Cantilever Bathroom Addition

Cantilever Bathroom Addition

Piezoelectric energy harvesting from pavement

Hao Wang , Abbas Jasim , in Eco-Efficient Pavement Construction Materials, 2020

14.3.1 Cantilever beam transducer

Bimorph piezoelectric cantilever beam is assembled with beam material and two piezoelectric ceramic strips, conductive adhesive, and two piezoelectric films by series or parallel connection, Toward the cantilever end is a block that can enhance the vibration amplitude of cantilever beam as well as alter the vibration frequency. Fig. 14.2 presents the illustration of a bimorph cantilever beam.

Figure 14.2. Illustration of a bimorph cantilever beam with PZT.

Many factors can affect the energy output of cantilever beam, including mainly coupling mode, frequency, thickness and type of material used for each layer of the cantilever beam, total length of cantilever beam, and tip mass. Jiang et al. [6] studied the effects of material thickness and tip mass of cantilever beam. They concluded that by reducing the cantilever thickness and increasing the tip mass, the maximum harvested power was achieved. The length and width of tip mass were also found affecting the output of the harvested power [7].

Kim et al. [8] assessed piezoelectric cantilever transducer and placed them on speed bumps and underground, respectively. They observed that the electric power generated from energy harvester on speed bump that was much less compared to the underground one. The harvested energy changed with vehicle speed and the cantilever vibration method.

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TRANSDUCERS AND DATA ACQUISITION

RICHARD HATHAWAY , KAH WAH LONG , in Fatigue Testing and Analysis, 2005

1.7.2 CANTILEVER BEAM IN BENDING WITH AXIAL LOAD COMPENSATION

The cantilever beam shown in Figure 1.24 has gages on both surfaces. Because of the configuration of the bridge, the gages compensate for any applied axial load and compensate for uniform temperature changes while having increased sensitivity over the SYSTEM shown in Figure 1.24. For the cantilever beam, Equation 1.7.2 is used, which is derived from Equation 1.4.4:

FIGURE 1.24. A multigage cantilever beam system.

(1.7.2) $\frac{e_{ocv}}{E_{ex}}=\left[\frac{(2\times R\times GF\times \varepsilon +2\times R\times GF\times \varepsilon )}{4R}\right]=\left[GF\times \varepsilon \right]$

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Mechanical Fastening

In Handbook of Plastics Joining (Second Edition), 2009

18.6.3 Types of Snap-fit

There are two main types of snap-fit: the cantilever beam and the cylindrical (annular) snap-fit. Other snap-fits include torsional, ball-and-socket, U-shaped, bayonet finger, snap-on or snap-in and the combination snap-fit.

18.6.3.1 Cantilever Beam

Cantilever beam snap-fits consist of a hook-and-groove joint in which a protrusion from one part interlocks with a groove on the other part (Fig. 18.18). The design of the beam can permit great flexibility to suit the material being joined, resulting in a tight fit with low stress induced into the system. This snap-fit design is the most common; it is easy to assemble, provides good retention, and is usually stress free [10, 28].

Figure 18.18. A cantilever beam snap-fit: (a) cantilever beam and mating piece before assembly, (b) latch is partially deflected as initial contact is made, (c) latch approaches maximum deflection, and (d) latch locks into the hole in the mating part and returns to its undeflected position.

18.6.3.2 Annular

Annular, or cylindrical, snap-fits are used to join spherical or elliptical parts, such as pens and bottles with caps. One of the parts contains a lip or protrusion around the part circumference that engages with a protrusion or a groove on the mating part (Fig. 18.19). After assembly, parts are stress-free. Multiaxial stresses are produced during annular snap-fit assembly, making design calculations difficult.

Figure 18.19. Annular snap-fits.

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Error Estimation Based on Finite Difference Smoothing

John O. Dow , in A Unified Approach to the Finite Element Method and Error Analysis Procedures, 1999

Cantilever Beam

The cantilever beam problem is chosen as an example for several reasons. In Lesson 12, we analyzed the strain modeling behavior of the individual elements in detail and we studied this problem closely. Thus, we know that in a model consisting of a single row of elements, the cantilever beam model representation cannot be accurate. By solving this problem, we can directly compare the error analysis results presented in the next section to our previous experience. Furthermore, this model contains several geometries commonly found in practice.

The problem contains an intersection of a fixed boundary with a free surface to produce a singularity point. It also contains a cross section with well-known stress conditions. The shear stress is zero at the boundaries and distributed across the section as a parabola. The normal stress is zero on the surface and the distribution of flexural stress is well understood. Furthermore, this example demonstrates the ability to model fixed and loaded boundary conditions for both straight edges and corners.

We see that the nodal averaging approach produces low estimates for the error at all levels of refinement. We see that the underestimation is especially severe when the problem is dominated by a free or loaded surface and when high errors are present. Furthermore, we see that as the stresses converge to the correct value on the free surfaces that the errors are dominated by the errors on the slow-to-converge fixed boundary. This shows the need for modifying the smoothed solution on fixed boundaries. Finally, we see that the finite difference approach is significantly better at identifying errors at singular points than the other approaches.

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Structural Design Sensitivity Analysis

Kuang-Hua Chang , in Design Theory and Methods Using CAD/CAE, 2015

4.6.2 Two-Dimensional Cantilever Beam Example

A cantilever beam with a fixed left end and a vertical load applied at the midpoint of the free end, as shown in Figure 4.34a , is used to illustrate the density method for topology optimization. The material properties are modulus of elasticity E = 2.07 × 10⁵ psi and Poisson's ratio ν = 0.3.

FIGURE 4.34. The two-dimensional topology optimization example. (a) Analysis model (dimensions are in inches and force is in pounds). (b) Optimal topology design using modified feasible direction. (c) Optimal topology design using sequential linear programming. (d) Optimal topology design using sequential quadratic programming.

The finite element model contains 32 × 20 mesh with four-node quad elements. There are 640 elements in the model and the density of each element is selected as the design variable. The design problem is to minimize the compliance with the area constraint of 25% imposed on the domain. Figure 4.34b shows the material distribution using the modified feasible direction after 10 iterations. Figure 4.34c shows the material distribution using the sequential linear programming after 36 iterations. Figure 4.34d shows the material distribution using the sequential quadratic programming after 28 iterations. They converge to slightly different topologies.

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Small-scale surface engineering problems

F.W. Delrio , ... R. Maboudian , in Tribology and Dynamics of Engine and Powertrain, 2010

33.3.1 In-plane adhesion

The cantilever beam geometry has long been used to measure adhesion of mica and glass (Obreimoff, 1930; Bailey, 1961; Wiederhorn, 1967; Michalske and Freiman, 1983; Roach et al., 1988 ). In these experiments, a crack is propagated along a cleavage plane using a wedge or universal testing machine and the resulting equilibrium crack geometry is used to calculate the surface energy. A similar technique, which involves an array of micromachined cantilever beams, can be used to measure the adhesion energy between micromachined surfaces. The structures are fabricated using standard surface micromachining techniques, which consist of depositing and patterning alternating layers of structural and sacrificial materials as discussed above. At the conclusion of this process, the sacrificial layers are selectively etched and the resulting structures are rendered freestanding using a variety of drying and coating techniques. A schematic representation of a cantilever beam in contact with the substrate is shown in Fig. 33.4(a); an SEM image of a cantilever beam array is shown in Fig. 33.4(b). The critical dimensions include gap height h, thickness t, width w and length L.

33.4. (a) Schematic representation of a cantilever beam in contact with the substrate

(reprinted with permission from DelRio F W, Dunn M L, Phinney L M, Bourdon C J, and de Boer M P (2007), Appl Phys Lett, 90, 163104) (b) SEM image of a cantilever beam array (reprinted with permission from Houston M R, Howe R T, and Maboudian R (1997), J Appl Phys, 81(8), 3474–3483).

The cantilever beam is forced into contact with the substrate via electrostatic (Houston et al., 1997; de Boer et al., 2000) or mechanical (Jones et al., 2003) loading, and the shape of the deformed structure can be related to the adhesion energy at the interface (Mastrangelo and Hsu, 1992). Using fracture mechanics terminology, the non-adhered length from the support post to the point where the beam meets the substrate is defined as the crack length s. Longer beams are adhered over a large portion of the length and bent into an S-shape. In this configuration, the crack length s is significantly shorter than the length of the beam L and the adhesion energy Γ is given by

(33.13) $\Gamma =\frac{3}{2}\left(\frac{E{t}^3{h}^2}{s^4}\right)$

where E is the Young's modulus. Shorter beams come into contact with the substrate only at their tip, bending the beam into an arc-shape. In this case, the crack length is approximately equal to the length of the beam and the adhesion energy is given by

(33.14) $\Gamma =\frac{3}{8}\left(\frac{E{t}^3{h}^2}{L^4}\right)$

De Boer and Michalske (1999) later modelled the two configurations from a linear elastic fracture mechanics perspective. Interestingly enough, the apparent adhesion energy calculated from the experimentally measured shortest arc-shaped beam can be different from that for the S-shaped beam. The discrepancy was resolved by analysing the total system energy of the cantilever in the adhered state as a function of the crack length. For the S-shaped geometry, there is a deep energy well at the equilibrium position, resulting in an accurate adhesion energy measurement (immune to small disturbances that may temporarily force the system out of equilibrium). In the arc-shaped configuration, however, the energy well approaches zero for the shortest beam. As a result, the shortest beam will often pop off the surface and a longer beam will be used in the calculations, yielding an inaccurate measure of the adhesion energy. It is only fair to mention, however, that the analyses based on S-shaped beams require an out-of-plane measurement technique (e.g. interferometry) to measure the crack length s, whereas the arc-shaped technique only requires a high-powered objective on an optical microscope to observe the shortest adhered beam within an array of different lengths.

Mastrangelo and Hsu (1992, 1993a,b) fabricated an array of cantilever beams with various lengths to investigate release-induced adhesion , or adhesion that occurs during the final stages of the fabrication process. On the wet etching of the sacrificial layer and the ensuing drying process, capillary menisci formed between the cantilever beams and the underlying substrate, resulting in strong attractive forces that pulled the surfaces together. The transition from adhered to free beams was detected by means of an optical microscope with a Michelson interferometeric attachment. Using Equation33.14, the adhesion energy for both hydrophobic (HF etch, deionised water rinse) and hydrophilic (HF etch, H₂SO₄:H₂O₂ clean, deionised water rinse) surfaces was evaluated (see also Chapter 3). It was found that the adhesion energy is almost equal for the two systems, suggesting that a thin hydrophilic oxide may be present on both samples. This is consistent with results from Gräf et al. (1989), which indicate that HF-treated silicon surfaces quickly oxidise in water due to (1) the development of surface OH groups, (2) the rupture of Si–Si bonds and (3) the formation of Si–O–Si bridges. In addition, the adhesion energy values were found to be about 140   mJ/m², which is roughly equivalent to twice the liquid–vapour surface energy for water, again suggesting that both surfaces thus produced were hydrophilic. Experimentally, this issue was resolved by Houston et al. (1997), who showed that by a careful release process, H-termination of silicon surfaces can be preserved, leading to a large reduction in adhesion in comparison to hydrophilic oxide-passivated surfaces. This will be further discussed in Section 33.5.

A number of techniques have been developed to minimise release-induced adhesion, such as freeze–dry sublimation (Guckel et al., 1989), supercritical CO₂ drying (Mulhern et al., 1993), vapour-phase HF etching (Lee et al., 1997), self-assembled monolayers (Srinivasan et al., 1998) and polystyrene microspheres (Mantiziba et al., 2005). After successful drying of the cantilever beams, these test structures can also be used to quantify in-use adhesion , or adhesion that occurs during device operation when the surfaces may come into contact. Houston et al. (1997) noted that the surface roughness leads to statistical variation in contact area, and hence a statistical variation in the measured adhesion. To take this into account, they used a statistical method to determine an average crack length and hence an average work of adhesion. For a SiO₂-coated polysilicon surface with an rms roughness of 14   nm, they reported a work of adhesion of 20   mJ/m² at 50% RH. At RH approaching 100%, the work of adhesion was found to increase to 140   mJ/m², which, as expected, is close to twice the surface tension of water.

DelRio et al. (2007a) performed cantilever experiments on hydrophilic SiO₂-coated polysilicon surfaces as a function of surface roughness and relative humidity. Prior to adhesion testing, the samples were cleaned using a glass DC plasma generator with oxygen gas and water vapour, which resulted in a water contact angle of <   10°, and moved to an environmental interferometer without intermediate exposure to the ambient. The cantilevers were brought into contact with the substrate at 0% RH by applying a voltage to the actuation pad. The RH within the environmental chamber was then increased from 0% RH to 95% RH in 5% RH increments. Using a long-working distance interference microscope (Sinclair et al., 2005) with a phase shifting algorithm (Hariharan et al., 1987), the deflection profiles of the cantilevers (surface roughnesses ranging from 2.6   nm rms to 10.3   nm rms) were recorded as a function of RH as shown in Fig. 33.5(a). The experimental data indicated a strong correlation between surface roughness and capillary condensation. As the landing pad roughness increases, the RH at which the adhesion initially jumps due to capillary condensation also increases. Once the initial jump occurs, the adhesion increases towards the upper limit of 144   mJ/m². A detailed model based on the measured surface topography qualitatively agrees with the experimental data only when the topographic correlations between the upper and lower surfaces are considered as illustrated in Fig. 33.5(b).

33.5. (a) Interferograms of cantilever beams with a landing pad roughness of 2.6   nm rms as a function of relative humidity (b) adhesion results from the experiments (data points with dashed lines to guide the eye) and model (solid lines) for landing pad roughnesses ranging from 2.6   nm rms to 10.3   nm rms. Solid vertical lines indicate RH at which capillary forces are included in the model. The maximum adhesion energy due to capillary condensation 2 γ  cos θ = 144   mJ/m² is shown for reference

(reprinted with permission from DelRio F W, Dunn M L, Phinney L M, Bourdon C J, and de Boer M P (2007), Appl Phys Lett, 90, 163104).

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EVALUATION OF DAMPING BEHAVIOR OF SPRAY DEPOSITED SiC PARTICULATES REINFORCED A1 COMPOSITES

Abo El-Nasr A.A. , in Current Advances in Mechanical Design and Production VII, 2000

2.4 Damping Measurements

The cantilever beam technique was used in the present study to characterize the damping behavior of the deposited materials. In this technique, one end of a rectangular specimen was fixed while the opposite end was allowed to move freely in response to a mechanically induced displacement on the dynamic mechanical thermal analyzer (DMTA). The damping capacity of the materials was then determined from the resulting displacement spectrum, by utilizing the logarithmic decrement and the half-power bandwidth analysis methodologies. In this method, a history of amplitude versus time during free vibration of the cantilever beam specimen was recorded by an oscilloscope through an optical displacement transducer. Fig. 2 shows a typical example of the response obtained in time domain. By measuring the free amplitude decay after excitation, the logarithmic decrement (δ) can be calculated from [10]

Fig. 2. A typical free decay response of a 12% SiCp-Al MMC sample

(4) $\delta \equiv \frac{1}{n}ln\left(\frac{A_i}{A_{i+n}}\right)$

where A _i and A_(i+n) are the amplitudes of the i ^th cycle and the (i+n) ^th cycle, respectively, separated by "n" periods of oscillation (Fig. 2). Finally, experimental data for the logarithmic decrement δ and the loss factor (Q ⁻¹) can be checked by using the relationship [4,10]

(5) $\delta =\pi Q^{-1}$

The damping capacity (tan ϕ) and loss factor (η) are related to the real and imaginary components (denoted E' and E", respectively) of the complex dynamic modulus by

(6) $tan\varphi =\eta =Q^{-1}=E^{″}/E^{\prime }$

The force and displacement curves were used to calculate the magnitude of E'. This value is referred to as the storage modulus and corresponds to Young's modulus for the nondynamic case. All the samples were displaced to a maximum strain of 2.6×10⁻⁴ and were placed in a furnace to investigate the effect of temperature on damping capacity. The temperature was increased at a rate of ~3 °C/min from 30 to 300 °C and was monitored through a platinum resistor adjacent to the test specimen. During the temperature cycle, the sample was oscillated at three different frequencies of 0.1, 1, 30 Hz.

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Vibration Fundamentals

Victor Giurgiutiu , in Structural Health Monitoring with Piezoelectric Wafer Active Sensors (Second Edition), 2014

3.4.2.2 Cantilever Beam

A cantilever beam is fixed at one end and free at the other end; it has fixed-free boundary conditions described as

(414) $\begin{array}{lllll}w(0,t)=0\hfill & & w\prime (0,t)=0& & \mathrm{fixed}(\text{built}-\text{in})\\ M(l,t)=0& & V(l,t)=0& & \mathrm{free}\end{array}$

Using Eqs. (322), (323), (327) into Eq. (414) yields the boundary conditions in terms of displacement and its derivatives, i.e.,

(415) $\begin{array}{ccc}\hat{w}(0)=0& & \hat{w}\prime (0)=0\\ \hat{w}″(l)=0& & \hat{w}‴(l)=0\end{array}$

Substitution of the general solution (336) into the boundary conditions (415) yields

(416) $\begin{array}{c}{C_1\sin \gamma x+C_2\cos \gamma x+C_3\sinh \gamma x+C_4\cosh \gamma x|}_{x=0}=0\\ {\gamma \left[C_1\cos \gamma x-C_2\sin \gamma x+C_3\cosh \gamma x+C_4\sinh \gamma x\right]|}_{x=0}=0\\ {{\gamma }^2\left[-C_1\sin \gamma x-C_2\cos \gamma x+C_3\sinh \gamma x+C_4\cosh \gamma x\right]|}_{x=l}=0\\ {{\gamma }^3\left[-C_1\cos \gamma x+C_2\sin \gamma x+C_3\cosh \gamma x+C_4\sinh \gamma x\right]|}_{x=l}=0\end{array}$

Upon simplification, we get

(417) $\begin{array}{c}{C}_2+C_4=0\\ C_1+C_3=0\\ -C_1\sin \gamma l-C_2\cos \gamma l+C_3\sinh \gamma l+C_4\cosh \gamma l=0\\ -C_1\cos \gamma l+C_2\sin \gamma l+C_3\cosh \gamma l+C_4\sinh \gamma l=0\end{array}$

Further simplification yields

(418) $\begin{array}{l}{C}_4=-C_2\\ C_3=-C_1\\ C_1(\sin \gamma l+\sinh \gamma l)+C_2(\cos \gamma l+\cosh \gamma l)=0\hfill \\ C_1(\cos \gamma l+\cosh \gamma l)+C_2(-\sin \gamma l+\sinh \gamma l)=0\end{array}$

The last two lines in Eq. (418) form a homogeneous linear algebraic system in $C_1$ and $C_2$

(419) $\begin{array}{c}(\sin \gamma l+\sinh \gamma l)C_1+(\cos \gamma l+\cosh \gamma l)C_2=0\\ (\cos \gamma l+\cosh \gamma l)C_1+(-\sin \gamma l+\sinh \gamma l)C_2=0\end{array}$

The homogeneous algebraic system (419) accepts nontrivial solutions only if its determinant has value zero, i.e.,

(420) $\left|\begin{array}{cc}\sin \gamma l+\sinh \gamma l& \cos \gamma l+\cosh \gamma l\\ \cos \gamma l+\cosh \gamma & -\sin \gamma l+\sinh \gamma l\end{array}\right|=0$

Expansion and simplification of Eq. (420) yields

(421) $\cos \gamma l\cosh \gamma l+1=0$

Introduce the notation

(422) $z=\gamma l$

Substitution of Eq. (422) into Eq. (421) yields the eigenvalue equation

(423) $\cos z\cosh z+1=0$

Equation (423) is a transcendental equation that yields the eigenvalues $z$ corresponding to the free flexural vibration of a cantilever beam. Numerical values of the first five eigenvalues $z$ are shown in Table 3.6. As $z$ becomes large, the numerical values approach a rational sequence. From the sixth eigenvalue onwards, they can be approximated to reasonable accuracy by

Table 3.6. Eigenvalues $z=\gamma l$ , natural frequencies $\omega$ , and the modal parameter $\beta$ for the flexural vibration of a cantilever beam

$\mathit{j}$	$z_j={(\mathit{\gamma }l)}_{\mathit{j}}$	${\mathit{\omega }}_{\mathit{j}},\mathbf{rad}/\mathbf{s}$	${\mathit{f}}_{\mathit{j}},\mathbf{Hz}$	${\mathit{\beta }}_{\mathit{j}}$
1	$1.87510407$	$3.51601527\sqrt{\frac{EI}{m{l}^4}}$	$0.55959121\sqrt{\frac{EI}{m{l}^4}}$	$0.73409551$
2	$4.6940911$	$22.0344916\sqrt{\frac{EI}{m{l}^4}}$	$3.5068983\sqrt{\frac{EI}{m{l}^4}}$	$1.01846732$
3	$7.8547574$	$61.697214\sqrt{\frac{EI}{m{l}^4}}$	$9.8194166\sqrt{\frac{EI}{m{l}^4}}$	$0.99922450$
4	$10.9955407$	$120.901916\sqrt{\frac{EI}{m{l}^4}}$	$19.2421376\sqrt{\frac{EI}{m{l}^4}}$	$1.00003355$
5	$14.1371684$	$199.859530\sqrt{\frac{EI}{m{l}^4}}$	$31.808632\sqrt{\frac{EI}{m{l}^4}}$	$0.99999855$
6,7,8, …	$(2j-1)\frac{\pi }{2}$	${\left[(2j-1)\frac{\pi }{2}\right]}^2\sqrt{\frac{EI}{m{l}^4}}$	${\left[(2j-1)\right]}^2\frac{\pi }{8}\sqrt{\frac{EI}{m{l}^4}}$	$1.00000000$

(424) $z_j=(2j-1)\frac{\pi }{2},j=6,7,8,\dots$

Recalling Eq. (330), we calculate the corresponding natural frequencies as

(425) ${\omega }_j=z_j^2\sqrt{\frac{EI}{m{l}^4}},f_j=\frac{1}{2\pi }{z}_j^2\sqrt{\frac{EI}{m{l}^4}},j=1,2,3,\dots$

where the values $z_j$ are given in Table 3.6. Also given in Table 3.6 are the corresponding values for ${\omega }_j$ and $f_j$ .

Solution of the algebraic system (419) and substitution into the general solution (336) yields the modeshapes for flexural vibration of a cantilever beam, i.e.,

(426) $W_j(x)=A_j[(\cosh {\gamma }_{j}x-\cos {\gamma }_{j}x)-{\beta }_j(\sinh \gamma x-\sin \gamma x)],j=1,2,3,\dots$

where ${\beta }_j$ is a modal parameter given by the solution of either the first or the second line in Eq. (419), i.e.,

(427) ${\beta }_j=-{\left(\frac{C_1}{C_2}\right)}_j=\frac{\cosh {\gamma }_{j}l+\cos {\gamma }_{j}l}{\sinh {\gamma }_{j}l+\sin {\gamma }_{j}l}=\frac{\sinh {\gamma }_{j}l-\sin {\gamma }_{j}l}{\cosh {\gamma }_{j}l+\cos {\gamma }_{j}l}$

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Structural Analysis

Kuang-Hua Chang , in e-Design, 2015

7.6.2.1 Cantilever Beam

A cantilever beam of 1 in. × 1 in. × 10 in. was modeled in both Simulation and Pro/MECHANICA. As discussed in Section 7.4.6, Pro/MECHANICA created 12 tetra-elements and employed a p-version solver to create results. Instead of using the default unit system in.-lb _m -sec, a more commonly employed system, in.-lb _f -sec, was chosen. The force added at the front top edge was 1000 lb _f ; the boundary conditions (restrained at the rear end face) and material properties (with modulus E = 1.06 × 10⁷ psi and Poisson's ratio ν = 0.33) remained the same. A multipass analysis with 1% strain energy was defined as the convergence criterion. The analysis took five passes (with maximum p-order = 5) and solved 1,017 equations to achieve a 0.8% convergence in strain energy. Essential solution-related information can be found in the Run Status window shown in Figure 7.39(a). The convergence graph shown in Figure 7.39(b) indicates that an excellent convergence was achieved in five passes. The maximum von Mises stress was 5,776 psi at the constrained end, as shown in Figure 7.39(c). Note that the maximum bending stress and vertical displacement were 7,353 psi and 0.03742 in., respectively (not shown in the figure). As before, the displacement result can be verified using classical beam theory. The bending stress would be close to 6,000 psi if Poisson's ratio were set to 0.

Figure 7.39. Simple beam modeled with Pro/MECHANICA: (a) the Run Status window, (b) the strain energy convergence graph, and (c) the von Mises stress fringe plot.

The same example was analyzed in SolidWorks Simulation. About 7,500 tetra-elements, as shown in Figure 7.40(a), were created using the default mesh setting, where the global size of the element was 0.2155 in. There were about 36,000 DOF in this model—about 35 times more than in the Pro/MECHANICA model. The maximum vertical displacement was 0.0382 in. (Figure 7.40(b)), which was very close to that of Pro/MECHANICA (0.03742 in.) and the analytical solution. The maximum bending stress shown in Figure 7.40(c) was 6,088 psi (with Poisson's ratio set to 0), which was also close to the analytical solution.

Figure 7.40. Simple beam modeled with SolidWorks Simulation: (a) the finite element mesh, (b) the displacement Ux (vertical) fringe plot, and (c) the von Mises stress fringe plot.

The 3D beam was idealized to a 1D finite element model in Simulation, as shown in Figure 7.41(a). The same load and boundary conditions were applied at respective ends where joints were created when the solid beam was converted to a 1D beam. The auto mesh generator created 46 beam elements (Figure 7.41(b)) with 276 DOF (as opposed to the 36,000 employed in the 3D model of Simulation). Note that for this simple example, one beam element gave exact solutions since the exact solution of the beam displacement was a cubic function and cubic shape functions were employed in element formulation. The bending stress and maximum vertical displacement were 6,000 psi (Figure 7.41(c)) and 0.03778 in., respectively. These results were excellent. Bending stress was identical to that of the analytical solution.

Figure 7.41. One-dimensional beam modeled in SolidWorks Simulation: (a) the beam with two end joints, (b) the finite element mesh with load and boundary conditions, and (c) the bending stress fringe plot.

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Cantilever Bathroom Addition

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