Determining Anchor Span Strand Tensions in the Completed State of a Suspension Bridge: An Analytical Algorithm

Data Availability Statement

Acknowledgments

Notation

A

anchor point of the strand in the anchorage;

a_i

parameter of the catenary equation;

b_i

parameter of the catenary equation;

C₁

center of the first vertical bend arc of the splay saddle;

C₂

center of the horizontal bend arc of the splay saddle;

c_i

parameter of the catenary equation = −H_ai/q;

e_1,i

moment arm of the horizontal component of strand tension at Q with respect to J;

e_2,i

moment arm of the vertical component of strand tension at Q with respect to J;

e₃

moment arm of the horizontal component of main cable tension in the side span at S with respect to J;

e₄

moment arm of the vertical component of main cable tension in the side span at S with respect to J;

e₅

moment arm of the gravity of the splay saddle with respect to J;

G

gravity of the splay saddle;

H

horizontal component of main cable tension in the side span;

H_ai

horizontal component of the tension of the ith strand;

IP

intersection point of the catenary extension line of side span and anchor span;

J

rotation center of the splay saddle;

l_ai

projected length between Points A and Q onto the x-axis of the ith strand;

M_ai

bending moment exerted by the tension in each strand on the splay saddle;

M_s

bending moment exerted by the main cable tension in the side span on the splay saddle;

M_P

bending moment resulting from the gravity of the splay saddle;

P

point of tangency to the horizontal bend of the strands;

Q

point of tangency to the vertical bend of the strands;

q

dead weight of the strand per linear meter;

R₁

distance from Point IP to Point C₁;

R₂

distance from Point IP to Point J;

R₃

distance from the center of gravity of the splay saddle to Point J;

r₁

radius of the first vertical bend arc of the splay saddle;

r₂

radius of the second vertical bend arc of the splay saddle;

r₃

radius of the third vertical bend arc of the splay saddle;

r₄

radius of the fourth vertical bend arc of the splay saddle;

r₅

radius of the horizontal bend arc of the splay saddle;

S

point of tangency of the main cable close to the side span;

T_i

tension of the ith strand in the anchor span;

U

starting point of the horizontal bend of the strands;

X_C1

longitudinal coordinate of Point C₁ in the global coordinate system;

X_J

longitudinal coordinate of Point J in the global coordinate system;

x_Pi

longitudinal coordinate of Point P in the plane coordinate system of the ellipse;

Y_C2

transverse coordinate of Point C₂ in the global coordinate system;

Y_J

transverse coordinate of Point J in the global coordinate system;

y_Pi

transverse coordinate of Point P in the plane coordinate system of the ellipse;

Z_C1

vertical coordinate of C₁ in the global coordinate system;

Z_J

vertical coordinate of J in the global coordinate system;

α

tangential angle of the main cable close to the side span;

β_i

tangential angle to the vertical bend at Q of the ith strand;

${\beta }_i^{\mathrm{\prime }}$

tangential angle to the vertical bend at P of the ith strand;

γ_i

tangential angle to the horizontal bend of the ith strand;

ΔX_a,1

longitudinal distance between the center of the first arc segment and the center of the second arc segment of the splay saddle;

ΔX_a,2

longitudinal distance between the center of the second arc segment and the center of the third arc segment of the splay saddle;

ΔX_a,3

longitudinal distance between the center of the third arc segment and the center of the fourth arc segment of the splay saddle;

ΔX_ai,4

longitudinal distance between the center of the fourth arc segment of the splay saddle and Point P;

$\mathrm{\Delta }{X}_{ai,4}^{\mathrm{\prime }}$

longitudinal distance between the center of the fourth arc segment of the splay saddle and Point Q;

ΔX_AJi

distance in the longitudinal direction between Points A and J of the ith strand;

ΔX_C2J

distance in the longitudinal direction between Points C₂ and J;

ΔX_PJi

distance in the longitudinal direction between Points P and J of the ith strand;

ΔX_QPi

distance in the longitudinal direction between Points Q and P of the ith strand;

ΔY_AJi

distance in the transverse direction between Points A and J of the ith strand;

Δy_i

distance in the transverse direction between Points A and Q of the ith strand;

ΔY_PJi

distance in the transverse direction between Points P and J of the ith strand;

ΔY_QPi

distance in the transverse direction between Points Q and P of the ith strand;

ΔZ_a,1

elevation difference between the center of the first arc segment and the center of the second arc segment of the splay saddle;

ΔZ_a,2

elevation difference between the center of the second arc segment and the center of the third arc segment of the splay saddle;

ΔZ_a,3

elevation difference between the center of the third arc segment and the center of the fourth arc segment of the splay saddle;

ΔZ_ai,4

elevation difference between the center of the fourth arc segment of the splay saddle and Point P;

$\mathrm{\Delta }{Z}_{ai,4}^{\mathrm{\prime }}$

elevation difference between the center of the fourth arc segment of the splay saddle and Point Q;

ΔZ_AJi

elevation difference between Points A and J of the ith strand;

Δz_i

elevation difference between Points A and Q of the ith strand;

ΔZ_PJi

elevation difference between Points P and J of the ith strand;

ΔZ_QPi

elevation difference between Points Q and P of the ith strand;

η

inclination angle of the splay saddle;

θ₀

included angle between the radius of the first arc segment of the splay saddle (close to the side span) and the vertical line;

θ₁

central angle of the first vertical bend arc of the splay saddle;

θ₂

central angle of the second vertical bend arc of the splay saddle;

θ₃

central angle of the third vertical bend arc of the splay saddle;

θ₄

central angle of the fourth vertical bend arc of the splay saddle;

φ₁

included angle between the line connecting Points IP and C₁ and the vertical line;

φ₂

included angle between the line connecting Points IP and J and the vertical line; and

φ₃

included angle between the line connecting the center of gravity and Point J and the vertical line.

References