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Enhancing the teaching of linear structural analysis using additive manufacturing
Engineering Structures 150 (2017) 135–142
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Engineering Structures journal homepage: www.elsevier.com/locate/engstruct
Enhancing the teaching of linear structural analysis using additive manufacturing Lawrence Virgin School of Engineering, Duke University, Durham, NC 27708, USA
a r t i c l e
i n f o
Article history: Received 7 March 2017 Revised 17 June 2017 Accepted 17 July 2017
Keywords: Structural analysis Flexural stiffness Thermoplastic 3D printing Education
a b s t r a c t Structural analysis forms a key component in many courses in civil, mechanical and aerospace engineering. Conventionally, the matrix stiffness method, a subset of finite element analysis, tends to occupy a central position in a typical syllabus, with a special focus on plane frames providing a bridge between basic structural components with pedagogical clarity and real-world structures. Equations of equilibrium are set-up and the full force of linear algebra brought to bear using the capabilities of Matlab or more specialized FEA packages. Such classes have a tendency to become a little dry and suffer from the usual shortcomings of numerical analysis and a black box approach - shortcomings in the sense of conceptual understanding as opposed to usefulness in the hands of experienced practitioners. The relatively recent advent of additive manufacturing is an exciting opportunity to incorporate a practical aspect to structural analysis. This paper describes the use of 3D printing, via the flexural stiffness of plane frames, to develop a structural feel for students, augmenting theoretical analyses. In addition to directly addressing the role of modeling, approximation, applicability of the underlying theory, and measurement uncertainty, it is thoroughly hands-on and initial anecdotal evidence suggests a higher degree of student buy-in. Ó 2017 Elsevier Ltd. All rights reserved.
1. Introduction The bending stiffness of a structural member (based on standard beam theory) is a function of loading, boundary conditions, material and geometry [1]. For a given prismatic member the stiffness typically scales with EI=L3 , and it is easy to demonstrate the length dependency with a simple ruler for example. Analytically, given a set of forces and boundary conditions, we can integrate the governing expressions and obtain deflection, and hence stiffness. This process is not so straightforward for frames, consisting of an arrangement of beams (and columns) in typically rectangular combinations. However, the overall stiffness of such structures is very important and perhaps a key teaching opportunity occurs when we seek to shed light on how stiffness depends on these more realistic geometrical arrangements. This is where the stiffness method comes in. Despite the liberating effect of Matlab [2] and the ease of numerical methods [3], the stiffness method becomes a decreasingly hands-on approach for all but the simplest examples. Many textbooks throughout the last 50 years or so have included chapters on the stiffness method including plane frames; a representative selection is [4–10]. However, it is possible to exploit versatile
E-mail address: [email protected] http://dx.doi.org/10.1016/j.engstruct.2017.07.054 0141-0296/Ó 2017 Elsevier Ltd. All rights reserved.
3D printing for deepening an appreciation for structural behavior, and specifically in terms of printing a range of elastic, relatively slender, plane frames. Additionally, with relatively simple loading and boundary conditions it is then possible to assess the role of geometry and its influence on certain stiffness properties of the frames using measured data. This paper considers the lateral stiffness of a baseline plane frame and some basic variations with three main foci: use of 3D printing in the teaching of structural analysis, exploring the influence of parameters on stiffness, assessing the role of simplifying assumptions. The overall goal of the paper is to provide a systematic approach to developing a deeper student understanding of structural stiffness.
2. 3D printing of slender structures 3D printing has already revolutionized the teaching of mechanical engineering, via the rapid-prototyping of components like gears, for example. And given increasingly higher resolution it is now relatively easy to produce slender elastic elements and structures, and providing deflections and stresses are kept within
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acceptable limits, the behavior of such elements and structures is linear and elastic. Thus, 3D printing has potential as a teaching tool in the realm of deformable bodies and structures. Furthermore, in terms of the capabilities of 3D printing and elementary testing configurations, we shall focus attention on planar structural frames in which (i) the boundary conditions are either essentially fixed or free, and (ii) loads are applied at specific locations (point loads). This provides the context in which students can simply print and test structures and thus assess the important features of the flexural behavior in general, and aspects of the stiffness method in particular. We shall focus attention on relatively flexible right-angled plane portal frames with clamped boundary conditions, momenttransmitting joints, and single point loading. This configuration was chosen partly to facilitate 3D printing and simple experimental testing, but it also represents a class of structure that is more instructive than a single structural element like a cantilever, and also can be analyzed using back-of-the-envelop (sway) calculations in some cases. Thus, in order to maintain a reasonable balance between hand-calculation and the fully fledged stiffness method we develop a focus on some simple portal frames, and compare theory (essentially the linear stiffness method using elastic beam elements) and experimental stiffness measurements. More specific details will be given later, but by way of introduction, consider the middle frame shown in Fig. 1. If we clamp the lower edge, and subject the top corner to a lateral force, we can extract the stiffness of the frame in terms of lateral deflection. The stiffness of the columns scales with EI=L3 , and thus in moving to the frame shown on the left we expect a (much) greater stiffness, all other things being equal. Furthermore, if the cross-beam is relatively stiff (the frame on the right), we also expect an increase in stiffness, but in terms of reducing the rotation of the corners, and a deflected shape that resembles a lateral sway. This
paper uses the 3D printer to produce frames that can be used to directly examines these effects.
3. Flexural stiffness analysis of a simple portal frame Historically, the teaching of structural analysis proceeded from pin-jointed trusses. This was partly justified by the ubiquity of riveted joints, but mostly because very often simplified analytical techniques, such as the methods of joints or sections, could be brought to bear [11]. But now most structural connections are rigid (e.g., welded) and computational techniques, for example the stiffness method, dominate. Thus frames rather than trusses now form the backbone of most courses on structural analysis (see the list of references). In order to explore this a little further consider a simple plane portal frame, consisting of two vertical columns that are clamped at their bottom ends and connected (via moment transmitting joints) to a horizontal beam as shown in Fig. 2(a). Assuming the lengths L and flexural rigidity EI are the same for all members, and that the frame is subject to a single horizontal load at one of the corners, we can write down the set of equilibrium equations for the structure: 2 2 3 ðAL2 =I þ 12Þ 0 6L AL2 =I 0 F 6 6 7 6 2 607 6 0 12 ðAL =I þ 12Þ 6L 6 7 6 6 7 6 6 7 6 2 607 6 8L 0 6L 6 7 6 6 7 ¼ EI=L3 6 6 7 6 2 607 6 ðAL =I þ 12Þ 0 6 7 6 6 7 6 6 7 6 2 607 6 sym: ðAL =I þ 12Þ 4 5 6 4 0
3 2 3 X 7 1 76 7 6 7 6L 7 76 Y 1 7 76 7 76 7 2 76 2L 76 h1 7 7 76 7; 76 7 6 X2 7 6L 7 76 7 76 7 76 7 6Y 7 6L 7 74 2 5 5 2 h2 8L 0
ð1Þ
in which ðX; Y; h) are global coordinates, relative to the bottom left corner of the frame. The element stiffness matrix for a beamcolumn in global coordinates can be found in Appendix A. If we now assume an overall frame dimension (relevant to the physical dimensions to be 3D printed later) of L ¼ 0:1 m, and a rectangular cross-section area ðb dÞ ¼ ð0:01 0:002Þ, gives an area A ¼ 20 106 m2, second moment of area I ¼ 6:67 1012 m4,
Fig. 1. Simple 3D-printed plane frames.
and thus AL2 =I ðL=rÞ2 ¼ 30 103 , where r is the radius of gyration. These parameters relate to a geometry that may be considered highly flexible, and this facilitates relatively easy measurements. In analysis, the effective degrees of freedom can be reduced if we examine the relative magnitudes of each element and exploit certain symmetry conditions.
Fig. 2. (a) A simple square portal frame with identical beam and columns, (b) the baseline configuration.
L. Virgin / Engineering Structures 150 (2017) 135–142
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Solving the set of linear simultaneous Eq. (1) we obtain the dominant sway deflection: X 1 ¼ 4:466F with essentially no difference between this and X 2 . Furthermore, the vertical Y deflections are effectively zero. Given the specific values used in the example we get a sway deflection of about FL3 =ð16:79EIÞ. Solving a 6 6 set of simultaneous equations is a relatively simple task (thanks to Matlab) but not a hand-calculation, despite the fact that some of the resulting deflections are quite small (h) and very small (Y). Let’s simplify things by assuming that the frame elements are axially stiff (relative to bending stiffness), and that the stiffness of the horizontal beam is relatively large (compared to the columns, we shall investigate this further later), then the sway deflection of the frame is effectively a single-degree-of-freedom (SDOF) spring characterized by: F ¼ ð2 12EI=L3 Þd, that is, eight times stiffer than the familiar cantilever result (now having two members each effectively restrained against rotation at both ends), and about 43% stiffer than our previous analysis in which the beam had the same flexural rigidity (and length) as the columns. But now the reduction to a SDOF allows a more systematic investigation of stiffness in terms of ‘easily set or measured’ parameters of a structural system. This is an ideal context for 3D-printed specimens to provide a practical hands-on augmentation to the introduction of elastic flexural modeling including the stiffness method. For our square portal frame we can assess the accuracy of the pure sway approximation by varying the stiffness of the crossbeam relative to the columns. In the general case this would involve the length of the beam, but for the square frame we can make a valid comparison using the ratio Ib =Ic . Fig. 3 summarizes the resulting sway stiffness. The stiffness is nondimensionalized by the Ib ¼ Ic case - we could have just as easily used the pure sway case. In any case, we see that pure sway effectively occurs for Ib =Ic ’8, corresponding to a rectangular cross-section in which the beam is twice as thick as the columns. Similarly a frame with equal flexural rigidity ðEIÞ in the beams and columns can also be used to assess the influence of Lb =Lc on the sway behavior. The general structure stiffness matrix for a portal frame can be found in Appendix B. 4. The baseline case In order to later conduct a systematic parametric study to shed light on flexural stiffness we start by defining a standard ‘baseline’ case. Referring back to Fig. 2(b) (and the middle frame in Fig. 1) we initially fix the following parameters: The ‘width’ b of the whole frame is initially 20 mm, and the thickness d ¼ 2 mm for all frame elements (including H, initially); the overall dimensions of the frame are L ¼ 100 mm; and W ¼ 50 mm. The Young’s modulus for ABS thermoplastic for the 3D printer used in this study was taken as E ¼ 2:0 GPa.1 The frame was printed onto an integral rigid base, and a measured force F was then applied via hanging weights (in a horizontally clamped frame orientation), or using a load cell, and the lateral deflection d measured using a standard proximity laser (see Fig. 4), thus furnishing a force-deflection relation (linear for relatively small deflections), the slope of which characterizes the stiffness of the frame. According to the simple theory we should expect F ¼ kd, where k 24EI=L3 , depending on the validity of the SDOF assumption, otherwise the regular stiffness method can be used. The baseline frame was printed using a Stratsys printer [12] using ABSplus - P430 production-grade thermoplastic, with a specification Young’s modulus in the range 1.65–2.1 GPa, but in the computations we use E ¼ 2:0 GPa [13]. The Solidworks STL files 1 There are some relatively minor, but quite interesting uncertainties in this estimate based on the subtleties of the printing process [14].
Fig. 3. The effect of the relative beam-to-column stiffness on the sway deflection of a rectangular (square) frame.
were developed according to the baseline measurements and relative to the absolute resolution of the 3D printer. Although the resolution of most 3D printers continues to improve, we typically focus most attention on the accuracy of the thicknesses, since the thickness dimensions (which are typically small) also have a strong influence on the stiffness (via I). The specimen was clamped by a rigid vice at one end and the free end subject to a point load as shown in Fig. 4. The force-deflection relation is shown within Fig. 6(b) (later to be identified as frame B), in which some of the measurements were repeated (e.g., loading and unloading, and then turning the frame around and repeating) to provide a better linear least-squares fit. The slope of this experimental line is 0.63 N/mm. Solving for the lateral deflection we make use of the stiffness method (for which the stiffness matrix is given explicitly in Appendix B) resulting in a lateral stiffness of 0.52 N/mm. According to the simplified (pure sway) theory we would expect k ¼ 0:67 N/mm. Assuming there is a reasonably high accuracy in measuring the various parameters, the discrepancy could most likely be ascribed to the Young’s modulus or rotation at the top corners. However, we also chose to examine the validity of the SDOF model, and we did this by changing the H; L and b parameters. For example, it is likely that there is a little rotation at the corners of the portal frame, and thus we might expect an experimental stiffness that is slightly lower than the (pure sway) assumption of no rotation at the corners. 5. Plane frame variations We then depart from the baseline case in order to assess the effect of various parameters on both the stiffness and the assumption on which the very simple (pure sway) stiffness relation is based. We thus chose variations to the baseline with beam thickness H increased from 2 mm to 8 mm; width b reduced from 20 mm to 10 mm; and the length of the columns L increased to 150 mm and also reduced to 50 mm (see Fig. 5). In typical multistory buildings the stiffness of the horizontal members (floors) are somewhat stiffer than typical columns, and hence we tend to be less interested in cases for which ðEb Ib =L3b Ec Ic =L3c Þ. This provides a permutation of frame geometries for which stiffness can be compared systematically. Note that the 100 mm 50 mm overall dimension was chosen (rather than the square frame) due to practical limitations associated with the resolution of the 3D
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Fig. 4. (a) The baseline frame, (b) in the test configuration, (c) a frame with an intermediate member.
Fig. 5. Printed frames - variations to the baseline frame B.
printer and experimental measurements, i.e., these frames are relatively flexible. One of the advantages of this kind of parametric study, based on comparisons, is that, for example, the exact value of E is less problematic because all the frames are made of the same material, printed in the same manner, and tested under nominally the same conditions [14]. It is also a not-uncommon student mistake to change more than one parameter at a time, thus making it difficult to assess sensitivities. Six more frame geometries were then considered (and were later printed) with ðH; LÞ = (2, 50), (2, 150), (8, 50), (8, 100), (8, 150), (2,100), all values in mm, and where the last frame (later to be labeled G), has b ¼ 10. Increasing (decreasing) L decreases (increases) the overall lateral flexural stiffness of the frame; while increasing H also increases the lateral flexural stiffness but in terms of influencing the rotation at the free end(s), and thus the justification for the simple SDOF assumption. The analytical results are summarized in Table 1. Let’s use column 7 of Table 1 (k=kB ) to compare and contrast the stiffness between frames (relative to the baseline case, kB .):
The columns in Frame A are half the length of the columns in Frame B, and thus we would expect this frame to be roughly ð0:5Þ3 ¼ 8 times stiffer. The columns in Frame C are one and a half times the length of the columns in Frame B, and thus we would expect this frame to be roughly 1=ð1:5Þ3 ¼ 0:296 the lateral stiffness. The width of all the frame elements in Frame G are one half of those in Frame B, and since I scales linearly with b we would expect half the lateral stiffness. Frames A and D are both squares, and thus these specific results can be located on Fig. 3 with coordinates: ðIb =Ic ¼ 1; k=kIb ¼Ic Þ and ðIb =Ic ¼ 64; k=kIb ¼Ic ! 1:43Þ, respectively, and using the appropriate non-dimensionalization for stiffness. We can also assess the SDOF sway approximation (referencing column 8 of Table 1), where we use C ¼ 24 (pure sway) to normalize the results, with k=k0 ! 1 thus indicating the pure sway case: Frames D, E, and F (with their thick cross-beams) all conform very closely to the pure sway case.
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L. Virgin / Engineering Structures 150 (2017) 135–142 Table 1 Frame geometries (with reference to Figs. 2(b) and 5). Frame B is the baseline case; k0 is for pure sway. Frame #
H (mm)
L (mm)
b (mm)
k (N/mm)
FL3 =dEI (pure sway = 24)
k=kB
k=k0
A B C D E F G
2 2 2 8 8 8 2
50 100 150 50 100 150 100
20 20 20 20 20 20 10
3.576 0.519 0.163 5.068 0.636 0.189 0.260
16.763 19.461 20.687 23.758 23.859 23.894 19.481
6.891 1 0.315 9.767 1.226 0.364 0.500
0.698 0.811 0.862 0.990 0.994 0.996 0.812
Comparing Frames B and G we see that they approximate similarly in sway deflection. Comparing Frames A, B, and C we observe that the sway approximation is better if the length of the columns L (relative to the length of the beam W ¼ 50 mm) is large. In summary, and based on the specific parameters we chose to vary in this study, we confirm that the lateral stiffness for a frame generally scales with Ec Ic =L3c , and that the SDOF sway approximation is increasingly justified with increasing Ib =Ic , but also with the overall slenderness of the frame L=W (since the stiffness of the beam relative to the column contributes to the joint rotation).
6. Experimental testing results We can plot the measured force-deflection relation for all seven frames (and their least-squares fits) superimposed in dimensional terms as in Fig. 6(a). We observe how the height L of the frame influences the stiffness, for example in going from frame A to frame B to frame C. Fig. 6(b) is a plot of the same data for frames D, E, and F, and scaled with respect to the column length (cubed), with EI consistent between the frames. These three frames have the stiff cross-beams and thus deform in a predominantly sway mode. Superimposed on this data is a straight line that is the theoretical pure sway relation FL3 ¼ 24EId, in which E ¼ 2:1 GPa. A summary of the experimental results is contained in Table 2. The relative stiffness effects can be extracted along the same lines as the theoretical results in Table 1. However, the interesting thing to note is the consistently higher stiffness of the 3D-printed frames as compared with the theoretical model. This suggests the following influences: The Young’s modulus for the 3D printed plastic is probably a little higher than the 2.0’ GPa assumed in the calculations (independent tests [14] suggest a value of 2.1 GPa might be a more accurate value, and still within the manufacturer’s specifications). Small tabs were included (printed) at the top corners to facilitate hanging the loads (see Fig. 4), and these might have a minor stiffening effect on the top corners. Fillets were used to mitigate the possibility of stress concentrations at corners, the lengths of the columns were measured to the underside of the cross-beam. The measured thickness of the printed columns was typically 2.03 mm, as opposed to the nominal value of 2 mm. A small difference, but the stiffness via the second moment of area is related to thickness in a cubic relation and would thus correspond to a roughly 5% increase in stiffness. All of the above would have a tendency to increase stiffness, and bring the theoretical and experimental results closer together. That is, the experimental results confirm the theoretical predictions to within the uncertainty in system parameters. However, as empha-
sized earlier it is the relative effect of geometric parameters that is the major focus of this paper. One of the key advantages of 3D printing is repeatability [15]. Three additional frame B’s were printed, tested, and the resulting stiffness was measured over the range 0:63 0:03 N/mm. The overall uncertainty in the measurements is due to variations in the clamping, the manner in which the weights are attached, etc., but, assuming Gaussian precision error distributions, the linear least squares fit gives a relatively accurate stiffness estimate. Printing multiple copies of frames and repeat testing is also a useful means of introducing students to experimental uncertainty. We assume that the self-weight of the frames is negligible.
7. Other stiffness enhancements Given the baseline configuration we can also make stiffness changes via the addition of further structural frame elements. Some other possibilities were shown in Fig. 7. It is again convenient to focus on moment-carrying connections and point loads but that still leaves considerable scope for conducting additional comparative stiffness assessments.2 The first example is simply to add a diagonal bracing member (e.g., Fig. 7(b)), that is oriented to ensure it carries tensile loads. In this case, and using the square frame where both the beam and columns have equal flexural rigidity, the lateral stiffness is increased by a factor of about 370. This massive increase in stiffness (which may make force-deflection measurements more challenging, at least using the same loading approach) is clearly due to a high axial stiffness contribution, i.e., adding the appropriately transformed diagonal stiffness contribution to the top right corner of the frame, i.e., in the lower 3 3 elements in the structure stiffness matrix in which the unbraced frame was relatively flexible. However, on a practical issue we note that this type of frame tends to induce compressive axial loading in other frame elements and thus buckling may become an issue. But generally we find the triangulation considerably increases lateral stiffness despite only a relatively small increase in the weight of the structure. The benefits of diagonal bracing are well-known of course, but it is easy enough to add members in the 3D print. Adding an opposite diagonal provides added stiffness in spite of the direction of loading. We choose the longest frame (F) and add an additional crossbeam member (Fig. 7(c)), in which the new beam member is also 8 mm thick, located at a distance Z from the base (and Z can also be conveniently varied according to student ID numbers), and we call this frame I, as shown in Fig. 8(a). Now the frames act as a two-story sway frame. Again applying a corner load and measuring the corresponding sway deflection at the top level (d) results in a stiffness of 0.96 N/mm. Based on the simplified sway assumption we would expect this stiffness to be ðLF =LI Þ3 =2 ¼ 4 times the stiff2 In class I have used the last couple of digits of student ID’s to produce a spread of geometries within certain bounds. Depending on the size of the class this will also typically result in a few identical frames, and these have their use for assessing measurement uncertainty [15].
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Fig. 6. (a) Measured force-deflection data and least squares fit stiffness for the seven frames, (b) nondimensionalized data for the sway frames (with stiff cross-beams).
Table 2 A comparison between experiment and theory of the frame geometries tested. Frame #
ktheo (N/mm)
kexp (N/mm)
ðk=kB Þexp
kexp =ktheo
A B C D E F G
3.576 0.519 0.163 5.068 0.636 0.189 0.260
4.08 0.63 0.21 6.67 0.73 0.23 0.30
6.80 1 0.35 11.12 1.22 0.38 0.50
1.14 1.21 1.29 1.32 1.15 1.22 1.15
Fig. 7. Additional stiffness enhancements to the baseline frame.
Fig. 8. Additional stiffening effects, (a) a single additional cross-beam (frame I); (b) plus an additional diagonal bracing element.
L. Virgin / Engineering Structures 150 (2017) 135–142
141
Fig. 9. Some non-simple 3D-printed plane frames.
ness of frame F (which was measured at 0.23 N/mm). We then also printed frame C with an additional mid-height horizontal frame element (2 mm thickness), and we call this frame H. This frame has greater flexibility due to not insignificant rotations at the joints, as seen in its loaded condition in Fig. 4(c). A similar experimental procedure results in a lateral stiffness measurement of 0.54 N/mm, i.e., somewhat less stiff than the frame with thicker cross-beams. However, now the simple sway theory is inappropriate, and the full stiffness (FEA) method used (within Solidworks [16]) to produce a stiffness of 0.52 N/mm, with Solidworks using data directly from the 3D-printed designs within the context of print resolution, material properties, etc. Taking this idea a little further, and allowing a degree of flexibility for demonstration purposes, we can consider the examples shown schematically in Fig. 7(d,e), with a photographic image of the frame in Fig. 7(e) shown for the lower-bay stiffened frame J in Fig. 8(b). Again, despite a very modest increase in mass (weight) of these frames, their lateral stiffness is very much greater, even to the extent that the bay within which the diagonals are placed are effectively rigid, and thus the sway of the frame is largely governed by the stiffness of the columns without the bracing, and this of course incorporates columns somewhat shorter than the overall height of the frame, i.e., based on Z rather than L. Again applying a corner load and measuring the corresponding deflection leads to a stiffness of 1.88 N/mm. This is essentially ðLF =LJ Þ3 ¼ 8, eight times stiffer than frame F (which was measured at 0.23 N/mm), and twice the stiffness of frame I (which was measured at 0.96 N/mm). A stiffness of 1.80 N/mm was also measured for the frame with a diagonal bracing element in the upper story: the braced story acts effectively like a rigid panel with the lateral frame stiffness governed by the columns in the unbraced (sway) story. Finally, we can take this whole approach further and consider (rigidly jointed) trusses of the type shown in Fig. 9. The lower two (red) frames in this figure were tested beyond their elastic limit, with the one on the left showing an overall permanent deflection somewhat resembling elastic deformation, and the one on the right showing a clear plastic deformation in a single (buckled) member. In these cases, with a relatively high number of joint degrees of freedom we necessarily resort to the capabilities of Solidworks or any other FEA package. Based on the earlier discussion it is again clear that a relatively modest increase in added weight can lead to a much enhanced lateral stiffness. These less
simple frames are ideal for longer term-length type student projects.
8. Discussion and other practical considerations As demonstrated here, 3D printing has a potentially useful role to play in the teaching of the strength of materials and structural analysis. Although 3D printing has focused quite naturally on essentially rigid mechanical components especially in the manufacturing context, the increasing resolution and use of various materials enables 3D-printed elastic slender structures to be used to complement the teaching of engineering mechanics. This paper has focused on the stiffness of slender plane frames: a key pedagogical stepping stone between single structural elements and geometrically complex elastic continua. By making isolated but consistent structural changes, which is easily accomplished in 3D printing, it is then possible to gain an appreciation for how stiffness depends on the geometric arrangements of members within a frame. Considerations concerning stress and ultimate limit states could also be addressed, but these are not considered here (other than using small corner fillets to avoid stress concentrations). Other extensions to this work can easily be envisioned, for example, confirmation of the MaxwellBetti law of reciprocal deflections in the two-story frame. Furthermore, it would be easy to set-up a student competition based on, perhaps, asking students to print a frame to exhibit a required stiffness, within certain geometric constraints, or developing a stiffness/weight measure a ratio that is especially important in aerospace engineering for example. Another aspect of 3D printing is that additional minor changes that facilitate testing can be conveniently incorporated into a design. For example, small ‘tabs’ with a hole were printed into the corners of the frames for ease of attaching (point) loads. Printing frames contiguous with relatively massive bases ensures fully clamped boundary conditions. It should be mentioned that ABS thermoplastic is not always ideal: there is a minor anisotropy due to the orientation of print, there may be small mechanical property variations due to material creep, age of specimen, thermal environment, speed of loading, etc. But again, since the emphasis in this paper has been on relative stiffness effects these minor issues do not play a significant role.
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Depending on the sophistication of the measuring device (an OPTO NCDT 1302 proximity laser was used in this study [17]) it is also relatively easy to measure linear natural frequencies, and thus stiffness, given an easy measurement of mass. This is a natural extension of the basic stiffness method but beyond the scope of the linear static behavior described here.
in which C ¼ cos h and S ¼ sin h, where h is the angle (measured counterclockwise) between the member in going from local to global coordinates. Appendix B. Structure stiffness matrix for a rectangular plane frame
9. Concluding remarks
The teaching of structural analysis for frame-type structures typically relies on the stiffness method, incorporating beam or beam-column elements, and utilizes a variety of techniques from linear algebra. This is, of course, a subset of finite element analysis. For non-simple geometries it may be difficult for students to retain a structural feel for how the stiffness of a structure is related to geometry. 3D-printed plane frames can provide a beneficial practical appreciation, especially in terms of the how geometry affects lateral stiffness, and the extent to which simplifying assumptions, e.g., pure sway, are justified. Making changes to a baseline configuration enables a direct comparative study to be conducted. Furthermore, a comparison between theory and experimental data is easily achieved, based on simultaneously testing and analyzing simple frame structures. Much of the behavior described in this paper is based on simple back-of-the-envelope type calculations, but even for moderately complex structures it is still straightforward to compare structural behavior of a physical specimen with any number of FEA packages, but especially Solidworks, since the printed specimens are drawn directly from Soliworks data files. Acknowledgement The author appreciates the assistance of Joe Giliberto and a number of undergraduates at Duke for assisting with the acquisition of data for this paper. Appendix A. Beam stiffness matrix in global coordinates
kð1; 1Þ ¼ kð4; 4Þ ¼ kð1; 4Þ ¼ kð4; 1Þ ¼ ðEA=LÞC 2 þ ð12EI=L3 ÞS2 kð1; 2Þ ¼ kð2; 1Þ ¼ kð1; 5Þ ¼ kð5; 1Þ ¼ kð2; 4Þ ¼ kð4; 2Þ ¼ kð4; 5Þ ¼ kð5; 4Þ ¼ ðEA=LÞCS ð12EI=L3 ÞCS kð1; 3Þ ¼ kð3; 1Þ ¼ kð1; 6Þ ¼ kð6; 1Þ ¼ kð3; 4Þ ¼ kð4; 3Þ ¼ kð4; 6Þ ¼ kð6; 4Þ ¼ ð6EI=L2 ÞS kð2; 2Þ ¼ kð5; 5Þ ¼ kð2; 5Þ ¼ kð5; 2Þ ¼ ðEA=LÞS2 þ ð12EI=L3 ÞC 2 kð2; 3Þ ¼ kð3; 2Þ ¼ kð2; 6Þ ¼ kð6; 2Þ ¼ kð3; 5Þ ¼ kð5; 3Þ ¼ kð5; 6Þ ¼ kð6; 5Þ ¼ ð6EI=LÞC kð3; 3Þ ¼ kð6; 6Þ ¼ 4EI=L kð3; 6Þ ¼ kð6; 3Þ ¼ 2EI=L
ð2Þ
K¼
k11
k12
k21
k22
ð3Þ
in which 2 k11
6 ¼4
ðAE=LÞB þ 12ðEI=L3 ÞC
0
6ðEI=L2 ÞC
0
ðAE=LÞC þ 12ðEI=L3 ÞB
6ðEI=L2 ÞB
6ðEI=L2 ÞB 0
4ðEI=LÞB þ 4ðEI=LÞC
2 T 6 k12 ¼ k21 ¼ 4
2 k22
6 ¼4
6ðEI=L2 ÞB ðAE=LÞB
0
0
12ðEI=L3 ÞB 2
6ðEI=L ÞB
0
ðAE=LÞB þ 12ðEI=L3 ÞC
3
2ðEI=LÞB 6ðEI=L2 ÞC
0 ðAE=LÞC þ 12ðEI=L ÞB
2
6ðEI=L ÞB
7 5;
7 6ðEI=L2 ÞB 5;
3
0
3
2
6ðEI=L ÞB
2
6ðEI=L ÞB
3 7 5:
4ðEI=LÞB þ 4ðEI=LÞC
in which the B and C subscripts refer to beam and column respectively. References [1] Timoshenko SP. Strength of materials. New Jersey: Van Nostrand; 1977. [2] MATLAB and Statistics Toolbox Release 2012b. The MathWorks, Inc, Natick, Massachusetts, United States. [3] Strang G. Introduction to linear algebra. Wellesley-Cambridge Press; 2009. [4] Rubinstein MF. Matrix computer analysis of structures. Prentice-Hall; 1966. [5] Przemieniecki JS. Theory of matrix structural analysis. Dover; 1968. [6] Laursen HI. Structural analysis. McGraw-Hill Education; 1988. [7] Weaver W, Gere JM. Matrix analysis of framed structures. 3rd ed. NY: van Nostrand Reinhold; 1990. [8] McGuire W, Gallagher RH, Ziemian RD. Matrix structural analysis. Wiley; 1999. [9] Reddy JN. An introduction to the finite element method. McGraw-Hill; 2006. [10] Kassimali A. Matrix analysis of structures. Cengage Learning; 2012. [11] Hibbeler RC. Structural analysis. Pearson Education; 2008. [12] http://www.stratasys.com/materials/fdm/absplus. [13] Tymrak BM, Kreiger M, Pearce JM. ‘Mechanical properties of components fabricated with open-source 3-D printers under realistic environmental conditions’. Mater Des 2014;58:242–6. [14] Virgin LN. On the flexural stiffness of 3D printer thermoplastic. Int J Mech Eng Educ 2017;45:59–75. [15] Wheeler AJ, Ganji AR. Introduction to engineering experimentation. New Jersey: Prentice-Hall; 1996. [16] http://www.solidworks.com/. [17] http://www.micro-epsilon.com/download/manuals/man-optoNCDT-1302-en. pdf.
Contents lists available at ScienceDirect
Engineering Structures journal homepage: www.elsevier.com/locate/engstruct
Enhancing the teaching of linear structural analysis using additive manufacturing Lawrence Virgin School of Engineering, Duke University, Durham, NC 27708, USA
a r t i c l e
i n f o
Article history: Received 7 March 2017 Revised 17 June 2017 Accepted 17 July 2017
Keywords: Structural analysis Flexural stiffness Thermoplastic 3D printing Education
a b s t r a c t Structural analysis forms a key component in many courses in civil, mechanical and aerospace engineering. Conventionally, the matrix stiffness method, a subset of finite element analysis, tends to occupy a central position in a typical syllabus, with a special focus on plane frames providing a bridge between basic structural components with pedagogical clarity and real-world structures. Equations of equilibrium are set-up and the full force of linear algebra brought to bear using the capabilities of Matlab or more specialized FEA packages. Such classes have a tendency to become a little dry and suffer from the usual shortcomings of numerical analysis and a black box approach - shortcomings in the sense of conceptual understanding as opposed to usefulness in the hands of experienced practitioners. The relatively recent advent of additive manufacturing is an exciting opportunity to incorporate a practical aspect to structural analysis. This paper describes the use of 3D printing, via the flexural stiffness of plane frames, to develop a structural feel for students, augmenting theoretical analyses. In addition to directly addressing the role of modeling, approximation, applicability of the underlying theory, and measurement uncertainty, it is thoroughly hands-on and initial anecdotal evidence suggests a higher degree of student buy-in. Ó 2017 Elsevier Ltd. All rights reserved.
1. Introduction The bending stiffness of a structural member (based on standard beam theory) is a function of loading, boundary conditions, material and geometry [1]. For a given prismatic member the stiffness typically scales with EI=L3 , and it is easy to demonstrate the length dependency with a simple ruler for example. Analytically, given a set of forces and boundary conditions, we can integrate the governing expressions and obtain deflection, and hence stiffness. This process is not so straightforward for frames, consisting of an arrangement of beams (and columns) in typically rectangular combinations. However, the overall stiffness of such structures is very important and perhaps a key teaching opportunity occurs when we seek to shed light on how stiffness depends on these more realistic geometrical arrangements. This is where the stiffness method comes in. Despite the liberating effect of Matlab [2] and the ease of numerical methods [3], the stiffness method becomes a decreasingly hands-on approach for all but the simplest examples. Many textbooks throughout the last 50 years or so have included chapters on the stiffness method including plane frames; a representative selection is [4–10]. However, it is possible to exploit versatile
E-mail address: [email protected] http://dx.doi.org/10.1016/j.engstruct.2017.07.054 0141-0296/Ó 2017 Elsevier Ltd. All rights reserved.
3D printing for deepening an appreciation for structural behavior, and specifically in terms of printing a range of elastic, relatively slender, plane frames. Additionally, with relatively simple loading and boundary conditions it is then possible to assess the role of geometry and its influence on certain stiffness properties of the frames using measured data. This paper considers the lateral stiffness of a baseline plane frame and some basic variations with three main foci: use of 3D printing in the teaching of structural analysis, exploring the influence of parameters on stiffness, assessing the role of simplifying assumptions. The overall goal of the paper is to provide a systematic approach to developing a deeper student understanding of structural stiffness.
2. 3D printing of slender structures 3D printing has already revolutionized the teaching of mechanical engineering, via the rapid-prototyping of components like gears, for example. And given increasingly higher resolution it is now relatively easy to produce slender elastic elements and structures, and providing deflections and stresses are kept within
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acceptable limits, the behavior of such elements and structures is linear and elastic. Thus, 3D printing has potential as a teaching tool in the realm of deformable bodies and structures. Furthermore, in terms of the capabilities of 3D printing and elementary testing configurations, we shall focus attention on planar structural frames in which (i) the boundary conditions are either essentially fixed or free, and (ii) loads are applied at specific locations (point loads). This provides the context in which students can simply print and test structures and thus assess the important features of the flexural behavior in general, and aspects of the stiffness method in particular. We shall focus attention on relatively flexible right-angled plane portal frames with clamped boundary conditions, momenttransmitting joints, and single point loading. This configuration was chosen partly to facilitate 3D printing and simple experimental testing, but it also represents a class of structure that is more instructive than a single structural element like a cantilever, and also can be analyzed using back-of-the-envelop (sway) calculations in some cases. Thus, in order to maintain a reasonable balance between hand-calculation and the fully fledged stiffness method we develop a focus on some simple portal frames, and compare theory (essentially the linear stiffness method using elastic beam elements) and experimental stiffness measurements. More specific details will be given later, but by way of introduction, consider the middle frame shown in Fig. 1. If we clamp the lower edge, and subject the top corner to a lateral force, we can extract the stiffness of the frame in terms of lateral deflection. The stiffness of the columns scales with EI=L3 , and thus in moving to the frame shown on the left we expect a (much) greater stiffness, all other things being equal. Furthermore, if the cross-beam is relatively stiff (the frame on the right), we also expect an increase in stiffness, but in terms of reducing the rotation of the corners, and a deflected shape that resembles a lateral sway. This
paper uses the 3D printer to produce frames that can be used to directly examines these effects.
3. Flexural stiffness analysis of a simple portal frame Historically, the teaching of structural analysis proceeded from pin-jointed trusses. This was partly justified by the ubiquity of riveted joints, but mostly because very often simplified analytical techniques, such as the methods of joints or sections, could be brought to bear [11]. But now most structural connections are rigid (e.g., welded) and computational techniques, for example the stiffness method, dominate. Thus frames rather than trusses now form the backbone of most courses on structural analysis (see the list of references). In order to explore this a little further consider a simple plane portal frame, consisting of two vertical columns that are clamped at their bottom ends and connected (via moment transmitting joints) to a horizontal beam as shown in Fig. 2(a). Assuming the lengths L and flexural rigidity EI are the same for all members, and that the frame is subject to a single horizontal load at one of the corners, we can write down the set of equilibrium equations for the structure: 2 2 3 ðAL2 =I þ 12Þ 0 6L AL2 =I 0 F 6 6 7 6 2 607 6 0 12 ðAL =I þ 12Þ 6L 6 7 6 6 7 6 6 7 6 2 607 6 8L 0 6L 6 7 6 6 7 ¼ EI=L3 6 6 7 6 2 607 6 ðAL =I þ 12Þ 0 6 7 6 6 7 6 6 7 6 2 607 6 sym: ðAL =I þ 12Þ 4 5 6 4 0
3 2 3 X 7 1 76 7 6 7 6L 7 76 Y 1 7 76 7 76 7 2 76 2L 76 h1 7 7 76 7; 76 7 6 X2 7 6L 7 76 7 76 7 76 7 6Y 7 6L 7 74 2 5 5 2 h2 8L 0
ð1Þ
in which ðX; Y; h) are global coordinates, relative to the bottom left corner of the frame. The element stiffness matrix for a beamcolumn in global coordinates can be found in Appendix A. If we now assume an overall frame dimension (relevant to the physical dimensions to be 3D printed later) of L ¼ 0:1 m, and a rectangular cross-section area ðb dÞ ¼ ð0:01 0:002Þ, gives an area A ¼ 20 106 m2, second moment of area I ¼ 6:67 1012 m4,
Fig. 1. Simple 3D-printed plane frames.
and thus AL2 =I ðL=rÞ2 ¼ 30 103 , where r is the radius of gyration. These parameters relate to a geometry that may be considered highly flexible, and this facilitates relatively easy measurements. In analysis, the effective degrees of freedom can be reduced if we examine the relative magnitudes of each element and exploit certain symmetry conditions.
Fig. 2. (a) A simple square portal frame with identical beam and columns, (b) the baseline configuration.
L. Virgin / Engineering Structures 150 (2017) 135–142
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Solving the set of linear simultaneous Eq. (1) we obtain the dominant sway deflection: X 1 ¼ 4:466F with essentially no difference between this and X 2 . Furthermore, the vertical Y deflections are effectively zero. Given the specific values used in the example we get a sway deflection of about FL3 =ð16:79EIÞ. Solving a 6 6 set of simultaneous equations is a relatively simple task (thanks to Matlab) but not a hand-calculation, despite the fact that some of the resulting deflections are quite small (h) and very small (Y). Let’s simplify things by assuming that the frame elements are axially stiff (relative to bending stiffness), and that the stiffness of the horizontal beam is relatively large (compared to the columns, we shall investigate this further later), then the sway deflection of the frame is effectively a single-degree-of-freedom (SDOF) spring characterized by: F ¼ ð2 12EI=L3 Þd, that is, eight times stiffer than the familiar cantilever result (now having two members each effectively restrained against rotation at both ends), and about 43% stiffer than our previous analysis in which the beam had the same flexural rigidity (and length) as the columns. But now the reduction to a SDOF allows a more systematic investigation of stiffness in terms of ‘easily set or measured’ parameters of a structural system. This is an ideal context for 3D-printed specimens to provide a practical hands-on augmentation to the introduction of elastic flexural modeling including the stiffness method. For our square portal frame we can assess the accuracy of the pure sway approximation by varying the stiffness of the crossbeam relative to the columns. In the general case this would involve the length of the beam, but for the square frame we can make a valid comparison using the ratio Ib =Ic . Fig. 3 summarizes the resulting sway stiffness. The stiffness is nondimensionalized by the Ib ¼ Ic case - we could have just as easily used the pure sway case. In any case, we see that pure sway effectively occurs for Ib =Ic ’8, corresponding to a rectangular cross-section in which the beam is twice as thick as the columns. Similarly a frame with equal flexural rigidity ðEIÞ in the beams and columns can also be used to assess the influence of Lb =Lc on the sway behavior. The general structure stiffness matrix for a portal frame can be found in Appendix B. 4. The baseline case In order to later conduct a systematic parametric study to shed light on flexural stiffness we start by defining a standard ‘baseline’ case. Referring back to Fig. 2(b) (and the middle frame in Fig. 1) we initially fix the following parameters: The ‘width’ b of the whole frame is initially 20 mm, and the thickness d ¼ 2 mm for all frame elements (including H, initially); the overall dimensions of the frame are L ¼ 100 mm; and W ¼ 50 mm. The Young’s modulus for ABS thermoplastic for the 3D printer used in this study was taken as E ¼ 2:0 GPa.1 The frame was printed onto an integral rigid base, and a measured force F was then applied via hanging weights (in a horizontally clamped frame orientation), or using a load cell, and the lateral deflection d measured using a standard proximity laser (see Fig. 4), thus furnishing a force-deflection relation (linear for relatively small deflections), the slope of which characterizes the stiffness of the frame. According to the simple theory we should expect F ¼ kd, where k 24EI=L3 , depending on the validity of the SDOF assumption, otherwise the regular stiffness method can be used. The baseline frame was printed using a Stratsys printer [12] using ABSplus - P430 production-grade thermoplastic, with a specification Young’s modulus in the range 1.65–2.1 GPa, but in the computations we use E ¼ 2:0 GPa [13]. The Solidworks STL files 1 There are some relatively minor, but quite interesting uncertainties in this estimate based on the subtleties of the printing process [14].
Fig. 3. The effect of the relative beam-to-column stiffness on the sway deflection of a rectangular (square) frame.
were developed according to the baseline measurements and relative to the absolute resolution of the 3D printer. Although the resolution of most 3D printers continues to improve, we typically focus most attention on the accuracy of the thicknesses, since the thickness dimensions (which are typically small) also have a strong influence on the stiffness (via I). The specimen was clamped by a rigid vice at one end and the free end subject to a point load as shown in Fig. 4. The force-deflection relation is shown within Fig. 6(b) (later to be identified as frame B), in which some of the measurements were repeated (e.g., loading and unloading, and then turning the frame around and repeating) to provide a better linear least-squares fit. The slope of this experimental line is 0.63 N/mm. Solving for the lateral deflection we make use of the stiffness method (for which the stiffness matrix is given explicitly in Appendix B) resulting in a lateral stiffness of 0.52 N/mm. According to the simplified (pure sway) theory we would expect k ¼ 0:67 N/mm. Assuming there is a reasonably high accuracy in measuring the various parameters, the discrepancy could most likely be ascribed to the Young’s modulus or rotation at the top corners. However, we also chose to examine the validity of the SDOF model, and we did this by changing the H; L and b parameters. For example, it is likely that there is a little rotation at the corners of the portal frame, and thus we might expect an experimental stiffness that is slightly lower than the (pure sway) assumption of no rotation at the corners. 5. Plane frame variations We then depart from the baseline case in order to assess the effect of various parameters on both the stiffness and the assumption on which the very simple (pure sway) stiffness relation is based. We thus chose variations to the baseline with beam thickness H increased from 2 mm to 8 mm; width b reduced from 20 mm to 10 mm; and the length of the columns L increased to 150 mm and also reduced to 50 mm (see Fig. 5). In typical multistory buildings the stiffness of the horizontal members (floors) are somewhat stiffer than typical columns, and hence we tend to be less interested in cases for which ðEb Ib =L3b Ec Ic =L3c Þ. This provides a permutation of frame geometries for which stiffness can be compared systematically. Note that the 100 mm 50 mm overall dimension was chosen (rather than the square frame) due to practical limitations associated with the resolution of the 3D
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Fig. 4. (a) The baseline frame, (b) in the test configuration, (c) a frame with an intermediate member.
Fig. 5. Printed frames - variations to the baseline frame B.
printer and experimental measurements, i.e., these frames are relatively flexible. One of the advantages of this kind of parametric study, based on comparisons, is that, for example, the exact value of E is less problematic because all the frames are made of the same material, printed in the same manner, and tested under nominally the same conditions [14]. It is also a not-uncommon student mistake to change more than one parameter at a time, thus making it difficult to assess sensitivities. Six more frame geometries were then considered (and were later printed) with ðH; LÞ = (2, 50), (2, 150), (8, 50), (8, 100), (8, 150), (2,100), all values in mm, and where the last frame (later to be labeled G), has b ¼ 10. Increasing (decreasing) L decreases (increases) the overall lateral flexural stiffness of the frame; while increasing H also increases the lateral flexural stiffness but in terms of influencing the rotation at the free end(s), and thus the justification for the simple SDOF assumption. The analytical results are summarized in Table 1. Let’s use column 7 of Table 1 (k=kB ) to compare and contrast the stiffness between frames (relative to the baseline case, kB .):
The columns in Frame A are half the length of the columns in Frame B, and thus we would expect this frame to be roughly ð0:5Þ3 ¼ 8 times stiffer. The columns in Frame C are one and a half times the length of the columns in Frame B, and thus we would expect this frame to be roughly 1=ð1:5Þ3 ¼ 0:296 the lateral stiffness. The width of all the frame elements in Frame G are one half of those in Frame B, and since I scales linearly with b we would expect half the lateral stiffness. Frames A and D are both squares, and thus these specific results can be located on Fig. 3 with coordinates: ðIb =Ic ¼ 1; k=kIb ¼Ic Þ and ðIb =Ic ¼ 64; k=kIb ¼Ic ! 1:43Þ, respectively, and using the appropriate non-dimensionalization for stiffness. We can also assess the SDOF sway approximation (referencing column 8 of Table 1), where we use C ¼ 24 (pure sway) to normalize the results, with k=k0 ! 1 thus indicating the pure sway case: Frames D, E, and F (with their thick cross-beams) all conform very closely to the pure sway case.
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L. Virgin / Engineering Structures 150 (2017) 135–142 Table 1 Frame geometries (with reference to Figs. 2(b) and 5). Frame B is the baseline case; k0 is for pure sway. Frame #
H (mm)
L (mm)
b (mm)
k (N/mm)
FL3 =dEI (pure sway = 24)
k=kB
k=k0
A B C D E F G
2 2 2 8 8 8 2
50 100 150 50 100 150 100
20 20 20 20 20 20 10
3.576 0.519 0.163 5.068 0.636 0.189 0.260
16.763 19.461 20.687 23.758 23.859 23.894 19.481
6.891 1 0.315 9.767 1.226 0.364 0.500
0.698 0.811 0.862 0.990 0.994 0.996 0.812
Comparing Frames B and G we see that they approximate similarly in sway deflection. Comparing Frames A, B, and C we observe that the sway approximation is better if the length of the columns L (relative to the length of the beam W ¼ 50 mm) is large. In summary, and based on the specific parameters we chose to vary in this study, we confirm that the lateral stiffness for a frame generally scales with Ec Ic =L3c , and that the SDOF sway approximation is increasingly justified with increasing Ib =Ic , but also with the overall slenderness of the frame L=W (since the stiffness of the beam relative to the column contributes to the joint rotation).
6. Experimental testing results We can plot the measured force-deflection relation for all seven frames (and their least-squares fits) superimposed in dimensional terms as in Fig. 6(a). We observe how the height L of the frame influences the stiffness, for example in going from frame A to frame B to frame C. Fig. 6(b) is a plot of the same data for frames D, E, and F, and scaled with respect to the column length (cubed), with EI consistent between the frames. These three frames have the stiff cross-beams and thus deform in a predominantly sway mode. Superimposed on this data is a straight line that is the theoretical pure sway relation FL3 ¼ 24EId, in which E ¼ 2:1 GPa. A summary of the experimental results is contained in Table 2. The relative stiffness effects can be extracted along the same lines as the theoretical results in Table 1. However, the interesting thing to note is the consistently higher stiffness of the 3D-printed frames as compared with the theoretical model. This suggests the following influences: The Young’s modulus for the 3D printed plastic is probably a little higher than the 2.0’ GPa assumed in the calculations (independent tests [14] suggest a value of 2.1 GPa might be a more accurate value, and still within the manufacturer’s specifications). Small tabs were included (printed) at the top corners to facilitate hanging the loads (see Fig. 4), and these might have a minor stiffening effect on the top corners. Fillets were used to mitigate the possibility of stress concentrations at corners, the lengths of the columns were measured to the underside of the cross-beam. The measured thickness of the printed columns was typically 2.03 mm, as opposed to the nominal value of 2 mm. A small difference, but the stiffness via the second moment of area is related to thickness in a cubic relation and would thus correspond to a roughly 5% increase in stiffness. All of the above would have a tendency to increase stiffness, and bring the theoretical and experimental results closer together. That is, the experimental results confirm the theoretical predictions to within the uncertainty in system parameters. However, as empha-
sized earlier it is the relative effect of geometric parameters that is the major focus of this paper. One of the key advantages of 3D printing is repeatability [15]. Three additional frame B’s were printed, tested, and the resulting stiffness was measured over the range 0:63 0:03 N/mm. The overall uncertainty in the measurements is due to variations in the clamping, the manner in which the weights are attached, etc., but, assuming Gaussian precision error distributions, the linear least squares fit gives a relatively accurate stiffness estimate. Printing multiple copies of frames and repeat testing is also a useful means of introducing students to experimental uncertainty. We assume that the self-weight of the frames is negligible.
7. Other stiffness enhancements Given the baseline configuration we can also make stiffness changes via the addition of further structural frame elements. Some other possibilities were shown in Fig. 7. It is again convenient to focus on moment-carrying connections and point loads but that still leaves considerable scope for conducting additional comparative stiffness assessments.2 The first example is simply to add a diagonal bracing member (e.g., Fig. 7(b)), that is oriented to ensure it carries tensile loads. In this case, and using the square frame where both the beam and columns have equal flexural rigidity, the lateral stiffness is increased by a factor of about 370. This massive increase in stiffness (which may make force-deflection measurements more challenging, at least using the same loading approach) is clearly due to a high axial stiffness contribution, i.e., adding the appropriately transformed diagonal stiffness contribution to the top right corner of the frame, i.e., in the lower 3 3 elements in the structure stiffness matrix in which the unbraced frame was relatively flexible. However, on a practical issue we note that this type of frame tends to induce compressive axial loading in other frame elements and thus buckling may become an issue. But generally we find the triangulation considerably increases lateral stiffness despite only a relatively small increase in the weight of the structure. The benefits of diagonal bracing are well-known of course, but it is easy enough to add members in the 3D print. Adding an opposite diagonal provides added stiffness in spite of the direction of loading. We choose the longest frame (F) and add an additional crossbeam member (Fig. 7(c)), in which the new beam member is also 8 mm thick, located at a distance Z from the base (and Z can also be conveniently varied according to student ID numbers), and we call this frame I, as shown in Fig. 8(a). Now the frames act as a two-story sway frame. Again applying a corner load and measuring the corresponding sway deflection at the top level (d) results in a stiffness of 0.96 N/mm. Based on the simplified sway assumption we would expect this stiffness to be ðLF =LI Þ3 =2 ¼ 4 times the stiff2 In class I have used the last couple of digits of student ID’s to produce a spread of geometries within certain bounds. Depending on the size of the class this will also typically result in a few identical frames, and these have their use for assessing measurement uncertainty [15].
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Fig. 6. (a) Measured force-deflection data and least squares fit stiffness for the seven frames, (b) nondimensionalized data for the sway frames (with stiff cross-beams).
Table 2 A comparison between experiment and theory of the frame geometries tested. Frame #
ktheo (N/mm)
kexp (N/mm)
ðk=kB Þexp
kexp =ktheo
A B C D E F G
3.576 0.519 0.163 5.068 0.636 0.189 0.260
4.08 0.63 0.21 6.67 0.73 0.23 0.30
6.80 1 0.35 11.12 1.22 0.38 0.50
1.14 1.21 1.29 1.32 1.15 1.22 1.15
Fig. 7. Additional stiffness enhancements to the baseline frame.
Fig. 8. Additional stiffening effects, (a) a single additional cross-beam (frame I); (b) plus an additional diagonal bracing element.
L. Virgin / Engineering Structures 150 (2017) 135–142
141
Fig. 9. Some non-simple 3D-printed plane frames.
ness of frame F (which was measured at 0.23 N/mm). We then also printed frame C with an additional mid-height horizontal frame element (2 mm thickness), and we call this frame H. This frame has greater flexibility due to not insignificant rotations at the joints, as seen in its loaded condition in Fig. 4(c). A similar experimental procedure results in a lateral stiffness measurement of 0.54 N/mm, i.e., somewhat less stiff than the frame with thicker cross-beams. However, now the simple sway theory is inappropriate, and the full stiffness (FEA) method used (within Solidworks [16]) to produce a stiffness of 0.52 N/mm, with Solidworks using data directly from the 3D-printed designs within the context of print resolution, material properties, etc. Taking this idea a little further, and allowing a degree of flexibility for demonstration purposes, we can consider the examples shown schematically in Fig. 7(d,e), with a photographic image of the frame in Fig. 7(e) shown for the lower-bay stiffened frame J in Fig. 8(b). Again, despite a very modest increase in mass (weight) of these frames, their lateral stiffness is very much greater, even to the extent that the bay within which the diagonals are placed are effectively rigid, and thus the sway of the frame is largely governed by the stiffness of the columns without the bracing, and this of course incorporates columns somewhat shorter than the overall height of the frame, i.e., based on Z rather than L. Again applying a corner load and measuring the corresponding deflection leads to a stiffness of 1.88 N/mm. This is essentially ðLF =LJ Þ3 ¼ 8, eight times stiffer than frame F (which was measured at 0.23 N/mm), and twice the stiffness of frame I (which was measured at 0.96 N/mm). A stiffness of 1.80 N/mm was also measured for the frame with a diagonal bracing element in the upper story: the braced story acts effectively like a rigid panel with the lateral frame stiffness governed by the columns in the unbraced (sway) story. Finally, we can take this whole approach further and consider (rigidly jointed) trusses of the type shown in Fig. 9. The lower two (red) frames in this figure were tested beyond their elastic limit, with the one on the left showing an overall permanent deflection somewhat resembling elastic deformation, and the one on the right showing a clear plastic deformation in a single (buckled) member. In these cases, with a relatively high number of joint degrees of freedom we necessarily resort to the capabilities of Solidworks or any other FEA package. Based on the earlier discussion it is again clear that a relatively modest increase in added weight can lead to a much enhanced lateral stiffness. These less
simple frames are ideal for longer term-length type student projects.
8. Discussion and other practical considerations As demonstrated here, 3D printing has a potentially useful role to play in the teaching of the strength of materials and structural analysis. Although 3D printing has focused quite naturally on essentially rigid mechanical components especially in the manufacturing context, the increasing resolution and use of various materials enables 3D-printed elastic slender structures to be used to complement the teaching of engineering mechanics. This paper has focused on the stiffness of slender plane frames: a key pedagogical stepping stone between single structural elements and geometrically complex elastic continua. By making isolated but consistent structural changes, which is easily accomplished in 3D printing, it is then possible to gain an appreciation for how stiffness depends on the geometric arrangements of members within a frame. Considerations concerning stress and ultimate limit states could also be addressed, but these are not considered here (other than using small corner fillets to avoid stress concentrations). Other extensions to this work can easily be envisioned, for example, confirmation of the MaxwellBetti law of reciprocal deflections in the two-story frame. Furthermore, it would be easy to set-up a student competition based on, perhaps, asking students to print a frame to exhibit a required stiffness, within certain geometric constraints, or developing a stiffness/weight measure a ratio that is especially important in aerospace engineering for example. Another aspect of 3D printing is that additional minor changes that facilitate testing can be conveniently incorporated into a design. For example, small ‘tabs’ with a hole were printed into the corners of the frames for ease of attaching (point) loads. Printing frames contiguous with relatively massive bases ensures fully clamped boundary conditions. It should be mentioned that ABS thermoplastic is not always ideal: there is a minor anisotropy due to the orientation of print, there may be small mechanical property variations due to material creep, age of specimen, thermal environment, speed of loading, etc. But again, since the emphasis in this paper has been on relative stiffness effects these minor issues do not play a significant role.
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Depending on the sophistication of the measuring device (an OPTO NCDT 1302 proximity laser was used in this study [17]) it is also relatively easy to measure linear natural frequencies, and thus stiffness, given an easy measurement of mass. This is a natural extension of the basic stiffness method but beyond the scope of the linear static behavior described here.
in which C ¼ cos h and S ¼ sin h, where h is the angle (measured counterclockwise) between the member in going from local to global coordinates. Appendix B. Structure stiffness matrix for a rectangular plane frame
9. Concluding remarks
The teaching of structural analysis for frame-type structures typically relies on the stiffness method, incorporating beam or beam-column elements, and utilizes a variety of techniques from linear algebra. This is, of course, a subset of finite element analysis. For non-simple geometries it may be difficult for students to retain a structural feel for how the stiffness of a structure is related to geometry. 3D-printed plane frames can provide a beneficial practical appreciation, especially in terms of the how geometry affects lateral stiffness, and the extent to which simplifying assumptions, e.g., pure sway, are justified. Making changes to a baseline configuration enables a direct comparative study to be conducted. Furthermore, a comparison between theory and experimental data is easily achieved, based on simultaneously testing and analyzing simple frame structures. Much of the behavior described in this paper is based on simple back-of-the-envelope type calculations, but even for moderately complex structures it is still straightforward to compare structural behavior of a physical specimen with any number of FEA packages, but especially Solidworks, since the printed specimens are drawn directly from Soliworks data files. Acknowledgement The author appreciates the assistance of Joe Giliberto and a number of undergraduates at Duke for assisting with the acquisition of data for this paper. Appendix A. Beam stiffness matrix in global coordinates
kð1; 1Þ ¼ kð4; 4Þ ¼ kð1; 4Þ ¼ kð4; 1Þ ¼ ðEA=LÞC 2 þ ð12EI=L3 ÞS2 kð1; 2Þ ¼ kð2; 1Þ ¼ kð1; 5Þ ¼ kð5; 1Þ ¼ kð2; 4Þ ¼ kð4; 2Þ ¼ kð4; 5Þ ¼ kð5; 4Þ ¼ ðEA=LÞCS ð12EI=L3 ÞCS kð1; 3Þ ¼ kð3; 1Þ ¼ kð1; 6Þ ¼ kð6; 1Þ ¼ kð3; 4Þ ¼ kð4; 3Þ ¼ kð4; 6Þ ¼ kð6; 4Þ ¼ ð6EI=L2 ÞS kð2; 2Þ ¼ kð5; 5Þ ¼ kð2; 5Þ ¼ kð5; 2Þ ¼ ðEA=LÞS2 þ ð12EI=L3 ÞC 2 kð2; 3Þ ¼ kð3; 2Þ ¼ kð2; 6Þ ¼ kð6; 2Þ ¼ kð3; 5Þ ¼ kð5; 3Þ ¼ kð5; 6Þ ¼ kð6; 5Þ ¼ ð6EI=LÞC kð3; 3Þ ¼ kð6; 6Þ ¼ 4EI=L kð3; 6Þ ¼ kð6; 3Þ ¼ 2EI=L
ð2Þ
K¼
k11
k12
k21
k22
ð3Þ
in which 2 k11
6 ¼4
ðAE=LÞB þ 12ðEI=L3 ÞC
0
6ðEI=L2 ÞC
0
ðAE=LÞC þ 12ðEI=L3 ÞB
6ðEI=L2 ÞB
6ðEI=L2 ÞB 0
4ðEI=LÞB þ 4ðEI=LÞC
2 T 6 k12 ¼ k21 ¼ 4
2 k22
6 ¼4
6ðEI=L2 ÞB ðAE=LÞB
0
0
12ðEI=L3 ÞB 2
6ðEI=L ÞB
0
ðAE=LÞB þ 12ðEI=L3 ÞC
3
2ðEI=LÞB 6ðEI=L2 ÞC
0 ðAE=LÞC þ 12ðEI=L ÞB
2
6ðEI=L ÞB
7 5;
7 6ðEI=L2 ÞB 5;
3
0
3
2
6ðEI=L ÞB
2
6ðEI=L ÞB
3 7 5:
4ðEI=LÞB þ 4ðEI=LÞC
in which the B and C subscripts refer to beam and column respectively. References [1] Timoshenko SP. Strength of materials. New Jersey: Van Nostrand; 1977. [2] MATLAB and Statistics Toolbox Release 2012b. The MathWorks, Inc, Natick, Massachusetts, United States. [3] Strang G. Introduction to linear algebra. Wellesley-Cambridge Press; 2009. [4] Rubinstein MF. Matrix computer analysis of structures. Prentice-Hall; 1966. [5] Przemieniecki JS. Theory of matrix structural analysis. Dover; 1968. [6] Laursen HI. Structural analysis. McGraw-Hill Education; 1988. [7] Weaver W, Gere JM. Matrix analysis of framed structures. 3rd ed. NY: van Nostrand Reinhold; 1990. [8] McGuire W, Gallagher RH, Ziemian RD. Matrix structural analysis. Wiley; 1999. [9] Reddy JN. An introduction to the finite element method. McGraw-Hill; 2006. [10] Kassimali A. Matrix analysis of structures. Cengage Learning; 2012. [11] Hibbeler RC. Structural analysis. Pearson Education; 2008. [12] http://www.stratasys.com/materials/fdm/absplus. [13] Tymrak BM, Kreiger M, Pearce JM. ‘Mechanical properties of components fabricated with open-source 3-D printers under realistic environmental conditions’. Mater Des 2014;58:242–6. [14] Virgin LN. On the flexural stiffness of 3D printer thermoplastic. Int J Mech Eng Educ 2017;45:59–75. [15] Wheeler AJ, Ganji AR. Introduction to engineering experimentation. New Jersey: Prentice-Hall; 1996. [16] http://www.solidworks.com/. [17] http://www.micro-epsilon.com/download/manuals/man-optoNCDT-1302-en. pdf.