The teaching of structural stiffness is one of the keystones of the undergraduate curriculum in mechanics and the strength of materials. Standard linear theory, going back to Hooke's law, has proven to be very successful in predicting the performance of elastic structures under load. Many courses in basic mechanics have a conventional laboratory component often involving a universal testing machine and extensometer. However, the advent of 3D printing presents an appealing pedagogical opportunity midway between theory and a formal lab experience. The material contained in this paper focuses on using the 3D printing of relatively simple, flexible cantilever structures. The relatively high resolution of modern 3D printers facilitates the production of slender (elastically deformable) structures, and thus provides an opportunity to exploit geometric parametric variations to enhance a practical understanding of fundamental mechanics concepts such as stiffness. This approach has proved successful in initial inclusion in both the classroom via demonstration models, as well as in the lab in which elementary facilities can be utilized to acquire data. The boundary conditions associated with a cantilever, and the application of a point force are especially simple to produce in practice, and provide an effective tactile demonstration of the influence of geometrical changes on the relation between force and deflection, i.e., stiffness.
I. INTRODUCTION
The ability of structural elements, and structures in general, to effectively withstand load is often achieved by flexural elastic deformation and the storage of strain energy. Provided the resulting stresses and deflections are kept within reasonable limits, this is often the basis of an economic design (minimal material use). Very often there is a linear relation between load and deflection, at least for relatively small deflections, broadly in the range of practical application.^{1} This is the case for beam theory, for example, and also allows an effective demonstration of the utility of using dimensionless variables.^{2–4} Coupled with the versatility of 3D printers to conveniently produce structures with specific geometry, and in an elastic material, the teaching of solid mechanics can be enhanced by simple handson demonstration models, which can also be used to acquire data.^{5}
3D printing has been introduced by a number of instructors to enhance the quality of instruction in a variety of contexts, including an appreciation of auxetic behavior,^{6} laboratory modules,^{7} mechanical behavior,^{8,9} general mechanical systems,^{10} and general education.^{11} The primary focus of this paper is on flexural behavior of linear elastic elements, for which we invoke Hooke's law, $F=kx$, i.e., force F causes a proportionate deflection x, characterized by the stiffness of coefficient k. The elastic flexural stiffness of a simple structure (here, a cantilever) is assessed, primarily through a subjective, tactile appreciation, and the ways in which the stiffness changes with geometry.^{12,13}
The key mechanism in this paper is a comparison based on parameter variation. Thus, the demonstration models can be used based purely on feel, where simply pushing down on the ends of cantilevers sequentially provides a direct assessment of stiffness. There is nothing unique about the dimensions chosen here, and students and instructors are encouraged to produce their own geometries. If measurements are made, then the results can be conveniently presented in terms of dimensionless quantities.
We shall focus on five fundamental cases, in which a change in geometry leads to a change in stiffness. Since the 3Dprinted specimens are all produced monolithically using the same material, we isolate a specific parameter, so that the effect on stiffness can be appreciated in a comparative sense. We shall focus on sets of tiploaded cantilevers (see Fig. 1), and specifically, we consider the following:

a simple cantilever, under variation of length: Fig. 1(a);

a simple cantilever, under variation of width: Fig. 1(b);

a cantilever with different linearly tapered widths: Fig. 1(c);

a cantilever with a longitudinal stiffener (rib) of varying depth: Fig. 1(d); and

a cantilever with a turnaround section with varying length: Fig. 1(e).
All of these cantilever sets can be printed with relative ease using any 3Dprinter (STL files are available from the author by request). We use the linear relation between force and deflection at the free end as a measure of stiffness.
In the classroom setting, students are in a position to assess stiffness by, for example, pushing down on each cantilever sequentially and experiencing the resistance, and then feeling the differences between each member of a cantilever set. In this way, they can verify how this stiffness depends on the change in geometry according to theory.
II. BACKGROUND BEAM THEORY
III. A SIMPLE CANTILEVER
In general, we shall examine the effect of systematically varying parameters away from a standard baseline geometry, as outlined in Table I. These values are generally used for nondimensionalization in later figures, i.e., parameters are typically divided by a baseline value and indicated by an overbar. Furthermore, E is typically a little lower than 2.1 GPa (the reference value) due to the 3Dprinting process.^{9} However, we are primarily interested in how geometrical changes influence stiffness in a relative sense with all cantilevers printed with the same thickness and using the same (ABS thermoplastic) material. Small weights were attached to the ends of the cantilevers resulting in vertical deflection. The deflections included later were generally measured using a proximity sensor, with an accuracy of about 1/40 mm. The Appendix includes some results based on a more sophisticated experimental approach (for example using a load cell), but in general, we simply observe how the cantilevers deflect under load, and this is tantamount to feeling the resistance to pushing on each cantilever by hand.
Length, L  180 mm 
Width, b  20 mm 
Thicknessa, d  1.68 mm 
Second moment of areab, I  7.9 mm^{4} 
Young's modulus, E  2.1 GPa 
Stiffness, k  8.54 N/m 
Mass applied to the tip, i.e., load, F = mg → deflection, y_{0}  29 g, 0.285 N, $\u2248$33 mm 
Length, L  180 mm 
Width, b  20 mm 
Thicknessa, d  1.68 mm 
Second moment of areab, I  7.9 mm^{4} 
Young's modulus, E  2.1 GPa 
Stiffness, k  8.54 N/m 
Mass applied to the tip, i.e., load, F = mg → deflection, y_{0}  29 g, 0.285 N, $\u2248$33 mm 
A careful measurement of thickness for the 3Dprinted specimens to be described later, gave an average value d = 1.68 mm (compared to a prescribed thickness of 1.5 mm), a relatively precise parameter with an enhanced sensitivity due to its effect on the second moment of area, and the small dimension most sensitive to the resolution of the 3Dprinter.
In the weaker direction, and about the (horizontal) neutral axis passing through the centroid.
As introductory examples, we will examine the effect of varying the width b and the length L. We subject the end to the same specific vertical load (a weight), whereas more formally (in a lab setting), the stiffness is characterized by how force and deflection are related over a range, with the slope of a linear fit providing an accurate estimate of stiffness.
Figures 3 and 4 shows the effect of varying these two parameters in terms of tip deflection and stiffness, including some photographic images. Here, the relations are normalized with respect to the aforementioned standard cantilever (indicated by the black circles in subsequent plots). In dimensional terms, we use $L=120,\u2009140,\u2009160,\u2009180,\u2009200$ mm and $b=10,\u200915,\u200920,\u200925,\u200930$ mm, with the specific cases indicated by the open circles points for experimental data in this, and later figures. The Appendix contains a detailed measured example for two different lengths under variable load, together with tabulated values corresponding to the datapoints shown in the figures. These parameter dependencies also underlie the notion of propagation of uncertainty and measurement error. For example, the thickness and stiffness are related cubically and hence any error in measuring the thickness (which is generally small) has a tendency to be propagated to a greater extent than say, the width, although this issue will not be addressed here.^{15,16}
It is important to recall some of the limitations of standard beam theory, including the restriction to relatively small deflections. For example, the least wide cantilever (10 mm) deflects vertically about 70 mm at the tip when subject to an end mass of 29 g, and this corresponds to $ y tip / L$ of about 35%, a value near the curvature limit of the linear theory.^{1}
A. A note on selfweight
We also mention that there is a small deflection associated with selfweight, at least where the cantilevers are oriented horizontally. Using the same linear theory, and an evenly distributed load, leads to a tip deflection $ y tip = f L 4 / 8 E I$, where f is the distributed load (per unit length).^{12} The density of the material used (ABS thermoplastic) in this study is typically close to 1050 Kg/m^{3}, corresponding to “solid” print settings, although the density of prints can be adjusted. Thus, for example, the standard cantilever weighs about 6 g, leading to a tip deflection of approximately 2.7 mm, an order of magnitude smaller that the deflection of close to 40 mm under the 29 g endweight (which we can choose to vary). In general, the weight scales with volume $ ( bdL )$, whereas the stiffness scales as $ ( b ( d / L ) 3 )$. Given the range of geometries and weights used in this study, selfweight is ignored. However, when assessing stiffness by hand, the models resist force about a natural equilibrium shape, and can also be “held” in a vertical orientation. Selfweight does provide an alternative opportunity to demonstrate comparative stiffness based on the “natural” shape of unloaded, relatively low stiffness cantilevers allowed to droop when oriented horizontally (but not so much as to violate the linear theory). The vertical orientation is also natural for assessing natural frequency, rather like a tuning fork.^{17}
IV. RIBSTIFFENED CANTILEVERS
V. TAPERED CANTILEVERS
It is often the case that a cantilever is tapered, for example, a fishing rod, or tree, or the wing of an aircraft. Referring to Fig. 2(b), suppose we have a rectangular cross section with a width $b=b(x)$ that varies linearly such that we have a second moment of area I_{A} at the left end and I_{B} at the right, i.e., $ I x= I A(1+Kx/L)$, with $K= I B/ I A\u22121$. Again using the $180\xd720\xd71.68$ mm geometry as the baseline case, we consider a set of four cantilevers: $( b A,\u2009\u2009 b B)=(20,\u200920),(15,\u200925),(10,\u200930),(5,\u200935)$ mm, see Fig. 2(b). An alternative approach would be to keep I_{B} constant, say 20 mm, and reduce I_{A}, resulting in a progressive reduction in stiffness. For example, if the baseline cantilever was tapered to 5 mm at the free end, we would obtain a stiffness reduction of about 20%. Given the relatively simple variation in $ I = I ( x )$, we can directly incorporate this expression when integrating Eq. (2), although this is not trivial. However, standard cases have been tabulated. For example, in Roark's formulas for stress and strain,^{19} which, for the four cases considered ( $ I B/ I A=1.0,\u20091.67,\u20093,7$), provides relative tip deflections of 1.0, 0.919, 0.846, and 0.752, and these are shown in Fig. 6 where the relations are evaluated numerically at the given values and then subject to a curve fit. The deflections for these four geometries represent a fairly modest change in tip deflection. However, the volumes (and hence mass) of each of these cantilevers are the same, and thus, the more tapered cantilever may represent a more economic design, at least for this specific (tiploaded) case. The theory also holds for the (much less practical) case of $ I B / I A < 1$, in which case the effect is more noticeable, although not included as an example here. A cantilever with a taper in depth also presents certain stiffness benefits. There may be nonstiffness reasons for preferring a tapered cantilever, and the linear theory is less appropriate for more extreme tapers and cases in which the behavior is more platelike, requiring a more appropriate twodimensional analysis.^{19}
VI. CANTILEVERS WITH A TURNAROUND
VII. CONCLUDING REMARKS
This paper introduces the notion of using sets of 3Dprinted cantilevers to provide a tactile demonstration of stiffness and how it is influenced by changes in geometry. Although the measured results are included, this is done mainly for completeness, with the primary emphasis based on relative behavior in which students directly experience the feel of changes in geometry. By simply pushing down on the ends of each cantilever in succession, the change in stiffness is immediately apparent. The focus on relative behavior is intimately related to nondimensionalization and generally deemphasizes the importance of units. Students can estimate the proportional magnitude of changes in deflection and stiffness as a function of changes in beam geometry.
The resolution of the printer and the material properties of the 3Dprinter thermoplastic are much less important than if they were being used to confirm theory against experiment in a conventional sense. 3Dprinting is now ubiquitous in science and engineering. We also mention a couple of subtle practical ways in which 3Dprinting also facilitates testing: the judicious use of “fillets” on interior corners to reduce stress concentrations; small “eyes” to facilitate the applications of loads (most clearly seen in Fig. 5(b)); and holes printed in the base to allow a convenient attachment to a teststand.
As mentioned earlier, ABS thermoplastic does not have precise mechanical properties but rather a range, e.g., $ 1010 < \rho < 1210$ kg/m^{3} and $ 1.2 < E < 2.9$ N/m^{2}. These are bulk estimates and the 3Dprinting process can lead to small voids with different density settings. Furthermore, the nominal values for geometry as printed depend on the resolution of the printer. The deflections are also measured on the top surface of the cantilever whereas the theory corresponds to the cantilever centerline. Thus, we should not be surprised by an imperfect theoryexperiment correlation. However, this reinforces the presentation in comparative terms—the cantilevers are all printed with the same material, in the same orientation, into a contiguous base, and to the same geometric tolerance, and hence their relative stiffness/frequency behavior is consistent (as confirmed in the Appendix A).
The (linear) relation between force and deflection is a dominant issue in understanding the mechanics of solids. The advent of 3Dprinting presents an appealing opportunity to produce demonstration models that provide an effective (literally) handson appreciation of behavior wellsupported by fundamental theory.
ACKNOWLEDGMENTS
This work was partially supported by the NSF under Award No. CMMI1926672. The authors thank the reviewers for their constructive comments.
APPENDIX A: DETAILED MEASUREMENT OF AN INDIVIDUAL CANTILEVER
Since this paper involves producing physical specimens, it is easy enough to take measurements with relatively rudimentary equipment. Stiffness, the slope of the forcedeflection relation, is typically measured by taking a set of deflection measurements under different levels of force, and fitting the data using least squares. For the tiploaded cantilever, measuring force and deflection is straightforward, especially where they are colocated. Either by using a cell load pushing on the end of the cantilever or simply hanging a weight (F = mg), the corresponding measured end deflection can be achieved using a proximity laser (an OPTO NCDT 1302 with a precision of 0.025 mm was used in this study). The latter is especially convenient since it does not contact the cantilever and can be aimed to measure deflection at any location. An example is shown in Fig. 9(a). Two cantilevers were subject to increasing levels of tip force. The cantilevers had identical cross sections $(25.4\xd72.5)$ mm, made from the same material (and printed in the same orientation), with the only difference in the length: cantilever A has a length of L = 98.6 mm and for cantilever B, L = 124 mm.
The experimental data (tip force vs deflection) are shown in Fig. 9(b). In each case, the deflection was measured under increasing and decreasing force using a load cell (an 5lbf Omega Dyne stainless steel S beam was used in this study), and then repeated, hence the slight spread in the data (in addition to a little electronic noise associated with the measurement devices). A least squares curve fit gave a stiffness (slope) of 0.242 N/mm for cantilever A and 0.126 N/mm for cantilever B. Given the ratio between their lengths (124/98.6 = 1.254) and the inverse cubic relation, we would expect all other things being equal, cantilever A to be about $ 1.254 3 = 1.97$ times stiffer than cantilever B (comparing the slopes in Fig. 9(a) confirms this). Given a deflection for the “unloaded” state ( $ F = y tip = 0$), the stiffness can be extracted using a single weight and single deflection, as effectively done earlier in this paper. While this is useful in the comparative examples provided in the main body of this paper, a conventional theory vs experimental data study benefits from a more thorough statistical treatment to reduce precision error.
It is instructive to consider these results in nondimensional terms. This provides a convenient basis for comparison, as shown in Fig. 9(c). Here, the axes are $ F L 2 / E I$ and $ y tip / L$, and we obtain a universal plot in which the slope is close to the theoretical prediction of three. A slight deviation from linearity can be observed for higher loads/deflections, a reminder that the linear theory relies on limiting restrictions.
APPENDIX B: NATURAL FREQUENCIES OF VIBRATION
These types of slender structures are also wellsuited for vibration studies, both in terms of a visual appreciation and actual measurements. When disturbed from equilibrium and set into motion, these lightly damped slender elements vibrate in a characteristic fundamental mode of vibration, the frequency of which can be related to simple geometry in much the same way as stiffness.
We briefly show some measured frequencies, data acquired with the same proximity laser (with a sampling rate of 750 Hz) as used for the deflection measurements. Figure 10 shows three sample time series, resulting from a sudden tip disturbance. The frequency is extracted using the Fast Fourier Transform (FFT), or by simply counting the number of oscillations over a given time duration. We also see the relatively light damping present in these systems. These oscillations decay exponentially, and we clearly see how the frequency changes with length. Some theoretical and measured frequencies are included in Table II.
L(mm) ( $ L / L 0$) .  120 (0.667) .  140 (0.778) .  160 (0.889) .  180 (1) .  200 (1.111) . 

$ k / k 0$ (Theory)  3.375  2.125  1.424  1  0.729 
$ k / k 0$ (Measured)  3.37  2.13  1.42  1  0.75 
$ y / y 0$ (Theory)  0.296  0.471  0.702  1  1.372 
$ y / y 0$ (Measured)  0.29  0.46  0.70  1  1.34 
$ \omega / \omega 0$ (Theory)  2.25  1.65  1.27  1  0.81 
$ \omega / \omega 0$ (Measured)  2.22  1.72  1.28  1  0.81 
L(mm) ( $ L / L 0$) .  120 (0.667) .  140 (0.778) .  160 (0.889) .  180 (1) .  200 (1.111) . 

$ k / k 0$ (Theory)  3.375  2.125  1.424  1  0.729 
$ k / k 0$ (Measured)  3.37  2.13  1.42  1  0.75 
$ y / y 0$ (Theory)  0.296  0.471  0.702  1  1.372 
$ y / y 0$ (Measured)  0.29  0.46  0.70  1  1.34 
$ \omega / \omega 0$ (Theory)  2.25  1.65  1.27  1  0.81 
$ \omega / \omega 0$ (Measured)  2.22  1.72  1.28  1  0.81 
APPENDIX C: TIP MEASUREMENTS OF THE CANTILEVER SETS
In the measurements of the cantilever sets, the load is applied in a vertical downwards direction (with weights, typically 29 g) and the corresponding deflections are (generally) vertically down and hence negative. A digital load cell is an alternative method of loading, with the advantage of not being restricted to a vertical downward direction and easily incremented. The deflections were measured using a noncontacting laser sensor (the signal is improved using reflective tape—with the small gray targets being apparent in the images of the cantilever sets). The nondimensional normalized stiffnesses and deflections are inverse to each other. The information provided in the following tables was used to populate Figs. 3–7. Theoretical results are based for example on Eq. (3) for Table II and Eq. (6) for Table III, and shown as continuous lines in the figures (with the measured values as the circular symbols). For the values listed in the table below, the data are presented relative to the standard cantilever case, and appear in boldface (Tables IV–VI). Note that the standard case values are typically slightly different in each set due to a small amount of experimental uncertainty and variability. Since the baseline values are divided by themselves, they are included as black data points in the figures.
b (mm) ( $ b / b 0$) .  10 (0.5) .  15 (0.75) .  20 (1) .  25 (1.25) .  30 (1.5) . 

$ k / k 0$ (Theory)  0.5  0.75  1  1.25  1.5 
$ k / k 0$ (Measured)  0.58  0.79  1  1.27  1.42 
$ y / y 0$ (Theory)  2  1.333  1  0.8  0.667 
$ y / y 0$ (Measured)  1.73  1.27  1  0.78  0.70 
b (mm) ( $ b / b 0$) .  10 (0.5) .  15 (0.75) .  20 (1) .  25 (1.25) .  30 (1.5) . 

$ k / k 0$ (Theory)  0.5  0.75  1  1.25  1.5 
$ k / k 0$ (Measured)  0.58  0.79  1  1.27  1.42 
$ y / y 0$ (Theory)  2  1.333  1  0.8  0.667 
$ y / y 0$ (Measured)  1.73  1.27  1  0.78  0.70 
D (mm) .  0 .  1 .  2 .  3 .  4 . 

$ k / ( k 0 ( D = 0 ) )$ (Theory)  1  1.45  2.70  5.10  8.95 
$ k / k 0$ (Measured)  1  1.52  3.06  5.77  10.76 
$ y / y 0$ (Theory)  1  0.69  0.37  0.20  0.11 
$ y / y 0$ (Measured)  1  0.66  0.33  0.17  0.09 
D (mm) .  0 .  1 .  2 .  3 .  4 . 

$ k / ( k 0 ( D = 0 ) )$ (Theory)  1  1.45  2.70  5.10  8.95 
$ k / k 0$ (Measured)  1  1.52  3.06  5.77  10.76 
$ y / y 0$ (Theory)  1  0.69  0.37  0.20  0.11 
$ y / y 0$ (Measured)  1  0.66  0.33  0.17  0.09 
b_{A}, b_{B} (mm) .  20, 20 .  15, 25 .  10, 30 .  5, 35 . 

$ I B / I A $  1  1.67  3  7 
$ k / k 0$ (Theory)  1  1.09  1.26  1.33 
$ k / k 0$ (Measured)  1  1.13  1.22  1.26 
$ y / y 0$ (Theory)  1  0.92  0.79  0.75 
$ y / y 0$ (Measured)  1  0.88  0.82  0.79 
b_{A}, b_{B} (mm) .  20, 20 .  15, 25 .  10, 30 .  5, 35 . 

$ I B / I A $  1  1.67  3  7 
$ k / k 0$ (Theory)  1  1.09  1.26  1.33 
$ k / k 0$ (Measured)  1  1.13  1.22  1.26 
$ y / y 0$ (Theory)  1  0.92  0.79  0.75 
$ y / y 0$ (Measured)  1  0.88  0.82  0.79 
a (mm) ( $ a / L 0$) .  0 (0) .  30 (0.167) .  60 (0.333) .  90 (0.5) .  120 (0.667) .  150 (0.833) . 

$ y tip / y 0$ (Theory)  1  0.75  0.5  0.25  0  −0.25 
$ y tip / y 0$ (Measured)  1  0.69  0.48  0.26  0.03  −0.15 
a (mm) ( $ a / L 0$) .  0 (0) .  30 (0.167) .  60 (0.333) .  90 (0.5) .  120 (0.667) .  150 (0.833) . 

$ y tip / y 0$ (Theory)  1  0.75  0.5  0.25  0  −0.25 
$ y tip / y 0$ (Measured)  1  0.69  0.48  0.26  0.03  −0.15 