Initial Stress Calculations The following are the initial hand stress calculations conducted during the early stages of the design process. Therefore, some of the material properties as well as dimensions are not up to date with the device design. Thread Calculations Adequate pressure generation is a key criterion in the design of the clamp. In order to ensure that the clamp exerts the necessary force on the bell, calculations must be made to predict the amount of force that will be generated from the threads in the bell. These equations can be used either to calculate the forced generated, or to solve for the needed thread angle in order to generate a given load. Firstly, the given specifications for loading, as well as thread dimensions, will be defined. Force necessary for threads to generate Force exerted on nut This value was determined based upon the average force that a human hand can generate by turning a knob. There is a great deal of variation in this number, so a conservatively low value was used for the force. Diameter of thread Length of lever arm (1/2 diameter of turning nut) coefficient of friction coefficient of friction of collar diameter of collar
Angle of acme thread (constant for all acme threads) Figure 1 Acme Thread Diagram Using the defined variables, the calculations can be made in order to find unknown terms. Applied torque (The torque applied to the screw by the defined hand force) [1, p. 118] (eq. 1) Using the governing equation, listed below, an expression for the lead (L) can be derived for acme threads with the given inputs, listed above. [2, p. 869] ( (eq. 2) The above equation can be rearranged into:
Then rearranged into: Finally, the maximum lead (L) can be isolated in the equation below With given geometry, and loading conditions, the lead must be less than or equal to 19.1 mm. This criteria will most definitely be satisfied. The actual lead used is 1.27 mm, less than 7% of the maximum possible lead. Calculations must be made to ensure that the screw will not back out. Due to the fact that the clamp will be tightened and then left to set for several minutes, it is crucial that the screw does not back off at all, since that would have a catastrophic effect upon the pressure distribution, as well as negatively effecting adequate pressure generation. Figure 2 - Thread Diagram [2, p. 869] Define lead (L) for thread Additionally, using the previously defined L, the minimum coefficient of friction needed in order to maintain the self-locking criteria, is defined as: [2, p. 870] (eq. 3)
The coefficient of friction criteria will be met, as the estimated coefficient of friction will be approximately 0.15, 7.5 times larger than the minimum criteria. In order to ensure no back off for the thread, the following relationship must be met [2, p. 870] (eq. 4) Clearly, there will be no issue designing the actual device to meet the 77.65 maximum requirement. In order to ensure that the threads will not have a shear failure, which would be catastrophic to the clamp, the following equations must be satisfied. The stress analysis of the threads will take into account material properties, something that is very important due to the utilization of unique materials in the construction of the clamp. Firstly, a list of variables used in the following calculations will be shown (applied load) (basic diameter) (thread pitch) (ultimate strength of material) (height, or thickness, of nut) (area of rod) [1, p. 656] (eq. 5) (area of thread) [1, p. 656] (eq. 6) (minimum thread engagement) [3] (eq. 7) (stress in rod) [2, p. 31] (eq. 8) (FOS of loaded rod) [2, p. 260] (eq. 9)
Shear failure of the threads was also analyzed, since this type of failure would be a catastrophic failure for the device. Due to the fact that the diameter of the threads is less than 25.4 mm, combined with the fact that the threads are a standard Acme thread, the minimum nut length required to ensure that the threads will have strip strength in excess of the screw s tensile strength is 10.03 mm [3, p. 876]. Since the defined nut height is 12.7 mm, it can be concluded that the threads will not fail in shear. Bell calculations Due to the complexity of the loading on the bell, which is exacerbated by the complexity of the geometry, the only truly accurate method for conducting stress analysis on the bell is through a finite element analysis. Since the finite element analysis will not be complete until the second semester, the dimensioning and material selection for the prototype bell will be based upon the results from competitive product analysis. In addition to failure analysis, the appropriate material hardness must be achieved, in order to ensure the bell does not have surface deformation that could affect the even pressure distribution. By using design requirements from competitive analysis, the initial prototype can be created, and then the results from the finite element analysis can be used to optimize and finalize the design. Base Plate Deflection Calculations The base plate and bell pushing against each other cause a pressure distribution on the skin. As this system is basically a clamp, there is an upward force at the center of the plate, which causes deflection. The maximum deflection will be located at the center of the plate. As this deflection increases, the pressure distribution on the skin will begin to vary. The less uniform this pressure distribution becomes, the more likely it is for incomplete hemostasis, complicating the surgery. To prevent this, the base plate deflection must be minimized as much as possible. The deflection is a function of the design, the force applied, and the modulus of elasticity of the material. For the initial deflection calculations, some simplifying assumptions will be made. It will be assumed that the base plate can be modeled by two different beams attached to each other (see descriptive figures, below). The first will be attached to a rigid post. The only load on this first beam will be the reaction forces from the second beam. One end of the second beam will be attached to the first beam, while the other end will be hanging in space. The second beam will experience the reaction load of the pressure distribution over its entire length.
Figure 3 - Side view of actual design Figure 4 - Side view of simplified model Figure 3, above, is picture of the base plate from the side view. Figure 4, above, is the simplified design. The smallest dimensions from the actual model were used to define the simplified model. Since the smallest values are used, a factor of safety is already built into the calculations. Figure 5 - Top view of actual design Figure 6 - Top view of simplified model
Figure 5, above, is the actual base plate, orientated so the view is from the top. Figure 6, above, is the simplified design. The dimensions of the simplified design are the thinnest sections of the actual design, thus maintaining a conservative factor of safety. The dimensions labeled in the figures are the dimensions that will be used throughout the analysis of the base plate. Using the simplified design, the base plate deflection is calculated below. Bending of beam 1 [2, p. 996] (eq. 11) Bending of beam 2 [2, p. 996] (eq. 12) Moment of inertia [4] (eq. 13) Where F is the force at the end of the beam L is the length of the beam E is the modulus of elasticity (material constant) I is the moment of inertia (design constant) W is the load (F) distributed across the beam B is the base length of the beam H is the height of the beam These equations will be slightly conservative since the smallest dimensions are used. Therefore, a factor of safety is already built into the calculations. Table 1 - Base plate dimensions and properties Beam 1 Beam 2 Base 30.48 mm 8.13 mm Height 3.05 mm 8.13 mm Length 3.81 mm 16.76 mm Properties (both bars) E = 10,342.14 MPa Force = 1,290.0 N The loading has been calculated by the force applied by the screw. As E is a material constant, it will change as different materials are tested. This value is most likely to be the variable modified to output the desired maximum deflection
Using a value of 10,342.14 MPa for the modulus of elasticity, the final resulting maximum deflection in the base plate is 0.12 mm. Base Plate Factor of Safety As a check to make sure the device does not break, the factor of safety is also found for both of the beams. The smallest factor of safety for the two beams will be roughly that of the base plate, and as such, should be a good comparison. For these calculations, the base plate was modeled the same way as the max deflection calculations. Figures 11 through 14 represent the assumptions made and the dimensions used for the following calculations. Stress in beams [2, p. 156] (eq. 14) Where σ is the stress M is the maximum moment c is half the height I is the moment of inertia For beam 1 the max moment will be calculated by [2, p. 996] For beam 2 the max moment will be calculated by [2, p. 996] (eq. 15) (eq. 16) Where F is the point force downward L is the length of the beam w is the loading per unit length As the two beams are the same beams for which the deflection was found, the dimensions will be the same. These can be found in table 1, above. Once the max moment is obtained the factor of safety can be found using the following equation [2, p. 260] (eq. 17)
Where Sfa is the factor of safety Sy is the ultimate shear stress σmax is the maximum stress found of either beam. When the numbers are inserted into these equations, the maximum stress calculated to be 95.49 MPa. Using 151.68 MPa as the ultimate shear strength, the factor of safety found for the base plate is 1.59. A factor of safety of 1.5 is more than adequate for a single use device. Therefore, under given loading conditions, the base plate will not fail. Column Factor of Safety The factor of safety for the columns must be calculated, to ensure that the device does not break while it is being used. As the columns are of equal diameter over their entire length, they will break at the point where the curvature, and stress concentration, is the greatest. The dimensions of the beam where the curvature is the greatest are as follows Table 2 - Column dimensions Outer radius of curvature of 19.69 mm beam Inner radius of curvature of beam 5.72 mm Diameter of column 13.97 mm D = 13.97 mm Figure 7 - Cross section of column
Figure 8 - Side view of column curve Figure 8, above, depicts the curved portion of the beam, where the beam is most likely to fail. The three arrows show the 3 radiuses of curvature, the smallest being the inner, the middle the average, and the largest is the outer radius of curvature. The following equations can be used to find the factor of safety for curved beams. Several of the constants needed in the equations are and e. These constants are calculated below. [4], the area of the cross section, co, ci, (eq. 18) (eq. 19) (eq. 20) (eq. 21) (eq. 22) Where ro is the outer radius of curvature ri is the inner radius of curvature d is the diameter of the column rc is the average radius of curvature The stress along the inner and outer radius can now be found using the following equations. [4]
(eq. 23) (eq. 24) To find the factor of safety, the ultimate tensile strength for the material will also be needed. This is a known constant, and can be determined from material constant tables. The factor of safety can be found for the columns using the equation below. [4] (eq. 25) When the numbers are inserted into the equations, the maximum stress within the beams is calculated to be 97.91 MPa. With this max stress, and using 151.68 MPa as the ultimate tensile strength, the resulting factor of safety for the base plate is 1.55. As seen, the columns have a factor of safety well above 1. From this, it can be concluded that columns of the device will not fail when the target load is applied to the device. Column Deflection There are two straight sections to each of the columns; these are the sections that will analyzed for deflection. The vertical deflection in both strait sections of the column will be found and added together. Despite one of the sections being angled heavily from vertical, this angle was neglected in order to add a conservative factor of safety. Figure 9 - Side view of the column
The equations used to find the approximate value for these deflections are as follows. [2, p. 31] (eq. 26) (eq. 27) (eq. 28) Where σ is the stress P is the load A is the area ε is the strain E is the modulus of elasticity l is the length after the load is applied lo is the length before the load is applied Using a modulus of Elasticity of 10,342.14 MPa, the vertical elongation of the column is calculated to be 0.04 mm. Material Selection At the time that these initial hand calculations were conducted, a material had not been selected for the device. Therefore, in order to complete the analysis material properties were chosen. The ultimate tensile strength was set at 151.68 MPa and the modulus of elasticity was set at 10,342.14 MPa. However, these properties can be modified according to the results from the analysis. If a material is used that has lower material property values than these, the device will be more likely to have a deflection greater than desired. In addition the device will also have a factor of safety below the desired factor of safety of 1.5. As such, these are the lowest material property values that should be used. Materials with properties above these values can be used, since they will only improve upon the already acceptable deflections and factor of safety. However, material cost must be taken into consideration, limiting some high strength materials.
Works Cited [1] R. C. Hibbeler, "Chapter 4," in Statics, Upper Saddle River, Pearson, 2010. [2] R. Norton, Machine design an integrated approach, 4th ed., Upper Saddle River, New Jersey: Prentice Hall, 2011, p. 145. [3] "RoyMech," [Online]. Available: www.roymech.co.uk/useful_tables/screws/thread_calcs.html. [Accessed 26th November 2013] [4] "Curved beams, derivation of stress equations," [Online]. Available: http://courses.washington.edu/me354a/curved%20beams.pdf. [Accessed 15 November 2013].