibration exists.The 2nd mode is, thus, only captured when the mesh isrefined to two or more elements. Secondly, the error inthe 2nd mode is significantly greater than that for the 1stmode. This is seen particularly for the coarser meshesbut it persists with mesh refinement with the error forthe 2nd mode being consistently about one order ofmagnitude (10 times) that of the 1st mode.

The reason why the 2nd mode is captured less accuratelythan mode 1, for a given mesh, is simply that the modeshape is more complicated and with the bar element onlyhaving linear displacement capability presents a greaterchallenge for the finite element model. The finiteelement displacements from the four element mesh havebeen added to the figure for mode 2. The nodal values ofdisplacement are accurate but the displacement betweennodes is highly inaccurate. From the figure it can beseen that in order to capture the second mode to thesame (or better) accuracy of the first mode then twoadditional levels of mesh refinement are required – onlytwo elements are required to capture the 1st mode within5% of the true value whereas eight elements are requiredfor the 2nd mode.

Further investigation of the manuals for CS1 and CS2reveal that whilst the stiffness matrices are identical, themass matrices are defined differently. CS1 uses aconsistent mass matrix whereas CS2 uses a lumpedmass matrix. As observed from figure 1 the difference inthe frequencies for these two different forms of massmatrix is very significant for coarse meshes butwhichever form is used both appear to converge to thesame solution with increasing mesh refinement. Whilst

Figure 2: Finite element approximations of the first two modes for the test structure (CS1)

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the default position for CS1 is consistent mass, there isan option to switch to lumped mass and if this is donethen identical results to CS2 are produced indicating thatthe default position for CS2 is lumped mass.

Solution through Constraint EquationsIt would be naive to hope that simple mesh refinement ofthe truss structure would produce sensible results sincefor each additional intermediate node will introduce anadditional mechanism whereby the node can move freelynormal to the member axis. This is indeed the case ifone performs this sort of analysis and whilst elasticmodes do still occur, the multiple rigid-body modesappear to pollute these so that the elastic modes are nolonger realistic modes for the structure. As such thesemechanisms need to be prevented and this can be donethrough the application of constraints. Unlike the simple(single point) constraints applied to the test structure toprevent lateral displacement, for the truss structure,where each of the members should be allowed to rotatefreely in a rigid manner, this rotation will need to beconsidered and will lead to the use of multi-pointconstraints.

A multi-point constraint is a linear constraint equationlinking the amplitudes of various degrees of freedom inthe model and, in this case, the coefficients andconstants of the equation enforce any intermediate nodesto remain on the member centre line. Thus there will beone constraint equation per intermediate node and theform of this equation is given in figure 3.

The way in which the constraint equation is implementedwill be software dependent; however, the constraintequation requires that the coordinate system for theintermediate nodes is rotated into the local membercoordinate system. As there is potential for not correctlyapplying the constraint equation, it is sensible to performsome verification checks on the constrained model. Twosensible checks are to check for under or over constraint.If the model remains under-constrained then this shouldshow up in a modal analysis through the presence ofmechanisms with corresponding zero (or near zero)frequencies. If the model is over-constrained then astatic analysis where the structure is loaded with a rigid-body displacement might lead to stresses which for aproperly constrained structure should be zero.

Constraint equations were applied to the intermediatenodes of the truss structure for uniformly refined mesheswith 2 to 16 elements per member. The convergence ofthe first two frequencies is shown in figure 4 togetherwith the mode shapes for two elements per membermesh. The frequencies for the 16 element per membermesh (the most accurate values available) have beenused for normalising the frequency. As seen for the teststructure, figure 1, the higher the mode, the greater theerror in predicted frequency. Interestingly, though, forthe first mode the error is small even when a oneelement per member mesh is used. Whereas for the testcase the error for the single element was about 10%, forthe truss structure it is less than 1%. An explanation forthis might be found by looking in further detail at themode shapes.

Consider a member of a truss structureand how it moves from its initial(unloaded) configuration to its final(loaded) position. Such a member, isdefined by its end nodes i and j. Anintermediate node k is also shown and aconstraint equation needs to be set up toforce this node to remain on the membercentre line. The member in its initialposition lies at an angle α to the global Xaxis and has a local Cartesian coordinatesystem placed at the centre of theelement with the local x axis parallel tothe member. The constraint equation atthe bottom of figure 5 expresses thetransverse displacement of theintermediate node (in the local membercoordinate system) as a sum of four termsinvolving the global coordinates of the twoend nodes. The coefficients multiplyingthese terms are simply determined fromthe position of the intermediate node andthe angle of the member.

Figure 3: Constraint equation for an intermediate node

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Challenge Problem 5 - The Solution - Dynamic Characteristics of a Truss Structure

Whilst the constraint equations have been checked forthe models of the truss structure, it would be useful tofind some independent way of verifying these resultsbefore accepting them as truth. One approach might beto model the structure using planar beam elements andreleasing the moments at the joints of the structure.Such models would not require constraint equations andis considered further in the next section.

Solution through Beam ElementsThe planar beam element in CS1 has three degrees offreedom per node (two in-plane displacements and theout of plane rotation) and it allows the cross-sectionalproperties (area and second moment of area) to bespecified independently. It is also possible to switch offthe influence of shear deformation and for the purpose ofthis study this will be done. With the addition of thebending capability, the number of potential modes ofvibration is increased over the same model using barelements. However as the element formulation includesthat of the bar element (and this may be confirmed byanalysing the test structure with the beam element, one

might expect, in addition to the bending modes, torecover the axial modes predicted by the bar elementmodel of the truss structure.

The truss structure was modelled with 200 beamelements per member which ensured accuracy on thefrequency of within 1% for modes up to 1600Hz. In thisfrequency range the model using bar elements revealedonly two modes. However the beam model produces 22modes of vibration as shown in figure 6 where the twomode shapes for the bar model are shown (from the twoelement per member model) together with the beammodes either side of the bar modes.

In addition to producing more modes of vibration (in thefrequency range considered), the results for the beammodel do not appear to include the pure axial modesseen in the bar model with the closest modes indicatingshort wavelength bending modes. Such ‘mixing’ ofmodes is common in structural analysis but it shows thatthe modes found from the analysis using bar elements donot occur in practice and not only that, but the idea thatthe bar element model would capture the fundamental

In figure 5 the finite element axialdisplacement along the twomembers has been plotted forthe first three modes for the mostrefined mesh (16 elements permember). It is interesting to seethat for the first mode the axialdeflections are almost linear andthis gives an explanation as towhy the frequency for this modeis captured with good accuracyeven for the coarsest mesh. Thesecond mode has axialdeflections that approximate thequarter sine wave and the thirdmode the three quarter sine waverather like those for the first andsecond modes of the trussstructure.

The first natural frequencies forthe two individual members fixedat one end are 1297Hz and 917Hzrespectively for the horizontaland angled members (members1 and 2). The correspondingmode shapes are the quarter sinewave shown earlier in figure 2.The second mode of the trussstructure has a frequency thatlies between the fundamentalfrequencies for the individualmembers and exhibits similarmode shapes. What isinteresting, however, for the trussstructure is that a newfundamental mode with a muchlower frequency has appeared.

Figure 4: Convergence of frequency for truss structure modelled with constraint equations

Figure 5: Member axial displacements for the 16 element permember mesh of the truss structure

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mode is mistaken; the lowest frequency for the beamelement model is significantly lower than that predicted bythe bar element model. It is clear then that whilst the barelement model might be satisfactory for determiningmember forces under a static load, it is totally inappropriatefor doing so under a dynamic loading scenario.

Having gone down the line of using bar elements andconstraint equations it would still be good to verify that thismodel, the bar element model, was indeed correct. One waythat this might be done is to force the beam model to recoverthe axial modes and this may be done by increasing thebending stiffness so that the bending modes havefrequencies higher than the axial modes. As mentioned thecross-sectional area and the second moment of area may beinput to CS1 independently and so this should be a simplematter of increasing the second moment of area whilstholding the cross-sectional area constant. A parameter thatrelates the two section properties is the radius of gyrationand in figure 7 the way in which the first two frequencies varywith radius of gyration is shown. It should be noted, whenlooking at figure 7, that the mode shapes will be varying withradius of gyration.

In considering how the frequencies of the trussstructure varied with radius of gyration, it wasdiscovered that when using consistent mass, the axialmodes could not be recovered with the frequenciesrising to that of the axial modes but then decreasingas the radius of gyration was further increased.When, however, lumped mass was used, thefrequencies and indeed mode shapes did converge,with increasing radius of gyration, to those producedby the bar element model. Thus the bar elementmodel using constraint equations has been verified.The reason why the consistent mass beam model didnot converge to the results produced by the barelement model is that the consistent mass matrixincludes terms to model the rotary inertia of thebeam and whilst the stiffness of the beam increaseswith radius of gyration, so does the inertia associatedwith the bending modes but it does so at a differentrate and thus leads to the quadratic curve shapeshown in the figure. The lumped mass matrix forCS1 is diagonal and includes no attempt to model therotary inertia of the beam element.

Figure 6: Frequencies and modes for truss structure modelled with beams and bars

Figure 7: Frequency variation with radius of gyration for beam model of truss structure

ClosureTo a practising structural engineer who has been involvedin the design and analysis of pin-jointed structures, thischallenge might be considered somewhat naive sincehe/she will probably realise that whilst a static analysis ofa truss structure using bar elements is adequate forpredicting the structural response to static loading, thisis not generally the case where the loads are dynamic;for the dynamic case, the axial modes of vibration formonly a part of the whole story and indeed are generallynot distinct when member bending is considered.Nonetheless working through the challenge has providedthe sort of learning experience a graduate engineermight find valuable in that it clearly exposes the reasonwhy the correct dynamic behaviour of pin-jointedstructures should not be expected to be predicted usingsimple bar element models.

The challenge has also revealed the effect of differentmass approximations on the results of a modal analysis.Whereas the consistent mass approach is exact, thelumped mass is only an approximation of the truth andthis approach probably hails from days, now past, wherethe computational advantage of working with a diagonalmass matrix where considered valuable. Whilst the

lumped mass approximation was useful in this challengeresponse to provide verification that the constraintequations in the bar element model of the truss structurewere correctly applied, the true behaviour would be thatobtained using a consistent mass matrix. It is also worthnoting that, whilst the consistent mass matrix is uniquelydefined, different finite element systems may adoptdifferent approaches to mass lumping. Indeed, whereasthe lumped mass matrix for the beam in CS1 does notinclude rotary inertia terms, the same element in CS2does.

It should, of course, be pointed out that the term massmatrix should generally include both translational inertiaand rotational inertia terms, unless the analyst issatisfied that the rotation terms are insignificant. Itmight be that this inaccuracy has something to do with acommonly seen mistake made in finite element analysiswhere the analyst fails to account for the rotary inertia ofa non-structural component whilst including the mass ofthis component in the finite element model. This is,clearly, a risky form of approximation since any modes ofvibration that involve rotation of the part will thengenerally have frequencies that are over-predicted.

Challenge Problem 5 - The Solution - Dynamic Characteristics of a Truss Structure

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