Increasing bending strength of tibial locking screws: Mechanical tests and finite element analyses

1. Introduction 

Interlocking nailing with its advantages of minimal tissue injury and stable fracture fixation is widely accepted for treatment of tibial shaft fractures (Alho et al., 1990, Brumback, 1996, Greitbauer et al., 1998). Unfortunately, it is potentially threatened by failure of the distal locking screws, especially in patients with metaphyseal fractures, open fractures, comminuted fractures, fractures with delayed union, and fractures treated with small nails (Hutson et al., 1995, Larsen et al., 2004, Lin and Hou, 2001, Whittle et al., 1992). Screw failure may cause loss of fracture fixation, impairment of fracture healing, and difficulty in retrieving the screw fragments. The mechanical failure of the screws results mainly from a bending force caused by the loading of the nail at the middle of the screws. The screw failure mode may be either single-load plastic yielding or cyclic-load fatigue fracture. In spite of different failure modes, the mechanical strength of the screw is closely related to its material and structure (Evans et al., 1990, Krag, 1991, Lin et al., 2001). In particular, screw threads acting as the stress risers may increase the screw stress markedly and decrease the fatigue life exponentially.

The hypothesis of this study was that locking screws’ material and geometry significantly affect their mechanical strength. To test this hypothesis, mechanical tests and finite element analyses were conducted on five types of commercially available screws with full threads and two types of specially manufactured screws with limited threads. The results of the mechanical tests were correlated to that of finite element analyses. Parametric analysis and design optimization were done according to the Taguchi robust design method (Dar et al., 2002, Fowlkes and Creveling, 1995, Yang and Tarng, 1998). The stress concentration effect of the threads on the bending strength was subsequently investigated in screws made from the most commonly used metals.

2. Methods 

2.1. Structures of the tested screws 

The screw design factors including outer diameter, inner diameter, root radius, pitch, proximal half angle, distal half angle, and thread width (Fig. 1) were measured on five types of commercially available fully threaded stainless steel tibial locking screws (Synthes, Paoli, PA, USA; Howmedica, Rutherford, NJ, USA; Richards, Memphis, TN, USA; Osteo AG, Selzach, Switzerland; Zimmer, Warsaw, IN, USA) and one type of both-ends-threaded screw and one type of unthreaded bolt (Fig. 2) with a Surface Roundness RA-100 (Mitutoyo, Kawasaki, Japan) (Table 1). The both-ends-threaded screw and the unthreaded bolt were each made from stainless steel (Carpenter Technology, Reading, PA, USA) according to the specification of ASTM F138 with the yield strength of 786MPa and failure strength of 1000MPa. The length of the threaded part of the both-ends-threaded screw was 5mm at the proximal end and 20mm at the distal end. The unthreaded bolt was completely smooth without any thread and had a 3-mm oblique set screw at the screw cap to prevent back-out. The nominal length of these screws was 50mm.

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Fig. 1. Schematic drawing showing the design factors of the locking screw.

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Fig. 2. Tested screws include Synthes, Howmedica, Richards, Osteo AG, Zimmer, both-ends-threaded screw, and unthreaded bolt (left to right).

Table 1. Geometry, results of mechanical tests, and results of finite element analysis

	Geometry and results	Screw type
		Synthes	Howmedica	Richards	Osteo AG	Zimmer	Both-ends-threaded	Unthreaded
	Outer diameter (mm)	4.88	4.46	4.51	4.97	4.45	4.48a 5.97b	4.0
	Inner diameter (mm)	4.32	3.80	3.71	3.52	3.92	3.59a 4.97b	4.0
	Proximal root radius (mm)	0.3	0.4	0.1	0.2	0.2	0.1	–
	Distal root radius (mm)	0.3	0.4	0.4	1.0	1.0	0.1	–
	Pitch (mm)	2.75	1.48	1.26	1.74	1.82c	1.50	–
	Proximal half angle (°)	30	67.5	45	35	30	40	–
	Distal half angle (°)	30	22.5	10	10	55	30	–
	Thread width (mm)	0.50	0.20	0.30	0.10	0.4	0.17	–
	Section modulus (mm3)	7.91	5.38	5.01	4.28	5.91	8.82	6.28
	Yielding strength (N)	1853 (SD 134)	1395 (SD 81)	1376 (SD 103)	999 (SD 93)	1412 (SD 121)	1700 (SD 68)	1380 (SD 45)
	Fatigue life (103 cycles)	111 (SD 21.3)	28.7 (SD 7.9)	18.1 (SD 4.0)	10.2 (SD 4.6)	32.8 (SD 16.9)	60.7 (SD 25.7)	>1000
	Total strain energy (J)	19.1	27.5	29.7	33.1	26.0	20.5	30.0
	Maximal deflection (mm)	0.081	0.117	0.126	0.142	0.110	0.086	0.128
	Maximal tensile stress (MPa)	628	1000	1203	1140	933	800	566

The both-ends-threaded screw had a diameter of 4.48mm at the smooth part.

a Distal threads. b Proximal threads. c Double threaded screw.

2.3. Finite element analysis 

The finite element analyses of the locking screws were conducted with the use of commercial software ANSYS 5.8 (Canonsburg, PA). As determined by the measured geometry and dimension of the screws, three-dimensional finite element models were generated with Ansys Parametric Design Language (APDL) and a Boolean operation (Supplementary A; www.sciencedirect.com). The locking screws were 40-mm long and were surrounded at each end by a polyethylene cuff with an outer diameter of 7mm and a width of 5mm to simulate the mechanical tests (Fig. 3). The Young’s modulus was 230GPa for stainless steel and 2.6GPa for polyethylene. The Poisson’s ratio was 0.3 for both materials. The core of the locking screws was map-meshed and the surface including screw threads with curved boundaries was free-meshed with high order 20-node hexahedral elements (Solid 95). Similarly, the polyethylene cuff was also map-meshed except for the layer surrounding the screws. The overall element size was 0.4mm. Surface-to-surface contact elements were used for the interface between the locking screw and polyethylene, CONTA 174 for contact surfaces of the locking screw and TARGE 170 for the contact surfaces of the polyethylene. Axial rotation of the screws was not allowed. The loading condition was a point load of 475N (mean load of the fatigue tests) at the middle of the screws. The lower half of the surface of both bone cuffs was fully constrained to simulate the boundary condition of the mechanical tests. After processing, the total strain energy, maximal deflection, and maximal tensile stress of the locking screws were recorded. The total strain energy was the sum of the strain energy of all the elements of the screw. Because of the discretization errors at the thread valley, the mesh in areas with peak stress was refined by increase of mesh density, and numerical convergence was considered accomplished when the variation of the result of the sequential analysis was less than 3%. The results of the finite element models and the mechanical testing were correlated with linear regression tests.

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Fig. 3. Loading condition and boundary condition of the finite element model. The lower half (gray zone) of the polyethylene cuff is fully constrained.

3. Results 

3.2. Finite element analysis 

Total element number of the finite element models ranged from 90 to 280 thousands. Total node number ranged from 130 to 440 thousands, and the computer solution time ranged from 5 to 50h under the Microsoft Windows NT system. Maximal deflection was found at the middle of all the locking screws. The point with maximal tensile stress (Fig. 4) corresponded to the crack initiation site in mechanical tests. For fully threaded screws and unthreaded bolts, maximal tensile stress was observed at the root of the thread just opposite the loading point. For both-ends-threaded screws, it was observed at the root of the thread just between the smooth part and the threaded part. In the convergence test, the total strain energy and maximal deflection converged very rapidly without further mesh refinement. Total strain energy and maximal deflection showed a parallel relationship with a correlation coefficient approximating 1. However, maximal tensile stress converged more slowly in screws with smaller root radii—0.02mm refined element size in Richards and Osteo AG screws, as compared with 0.06mm in unthreaded bolts and in Synthes and Howmedica screws. Total strain energy, as well as maximal deflection, was inversely related to yielding strength of the screws with a high correlation coefficient of −0.95 (P<0.01) (Table 1). Maximal tensile stress was also inversely related to the logarithm of the fatigue life with a high correlation coefficient of −0.90 (P<0.01).

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Fig. 4. The maximal tensile stress in finite element analyses is located at the stress concentration points in the fully threaded screw and the both-ends-threaded screw.

3.3. Taguchi robust design analysis 

Only fully threaded screws were considered in this factorial analysis. As shown in the S/N graphs (Fig. 5), either increasing the outer diameter, inner diameter, root radius, half angle, and thread width or decreasing the pitch could reduce the total strain energy and increase the yielding strength. On the other hand, increasing the inner diameter, root radius, and pitch or decreasing the outer diameter, half angle, and thread width could reduce the maximal tensile stress and increase the fatigue life. According to ANOVA tables (Table 2), total strain energy was mainly determined by inner diameter alone, with a contribution of 88.5%. Inner diameter and root radius were the main determining factors of the screw stress with contributions of 63.8% and 27.8%, respectively. A verification test using optimal levels of each design factor for the lowest bending stress showed that optimal total strain energy was 28.1J and optimal screw stress was 470MPa. These values were consistent with the optimal results, 26.2J and 424MPa, computed by the additive model predictive equations. The additive model was (S/Npred: the predicted S/N, n: number of design parameters, : mean of S/N at optimum level of each design parameter).

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Fig. 5. S/N graphs demonstrate the effects of each design factor on the yielding strength (total strain energy) and fatigue life (maximal tensile stress) of the screws.

Table 2. ANOVA tables for the Taguchi analytical results

	Design factor	Sum of squares	Degrees of freedom	Mean square	F value	Contribution (%)
	Total strain energy
	Outer diameter	1.81	2	0.91	21.57	3.57
	Inner diameter	44.87	2	22.44	534.36	88.50
	Root radius	0.25	2	0.13	3.00	0.50
	Pitch	2.24	2	1.12	26.62	4.41
	Half angle	0.79	2	0.40	9.46	1.57
	Thread width	0.53	2	0.26	6.29	1.04
	Error	0.21	5	0.04	–	0.41
	Total	50.702	17	–	–	100
	Maximal tensile stress
	Outer diameter	0.82	2	0.41	5.80	0.91
	Inner diameter	57.48	2	28.74	404.88	63.84
	Root radius	25.06	2	12.53	176.55	27.84
	Pitch	0.10	2	0.05	0.73	0.12
	Half angle	2.66	2	1.33	18.71	2.95
	Thread width	3.55	2	1.78	25.04	3.95
	Error	0.35	5	0.07	–	0.39
	Total	90.04	17	–	–	100

4. Discussion 

The bending strength of tibial locking screws is closely related to their structures (Hou et al., 2004, Hou et al., 2002, Lin et al., 2001, Whittle et al., 1992). In particular, the threads and their geometry may markedly affect the fatigue strength of the screws (Greitbauer et al., 1998, Hou et al., 2002, Lin et al., 2001). As indicated in the report of Hutson et al. (1995), the three-point bending tests used in the current study could simulate the real loading condition of the locking screws in tibial fractures. The screw failure patterns were compatible with those observed in clinical situations. The present study investigated how the screw design improved the mechanical strength of the screw. Although it was also affected by the material properties of the metal such as the degree of cold working (Whittle et al., 1992), the single-load yielding strength of the screws was closely related to the section modulus of the inner diameter with a correlation coefficient of 0.89. This finding supports the use of simple beam theory to predict the yielding strength of the screws (Brown et al., 2000). However, simple beam theory, which does not consider the stress concentration effect and can only be applied to homogenous beams (Brown et al., 2000, Hou et al., 2004, Hou et al., 2002), is not suitable for predicting fatigue strength of screws with inhomogeneous geometry. These circumstances explain why the correlation coefficient between the logarithm of the fatigue life and section modulus noticeably dropped to 0.47. Although the material’s quality, the manufacturing process, and the surface treatment can also affect the fatigue life of the screws (Greitbauer et al., 1998, Liu et al., 1990), such factors had minimal effects on the testing results because these locking screws were made from high quality materials and had no metallurgical or manufacturing defects according to ASTM standards (Hou et al., 2002).

Besides simple beam theory, the finite element method—a powerful tool for computing the stress and strain inside an arbitrary complex structure—can be an alternative theoretical model for predicting the mechanical strength of the screws, especially in screws with irregular thread patterns or core diameter. The finite element method can appreciably save the expense, time, and effort involved in repeated implant manufacture and mechanical tests. The finite element models incorporate all the screw design factors that potentially affect the bending strength, and sensitivity analysis can further investigate the effects of each design factor independently. In this study, the three-dimensional structures of the screws could be accurately generated and meshed with the finite element method. The mesh around the thread valleys with stress concentration was refined for numerical stability. The influence of mesh density was found to be lower on total strain energy or maximal deflection (Marks and Gardner, 1993), but it was relatively higher on maximal tensile stress (Keyak and Skinner, 1992), especially in screws with a smaller root radius. The point with maximal deflection or maximal tensile stress was consistent with the failure site observed both in static-loading and cyclic-loading tests. The parallel relationship between the total strain energy and the maximal deflection was compatible with the linear part of the load–deformation curve. The current study demonstrated a close relationship between the results of finite element analyses and mechanical tests in both yielding and cyclic-loading tests with correlation coefficients of −0.95 and −0.90, respectively. The finite element models predicted mechanical strength better than simple beam theory did, especially for fatigue life (Brown et al., 2000, Kasman and Chao, 1984).

The Taguchi robust design method, which lends itself well to finite element analysis (Dar et al., 2002), is a powerful tool for improving the quality of engineering designs. It enables estimation of the sensitivity of a system to variation in numerous input parameters while reducing the number of experimental efforts. For a full factorial design, the present study required 36 finite element analyses. However, with a fractional factorial design using an L18 orthogonal array, only 18 runs of the analyses were sufficient. L18 evenly distributes the effect of the interaction between any pair of factors over the columns and gives the most realistic simulation of the actual effects of the parameters (Fowlkes and Creveling, 1995). According to the results of factorial analysis, among all the structural parameters, the inner diameter was the most important determining factor for total strain energy. The high contribution of inner diameter to total strain energy was compatible with the high correlation coefficient between the section modulus and the yielding strength. Root radius as a significant stress riser has minimal effects on yielding strength, but it can substantially reduce fatigue strength. Because this finding has rarely been reported before, a validation mechanical study on the effects of this root radius was conducted subsequently.

Validation of the numerical models is critical before their application (Editorial, 2005). The mathematical models in the present study were well validated by the mechanical tests on screws made from different metals. Irrespective of stainless steel or titanium, the screws with larger root radii consistently had higher stiffness and fatigue strength. However, the stainless steel screws with higher elastic modulus had significantly higher cyclic stiffness. On the other hand, the titanium screws with higher endurance limit had significantly higher fatigue life (Greitbauer et al., 1998). The increases of fatigue life in stainless steel screws were about 24% and 30% when the root radius was increased from 0.1mm to 0.3mm and 0.5mm. These increases were much smaller than those in titanium screws, which were about 3- and 5-fold, respectively. Because of the property of notch sensitivity in titanium alloy (Dick and Bourgeault, 2001), the difference between the fatigue life of titanium screws and that of their counterpart stainless steel screws was less in situations with smaller root radius. This finding implies that titanium alloy screws might lose their advantages of high fatigue strength if the screws had relatively sharp notches. Sharp notches might be an important factor accounting for the high risk of locking screw failure (8%), according to the report of Im and Shin who used titanium nails to treat femoral shaft fractures (Im and Shin, 2002). For design consideration, the root radius with minimal effects on the pullout strength of the screws (Hou et al., 2004) can be as large as desired to achieve a higher bending strength. On the other hand, increasing the inner diameter to increase the bending strength may jeopardize the pullout strength (Brown et al., 2000). The surface property of the screws might be another factor causing fatigue fracture especially in titanium screws, even though they had high quality materials and microstructures. This surface property could explain the high variation of the fatigue life of the titanium screws with root radii of 0.5mm.

The present study had potential limitations. The boundary conditions and loading conditions of the locking screws were based on certain presumptions. Using three-point bending instead of four-point bending to simulate the physiological loading might overestimate the bending load. Other factors that might also affect the absolute values of the results included the material properties of the bone and screws, the interface properties between the bone and the screw, the loading condition and boundary condition of the constructs, and the mesh density used for finite element analysis (Keyak and Skinner, 1992). Nevertheless, the results of the mechanical testing and finite element analysis mainly reflected the relative scales (Brown et al., 2000, Hou et al., 2004, Marks and Gardner, 1993) that were minimally affected by the aforementioned factors. Theoretically, the polyethylene might have some material changes during cyclic loading, but this was not simulated in finite element analyses, because the overall material integrity of the polyethylene was still preserved throughout the test, as shown in the cycle–displacement curves. In the Taguchi method, the chosen range may affect the contribution of the design factors. In this study, the selected range covered the majority of the most commonly used ranges for screw design and thus could reflect real conditions. Although L18 orthogonal array could not detect the interactions between factors, the interactions were minimal in this study because the predicted optimal results were consistent with those predicted by additive models (Dar et al., 2002), and the interactions were evenly distributed among all the columns.

In conclusion, increasing inner diameter and root radius or eliminating the screw threads could effectively prevent fatigue failure, the most commonly seen screw failure mode clinically (Hutson et al., 1995). Finite element analyses could reliably predict the relative yielding strength and fatigue life even in screws with heterogeneous thread patterns. This method could assist surgeons in selecting suitable devices during surgery and help manufacturers in designing implants without the necessity of conducting repeated mechanical tests. Titanium screws were much superior to stainless steel screws in fatigue strength if the notch sensitivity effect was minimized. Further clinical studies to verify the findings in the present study are warranted.