New Method for Calculating the Preoffsetting Value of the Saddle on Suspension Bridges Considering the Influence of More Parameters

Wang, Shaorui; Zhou, Zhixiang; Wen, Dong; Huang, Yayi

doi:10.1061/(ASCE)BE.1943-5592.0000956

Open access

Technical Notes

Aug 26, 2016

New Method for Calculating the Preoffsetting Value of the Saddle on Suspension Bridges Considering the Influence of More Parameters

Authors: Shaorui Wang [email protected], Zhixiang Zhou, Dong Wen, and Yayi HuangAuthor Affiliations

Publication: Journal of Bridge Engineering

Volume 21, Issue 12

https://doi.org/10.1061/(ASCE)BE.1943-5592.0000956

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Abstract

To seek a proper way to calculate the preoffsetting value of saddle on a suspension bridge, some influences of structural parameters on the preoffsetting are considered, then a new method for the preoffsetting value of saddle is proposed. On the basis of the relationship of the main cable and saddle, equilibrium equations, geometric equations, and physical equations are deduced. Then, the equations are solved by the Newton-Raphson iteration method to derive the Jacobian influence matrix of preoffsetting. For convergence of the numerical calculation, constraint conditions and the range of initial values are given. Through comparison of the measured value in a 1/30-scale model test and the calculated value, the reliability of the new method is confirmed. Finally, parametric analysis of preoffsetting shows that (1) with an increase of the saddle radius, saddle preoffsetting increases in the middle-span direction; (2) with increases of the side/middle-span and rise/span ratios, saddle preoffsetting increases in the side-span direction; (3) the influence of cable self-weight on saddle preoffsetting is negligible; (4) with an increase of unstressed cable length in the side span, saddle preoffsetting increases in the middle-span direction; and (5) with an increase of unstressed cable length in the middle span, saddle preoffsetting increases in the side-span direction.

Introduction

The final state of pylons on a suspension bridge should satisfy the following requirements: (1) no deviation on the top of the pylon; (2) no bending moment on the bottom of the pylon; and (3) the horizontal components of cable are equal on both sides of the pylon. At the free-cable stage, unstressed cable lengths on both sides of the pylon are unequal, resulting in inequality of the horizontal components of the cable. If the saddle stays in the position of final state, the pylon will be subject to a large unbalanced force, which can cause pylon deformation. However, it is not realistic to change the bridge span or reduce the unbalanced force simply depending on pylon deformation. Therefore, the cable force needs to be kept in balance by offsetting or deflecting the saddle, and the value of offsetting or deflecting is called the preoffsetting value of the saddle.

Methods for calculating the preoffsetting value of the saddle are found infrequently in the literature. Gil and Choi (2001) presented the results of mock-up cable-erection tests, which were applied successfully for erecting the main cables of the Yong Jong Grand Bridge, the first spatial self-anchored suspension bridge in the world. Hasegawa et al. (1995) used a horizontal friction plate between the cable and saddle and considered the frictional behaviors under normal and seismic loads, even a violent variation of load resulting from an earthquake. Takena et al. (1992) studied the slip behavior of cable against saddle in suspension bridges. Govindjee (2010) analyzed the stability of the Bay Bridge saddle configuration. Tang (2003) assumed an initial preoffsetting value and then calculated the cable curve and unbalanced force along the sliding surface of the saddle, used slip stiffness to adjust the saddle preoffsetting, and finally achieved the balance of force on both sides of the saddle. Li (2006) assumed an initial preoffsetting value and then calculated the coefficient of unbalanced force, selected different increments for the preoffsetting value, and adjusted the coefficient according to the accuracy requirement, which is the error range of the coefficient of unbalanced force.

In this paper, the authors hold the following views: (1) the tangent point coordinate of the saddle is related to saddle preoffsetting, so the influence of the moving tangent point cannot be ignored; (2) cable shape between the peak of the saddle (as a fixed point) and two tangent points of the saddle is not a catenary, so unstressed cable length should be corrected in these segments, and these cable segments do not participate in cable-shape calculation; (3) on the basis of trial calculation, if the initial preoffsetting value is not considered in either the influence matrix or correction of the saddle preoffsetting, the calculation would be inefficient; and (4) friction between the cable and the saddle should be considered to estimate the correction of unstressed cable length.

According to these views, the authors propose a new method for calculating the preoffsetting value of the saddle on a suspension bridge, which can correct unstressed cable length by considering the influences of various tangent points, the noncatenary segments, and the friction between the cable and the saddle. Furthermore, the initial preoffsetting value is introduced into the influence matrix to consider its influence on the correction of saddle preoffsetting in this method.

Analysis of the Saddle

The following assumptions are made in this analysis of the saddle: (1) the material property of the main cable obeys Hooke’s law; (2) the influence of the main cable diameter is neglected; and (3) the influence of cable elongation on the cable section area is neglected.

As shown in Fig. 1, a Cartesian coordinate system is established on the saddle, with the origin located at the fixed point of the saddle. The parameters in Fig. 1 are shown in the “Notation” section.

Fig. 1. Schematic diagram of saddle calculation at the free-cable stage

In the side span, unstressed cable length between the cable tangent point and saddle fixed point is modified by

S_{L} = S_{1} - arcsin (\frac{X_{0} - X_{1}}{R}) R + \frac{R \sqrt{H_{L}^{2} + V_{L}^{2}}}{E A μ} (1 - e^{- μ arcsin [\frac{X_{0} - X_{1}}{R}]})

(1)

where S₁ is unstressed cable length between the saddle fixed point and cable anchoring point in the side span.

μ

(BSI 1993) is the friction coefficient between the cable and saddle.

The geometric equations of the main cable can be deduced as follows:

l_{L} - Δ X - (X_{0} - X_{1}) = \frac{H_{L} S_{L}}{E A} + \frac{H_{L}}{q} {ln (V_{L} + \sqrt{H_{L}^{2} + V_{L}^{2}}) - ln (V_{L} - q S_{L} + \sqrt{H_{L}^{2} + {(V_{L} - q S_{L})}^{2}})}

(2)

h {}_{L}- (Y_{0} - Y_{1}) = \frac{q S_{L}^{2} - 2 V_{L} S_{L}}{2 E A} - \frac{1}{q} [\sqrt{H_{L}^{2} + V_{L}^{2}} - \sqrt{H_{L}^{2} + {(V_{L} - q S_{L})}^{2}}]

(3)

Similarly, in the middle span, unstressed cable length between the cable tangent point and saddle fixed point is modified by

S_{R} = S_{2} - arcsin (\frac{X_{2} - X_{0}}{R}) R + \frac{R \sqrt{H_{R}^{2} + V_{R}^{2}}}{E A μ} (1 - e^{- μ arcsin [\frac{X_{2} - X_{0}}{R}]})

(4)

where S₂ is unstressed cable length between saddle fixed point and cable anchoring point in middle span.

Therefore, the geometric equations of the main cable can be deduced as follows:

l_{R} + Δ X - (X_{2} - X_{0}) = \frac{H_{R} S_{R}}{E A} + \frac{H_{R}}{q} {ln (V_{R} + \sqrt{H_{R}^{2} + V_{R}^{2}}) - ln (V_{R} - q S_{R} + \sqrt{H_{R}^{2} + {(V_{R} - q S_{R})}^{2}})}

(5)

(Y_{0} - Y_{2}) - h_{R} = \frac{q S_{R}^{2} - 2 V_{R} S_{R}}{2 E A} - \frac{1}{q} [\sqrt{H_{R}^{2} + V_{R}^{2}} - \sqrt{S_{R}^{2} + {(V_{R} - q S_{R})}^{2}}]

(6)

Because the intersections of the cable and saddle are located on the arc surface of the saddle, the geometric equations can be written as

{(X_{1} - X_{3})}^{2} + {(Y_{1} - Y_{3})}^{2} = R^{2}

(7)

{(X_{2} - X_{3})}^{2} + {(Y_{2} - Y_{3})}^{2} = R^{2}

(8)

According to the vector relationship, the equilibrium equations at the intersections of the cable and saddle can be written as (Li et al. 2005; Tang et al. 2001; Sun et al. 2004)

(X_{1} - X_{3}) H_{L} + (Y_{1} - Y_{3}) (V_{L} + q S_{L}) = 0

(9)

(X_{2} - X_{3}) H_{R} + (Y_{2} - Y_{3}) V_{R} = 0

(10)

Remembering the definition of saddle preoffsetting, the equation of horizontal components can be obtained as

H_{L} = H_{R}

(11)

On the basis of the 11 known quantities, S₁, S₂, R, μ, E, A, q, X₀, Y₀, X₃, and Y₃, the 11 unknown quantities, ΔX, H_L, V_L, H_R, V_R, S_L, S_R, X₁, Y₁, X₂, and Y₂, can be solved by Eqs. (1)–(11).

Numerical Iteration Method

Eqs. (1)–(11) can be rewritten as

F_{i} (x_{1}, x_{2}, \dots, x_{10}, x_{11}) = 0 (i = 1, 2, \dots, 11)

(12)

where x_i(i = 1,2,…,11) represents ΔX, H_L, V_L, H_R, V_R, S_L, S_R, X₁, Y₁, X₂, and Y₂, respectively.

Assume F_i is continuous in the neighborhood space of

(x_{1}^{°}, x_{2}^{°}, x_{3}^{°}, x_{4}^{°}, x_{5}^{°}, x_{6}^{°}, x_{7}^{°}, x_{8}^{°}, x_{9}^{°}, x_{10}^{°}, x_{11}^{°})

, then F_i can be developed by Taylor series expansion as

F_{i} ({x °}_{1}, \dots, {x °}_{11}) + \sum_{j = 1}^{11} \frac{\partial F_{i} (x_{1}^{°}, \dots, x_{11}^{°})}{\partial x_{j}} Δ x_{j} + R_{11} = 0

(13)

where R₁₁ = high-order term of the Taylor series; and Δx_j = increment of the independent variable.

Define the Jacobian influence matrix

J_{i j}

as

J_{i j} = \sum_{j = 1}^{11} \frac{\partial F_{i} (x_{1}^{°}, \dots, x_{11}^{°})}{\partial x_{j}}

(14)

Then, Eq. (13) can be deduced as

J_{i j} Δ x_{j} = - F_{i}

(15)

Hence, the iteration equation is expressed as

x^{K} = x^{K - 1} + Δ x

(16)

According to Eq. (14), the Jacobian influence matrix,

J_{i j}

, can be deduced as

J_{11 \times 11} = [\begin{matrix} J_{12} & J_{13} & J_{16} & J_{18} \\ J_{21} & J_{22} & J_{23} & J_{26} & J_{28} \\ J_{32} & J_{33} & J_{36} & J_{39} \\ J_{44} & J_{45} & J_{47} & J_{4, 10} \\ J_{51} & J_{54} & J_{55} & J_{57} & J_{5, 10} \\ J_{64} & J_{65} & J_{67} & J_{6, 11} \\ J_{78} & J_{79} \\ J_{8, 10} & J_{8, 11} \\ J_{92} & J_{93} & J_{96} & J_{98} & J_{99} \\ J_{10, 4} & J_{10, 5} & J_{10, 10} & J_{10, 11} \\ J_{11, 2} & J_{11, 4} \end{matrix}]

Except for the marked matrix elements above, the other elements are 0.

The procedure of the Newton-Raphson iteration method is given as follows: (1) assume the initial values of X_i(

i = 1, 2, \dots, 11

) as

(x_{1}^{°}, x_{2}^{°}, x_{3}^{°}, x_{4}^{°}, x_{5}^{°}, x_{6}^{°}, x_{7}^{°}, x_{8}^{°}, x_{9}^{°}, x_{10}^{°}, x_{11}^{°})

; (2) check whether the equations satisfy the accuracy requirement of

| F_{i} | \leq ε

, where

ε

is a constant (e.g., 0.1) that reflects calculative accuracy; if

| F_{i} | \leq ε

, end the calculation process, but otherwise, continue to Step 3; (3) deduce the Jacobian influence matrix

J_{i j}

, and calculate

Δ X_{j}

by using Eq. (15); and (4) modify the initial values of X_i by using Eq. (16), and then repeat Step 2 with the modified values until all equations satisfy the accuracy requirements.

Initial Values

The calculative convergence of the Newton-Raphson iteration method is influenced by the initial values, which are a vector of 11 dimensions. It is very important to identify the reasonable range of the initial value. Meanwhile, these equations are also the constraint conditions of the Newton-Raphson iteration method, so that X₁, X₂, Y₁, and Y₂ should meet these requirements in the iteration procedure.

Test Verification and Comparison with Conventional Method

A test model of the Taohuayu Yellow River Bridge was created to investigate the force-bearing conditions of the construction stage and mechanical properties of the final state. In this test model, the geometrical scale is 1/30, and the mechanical scale is 1:1 (Fig. 2). The related parameters of the test model are provided in Table 1, and the process of the model test is the same as that by Wang et al. (2014).

Table 1. Parameters of Test Model

Parameter	Value
Side span [l_L (m)]	5.3002
Elevation difference in side span [h_L (m)]	2.5838
Unstressed cable length in side span [S₁ (m)]	5.9249
Area of cable section [A (mm²)]	0.1279
Saddle radius [R (m)]	0.1693
Fixed point (X₀, Y₀)	0, 0
Middle span [l_R (m)]	13.5196
Elevation difference in middle span [h_R (m)]	0
Unstressed cable length in middle span [S₂ (m)]	14.4763
Cable self-weight [q (kN/m)]	10.0399
Young’s modulus of cable [E (kPa)]	1.95 × 10⁸
Circle center [(X₀, Y₀) (m)]	0, −0.1693

As shown in Tables 2 and 3, the saddle preoffsetting obtained by numerical analysis meets the requirement for the horizontal forces of cable to be equal on both sides of the pylon; the horizontal forces are 106.01 kN. The saddle preoffsetting calculated by using the Newton-Raphson iteration method is 2.343 cm, and the saddle preoffsetting measured by using the model test is 2.350 cm. At the free-cable stage, we remove the saddle constraint and allow the saddle to slide freely on the top of the tower, and then the sliding distance of the saddle is preoffsetting, which is 2.350 cm. Then, the differential value between the numerical analysis and model test is calculated as 0.007 cm, and the error ratio is 2.98‰, which indicates that the numerical results are in good agreement with the test results. Therefore, the authors conclude that the new method is reliable.

Table 2. Iteration Procedure of Numerical Analysis with the New Method

Number of iterations	ΔX (m)	Tangent point in side span [(X₁, Y₁) (m)]	Tangent point in middle span [(X₂, Y₂) (m)]
0 (initial value)	0.8	(−0.1, −0.1)	(0.1, −0.1)
1	−1.369 690	(−0.140 680, −0.058 671)	(0.087 089, −0.053 495)
2	−0.516 360	(−0.096 538, −0.017 751)	(0.177 910, −0.060 112)
3	−0.288 620	(−0.097 687, −0.030 433)	(0.115 540, −0.051 059)
8	0.008 360	(−0.103 510, −3.500 220)	(0.097 116, −0.030 667)
13	0.028 862	(−0.103 930, −0.035 650)	(0.095 167, −0.029 273)
18	0.023 410	(−0.103 950, −0.035 662)	(0.095 056, −0.029 197)
24	0.023 430	(−0.103 950, −0.035 663)	(0.095 050, −0.029 193)

The procedures of the conventional method are as follows (Fig. 3): (1) assume the initial value of preoffsetting; (2) calculate the shape and internal force of the main cable, and then calculate the tangent point of the saddle; (3) if the unbalanced force of the main cable is less than a given value, then end the calculation process, or otherwise add a certain increment to the initial value and repeat. As shown in Tables 3 and 4, the saddle preoffsetting calculated by the conventional method is 2.263 cm. The differential value between the numerical analysis and the model test is 0.087 cm, and the error ratio is 3.7%, which is more than 2.98‰. The number of iterations in the new method is 24, which is less than the 37 iterations in the conventional method.

Fig. 3. Contrast of the new and conventional methods

Table 3. Comparison of Test and Numerical Results

Parameter	Value
Horizontal cable force in side span [H_L (kN)]	106.01
Vertical cable force in side span [V_L (kN)]	24.08
Unstressed cable length cable in side span [S_L (m)]	5.812 9
Preoffsetting value by calculation (new/old method) [ΔX_c (cm)]	2.343/2.263
Horizontal cable force in middle span [H_R (kN)]	106.01
Vertical cable force in middle span [V_R (kN)]	71.90
Unstressed cable length in middle span [S_R (m)]	14.375 4
Preoffsetting value by test model [ΔX_t (cm)]	2.35

Table 4. Iteration Procedure of Numerical Analysis with the Conventional Method

Number of iterations	ΔX (m)	Tangent point in side span [(X₁, Y₁) (m)]	Tangent point in middle span [(X₂, Y₂) (m)]
0 (initial value)	0.8	(−0.1, −0.1)	(0.1, −0.1)
1	−2.134 792	(−0.144 425, −0.006 407)	(0.0915 71, −0.059 958)
7	−0.819 034	(−0.102 505, −0.022 062)	(0.122 997, −0.067 239)
13	−0.203 885	(−0.097 588, −0.030 921)	(0.113 488, −0.045 880)
19	0.047 404	(−0.101 918, −0.034 087)	(0.102 596, −0.034 829)
25	0.037 231	(−0.103 385, −0.035 222)	(0.097 743, −0.031 063)
31	0.017 953	(−0.103 794, −0.035 408)	(0.095 946, −0.029 800)
37	0.022 628	(−0.103 929, −0.035 645)	(0.095 209, −0.029 300)

Compared with conventional methods, the new method has two advantages. (1) By considering the influence of friction between the saddle and main cable and the correction for unstressed length of main cable, the new method is more appropriate for the actual situation. (2) By adopting the Newton-Raphson iteration method and establishing the Jacobian influence matrix, the calculation process of the new method is more efficient.

Parametric Analysis of Saddle Preoffsetting

A 1/30 scale test model is adopted to study the influences of structural parameters on the preoffsetting value of the saddle. The saddle radius, rise/span ratio, side/middle-span ratio, self-weight of the main cable, and the unstressed cable length in the side or middle span are considered variable when investigating their influences on the saddle preoffsetting.

Influence of Saddle Radius

On the basis of the saddle size of the actual bridge and the scale of the test model, the saddle radius is selected with a range of approximately 0.2–1.1 m, whereas the other structural parameters remain unchanged. The saddle preoffsetting is assumed to be positive in the side-span direction and negative in the middle-span direction. As shown in Fig. 4, the variation in the saddle radius has a certain influence on the saddle preoffsetting.

Influence of Rise/Span Ratio

The rise/span ratio of a self-anchored suspension bridge is commonly determined with a range of approximately 1/9 to 1/5. Therefore, seven different rise/span ratios, 1/10, 1/9, 1/8, 1/7, 1/6, 1/5, and 1/4, are used to study the variation of saddle preoffsetting, whereas the other structural parameters remain unchanged. As shown in Fig. 5, the rise/span ratio has a certain influence on the preoffsetting value.

Influence of Side/Middle-Span Ratio

In suspension bridge design, the side/middle-span ratio is often considered by topographical limitation or geological condition. When the middle span is limited, it is important to choose a suitable side span for bridge design. The side/middle-span ratio varies within a range of approximately 0.18 to 0.48. As shown in Fig. 6, the side/middle-span ratio has a great influence on saddle preoffsetting.

Influence of Cable Self-Weight

To investigate the influence of cable self-weight on saddle preoffsetting, self-weight is selected with a range of approximately 5–15 kN/m, which is based on actual cable self-weight of the suspension bridge, and the other structural parameters remain unchanged. As shown in Fig. 7, the influence of cable self-weight on saddle preoffsetting is so small that the saddle preoffsetting remains almost unchanged.

Influence of Unstressed Cable Length in Side Span

To investigate the influence of unstressed cable length in the side span on saddle preoffsetting, various unstressed cable lengths in the side span are selected, and the other structural parameters remain unchanged. As shown in Fig. 8, unstressed cable length has a great influence on saddle preoffsetting (in the range of approximately −1.2 to 1.6 m).

Influence of Unstressed Cable Length in Middle Span

To investigate the influence of unstressed cable length in the middle span on saddle preoffsetting, various unstressed cable lengths in the middle span are selected, and the other structural parameters remain unchanged. As shown in Fig. 9, unstressed cable length has a great influence on saddle preoffsetting (in the range of approximately −0.8 to 0.4 m).

Conclusions

From the analysis presented here, the following conclusions can be drawn.

1.

A more accurate and reliable numerical analysis method is proposed to investigate the preoffsetting value of the saddle. In this method, the combined influences (i.e., moving tangent point of saddle, correction of unstressed cable length on noncatenary segments, friction between the cable and saddle, etc.) on preoffsetting are considered. Compared with the conventional methods, the new method is more accurate, and the calculation process is more efficient.

2.

The calculation convergence of the numerical analysis method for saddle preoffsetting is influenced by the initial values. On the basis of the geometric relation of the saddle and simplified calculation method for the cable, the appropriate range of the initial values and constraint conditions are given to ensure convergence. By comparing the numerical results with the test results, a difference of only 0.007 cm is observed, and the error ratio is 2.98‰, which indicates that the numerical results are in good agreement with the test results.

3.

Through parametric analysis of the preoffsetting value of the saddle, the following is found: (1) with an increase in the saddle radius, saddle preoffsetting increases in the middle-span direction; (2) with an increase in the side/middle-span ratio and rise/span ratio, saddle preoffsetting increases in the side-span direction; (3) the influence of cable self-weight on preoffsetting is negligible; (4) with an increase in unstressed cable length in the side span, saddle preoffsetting increases in the middle-span direction; and (5) with an increase in unstressed cable length in the middle span, saddle preoffsetting increases in the side-span direction.

Notation

The following symbols are used in this paper:

A: area of cable section;
E: Young’s modulus of main cable;
H_L: horizontal component of T_L;
H_R: horizontal component of T_R;
h_L: cable sag between cable fixed point and cable anchoring point;
h_R: cable sag between two cable fixed points;
l_R: cable span between two cable fixed points;
q: uniform distribution of cable self-weight;
R: saddle radius;
S_L: unstressed cable length between cable anchoring point and saddle tangent point in left span;
S_R: unstressed cable length between cable tangent point and cable midpoint in right span;
T_L: cable force in left span;
T_R: cable force in right span;
V_L: vertical component of T_L;
V_R: vertical component of T_R;
(X₁,Y₁): coordinates of tangent point of main cable in left span;
(X₂,Y₂): coordinates of tangent point of main cable in right span;
(X₃,Y₃): coordinates of circle center of the saddle; and
ΔX: preoffsetting value of the saddle toward the left side at free-cable stage (e.g., for a three-span suspension bridge, ΔX is the sum of the preoffsetting value of two saddles).

Acknowledgments

This research was supported by the National Basic Research Program of China (Project 2012CB723300) and the National Natural Science Foundation of China (Project 51308571). The authors also thank the Ministry of Transport of China (MOT) for funding this research (Project 2010-353-341-230).

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Information & Authors

Information

Published In

Journal of Bridge Engineering

Volume 21 • Issue 12 • December 2016

Copyright

This work is made available under the terms of the Creative Commons Attribution 4.0 International license, http://creativecommons.org/licenses/by/4.0/.

History

Received: Jul 16, 2015

Accepted: May 9, 2016

Published online: Aug 26, 2016

Published in print: Dec 1, 2016

Discussion open until: Jan 26, 2017

Authors

Affiliations

Shaorui Wang [email protected]

Ph.D. Candidate, State Key Laboratory Breeding Base of Mountain Bridge and Tunnel Engineering, Chongqing 400074, China; and School of Civil Engineering, Chongqing Jiaotong Univ., Chongqing 400074, China (corresponding author). E-mail: [email protected]

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Zhixiang Zhou

Professor, School of Civil Engineering, Chongqing Jiaotong Univ., Chongqing 400074, China.

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Dong Wen

Master Candidate, School of Civil Engineering, Chongqing Jiaotong Univ., Chongqing 400074, China.

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Yayi Huang

Lecturer and Ph.D. Candidate, Editorial Dept. of Applied Mathematics and Mechanics, Chongqing Jiaotong Univ., Chongqing 400074, China.

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New Method for Calculating the Preoffsetting Value of the Saddle on Suspension Bridges Considering the Influence of More Parameters

Abstract

Introduction

Analysis of the Saddle

Numerical Iteration Method

Initial Values

Test Verification and Comparison with Conventional Method