Analysis of nonlinearly elastic cantilever snap beams for assembly of plastic parts.
A snap joint [ILLUSTRATION FOR FIGURE 1 OMITTED] between two plastic parts is a common type of lock pair, which consists of a cantilever beam with a projecting bead at its free end as a locking feature. The cantilever is an integral component of one of the two plastics parts. The locking feature in the other part is a groove matching the projecting bead at the end of the cantilever. During assembly/disassembly as the projecting bead slides over the matching part, the cantilever deflects in the direction opposite to the bead. Once the assembly is complete, the bead at the free end of the cantilever fits into the matching groove and the cantilever returns to its original shape. Because of the ease of assembly/disassembly and the reduction in the number of parts (since it eliminates the use of a fastener or an adhesive) integral attaching features, such as a cantilever snap joint, are often used to reduce the cost of plastic products. Other advantages of a snap joint include the possibility of joining parts with different materials, feasibility of design for different assembly and disassembly forces, time-independent joint strength, and a vibration-proof assembly. However, owing to its irreparability, failure of the locking features in a snap joint results in complete component failure. Therefore, proper design of the locking features is essential to avoid a premature product failure.
The traditional approach for designing a snap joint employs elastic beam equations (1-3) to determine the maximum stress, and the load versus deflection behavior of the cantilever. based upon these elastic beam equations, many plastic manufacturers such as BASF, Dow, DuPont, and General Electric provide guidelines and the corresponding software for designing snap joints (4-11). However, the significantly nonlinear stress-strain behavior of plastics along with the large strain in the cantilever snap beam during the assembly may result in large design errors if the classical beam theory is used for the analysis (12). Therefore, for a good snap-fit design, accounting for the nonlinear behavior of plastics is important. Many recent publications (13-17) have used the finite-element method for a nonlinear analysis of snap joints. The finite-element analysis of a snap joint is important because it also accounts for geometric complexities such as fillet radius at the cantilever base and the effect of wall dimensions on the snap-fit design (16). However, the finite-element analysis is typically performed only at an advanced stage of the design process and simple analytical equations, such as those from the elastic beam theory, are used to perform preliminary analysis at the conceptual design stage. The large discrepancy between the predictions from the preliminary calculations and the finite-element analysis may necessitate major modifications at an advanced stage of the design process, in turn, significantly increasing the product development time. Therefore, simple analytical equations for design of a snap joint with consideration to the nonlinear mechanical behavior of plastics are important to accelerate the product development. Such equations are presented in this paper. A nonlinear constitutive equation for plastic is introduced in the next section. based upon this nonlinear constitutive equation, the expressions for the stress, strain and deflection of a cantilever beam are developed in the following section.
For materials exhibiting nonlinear stress-strain behavior, different approaches have been used in the literature to approximate the stress-strain curve. The simplest generalization of the Hooke's law ([Sigma] = E[Epsilon]) is the power-law relation, [Sigma] = K[[Epsilon].sup.n], where [Sigma] and [Epsilon] denote stress and strain, respectively, and E, K, and n are material parameters (18). The power-law model has a limited application because it does not correlate the stress-strain behavior for a broad range of stress and strain. To alleviate this shortcoming, different values of K and n are often used over different portions of the stress-strain curve. Furthermore, for plastics, which initially exhibit a linearly elastic behavior, the powerlaw model is not an appropriate representation, because it predicts an infinitely large elastic modulus (since n is less than 1) as the strain goes to zero. For such materials the Ramberg-Osgood equation (18, 19), [Epsilon] = [Sigma]/E + K[([Sigma]/E).sup.n], provide a better representation of the stress-strain behavior. However, the Ramberg-Osgood equation, which contains three material parameters, is not very convenient for calculating the maximum stress and the load-deflection characteristic of a beam. In this work, we have represented the stress-strain behavior of plastics with a nonlinear curve corresponding to a linearly decreasing secant modulus. The secant modulus is defined as the ratio of stress to strain ([Sigma]/[Epsilon]) for any point on the stress-strain curve. For a linearly decreasing secant modulus, the stress is a quadratic function of strain.
[Mathematical Expression Omitted] (1)
where [E.sub.0] and [E.sub.1] are material parameters. It should be noted that in Eq 1 the secant modulus, [E.sub.s] [equivalent to] [Sigma]/[Epsilon] = [E.sub.o] - [E.sub.1][absolute value of [Epsilon]], decreases linearly with strain. The stress-strain curves for three different commercial-grade plastics, acetal (Delrin 500 from DuPont), PBT (Pocan B1505 from Bayer Corporation), and polycarbonate (Calibre 200-10 from Dow Plastics), have been given in Fig. 2. The experimental values of stress and strain in Fig. 2 were obtained from Ref. 20 for acetal and PBT, and from Ref. 21 for polycarbonate. For the three commercial-grade plastics, the material parameters [E.sub.0] and [E.sub.1], which have been obtained by a least-square fit to the experimental data, are given below.
Material [E.sub.1] (Mpa) [E.sub.1] (Mpa) Calibre 200-10 2520.7 28,231.0 Delrin 500 3106.6 41,669.0 Pecan B1505 2889.2 38,494.0
For each of the three plastics, the portion of the stress-strain data between 0% and 4% strain was considered to obtain the least-square fit. For all the results presented later in this paper, strain is within this range.
CANTILEVER BEAM WITH LINEARLY DECREASING SECANT MODULUS
For the cantilever beam shown in Fig. 3, assuming that a plane section parallel to the yz plane before deformation remains plane after the deformation, the strain varies linearly in the y directions,
[Epsilon] = -y([d.sup.2]u/d[x.sup.2]) (2)
where u denotes the y deflection of a point on the beam. For such a strain distribution and the constitutive equation with a linearly decreasing secant modulus (Eq 1), the bending moment (M) at any cross section is given by
M = -[integral of] [Sigma]ydA = [E.sub.o]I [d.sup.2]u/d[x.sup.2] - [E.sub.1]J[([d.sup.2]u/d[x.sup.2]).sup.2] with limit A (3)
where A denotes the area of the cross section of the beam, I [equivalent to] [integral of] [y.sup.2]dA with limit A is the second moment of area and J [equivalent to] [integral of] [[absolute value of y].sup.3]dA with limit A. For a rectangular cross section, if t and w denote the beam dimensions in the y and z directions, respectively, then I = ([wt.sup.3]/12) and J = ([wt.sup.4]/32). Equation 3 can be easily solved to obtain
[d.sup.2]u/d[x.sup.2] = [E.sub.o]I/2[E.sub.1]J (1 - [-square root of 1 - 4[E.sub.1]JM/[([E.sub.o]I).sup.2])]. (4)
For a cantilever beam with length L and a force F acting at the free end, Eq 4 can be integrated twice to determine the beam deflection
u = A [[x.sup.2]/2 + 2[C.sup.3/3]x/3B - 4/15[B.sup.2] [((Bx + C).sup.5/2] - [C.sup.5/2])] (5)
A = [E.sub.0]I/2[E.sub.1]J (6)
B = 4[E.sub.1]JF/[([E.sub.o]I).sup.2] (7)
C = 1 - BL. (8)
For a beam with rectangular cross section A = (4[E.sub.o]/3[E.sub.1]t) and [Mathematical Expression Omitted]. For small deflections, specifically if BL [much less than] 1, Eqs 4 and 5 can be approximated by the simpler expressions given below.
[d.sup.2]u/d[x.sup.2] [approximately equal to] M/[E.sub.o]I + [E.sub.1]J[M.sup.2]/[([E.sub.0]I).sup.3] (9)
u [approximately equal to] F[x.sup.2]/6[E.sub.o]I (3L - x) + [F.sup.2][E.sub.1]J[x.sup.2]/12[([E.sub.o]I).sup.3] (6[L.sup.2] + [x.sup.2] - 4Lx). (10)
Besides assuming the planar-to-planar transformation of the beam cross sections, Eqs 4, 5, 9, and 10 are based upon the following simplifications:
(i) The beam is long (L/t [greater than or equal to] 10) such that the effect of shear stress can be neglected.
(ii) Deflection of the wall at the cantilever base is small.
(iii) Stress concentration and the effect of fillet radius at the cantilever base is neglected.
(iv) square of the slope at the free end of the beam is small compared to unity. With the maximum slope of the cantilever beam less than 13 [degrees], the error in the stress and the deflection predicted by the small deflection theory is less than 5% (13).
(v) The material behavior in tension and compression is the same.
RESULTS AND DISCUSSION
To analyze the effect of nonlinear material behavior on the deformation characteristic of a cantilever beam, various equations given in the last section were solved for a 4-mm-thick beam with three different aspect ratios, L/t = 10, 20, and 30. Qualitatively, the stress, strain and deformation predicted by using the linear (constant E) and nonlinear material behavior (Eq 1) are similar for all the three plastics discussed earlier. Therefore, only the results for acetal (Delrin 500) are presented in this section, those for other plastics exhibit similar trends. For the linear analysis, the elastic modulus (E) of 3.12 GPa was used for Delrin 500 (22). It should be noted that for all the equations presented in the last section, the force per unit width of the beam (F/w), and not the force and width independently, determine the deformation and stress in the cantilever beam.
The maximum strain at any cross section of the cantilever beam can be obtained by solving Eqs 2 and 4 with y = t/2 and M = F(L - x). For three different aspect ratios L/t = 10, 20, and 30 with force/width = 500 N/m, the variation of strain along the length of the beam is shown in Fig. 4. Since the secant modulus in the nonlinear analysis is smaller than the elastic modulus used in the linear analysis, for the same loading conditions the predicted strain from the nonlinear analyses is higher. Because of the larger strain, the effect of nonlinearity is more prominent at the higher aspect ratios. For F/w = 500 N/m, the predicted slope at the free end of the cantilever is 1.4, 5.7, and 12.9 [degrees] for L/t = 10, 20, and 30, respectively. Therefore, the small deflection theory is applicable in all the three cases.
Even though the strain given by the nonlinear analysis is higher, the corresponding maximum stress at any cross section of the beam is smaller [ILLUSTRATION FOR FIGURE 5 OMITTED]. The reason for the lower value of the maximum stress from the nonlinear analysis can be explained from Fig. 6, which gives the distribution of the stress across a cross section of the beam. In particular, Fig. 6 shows the stress at the cantilever base (x = 0), for two different aspect ratios and force per unit width applied at the free end of the beam. In comparison to the stress predicted by the linear analysis, the nonlinear analysis predicts a higher stress near the neutral axis (y = 0), and a lower stress near the surface (y = [+ or -]t/2). It should be noted that the bending moment, (FL), which is the moment of stress integrated over the cross-sectional area of the beam (Eq 3), is the same for the linear as well as the nonlinear analysis. With L/t = 10 and F/w = 4000 N/m, for which the discrepancy between the linear and nonlinear analysis is more evident, the maximum slope is 12.9 [degrees], and therefore, the small deflection theory is still applicable. The maximum strain corresponding to this case is 2.62%, which is within the range of stress-strain curve used to determine the material parameters [E.sub.0] and [E.sub.1] in Eq 1.
For the three different aspect ratios, deflection at the free end of the cantilever beam as a function of the load per unit width is shown in Fig. 7. As expected, the deflection predicted from the nonlinear analysis is larger than that from the linear analysis with the same load. The deflection given by Eq 10, which is an approximation to Eq 5, is also included in Fig. 7. The maximum value of [Mathematical Expression Omitted] for the range of F/w in Fig. 7 is 0.097, 0.194, and 0.291 for aspect ratios of 10, 20, and 30, respectively. Since the value of BL is small for the loading conditions in Fig. 7, the predictions from the approximate equation (Eq 10) is in good agreement with those from Eq 5. To show the discrepancy between Eqs 5 and 10 at a higher load, for L/t = 10 and F/w = 4000 N/m, Fig. 8 presents the stress distribution across the cross section at the base of the cantilever beam obtained by using the two equations. For the aspect ratio and the load per unit width used in Fig. 8, the value of BL is 0.777. The large discrepancy between the predicted stress from Eqs 5 and 10 shows that Eqs 9 and 10 should not be used to predict the strain and deflection, respectively, as the parameter BL approaches unity. It should also be noted that the magnitude of the predicted stress, strain and deflection obtained by using the approximate Equations 9 and 10 will always be smaller than the predictions from Eqs 4 and 5.
For a 10.2 cm (4[double prime]) long, 1.27 cm (1/2[double prime]) wide and 0.3175 cm (1/8[double prime]) thick cantilever beam molded with Delrin 500, the deflection at the free end of the cantilever for load increments of 0.227 kg (1/4 lb) was experimentally measured by Henry (13). Henry measured the deflection for an unannealed beam as well as for a beam annealed for 90 min at 160 [degrees] C. Except for the load of 0.227 kg (1/4 lb), the difference in the experimentally determined deflections for the two cases was less than 7.8%. For the 0.227 kg load, Henry attributed the large difference of 15.4% between the deflections of the annealed and unannealed beams to the pre-load stress induced into the beam as it is clamped at the cantilever base. The experimental data from Ref. 13, along with the predicted deflection from the linear and nonlinear analysis (Eq 5), is shown in Fig. 9. Henry measured the deflection up to the load of 2.27 kg (5 lb). However, only the data up to 0.454 kg (corresponding to F/w = 351 N/m) are included in Fig. 9. With F/w = 351 N/m for t = 3.175 mm and L/t = 32, the maximum slope is 12.95 [degrees] . Therefore, if the load exceeds 0.454 kg, the predictions from the small deflection theory used in the current work are expected to have large errors. As expected, the predicted deflection from the nonlinear analysis is closer to the experimental data. The nonlinear analysis gives about 50% less error than the predictions from the linear analysis. The discrepancy between the experimental data and the predictions from the nonlinear analysis may be due to the simplifying assumptions used in the previous section.
Because of the significantly nonlinear stress-strain behavior of plastics, for an application such as snap fit, in which the material undergoes a large strain, the prediction from a linear analysis can have large error. In this work, the constitutive equation with a linearly decreasing secant modulus is found to give a good approximation to the mechanical behavior of many plastics. For this newly introduced constitutive equation, the governing equations for stress, strain and deflection of a cantilever beam have been developed. The stress, strain and deflection predicted by the linear and nonlinear analysis have been compared. The predicted deflection for a cantilever beam has been compared with the experimental values from the literature. In comparison to the experimental results, the predicted deflection from the nonlinear analysis has about 50% less error than the corresponding predictions from the linear analysis.
The author would like to thank Prof. M. Vable for his helpful discussions during this work.
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|Title Annotation:||First Symposium on Oriented Polymers|
|Publication:||Polymer Engineering and Science|
|Date:||Nov 1, 1997|
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