© 2004 Kluwer Academic Publishers. Printed in the Netherlands.
Experimental and finite element analyses on stress intensity
factors of an elliptical surface crack in a circular shaft under
tension and bending
C.S. SHIN∗and C.Q. CAI
Department of Mechanical Engineering, National Taiwan University. No.1, Sec.4, Roosevelt Road, Taipei, 10617, Taiwan, Republic of China
∗Author for corresponding (E-mail address: [email protected]; Phone+886-2-2362-2160; Fax:
+886-2-2362-2160)
Received 30 September 2003; accepted in revised form 26 April 2004
Abstract. Experimental backtracking technique and finite element analysis have been employed to evaluate the stress intensities along the front of an elliptical surface crack in a cylindrical rod. The finite element solution covers a wide range of crack shapes loaded under end-free and end-constrained axial tension and pure bending. Convenient closed form stress intensity expressions along the whole crack front for each of the loading cases have been given in terms of the crack aspect ratio, crack depth ratio and place ratio.
The closed form solutions have been compared against a number of representative solutions collected from the literature. It has been found that different finite element results for the interior points are generally in good mutual agreement, while solutions derived from other methods may sometimes indicate different trends. At the surface interception point agreement is less good because of a complication in the interpretation of stress intensity there.
Experimental backtracking results on the end-constrained axial tension case corroborate well with the closed form solution presented. It suggests that the current closed form solution is adequate in describing the stress intensities along the whole crack front of real surface cracks in cylindrical rods.
Key words: Closed form stress intensity solution, experiment backtracking technique, stress intensity factor, surface crack.
1. Introduction
Cylindrical shaped components such as pins, bolts, reinforcement wires and shafts are com-monly used in engineering structures. Under repeated loading, cracks may develop at the surface and grow across the section. Recently, it has been shown that miniature surface cracked rod specimen offer a cost-effective way for evaluating fatigue crack propagation properties (Shin and Chen, 2004). To assess the crack growth behavior and structural integrity involving these cracks, their stress intensity factor solutions must be known. The three-dimensional nature of this kind of cracks results in a stress intensity that is not only varying along the crack front but is also highly sensitive to the crack shape. Numerical techniques or approximate analyses were often employed to estimate the stress intensity for this problem.
Early attempts used a straight edge (Daoud et al., 1978; Bush, 1981; Carpinteri, 1992a) or a circular arc to idealize the crack front (Wilhem et al., 1982; Mackay and Alperin, 1985; Daoud and Cartwright, 1985; Forman and Shivakumar, 1986; Raju and Newman, 1986). In some works, the angle of intersection of the crack front with the rod external surface was taken to be 90o to facilitate crack shape definition (Forman and Shivakumar, 1986; Raju and
Newman, 1986). These idealizations, though close to, do not as a matter of fact exactly agree with experimental observations. The above discrepancies may have impact on the correctness of the stress intensity solutions. This problem of crack shape description is largely solved by using an elliptical arc to model the crack front (Athanassiadis, 1981; Astiz, 1986; Shiratori et al., 1986; Murakami and Tsuru, 1987; Caspers et al., 1990; Carpinteri, 1992b; Carpinteri and Brighenti, 1996; Couroneau and Royer, 1998; Shih and Chen, 2002). It is well known that the singularity power at the intersection points is no longer−1/2 and is dependent on the angle of intersection and Poisson ratio (Bazant and Estenssoro, 1979; Hayashi and Abe, 1980; Carpinteri and Brighenti, 1996). The departure from the square root singularity was sometimes pragmatically overcome by discarding the numerical solution at the end point and replaced it with that for a neighboring interior point instead (Carpinteri, 1992b; Carpinteri and Brighenti 1996; Couroneau and Royer, 1998).
Available stress intensity solutions from the literatures for surface cracks in cylindrical rods have some limitations. For example, the reported solutions are often limited to the deepest interior point (Daoud et al., 1978; Daoud and Cartwright, 1985; Bush, 1981; Wilhem et al., 1982; Mackay and Alperin, 1985; Forman and Shivakumar, 1986; Murakami and Tsuru, 1987; Couroneau and Royer, 1998) or the deepest interior and the surface interception points on the crack front (Carpinteri, 1992b; Carpinteri and Brighenti, 1996; Astiz, 1986; Athanassiadis, 1981; Shiratori et al., 1986; Caspers et al., 1990). Sometimes, they are available for a limited number of discrete aspect ratios and crack depth ratios. A closed form solution for the whole crack front covering a wide range of crack geometry is essential for fatigue life evaluation and structural integrity assessment involving surface cracks because practical surface cracks may come in any aspect ratios and crack depths, and such parameters may change as the crack grows along. A closed form solution has recently been published (Shih and Chen, 2002), and covers a wide range of crack aspect ratios and depth ratios, but the present authors think that it is not completely valid (Cai and Shin, 2004a).
In the present work, the stress intensity factors along the crack front are computed for an elliptical surface crack in a cylindrical rod under tension and bending. A wide range of crack aspect ratios that should be able to cover most practical crack shapes is examined. Closed-form equations are given to facilitate use. Moreover, the end condition of practical cylindrical structures under axial tension is often somewhere between unconstrained and fully constrained. These two extreme conditions are dealt with. Finally, experimental verification of the current solution is presented.
2. Stress intensity factor calculation
In the present paper, stress intensity factors of the surface cracks in a rod are deduced from finite element stress analysis results via the virtual crack extension method (Parks, 1974; Hellen, 1975). The finite element package ABAQUS has been employed for analysis. Owing to symmetry, only one quarter of the rod has been analyzed. Collapsed quarter point elements (Barsoum, 1976) have been used along the surface crack tip. Figure 1 shows details of the finite element mesh. Axial tensile stress has been applied to the end of the rod. Two kinds of boundary conditions have been considered: an end-constrained case and an end-free case. In the former, the last layer of elements at the end of the rod has been constrained to move in the axial direction. The element dimensions, especially that in the axial direction, have been gradually refined until the solution converged to a consistent value. For the end-constrained
Figure 1. Finite element mesh employed in the current work.
Figure 2. Nomenclature used for an elliptical surface crack.
case, the length of constrained region also affected solution convergence. The constrained end length chosen is 1 mm for a half-rod length of 90 mm.
3. Stress intensity solutions
With the geometrical dimensions defined in Figure 2, the stress intensity (KI)of the surface
crack can be expressed in the following form:
KI = FIσ
√
π a, (1)
where σ is the uniform axial stress or maximum bending stress, a is the crack depth and FIis
the geometry correction factor. Since the stress intensity factor of a surface crack is dependent upon the crack shape and varies from point to point on the crack front, FI may be written as:
Figure 3. Comparison of some typical results for the end-free and the end-constrained case for selected crack
aspect ratios. (a) at the deepest interior point A (b) at the surface point G.
FI = FI a b, a D, x h , (2) where a
b represents the crack aspect ratio, a
D is the crack depth ratio and x
h is the place ratio
for identifying different points on the crack front. Surface cracks with the aspect ratio a
b ranging from 0 (straight edge) to 1 (semi-circular)
in increment of 0.1 have been analyzed. For each crack shape, the crack depth ratio a
D varied
from 0.067 to 0.8 in eleven equal increments. The place ratiox
hranged from 0 (deepest point A
in Figure 1) to 1 (surface intersection point G in Figure 1) in six equal increments. As a result, the stress intensities at 13 points on the crack front were evaluated for each crack front.
Detailed results are listed in Tables A1, A2 and A3 in the appendix for the free, end-constrained axial tension and the pure bending cases, respectively. Figure 3 compares some typical results for the end-free and end-constrained cases at the deepest point A (Figure 3a)
Figure 4. Geometry correction factor variation along the crack front. (a) under end-free axial tension; (b) under
Figure 4. Continued.
and the surface point G (Figure 3b) for selected crack aspect ratios. It is clear that, for the same crack geometry, the stress intensity for the end-constrained case is always smaller than that for the end-free case. The discrepancy is within 5% when the crack is small
a
D <0.33
. For deeper cracks, the discrepancy increases quickly to 25% at a
D = 0.6 and 67% at a
This is reasonable as end constraint will impose a bending effect to check the crack opening and such an effect increases with the crack depth.
It has been pointed out that the square root singularity does not in general exist at the sur-face point G (Bazant and Estenssoro, 1979; Hayashi and Abe, 1980; Carpinteri and Brighenti, 1996). The usual meaning of stress intensity factor breaks down here. A plot of the geometry correction factor against the place ratio (see Figure 4) shows typically that the raw finite results at the surface point deviate markedly from the trends indicated by the interior points. Carpinteri and Brighenti (1996) pointed out that the region where the stress intensity breaks down is very small and proposed to take the result at a point next to the surface in the finite element mesh
x
h = 0.9
. However, the present work showed that the stress intensities at
x
h = 0.9 may be up to 14% different from the stress intensities at the surface if the latter were
deduced by extrapolating the trends exhibited by the interior points. In the current work, it is proposed to use a quadratic curve fitting to the interior points and obtain a pragmatic stress intensity value for the surface point by extrapolation. The raw finite element results deviate from these extrapolation results by−27% to 43%. The two results are within 6% of each other whena
b = 0.6. Deviation increases as a
b moves away from 0.6 in both directions. Incidentally,
the square root singularity exists at the surface interception point of an elliptical crack with an aspect ratio of 0.6 when the Poisson ratio is around 0.3.
4. Closed form stress intensity equations
It has been pointed out before that a closed form equation that gives us the stress intensities at different points along the crack front for different crack geometries is very useful in evaluating practical fatigue life or stress corrosion life and assessing other structural integrity problems involving surface cracks in cylindrical rods.
By using a multi-parameter fitting technique, the geometry correction factor FI under
tension at different points on the surface cracks with different aspect ratios and depth ratios may be expressed as:
FI = FI a b, a D, x h = 2 i=0 7 j=0 2 k=0 Mij k a b ia D jx h k (3) The coefficients Mij k for the end-free and end-constrained axial tension conditions are
listed in Tables 1 and 2, respectively. For bending, the geometry correction factor FI may
likewise be expressed as:
FI = FI a b, a D, x h = 2 i=0 6 j=0 2 k=0 Nij k a b ia D jx h k (4)
The coefficients Nij kfor bending are listed in Table 3. The FI values at the surface point G
employed in the above fitting come from quadratic extrapolation as mentioned in the previous section. Figure 4 compares the closed form solution (dotted lines) with the raw finite element results (discrete symbols) for different values of place ratio, crack depth ratio and crack aspect ratio. The maximum deviation of the closed form values from the raw finite element data is within 3%, whereas most of the deviations are within 1%.
Table 1. Coefficients Mij kin Equation 3 for end-free axial tension condition k= 0 k= 1 k= 2 j i 0 1 2 0 1 2 0 1 2 0 0.220 −0.326 0.266 0.123 0.065 0.118 −0.409 1.011 −1.584 1 28.513 −3.780 −9.118 0.511 −6.878 −3.515 −9.764 −3.946 45.562 2 −354.782 79.489 85.381 −2.034 47.747 75.016 128.817 41.099 −552.891 3 2178.632 −571.094 −465.013 −19.569 −119.954 −587.594 −727.078 −316.682 3322.477 4 −7140.202 1976.255 1475.911 144.435 14.769 2197.404 2201.067 1284.860−10812.317 5 12957.447−3583.421 −2794.532 −359.284 423.169 −4264.810 −3732.813 −2563.292 19328.127 6 −12227.977 3256.770 2878.868 393.518 −661.610 4138.287 3343.521 2455.158−17829.715 7 4721.868−1163.158 −1261.348 −159.206 306.176 −1588.135 −1240.214 −880.302 6638.698
Table 2. Coefficients Mij kin Equation 3 for end-constrained axial tension condition
k= 0 k= 1 k= 2 j i 0 1 2 0 1 2 0 1 2 0 1.095 −1.177 0.725 0.113 0.271 −0.388 −0.896 0.904 0.008 1 −1.336 17.924 −17.427 1.824 −11.649 10.074 3.092 0.701 −4.883 2 13.108 −137.252 134.652 −21.709 98.358 −80.088 −4.197 −32.641 55.092 3 −43.689 545.816 −551.902 105.483 −415.027 328.165 −13.255 204.104 −305.079 4 134.868 −1223.334 1239.493 −271.225 982.713 −772.921 51.548 −568.407 916.962 5 −242.653 1541.587 −1548.537 387.470 −1329.634 1055.952 −59.329 857.543 −1545.428 6 254.093 −1006.656 969.388 −290.024 961.893 −784.581 13.481 −657.659 1372.595 7 −108.196 264.206 −227.132 88.387 −288.565 245.798 10.854 191.570 −485.556
5. Comparison with solutions from the literature
A number of stress intensity factor solutions for surface crack in cylindrical rod under tension and bending exist in the literature. These solutions have been derived using finite element analyses (Raju and Newman, 1986; Astiz, 1986; Shiratori et al., 1986; Carpinteri, 1992b; Couroneau and Royer, 1998; Shih and Chen, 2002), boundary integral method (Athanassiadis et al., 1981), general weight function (Caspers et al., 1990) or approximate analyses (Douad et al., 1978; Douad and Cartwright, 1985; Bush, 1976, 1981; Murakami and Tsuru, 1987). Most of these solutions are only available for a limited range of crack shapes and all are for the end-free condition in case of axial tension. Moreover, only the stress intensities for the deepest point (A in Figure 1) and/or the surface point (G in Figure 1) are reported. Nevertheless, a number of solutions for certain common crack geometries are herein compared.
5.1. END-FREE AXIAL TENSION RESULTS
Figure 5 compares some published solutions with the current results for the end-free axial tension case. Regarding the stress intensity at the deepest point A, for a straight crack front
(a
b = 0, see Figure 5a), all the solutions collected agree well with each other with two
excep-tions: beyond a crack depth ratio of 0.4, the result deduced from an experimental compliance measurement (Bush, 1981) starts to deviate from others, and gives from 3% to 20%
under-Table 3. Coefficients Nij kin Equation 4 under bending k= 0 k= 1 k= 2 j i 0 1 2 0 1 2 0 1 2 0 1.346 −0.640 −0.022 0.190 −0.347 0.175 −0.926 1.399 −0.454 1 −9.627 6.435 0.207 −1.323 2.839 −1.635 6.767 −10.348 2.400 2 82.244 −36.062 −22.436 8.317 −18.649 9.091 −42.734 71.260 −4.388 3 −360.650 102.765 148.962 −31.454 70.186 −32.253 162.595 −263.786 −18.246 4 841.678 −151.830 −426.773 66.389 −142.227 60.188 −345.453 531.560 110.187 5 −973.482 107.831 554.803 −71.557 144.956 −55.293 375.935 −544.306 −186.619 6 449.146 −27.262 −276.533 31.022 −58.870 19.041 −165.151 225.705 108.877
estimation. Secondly, beyond a crack depth ratio of 0.5, the solution derived from a general weight function method (Caspers et al., 1990) deviates abnormally from the group trend, and is probably not reliable there. The deviation of the compliance solution is understandable as the experimental approach measured an average bulk intensity for the whole crack instead of pinpointed the point of interest. Nevertheless, such an average value is a good approximation for stress intensity at the deepest point when the crack depth ratio is smaller than 0.4. For an
aspect ratio a
b = 0.6 (Figure 5b), the majority of the collected solutions are within 10% of
the present data. The largest discrepancies are only about 13% and occur in the finite element (FE) solution by Carpinteri (1992b) (at a/D= 0.3) and the weight function solution (Caspers et al., 1990) (at a/D= 0.533). For a circular-arc crack front (a
b = 1.0 Figure 5c), the weight
function solution (Caspers et al., 1990) is markedly above the other solutions. The FE solution by Carpinteri (1992b) is also higher, while that by Raju and Newman (1986) is lower than the current results. In all three crack shapes above, the FE results by Couroneau and Royer (1998) and by Shiratori et al. (1986) nearly coincide with the present results. It should be pointed out that in the above comparison, due to the difference in the idealized crack front geometry and the difference in mesh refinement, the same crack aspect ratio will not guarantee exactly the same crack shape. Nevertheless, in the face of these differences, the worst-case discrepancies among these three stress intensity factor solutions for the deepest point is about 13%.
Regarding the surface point G, there is considerably more scatter among solutions for a straight-edge crack (a
b = 0, Figure 5d). The compliance solutions (Daoud et al., 1978;
Bush, 1981) are markedly higher than other solutions. It has been pointed out above that the compliance of a cracked member only reflects the overall effect of a surface crack and so only gives us an average stress intensity over the whole crack front. Its deviation from other solutions indicates that more detailed analysis is needed to account for stress intensity variation along the surface crack front. Carpinteri (1992b) and Astiz’s (1986) FE solutions indicated higher stress intensities at small crack depths
a
D <0.2
. At larger crack depths, discrepancies reduced and these two solutions merge with the current results. These two FE solutions are within 8% of the current results. Solution from the weight function method (Caspers et al., 1990) crossed the present results at a
D = 0.5. It is up to 39% higher than the
present results for a
D <0.5, and is up to 15% lower for a
Figure 5. Comparison of different stress intensity solutions for an elliptical surface crack in a shaft under tension.
are lower than the current results for a
D <0.5 and higher than all other results for a
D = 0.6.
At aspect ratios of 0.6 and 1 (Figures 5e and 5f, respectively), most of the collected solutions are very close to the current solution. Astiz’s (1986) FE results are prominently higher than other solutions at small crack depths
a
D <0.2
. At a
D >0.5, the weight function solution
Carpinteri’s solution at a
D = 0.6 is lower than the current solution. For a
b = 0.6 and 1, this
difference amounts to 14%.
In conclusion, for the deepest interior point A, all the collected FE solutions (Raju and Newman, 1986; Astiz, 1986; Shiratori et al., 1986; Carpinteri, 1992b; Couroneau and Royer, 1998) agree well with the current results and the maximum discrepancy is 16%. It should be pointed out that, although Shih and Chen (2002) has provided a rather complete closed form solution for this problem, their results for interior points are low and may even go negative, and so have not been included for comparison (Cai and Shin, 2004a). For the surface point G, agreement among different results are in general very good ata
b = 0.6 where the square root
singularity exists. For other aspect ratios, the overall agreement between several FE solutions (Raju and Newman, 1986; Astiz, 1986; Shiratori et al., 1986; Carpinteri, 1992b) with the current results is less good, with the maximum discrepancy equal to 32%, which comes from Astiz’s (1986) FE solutions for a
D = 0.05.
5.2. BENDING RESULTS
Figure 6 compares current stress intensity results on bending with solutions collected from the literature. Regarding the stress intensity at the deepest point A, for a straight fronted crack (Figure 6a), considerable disagreement occurs at crack depth ratio a
D below 0.3, while all the
results are close together for a
D above 0.4. If only the solutions obtained through FE analysis
are considered, then good agreement occurs throughout, and the literature results (Carpinteri, 1992b; Couroneau and Royer, 1998) are within 7.5% of the current results. For an aspect ratio
a
b of 0.6 (Figure 6b), all the results agree well, with the maximum deviation from the current
results equal to 7.2%. For circular-arc crack front (a
b = 1, Figure 6c), most of the results are
within 8% of the current solution except some of Carpinteri’s (1992b) and Raju and Newman’s ( 1986), which can be differ by up to 13% with the current results. The deviation of Raju and Newman’s solution may be due to the fact that they used a circular instead of an elliptical crack. Shih and Chen (2002) ’s closed form solution has not been included for comparison for the same reason as pointed out above in the tension case.
Regarding the surface point G, for a straight fronted crack (Figure 6d), the compliance solutions are markedly above the current results. Carpinteri’s FE results are 26% higher at
a
D < 0.1 and 9.5% lower at a
D = 0.6 but otherwise they agree well with the current work.
The weight function results are close to the current solution at a
D 0.5 but are markedly
higher for smaller depth ratios. Shih and Chen (2002) are markedly lower than for a
D <0.5
and prominently higher than all other results for a
D = 0.6. For an aspect ratio a
b = 0.6
(Figure 6e), Carpinteri’s solution (1992b) is close to the current results. Shiratori et al.’s FE solution (1986) is close to the weight function solution (Caspers et al., 1990) and both are below the current results at a
D 0.5. Raju and Newman’s FE solution (1986) is available
for crack depth ratio below 0.4. Although Raju and Newman’s results do not differ too much from the current solution, they do not show much variation as crack depth increases and this is
Figure 6. Comparison of different stress intensity solutions for an elliptical surface crack in a shaft under bending.
different from the trend shown by other solutions. Raju and Newman’s solution has the same problem for a
b = 1.
For circular-arc cracks a
b = 1, Shih and Chen’s results (2002) are higher than all other
results. Shiratori et al.’s FE results (1986) are markedly above the current solutions. The weight function results (Caspers et al., 1990) and Carpinteri’s results (1992b) are quite close to each other and are also close to the current results except at a
From the above comparisons, it can be seen that, except for some cases in Raju and Newman’s (1986), Shiratori et al.’s (1986) and Shih and Chen’s (2002), all stress intensity solutions derived directly from FE analysis have consistent trends and agree reasonably well with each other. The discrepancy exhibited by Raju and Newman’s solution may be due to the circular crack shape they employed being different from the elliptical crack shapes used in other studies.
In conclusion, some remarks can be made. Firstly, different FE results for the interior points are in general in good mutual agreement. Secondly, the discrepancy among different solutions at the surface interception point is, in general, larger than that at the interior points. It has been pointed out that the square root singularity leading to the stress intensity concept does not in general exist at the surface interception point G of a surface crack. In the collected solutions, with the exception of Carpinteri (1992b), no special treatment to deal with this situation has been mentioned. Carpinteri computed the stress intensity for 0.0 x
h 0.9 and ignored
the results obtained for 0.9 x
h 1.0. Experimental backtracking in the following section
suggests that the stress intensity factor variation over the crack front follows a rather smooth curve in the near surface region. As a result, a quadratic fit to stress intensity values at interior positions is used in present study to obtain a pragmatic definition of surface stress intensity.
Lastly, Shih and Chen (2002) reported a closed form solution that is the most complete to date. It covers the whole crack front and for a wide range of crack aspect ratios and depth ratios. However, the present authors think that their solution is not completely valid (Cai and Shin, 2004a).
6. Experimental verification of the stress intensity solutions
In the above comparison with other numerical solutions, discrepancies to different degrees invariably exist. In order to assess more critically the current solution and, in particular, the validity of the present treatment at the surface point G, experimental analysis using the backtracking method (James et al., 1969; Schijve, 1985; Shin, 1990) has been attempted for the case of end-constrained axial tension. In this method, the crack growth rate (da/dN) versus the stress intensity range (K) has first been obtained using standard specimens. Due to the variation in K, the amount of fatigue crack growth increment at different points on the crack front varies. By comparing successive crack profiles obtained after loading for a designated number of cycles, the crack growth rates at different points on the surface crack front can be evaluated. Knowing the crack growth rate at a particular point on the crack front, the associated stress intensity can then be deduced from the baseline da/dN versus K data.
6.1. EXPERIMENTAL PROCEDURES
Standard compact tension (CT) specimens (50 mm wide×12 mm thick) conforming to ASTM E647 as well as rod specimens (15 mm in diameter and 200 mm long) were machined from an AISI 304 stainless steel plate stock. Semi-elliptical surface starter notches with dif-ferent aspect ratios were created on the rod specimen using electric discharge machining with a thin circular copper foil electrode. Before testing, all specimens were stress relieved by a solution treated in vacuum at 1050◦C for one hour followed by quick cooling in a continuous supply of argon stream.
Figure 7. A typical rod fracture surface showing beach markings.
The CT specimens were tested under a loading with constant amplitude sinusoidal wave-form at a frequency of 15 Hz. Crack length was monitored with a traveling microscope to a resolution of 0.01 mm.
The rod specimens were tested with a pair of rigid hydraulic grips to simulate the end-constrained condition. Surface crack length was monitored with a rotary traveling microscope to a resolution of 0.03 mm. On a few specimens, instantaneous crack fronts were recorded periodically by creating beach marking using a waveform that has the same maximum but with the amplitude reduced to 30% of the instantaneous baseline loading. Constant amplitude as well as load shedding histories have been employed as the baseline loading. A typical beach marked fracture surface is shown in Figure 7. These instantaneous crack front records allow the crack growth rates and hence the stress intensities along the crack front to be deduced via the backtracking method. A normalized area-compliance method (Cai and Shin, 2004b) has also been employed on a number of rod specimens to monitor the surface crack growth to allow the crack size to be measured at much closer intervals and with much less effort than the beach marking technique. This compliance method provides an accurate definition of crack shape and depth (Cai and Shin, 2004b). Pre-matured crack closure invariably occurs during fatigue crack propagation. The closure behavior in a CT specimen is likely to be different from that in a rod specimen. Moreover, the phenomenon of crack closure is highly complex in a surface crack (Fleck et al., 1983; Fleck, 1988; Ray et al., 1987; Troha et al., 1988; Dawicke et al., 1990). Crack closure is heavier on the surface than in the interior. To ensure a valid backtracking stress intensity solution, the phenomenon of pre-mature crack closure has to be properly accounted for. In the current work, the compliance method aided with the offset procedure (Fleck, 1988) was employed to measure closure. A back face strain
Figure 8. Typical development of the crack shape and the surface and interior closure responses in a surface
cracked rod specimen.
gage was employed to monitor the closure response in a CT specimen. For the rod specimen, a strain gage straddling the crack at the tip region was used to reflect the crack closure on the surface. A back face gage, sticking in the middle of the uncracked surface, was used to reflect the bulk closure. This bulk closure is more representative of the closure in the interior (Fleck et al., 1983; Fleck 1988). On unloading, the slope of the back face compliance trace starts to change as closure at the surface begins to occur, and is continuously changing as closure spreads from the surface to the interior. The point on the compliance trace that corresponds to a fully closed crack is taken to determine the crack closure load for the deepest interior point (Dawicke et al., 1990). Crack closure at intermediate positions is approximated by a linear interpolation between the surface and the interior values.
6.2. CRACK PROPAGATION RELATIONSHIP FROM STANDARDCTSPECIMENS
The baseline crack growth data from standard fatigue crack propagation testing on CT speci-mens may be expressed as:
da dN = 9.947 × 10 −9K eff 2.973 , (5) where da
dN is the crack growth rate in mm/cycle and Keff is the portion of applied stress
intensity range during which the crack stays open. The unit of Keff is MPa
√ m. 6.3. EXPERIMENTAL RESULTS
Figure 8 shows a typical crack shape development in one of the rod specimens under constant amplitude axial tension loading. The initial aspect ratio a
b of the semi-elliptical starter notch
was 0.6. As crack initiated and grew, the aspect ratio increased steadily to∼ 0.95 at a crack depth ratio a
D 0.35. Thereafter it fell steadily back to ∼ 0.6 again. Figure 8 also shows
Figure 9. Comparison of typical crack growth responses at the deepest interior point A and the surface point G.
Figure 10. Comparison of the surface and interior growth in terms of the effective stress intensity range (Keff).
surface than in the interior. As the crack grew, the degree of closure decreased. The surface crack became fully open everywhere during the whole loading cycle when the crack depth ratio reached 0.39.
Figure 9 compares the typical crack growth responses at the deepest interior point A and the surface point G in terms of K for one of the rod specimens. K values are computed using Equation 3. Tests have been repeated on four rod specimens with different initial crack shapes, and reproducibility of the results is quite good. At lower K values, crack growth in the interior is consistently faster than that at the surface. This may be attributed to the difference in crack closure responses as revealed in Figure 8. At higher Ks where the cracks were fully open everywhere during the whole loading cycle, crack growth responses at the two locations merge together. This consistency suggests the current interpolation treatment to obtain pragmatic stress intensity solutions at the surface point G is acceptable.
Figure 10 compares the typical surface growth responses with the interior growth responses correlated in terms of the effective stress intensity rangeKeff
for the rod specimens. Both the surface growth and interior growth fall in line with each other and with the CT specimen
Figure 11. Comparison between the experimental backtracked and numerical stress intensity results along the
surface crack front for a range of values of crack depth ratio and crack aspect ratio.
data throughout the whole range of Keff tested. Therefore, the current method to measure
crack closure seems to be acceptable.
Figure 11 shows some typical comparison between the stress intensity values along the surface crack front obtained by experimental backtracking and the numerical results from Equation 3, for a range of crack depth and crack aspect ratios. The solid triangles represent direct backtracking deduction of stress intensity while the open triangles have accounted for
the effect of plastic zone size correction on the stress intensity. For a
D smaller than 0.482, both
the solid and dashed lines virtually overlap with each other. For a
D between 0.257 and 0.41,
the numerical values are within 3% of the experimental values. For a
D from 0.482 to 0.593,
the discrepancy between numerical values and the experimental values without accounting for plastic zone size effect steadily increases from 4.6% to 11%, with experimental values consistently above numerical values. If a plastic zone size correction is applied during the backtracking calculation, the experimental results will move downward towards the numerical results (solid line), and the maximum error is only 8.5%. It should be noted that the practical crack fronts are unlikely to be exactly the same as that modeled in the numerical analysis. Discrepancy in crack front shape increases as crack grows, leading to an increasing difference between numerical and experimental results. From the above comparison, it seems that the current closed form solution is adequate in describing the stress intensities along the whole crack front from surface to the interior of real surface cracks in cylindrical rods.
7. Conclusion
Experimental backtracking and finite element analysis have been performed to evaluate the stress intensities along the front of an elliptical surface crack in a cylindrical rod. Loading under end-free, end-constrained axial tension and pure bending have been analyzed using the finite element method. The solution covers crack aspect ratios from 0 to 1 and crack depth ratios from 0.067 to 0.8. Convenient closed form stress intensity expressions for both loading cases have been given in terms of crack aspect ratio, crack depth ratio and place ratio. The maximum deviation of the closed form values from the raw finite element data is within 3%, whereas most of the deviations are within 1%.
The current closed form solution has been compared against a number of representative solutions collected from the literature. It has been found that, in general, different finite ele-ment results for the interior points are in good mutual agreeele-ment, while solutions derived from other methods may sometimes indicate different trends. Agreement is not so good at the surface interception point. This is believed to be caused by the different treatments or non-treatment to account for the absence of the square root singularity there. A quadratic extrapolation to the interior stress intensities to obtain a pragmatic stress intensity value for the surface point is adopted in the current work.
Experimental backtracking results corroborate well with the above closed form solution. It can be concluded that the present closed form stress intensity solution seems to be adequate in describing the stress intensities along the whole crack front from surface to the interior of real surface cracks in cylindrical rods.
Acknowledgement
The authors are grateful to the National Science Council, R.O.C. for financial support for this work through the projects NSC 91-2212-E-002-030 and NSC 92-2212-E-002-008.
Appendix. Numerical results for the geometry correction factor FI under axial tension
and bending
Table A1. Geometry correction factor FIfor end-free axial tension condition a/b x/ h a/D 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.067 1.054 1.045 1.018 0.960 0.912 0.861 0.825 0.774 0.726 0.682 0.641 0.133 1.082 1.074 1.049 1.012 0.965 0.913 0.859 0.805 0.754 0.706 0.662 0.200 1.166 1.158 1.133 1.094 1.044 0.987 0.926 0.864 0.804 0.748 0.698 0.267 1.308 1.299 1.271 1.228 1.172 1.106 1.033 0.957 0.883 0.815 0.752 0.333 1.520 1.509 1.477 1.427 1.360 1.280 1.190 1.095 1.000 0.912 0.831 0.400 1.827 1.814 1.776 1.715 1.633 1.533 1.418 1.295 1.170 1.051 0.944 0.467 2.274 2.259 2.211 2.135 2.031 1.902 1.752 1.587 1.417 1.253 1.104 0.533 2.948 2.928 2.867 2.768 2.631 2.460 2.257 2.030 1.791 1.556 1.340 0.600 4.013 3.988 3.906 3.771 3.584 3.347 3.061 2.735 2.385 2.033 1.705 0.667 5.830 5.798 5.684 5.490 5.222 4.875 4.450 3.955 3.410 2.849 2.314 0.733 9.306 9.277 9.121 8.834 8.410 7.847 7.140 6.306 5.377 4.399 3.448 0.800 17.478 17.370 17.039 16.488 15.724 14.718 13.456 11.930 10.156 8.194 6.185 0.167 0.067 1.045 1.036 1.010 0.954 0.907 0.857 0.821 0.771 0.724 0.681 0.641 0.133 1.075 1.067 1.043 1.006 0.960 0.909 0.856 0.803 0.752 0.705 0.662 0.200 1.160 1.152 1.127 1.089 1.040 0.984 0.924 0.862 0.803 0.748 0.698 0.267 1.303 1.294 1.267 1.224 1.169 1.104 1.031 0.956 0.883 0.815 0.754 0.333 1.515 1.505 1.473 1.424 1.359 1.280 1.190 1.096 1.002 0.913 0.834 0.400 1.822 1.810 1.772 1.712 1.632 1.534 1.420 1.298 1.173 1.055 0.948 0.467 2.270 2.254 2.208 2.133 2.032 1.905 1.757 1.594 1.424 1.261 1.111 0.533 2.943 2.924 2.864 2.767 2.634 2.466 2.266 2.041 1.803 1.569 1.352 0.600 4.007 3.982 3.903 3.771 3.588 3.356 3.075 2.753 2.406 2.055 1.726 0.667 5.820 5.789 5.679 5.491 5.229 4.890 4.472 3.985 3.445 2.886 2.351 0.733 9.286 9.260 9.108 8.829 8.418 7.867 7.176 6.355 5.437 4.465 3.515 0.800 17.420 17.316 16.995 16.461 15.717 14.738 13.506 12.009 10.260 8.316 6.313 0.333 0.067 1.016 1.008 0.986 0.934 0.891 0.844 0.809 0.762 0.718 0.678 0.641 0.133 1.052 1.044 1.022 0.988 0.945 0.897 0.846 0.796 0.748 0.704 0.663 0.200 1.141 1.133 1.110 1.074 1.029 0.975 0.917 0.857 0.801 0.748 0.701 0.267 1.285 1.277 1.252 1.212 1.160 1.098 1.028 0.955 0.883 0.817 0.758 0.333 1.499 1.489 1.461 1.415 1.353 1.278 1.191 1.098 1.006 0.920 0.842 0.400 1.807 1.795 1.761 1.705 1.630 1.537 1.427 1.307 1.185 1.068 0.962 0.467 2.254 2.241 2.198 2.129 2.034 1.914 1.772 1.613 1.446 1.283 1.135 0.533 2.926 2.909 2.854 2.765 2.641 2.483 2.292 2.075 1.841 1.608 1.391 0.600 3.986 3.964 3.891 3.770 3.602 3.384 3.118 2.809 2.470 2.122 1.792 0.667 5.789 5.761 5.661 5.490 5.249 4.934 4.542 4.077 3.554 3.003 2.467 0.733 9.225 9.205 9.069 8.815 8.441 7.933 7.287 6.509 5.625 4.672 3.724 0.800 17.245 17.151 16.862 16.380 15.700 14.805 13.663 12.259 10.594 8.707 6.725
Table A1. Continued. a/b x/ h a/D 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.5 0.067 0.964 0.958 0.941 0.897 0.862 0.821 0.789 0.748 0.709 0.674 0.642 0.133 1.008 1.002 0.983 0.954 0.917 0.875 0.829 0.784 0.741 0.701 0.666 0.200 1.104 1.097 1.078 1.047 1.007 0.958 0.904 0.849 0.797 0.749 0.705 0.267 1.253 1.246 1.224 1.190 1.144 1.087 1.021 0.952 0.885 0.822 0.767 0.333 1.469 1.461 1.436 1.396 1.342 1.273 1.192 1.104 1.016 0.932 0.857 0.400 1.778 1.768 1.739 1.691 1.625 1.541 1.439 1.325 1.206 1.092 0.988 0.467 2.225 2.214 2.178 2.118 2.036 1.929 1.798 1.648 1.487 1.327 1.179 0.533 2.894 2.880 2.834 2.758 2.652 2.512 2.339 2.135 1.911 1.681 1.464 0.600 3.946 3.928 3.868 3.766 3.623 3.433 3.195 2.909 2.586 2.245 1.914 0.667 5.730 5.709 5.626 5.485 5.282 5.011 4.665 4.241 3.749 3.214 2.678 0.733 9.116 9.105 8.999 8.793 8.482 8.050 7.484 6.785 5.964 5.048 4.104 0.800 16.952 16.877 16.642 16.250 15.691 14.946 13.967 12.736 11.225 9.451 7.510 0.667 0.067 0.877 0.874 0.866 0.838 0.814 0.783 0.758 0.725 0.695 0.667 0.643 0.133 0.934 0.929 0.916 0.897 0.872 0.840 0.803 0.765 0.730 0.698 0.669 0.200 1.041 1.036 1.023 1.001 0.971 0.932 0.886 0.838 0.792 0.750 0.713 0.267 1.196 1.191 1.176 1.151 1.116 1.069 1.012 0.950 0.888 0.832 0.781 0.333 1.416 1.410 1.393 1.363 1.321 1.265 1.195 1.115 1.032 0.954 0.883 0.400 1.727 1.720 1.699 1.664 1.613 1.545 1.458 1.354 1.243 1.133 1.032 0.467 2.173 2.165 2.140 2.096 2.034 1.949 1.838 1.704 1.553 1.398 1.251 0.533 2.838 2.828 2.796 2.741 2.662 2.553 2.410 2.231 2.023 1.801 1.584 0.600 3.878 3.866 3.825 3.753 3.649 3.504 3.311 3.065 2.771 2.445 2.113 0.667 5.631 5.619 5.565 5.469 5.324 5.124 4.852 4.496 4.057 3.553 3.018 0.733 8.937 8.941 8.881 8.750 8.538 8.225 7.788 7.214 6.496 5.647 4.717 0.800 16.515 16.469 16.319 16.067 15.696 15.185 14.473 13.521 12.267 10.694 8.845 0.833 0.067 0.722 0.724 0.732 0.735 0.734 0.716 0.701 0.681 0.665 0.650 0.637 0.133 0.800 0.799 0.799 0.801 0.800 0.788 0.763 0.736 0.711 0.688 0.668 0.200 0.927 0.925 0.922 0.918 0.909 0.891 0.860 0.823 0.785 0.751 0.721 0.267 1.093 1.091 1.086 1.078 1.064 1.039 1.001 0.952 0.898 0.847 0.802 0.333 1.318 1.316 1.310 1.298 1.280 1.249 1.201 1.138 1.064 0.990 0.922 0.400 1.631 1.629 1.621 1.606 1.583 1.545 1.486 1.405 1.306 1.201 1.100 0.467 2.076 2.073 2.064 2.046 2.018 1.971 1.899 1.795 1.662 1.514 1.365 0.533 2.732 2.729 2.719 2.698 2.663 2.606 2.514 2.380 2.202 1.991 1.770 0.600 3.751 3.750 3.738 3.713 3.671 3.598 3.481 3.304 3.059 2.754 2.416 0.667 5.456 5.458 5.445 5.419 5.367 5.278 5.123 4.879 4.526 4.065 3.526 0.733 8.636 8.663 8.671 8.653 8.594 8.467 8.230 7.850 7.293 6.537 5.617 0.800 15.841 15.837 15.815 15.773 15.695 15.542 15.237 14.704 13.835 12.540 10.796 1 0.067 0.545 0.572 0.594 0.612 0.626 0.637 0.643 0.645 0.643 0.637 0.626 0.133 0.656 0.677 0.694 0.706 0.715 0.719 0.720 0.716 0.708 0.696 0.680 0.200 0.787 0.805 0.817 0.824 0.824 0.819 0.807 0.790 0.767 0.738 0.703 0.267 0.986 1.004 1.014 1.017 1.012 1.001 0.982 0.955 0.922 0.880 0.832 0.333 1.218 1.242 1.255 1.258 1.251 1.233 1.205 1.168 1.119 1.061 0.992 0.400 1.507 1.543 1.565 1.571 1.561 1.537 1.497 1.442 1.372 1.286 1.186 0.467 1.931 1.986 2.019 2.031 2.021 1.990 1.936 1.862 1.766 1.648 1.508 0.600 3.550 3.670 3.750 3.788 3.786 3.743 3.659 3.534 3.369 3.163 2.916 0.667 5.109 5.308 5.445 5.519 5.530 5.478 5.363 5.185 4.945 4.641 4.275 0.733 8.082 8.432 8.680 8.827 8.873 8.817 8.660 8.402 8.043 7.582 7.020 0.800 14.608 15.205 15.651 15.945 16.089 16.080 15.920 15.609 15.146 14.532 13.767 0.533 2.579 2.658 2.709 2.730 2.722 2.685 2.618 2.523 2.399 2.245 2.062
Table A2. Geometry correction factor FIfor end-constrained axial tension condition a/b x/ h a/D 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.067 1.042 1.033 0.973 0.934 0.886 0.834 0.782 0.732 0.712 0.668 0.626 0.133 1.070 1.062 1.039 1.004 0.958 0.906 0.851 0.797 0.746 0.698 0.654 0.200 1.144 1.136 1.114 1.078 1.030 0.975 0.914 0.853 0.794 0.739 0.689 0.267 1.265 1.257 1.233 1.194 1.141 1.079 1.010 0.937 0.867 0.800 0.740 0.333 1.439 1.430 1.403 1.359 1.299 1.227 1.144 1.057 0.970 0.887 0.811 0.400 1.677 1.667 1.636 1.585 1.516 1.430 1.330 1.223 1.112 1.006 0.909 0.467 1.998 1.987 1.950 1.891 1.809 1.706 1.585 1.450 1.309 1.171 1.042 0.533 2.428 2.415 2.372 2.302 2.204 2.081 1.932 1.764 1.583 1.399 1.226 0.600 2.999 2.984 2.934 2.851 2.735 2.587 2.406 2.196 1.964 1.721 1.485 0.667 3.742 3.724 3.667 3.571 3.437 3.262 3.046 2.789 2.496 2.179 1.856 0.733 4.665 4.646 4.583 4.477 4.325 4.129 3.882 3.582 3.229 2.826 2.396 0.800 5.723 5.703 5.638 5.526 5.371 5.164 4.902 4.578 4.183 3.712 3.173 0.167 0.067 1.033 1.024 0.966 0.928 0.881 0.830 0.779 0.730 0.710 0.667 0.626 0.133 1.063 1.055 1.033 0.998 0.954 0.902 0.848 0.795 0.744 0.697 0.654 0.200 1.138 1.131 1.108 1.073 1.027 0.972 0.912 0.852 0.793 0.739 0.690 0.267 1.260 1.252 1.228 1.190 1.139 1.077 1.008 0.937 0.867 0.801 0.741 0.333 1.434 1.425 1.399 1.356 1.298 1.226 1.144 1.058 0.971 0.888 0.814 0.400 1.673 1.663 1.633 1.583 1.515 1.430 1.332 1.225 1.116 1.010 0.913 0.467 1.994 1.983 1.948 1.890 1.810 1.708 1.588 1.455 1.315 1.177 1.048 0.533 2.424 2.411 2.370 2.301 2.206 2.085 1.939 1.772 1.592 1.409 1.236 0.600 2.995 2.980 2.932 2.851 2.738 2.593 2.416 2.209 1.978 1.737 1.501 0.667 3.738 3.720 3.665 3.572 3.442 3.271 3.060 2.807 2.518 2.203 1.881 0.733 4.659 4.641 4.580 4.477 4.331 4.140 3.900 3.606 3.258 2.861 2.433 0.800 5.716 5.698 5.635 5.527 5.376 5.176 4.923 4.607 4.221 3.759 3.226 0.333 0.067 1.004 0.995 0.942 0.908 0.866 0.819 0.770 0.723 0.704 0.664 0.627 0.133 1.041 1.034 1.014 0.982 0.940 0.891 0.839 0.788 0.740 0.696 0.655 0.200 1.120 1.113 1.092 1.060 1.016 0.963 0.905 0.847 0.791 0.739 0.692 0.267 1.243 1.237 1.215 1.179 1.131 1.071 1.005 0.935 0.867 0.803 0.745 0.333 1.419 1.412 1.387 1.347 1.292 1.224 1.145 1.060 0.975 0.894 0.821 0.400 1.659 1.651 1.623 1.577 1.513 1.432 1.338 1.233 1.125 1.021 0.925 0.467 1.982 1.971 1.939 1.885 1.811 1.715 1.600 1.471 1.333 1.196 1.068 0.533 2.412 2.400 2.363 2.300 2.211 2.097 1.958 1.798 1.622 1.441 1.268 0.600 2.983 2.969 2.925 2.852 2.748 2.613 2.446 2.247 2.024 1.786 1.551 0.667 3.723 3.708 3.658 3.574 3.455 3.298 3.101 2.863 2.585 2.278 1.958 0.733 4.642 4.626 4.571 4.479 4.348 4.174 3.954 3.680 3.352 2.969 2.550 0.800 5.696 5.679 5.624 5.529 5.393 5.215 4.986 4.698 4.340 3.905 3.394 0.5 0.067 0.949 0.943 0.898 0.872 0.838 0.796 0.752 0.709 0.695 0.659 0.627 0.133 1.000 0.994 0.978 0.951 0.915 0.871 0.824 0.777 0.733 0.694 0.658 0.200 1.085 1.079 1.062 1.035 0.996 0.948 0.894 0.839 0.787 0.740 0.697 0.267 1.213 1.207 1.189 1.158 1.115 1.061 0.999 0.933 0.868 0.808 0.754 0.333 1.392 1.385 1.365 1.330 1.282 1.220 1.146 1.065 0.983 0.905 0.835
Table A2. Continued. a/b x/ h a/D 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.400 1.634 1.627 1.603 1.563 1.508 1.435 1.348 1.248 1.144 1.042 0.949 0.467 1.958 1.949 1.922 1.876 1.812 1.727 1.621 1.499 1.366 1.232 1.106 0.533 2.389 2.379 2.348 2.295 2.219 2.119 1.994 1.844 1.676 1.499 1.328 0.600 2.958 2.948 2.912 2.851 2.764 2.647 2.499 2.318 2.107 1.877 1.643 0.667 3.696 3.684 3.644 3.576 3.477 3.345 3.175 2.962 2.706 2.413 2.099 0.733 4.609 4.597 4.554 4.481 4.375 4.233 4.048 3.812 3.519 3.164 2.761 0.800 5.659 5.646 5.604 5.530 5.423 5.283 5.097 4.859 4.554 4.167 3.694 0.667 0.067 0.856 0.853 0.818 0.807 0.786 0.755 0.720 0.686 0.680 0.653 0.628 0.133 0.930 0.927 0.917 0.900 0.873 0.838 0.798 0.759 0.722 0.689 0.660 0.200 1.025 1.022 1.010 0.991 0.962 0.923 0.877 0.828 0.783 0.741 0.704 0.267 1.159 1.156 1.143 1.121 1.088 1.044 0.990 0.931 0.872 0.817 0.769 0.333 1.343 1.338 1.324 1.299 1.262 1.212 1.148 1.075 0.999 0.926 0.860 0.400 1.588 1.583 1.567 1.538 1.496 1.438 1.364 1.274 1.177 1.079 0.989 0.467 1.914 1.909 1.890 1.858 1.809 1.743 1.655 1.547 1.424 1.295 1.171 0.533 2.347 2.341 2.319 2.282 2.228 2.151 2.050 1.920 1.767 1.599 1.428 0.600 2.915 2.909 2.885 2.844 2.784 2.698 2.582 2.431 2.244 2.029 1.798 0.667 3.650 3.643 3.618 3.575 3.508 3.417 3.291 3.122 2.904 2.639 2.336 0.733 4.557 4.551 4.526 4.482 4.417 4.325 4.198 4.025 3.792 3.488 3.116 0.800 5.607 5.601 5.578 5.538 5.477 5.396 5.281 5.125 4.906 4.606 4.203 0.833 0.067 0.694 0.696 0.665 0.685 0.692 0.682 0.665 0.646 0.648 0.634 0.620 0.133 0.796 0.796 0.799 0.801 0.799 0.785 0.759 0.731 0.705 0.682 0.662 0.200 0.914 0.914 0.913 0.910 0.902 0.884 0.853 0.816 0.778 0.743 0.713 0.267 1.062 1.062 1.059 1.053 1.041 1.018 0.981 0.933 0.881 0.831 0.787 0.333 1.254 1.253 1.249 1.241 1.225 1.198 1.154 1.096 1.027 0.958 0.895 0.400 1.506 1.504 1.499 1.489 1.471 1.439 1.389 1.318 1.231 1.138 1.047 0.467 1.836 1.834 1.828 1.817 1.797 1.762 1.705 1.622 1.514 1.390 1.264 0.533 2.270 2.269 2.263 2.251 2.230 2.192 2.130 2.036 1.905 1.746 1.573 0.600 2.838 2.838 2.833 2.823 2.803 2.766 2.701 2.598 2.447 2.249 2.018 0.667 3.570 3.571 3.568 3.562 3.545 3.515 3.455 3.352 3.190 2.959 2.664 0.733 4.475 4.477 4.478 4.477 4.472 4.454 4.411 4.326 4.175 3.932 3.588 0.800 5.526 5.530 5.535 5.541 5.546 5.549 5.533 5.486 5.378 5.175 4.837 1 0.067 0.506 0.528 0.548 0.565 0.580 0.593 0.603 0.611 0.617 0.620 0.621 0.133 0.653 0.675 0.691 0.703 0.711 0.714 0.712 0.705 0.694 0.679 0.659 0.200 0.794 0.813 0.826 0.833 0.834 0.829 0.819 0.803 0.781 0.753 0.719 0.267 0.954 0.974 0.986 0.991 0.987 0.976 0.958 0.931 0.897 0.855 0.806 0.333 1.151 1.176 1.191 1.195 1.190 1.173 1.147 1.110 1.062 1.005 0.936 0.400 1.400 1.435 1.455 1.462 1.456 1.435 1.401 1.354 1.292 1.217 1.128 0.467 1.723 1.770 1.800 1.812 1.806 1.783 1.742 1.684 1.608 1.514 1.402 0.533 2.145 2.209 2.251 2.272 2.270 2.246 2.200 2.132 2.041 1.929 1.795 0.600 2.695 2.781 2.840 2.873 2.879 2.859 2.813 2.740 2.641 2.515 2.363 0.667 3.402 3.511 3.591 3.642 3.663 3.656 3.620 3.554 3.460 3.337 3.184 0.733 4.287 4.412 4.511 4.583 4.627 4.645 4.636 4.601 4.538 4.449 4.333 0.800 5.348 5.467 5.570 5.657 5.729 5.784 5.824 5.849 5.857 5.849 5.826
Table A3. Geometry correction factor FIfor bending a/b x/ h a/D 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.067 0.963 0.954 0.929 0.878 0.834 0.786 0.739 0.692 0.649 0.609 0.576 0.133 0.897 0.890 0.870 0.840 0.801 0.757 0.710 0.662 0.618 0.576 0.537 0.200 0.872 0.866 0.848 0.820 0.783 0.739 0.690 0.640 0.592 0.547 0.506 0.267 0.879 0.873 0.856 0.828 0.790 0.743 0.692 0.637 0.583 0.532 0.486 0.333 0.917 0.911 0.893 0.863 0.823 0.773 0.716 0.654 0.592 0.532 0.478 0.400 0.991 0.984 0.964 0.932 0.888 0.832 0.767 0.695 0.621 0.549 0.482 0.467 1.112 1.104 1.082 1.045 0.994 0.930 0.854 0.768 0.678 0.588 0.504 0.533 1.302 1.294 1.268 1.224 1.164 1.087 0.995 0.889 0.775 0.659 0.550 0.600 1.609 1.599 1.566 1.512 1.437 1.341 1.224 1.088 0.938 0.783 0.634 0.667 2.126 2.113 2.070 1.998 1.899 1.771 1.614 1.429 1.222 1.002 0.787 0.733 3.082 3.063 3.002 2.899 2.755 2.570 2.342 2.069 1.758 1.421 1.083 0.800 5.140 5.110 5.011 4.841 4.606 4.302 3.923 3.466 2.934 2.344 1.737 0.167 0.067 0.953 0.945 0.922 0.872 0.829 0.783 0.736 0.691 0.648 0.609 0.577 0.133 0.890 0.883 0.865 0.836 0.799 0.755 0.709 0.662 0.618 0.577 0.539 0.200 0.866 0.861 0.844 0.817 0.781 0.738 0.690 0.641 0.594 0.550 0.509 0.267 0.874 0.869 0.852 0.825 0.789 0.744 0.693 0.639 0.586 0.536 0.490 0.333 0.913 0.907 0.890 0.862 0.823 0.774 0.718 0.657 0.596 0.537 0.483 0.400 0.987 0.981 0.962 0.931 0.888 0.834 0.771 0.700 0.627 0.555 0.489 0.467 1.108 1.101 1.080 1.044 0.995 0.933 0.859 0.775 0.685 0.596 0.513 0.533 1.299 1.291 1.266 1.223 1.165 1.091 1.001 0.898 0.785 0.670 0.561 0.600 1.605 1.595 1.564 1.511 1.439 1.346 1.232 1.099 0.951 0.798 0.650 0.667 2.121 2.108 2.067 1.998 1.901 1.778 1.625 1.444 1.240 1.023 0.808 0.733 3.074 3.056 2.997 2.897 2.758 2.579 2.356 2.090 1.784 1.450 1.113 0.800 5.124 5.095 4.999 4.835 4.607 4.311 3.943 3.496 2.973 2.391 1.787 0.333 0.067 0.925 0.918 0.899 0.854 0.816 0.772 0.728 0.685 0.645 0.608 0.580 0.133 0.868 0.863 0.847 0.823 0.790 0.749 0.705 0.661 0.619 0.580 0.545 0.200 0.849 0.844 0.830 0.807 0.776 0.736 0.691 0.645 0.599 0.556 0.518 0.267 0.859 0.855 0.841 0.818 0.786 0.745 0.698 0.646 0.595 0.546 0.502 0.333 0.900 0.895 0.880 0.856 0.822 0.778 0.726 0.668 0.609 0.551 0.499 0.400 0.975 0.970 0.954 0.927 0.889 0.841 0.782 0.715 0.644 0.575 0.510 0.467 1.097 1.091 1.073 1.042 0.999 0.942 0.874 0.795 0.709 0.622 0.540 0.533 1.288 1.281 1.259 1.222 1.170 1.103 1.021 0.923 0.816 0.704 0.596 0.600 1.593 1.584 1.556 1.510 1.445 1.361 1.257 1.132 0.992 0.843 0.696 0.667 2.105 2.093 2.057 1.996 1.909 1.797 1.657 1.489 1.295 1.085 0.873 0.733 3.048 3.032 2.979 2.890 2.766 2.603 2.400 2.152 1.862 1.540 1.209 0.800 5.072 5.046 4.961 4.814 4.608 4.341 4.003 3.589 3.096 2.536 1.945 0.5 0.067 0.874 0.870 0.856 0.821 0.791 0.754 0.714 0.675 0.639 0.607 0.584 0.133 0.828 0.825 0.816 0.799 0.773 0.739 0.700 0.660 0.622 0.587 0.555 0.200 0.816 0.813 0.804 0.789 0.765 0.732 0.692 0.650 0.608 0.569 0.533 0.267 0.831 0.829 0.819 0.803 0.779 0.746 0.705 0.658 0.611 0.565 0.524 0.333 0.875 0.872 0.862 0.845 0.819 0.784 0.739 0.687 0.632 0.578 0.527
Table A3. Continued. a/b x/ h a/D 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.400 0.953 0.950 0.938 0.918 0.890 0.851 0.801 0.742 0.676 0.610 0.547 0.467 1.076 1.072 1.058 1.035 1.002 0.957 0.900 0.830 0.751 0.668 0.588 0.533 1.267 1.262 1.246 1.217 1.177 1.123 1.054 0.969 0.870 0.765 0.660 0.600 1.569 1.563 1.542 1.506 1.456 1.388 1.300 1.192 1.064 0.923 0.780 0.667 2.074 2.066 2.037 1.990 1.921 1.830 1.713 1.567 1.392 1.195 0.990 0.733 3.000 2.987 2.947 2.876 2.776 2.644 2.474 2.260 2.001 1.701 1.380 0.800 4.978 4.958 4.890 4.774 4.607 4.389 4.107 3.751 3.313 2.796 2.226 0.667 0.067 0.790 0.790 0.786 0.770 0.753 0.725 0.693 0.661 0.632 0.606 0.591 0.133 0.763 0.763 0.762 0.757 0.744 0.721 0.690 0.657 0.625 0.596 0.570 0.200 0.762 0.762 0.760 0.756 0.744 0.723 0.693 0.658 0.622 0.588 0.558 0.267 0.785 0.785 0.782 0.777 0.765 0.745 0.715 0.677 0.635 0.595 0.558 0.333 0.835 0.834 0.830 0.823 0.811 0.790 0.758 0.716 0.668 0.619 0.573 0.400 0.916 0.915 0.910 0.901 0.887 0.863 0.829 0.782 0.725 0.665 0.607 0.467 1.041 1.039 1.033 1.022 1.004 0.977 0.937 0.883 0.815 0.741 0.665 0.533 1.231 1.229 1.221 1.206 1.184 1.151 1.103 1.037 0.955 0.860 0.760 0.600 1.529 1.526 1.515 1.495 1.466 1.424 1.363 1.281 1.175 1.050 0.914 0.667 2.022 2.017 2.002 1.975 1.934 1.877 1.796 1.686 1.543 1.369 1.174 0.733 2.919 2.913 2.889 2.847 2.786 2.701 2.584 2.425 2.215 1.954 1.652 0.800 4.822 4.812 4.772 4.701 4.596 4.454 4.259 3.995 3.644 3.199 2.671 0.833 0.067 0.643 0.649 0.663 0.680 0.686 0.671 0.651 0.631 0.613 0.598 0.594 0.133 0.650 0.654 0.666 0.682 0.694 0.690 0.672 0.650 0.627 0.607 0.588 0.200 0.669 0.672 0.681 0.694 0.706 0.707 0.694 0.670 0.642 0.615 0.590 0.267 0.704 0.707 0.714 0.725 0.736 0.738 0.728 0.705 0.673 0.639 0.606 0.333 0.762 0.764 0.771 0.780 0.789 0.792 0.783 0.760 0.724 0.682 0.640 0.400 0.849 0.851 0.856 0.864 0.872 0.874 0.866 0.841 0.800 0.749 0.694 0.467 0.976 0.978 0.983 0.990 0.996 0.998 0.988 0.960 0.913 0.850 0.779 0.533 1.164 1.166 1.171 1.177 1.182 1.182 1.169 1.138 1.081 1.003 0.909 0.600 1.454 1.456 1.460 1.465 1.469 1.467 1.450 1.410 1.340 1.238 1.111 0.667 1.928 1.930 1.933 1.937 1.938 1.933 1.910 1.857 1.764 1.625 1.444 0.733 2.779 2.782 2.784 2.786 2.783 2.771 2.735 2.659 2.525 2.319 2.043 0.800 4.564 4.567 4.569 4.565 4.554 4.528 4.464 4.338 4.116 3.769 3.288 1 0.067 0.486 0.523 0.553 0.578 0.596 0.609 0.616 0.618 0.613 0.603 0.587 0.133 0.510 0.548 0.579 0.604 0.623 0.635 0.641 0.640 0.633 0.619 0.599 0.200 0.557 0.596 0.629 0.654 0.673 0.684 0.689 0.686 0.677 0.660 0.637 0.267 0.600 0.640 0.673 0.699 0.717 0.728 0.732 0.728 0.717 0.699 0.674 0.333 0.654 0.695 0.729 0.755 0.774 0.784 0.788 0.783 0.771 0.751 0.724 0.400 0.742 0.786 0.822 0.850 0.869 0.880 0.882 0.877 0.862 0.840 0.809 0.467 0.877 0.926 0.966 0.996 1.017 1.028 1.029 1.022 1.004 0.978 0.941 0.533 1.062 1.118 1.163 1.197 1.220 1.231 1.232 1.222 1.200 1.168 1.124 0.600 1.326 1.393 1.446 1.485 1.511 1.524 1.524 1.510 1.483 1.443 1.389 0.667 1.755 1.838 1.904 1.953 1.985 2.000 1.998 1.979 1.943 1.890 1.820 0.733 2.544 2.659 2.749 2.816 2.859 2.878 2.873 2.844 2.791 2.714 2.614 0.800 4.138 4.317 4.458 4.560 4.625 4.651 4.639 4.589 4.500 4.374 4.209
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