Method for calculating temperature-dependent mid-span vertical displacement of girder bridge

Abstract

A method for calculating a temperature-dependent mid-span vertical displacement of a girder bridge includes: setting a joint rotation of a main girder at each support as an unknown quantity, and establishing an equation according to a bending moment equilibrium condition at the joint; then introducing a sequence to establish a quantitative relationship between each unknown quantity; substituting the relationship into the equation, to obtain an analytical formula for a rotation at each joint; establishing an analytical formula for a bending moment at each joint through a principle of superposition; and finally, establishing an analytical formula for a mid-span vertical displacement of each span girder through a principle of virtual work. This method provides an analytical formula with exact solutions for prismatic girder bridges which have equal side spans yet have any number of spans.

Claims

1. A method for calculating a temperature-dependent mid-span vertical displacement of a girder bridge, comprising the following steps: (1) setting a joint rotation of a main girder at each support as an unknown quantity, and establishing an equation according to a bending moment equilibrium condition at a joint: when a total number of a plurality of spans of the girder bridge is an odd number: $\begin{array}{cc}\{\begin{array}{cc}4i_0.Math.z_k+2i_0.Math.z_{k+1}+M_T=0& \mathrm{when}k=1\\ 2i_0.Math.z_{k-1}+8i_0.Math.z_k+2i_0.Math.z_{k+1}=0& \mathrm{when}k=2,3,Lu-1\\ 2i_0.Math.z_{k-1}+\left(4i_0+2i_1\right).Math.z_k=0& \mathrm{when}k=u\end{array};& \begin{array}{c}\begin{array}{c}\left(A1\right)\\ \left(A2\right)\end{array}\\ \left(A3\right)\end{array}\end{array}$ when the total number of the plurality of spans of the girder bridge is an even number: $\begin{array}{cc}\{\begin{array}{cc}4i_0.Math.z_k+2i_0.Math.z_{k+1}+M_T=0& \mathrm{when}k=1\\ 2i_0.Math.z_{k-1}+8i_0.Math.z_k+2i_0.Math.z_{k+1}=0& \mathrm{when}k=2,3,Lu-1\\ 2i_0.Math.z_{k-1}+\left(4i_0+4i_1\right).Math.z_k=0& \mathrm{when}k=u\end{array};& \begin{array}{c}\begin{array}{c}\left(B1\right)\\ \left(B2\right)\end{array}\\ \left(B3\right)\end{array}\end{array}$ wherein, z.sub.k, z.sub.k+1, and z.sub.k−1 are unknowns of the joint rotation, a clockwise rotation is set as a positive rotation; k is a subscript of each of the unknowns, k=1, 2, . . . , u; u is a number of the unknowns, u=┌n/2┐, u is a smallest integer not less than n/2; wherein n is the total number of the plurality of spans of the girder bridge; when the total number n of the plurality of spans is the odd number, a u-th span in a middlemost position is designated as a main span and has a length of l.sub.1, and the plurality of spans excluding the u-th span are side spans, wherein each of the side spans has a length of l.sub.0; when the total number n of the plurality of spans is the even number, the u-th span and a (u+1)-th span in the middlemost position are designated as the main span and each has the length of l.sub.1, and the plurality of spans excluding the u-th span and the (u+1)-th span are the side spans, wherein each of the side spans has the length of l.sub.0; i.sub.0 and i.sub.1 are a relative flexural stiffness of the main girder of the side spans and a relative flexural stiffness of the main girder of the main span, respectively, i.sub.0=(E.Math.I)/l.sub.0 and i.sub.1=(E.Math.I)/l.sub.1, wherein E is an elastic modulus of a material of the main girder, and I is an area moment of inertia of a section of the main girder; M.sub.T is a fixed-end bending moment of a single-span girder fixed at both ends, wherein the fixed-end bending moment is caused by a temperature difference between a top surface and a bottom surface of the main girder, a bending moment making a bottom of the main girder being stretched is assumed as a positive bending moment, M.sub.T=(α.Math.ΔT.Math.E.Math.I)/h, wherein α is a linear expansion coefficient of the material of the main girder; ΔT is the temperature difference between the top surface and the bottom surface of the main girder, a temperature of the top surface higher than a temperature of the bottom surface is set as positive; h is a depth of the section; (2) introducing a sequence to establish a quantitative relationship between the unknowns of the joint rotation: wherein the sequence {a.sub.k} has a general term of: $a_k=\frac{\lambda .Math.{\phi }_1^{k-1}+{\phi }_2^{k-1}}{\lambda .Math.{\phi }_1^k+{\phi }_2^k};$ wherein, φ.sub.1=2+√{square root over (3)}, φ.sub.2=2−√{square root over (3)} are constants; λ is also a constant; when the total number of the plurality of spans of the girder bridge is the odd number, λ=λ.sub.1=(√{square root over (3)}+ξ)/(√{square root over (3)}−ξ); when the total number of the plurality of spans of the girder bridge is the even number, λ=λ.sub.2=(√{square root over (3)}+2ξ)/(√{square root over (3)}−2ξ); wherein ξ is a side-to-main span ratio, and ξ=l.sub.0/l.sub.1; k=1, 2, . . . , u−1 represents a subscript of the sequence {a.sub.k}; by using the sequence {a.sub.k}, defining a relationship between z.sub.k and z.sub.k−1 as:
z.sub.k=−a.sub.u−k+1.Math.z.sub.k−1; wherein, the subscript k=2, 3, . . . , u; (3) substituting the relationship in step (2) into the equation in step (1), to obtain an analytical formula for the joint rotation: $z_k={\left(-1\right)}^k\frac{\alpha .Math.\Delta T.Math.l_0}{2\sqrt{3}.Math.h}.Math.\frac{\lambda .Math.{\phi }_1^{u-k}+{\phi }_2^{u-k}}{\lambda .Math.{\phi }_1^{u-1}-{\phi }_2^{u-1}};$ wherein, k=1, 2, . . . , u; for the girder bridge with the odd number of the plurality of spans, λ takes λ.sub.1, and according to a symmetry, z.sub.2u+1−k=−z.sub.k; and for the girder bridge with the even number of the plurality of spans, λ takes λ.sub.2; and according to the symmetry, z.sub.2u+2−k=−z.sub.k, z.sub.u+1=0; (4) using a result obtained from step (3) to obtain an analytical formula for a bending moment at each joint by a principle of superposition, wherein if the bending moment making the bottom of the main girder stretched is assumed as the positive bending moment, when the total number of the plurality of spans of the girder bridge is the odd number, a bending moment M.sub.k at a k-th joint is: $M_k=\left[1+{\left(-1\right)}^k.Math.\frac{{\lambda }_1.Math.{\phi }_1^{u-k}-{\phi }_2^{u-k}}{{\lambda }_1.Math.{\phi }_1^{u-1}-{\phi }_2^{u-1}}\right].Math.\frac{\alpha \mathrm{EI}.Math.\Delta T}{h},$ wherein, k=1, 2, . . . , u−1; a bending moment M.sub.u at a u-th joint is: $M_u=\left[1+{\left(-1\right)}^u\frac{\xi }{\sqrt{3}}.Math.\frac{{\lambda }_1+1}{{\lambda }_1.Math.{\phi }_1^{u-1}-{\phi }_2^{u-1}}\right].Math.\frac{\alpha \mathrm{EI}.Math.\Delta T}{h};$ according to the symmetry, a bending moment at a (2u+1−k)-th joint is equal to the bending moment M.sub.k at the k-th joint, namely
M.sub.2u+1−k=M.sub.k, wherein, k=1,2, . . . ,u; when the total number of the plurality of spans of the girder bridge is the even number, the bending moment M.sub.k at the k-th joint is: $M_k=\left[1+{\left(-1\right)}^k.Math.\frac{{\lambda }_2.Math.{\phi }_1^{u-k}-{\phi }_2^{u-k}}{{\lambda }_2.Math.{\phi }_1^{u-1}-{\phi }_2^{u-1}}\right].Math.\frac{\alpha \mathrm{EI}.Math.\Delta T}{h},$ wherein, k=1, 2, . . . , u; a bending moment M.sub.u+1 at a (u+1)-th joint is: $M_{u+1}=\left[1+{\left(-1\right)}^{u+1}\frac{\xi }{\sqrt{3}}.Math.\frac{{\lambda }_2+1}{{\lambda }_2.Math.{\phi }_1^{u-1}-{\phi }_2^{u-1}}\right].Math.\frac{\alpha \mathrm{EI}.Math.\Delta T}{h};$ according to the symmetry, a bending moment at a (2u+2−k)-th joint is equal to the bending moment M.sub.k at the k-th joint, namely
M.sub.2u+2−k=M.sub.k, wherein, k=1,2, . . . ,u; and (5) using a result obtained from step (4) to obtain an analytical formula for the temperature-dependent mid-span vertical displacement of the main girder of each of the main span and the side spans by a principle of virtual work and to guide layout of measuring points in a structural health monitoring system of the girder bridge, wherein if it is assumed that a mid-span downward movement of the main girder corresponds to a positive displacement, when the total number of the plurality of spans of the girder bridge is the odd number, a temperature-dependent mid-span vertical displacement ΔS.sub.k of the main girder of a k-th span is: ${\Delta }_k={\left(-1\right)}^k\frac{l_0^2.Math.\alpha .Math.\Delta T}{16h}.Math.\frac{\left({\lambda }_1.Math.{\phi }_1^{u-k-1}+{\phi }_2^{u-k}\right)\left(1+\sqrt{3}\right)}{{\lambda }_1.Math.{\phi }_1^{u-1}-{\phi }_2^{u-1}},$ wherein, k=1, 2, . . . , u−1; a temperature-dependent mid-span vertical displacement ΔS.sub.u of the main girder of the u-th span is: ${\Delta }_u={\left(-1\right)}^k\frac{l_0^2.Math.\alpha .Math.\Delta T}{8\sqrt{3}.Math.\xi .Math.h}.Math.\frac{{\lambda }_1+1}{{\lambda }_1.Math.{\phi }_1^{u-1}-{\phi }_2^{u-1}};$ according to the symmetry, a temperature-dependent mid-span vertical displacement ΔS.sub.2u−k of the main girder of a (2u−k)-th span is:
ΔS.sub.2u−k=ΔS.sub.k, wherein, k=1,2, . . . ,u−1; when the total number of the plurality of spans of the girder bridge is the even number, the temperature-dependent mid-span vertical displacement ΔS.sub.k of the main girder of the k-th span is: ${\Delta }_k={\left(-1\right)}^k\frac{l_0^2.Math.\alpha .Math.\Delta T}{16h}.Math.\frac{\left({\lambda }_2.Math.{\phi }_1^{u-k-1}+{\phi }_2^{u-k}\right)\left(1+\sqrt{3}\right)}{{\lambda }_2.Math.{\phi }_1^{u-1}-{\phi }_2^{u-1}},$ wherein, k=1, 2, . . . , u−1; the temperature-dependent mid-span vertical displacement ΔS.sub.u of the main girder of the u-th span is: ${\Delta }_u={\left(-1\right)}^u\frac{l_0^2.Math.\alpha .Math.\Delta T}{16\sqrt{3}.Math.\xi .Math.h}.Math.\frac{{\lambda }_2+1}{{\lambda }_1.Math.{\phi }_1^{u-1}-{\phi }_2^{u-1}};$ according to the symmetry, a temperature-dependent mid-span vertical displacement ΔS.sub.2u+1−k of the main girder of a (2u+1−k)-th span is:
ΔS.sub.2u+1−k=ΔS.sub.k, wherein, k=1,2, . . . ,u.

2. The method according to claim 1, wherein the girder bridge is a straight bridge, and the girder bridge has a constant cross section and made of an identical material; the temperature difference between the top surface and the bottom surface of the main girder is equal everywhere along the girder bridge; if the total number of the plurality of spans is the odd number, a middlemost span is the main span and the plurality of spans excluding the middlemost span are the side spans; if the total number of the plurality of spans is the even number, two middlemost spans are the main span and the plurality of spans excluding the two middlemost spans are the side spans; and the side spans of the girder bridge each have an identical length.

3. The method according to claim 1, wherein when the plurality of spans of the girder bridge each have an identical length, the temperature-dependent mid-span vertical displacement ΔS.sub.k of the main girder of the k-th span in the girder bridge with odd-numbered spans has a unified formula: ${\Delta }_k={\left(-1\right)}^k\frac{l_0^2.Math.\alpha .Math.\Delta T}{16h}.Math.\frac{\left({\phi }_1^{u-k}+{\phi }_2^{u-k}\right)\left(1+\sqrt{3}\right)}{{\phi }_1^u-{\phi }_2^{u-1}},$ wherein, k=1, 2, . . . , u; according to the symmetry, the temperature-dependent mid-span vertical displacement ΔS.sub.2u−k of the main girder of the (2u−k)-th span is:
ΔS.sub.2u−k=ΔS.sub.k, wherein, k=1,2, . . . ,u−1; the temperature-dependent mid-span vertical displacement ΔS.sub.k of the main girder of the k-th span in the girder bridge with even-numbered spans has a unified formula: ${\Delta }_k={\left(-1\right)}^k\frac{l_0^2.Math.\alpha .Math.\Delta T}{16h}.Math.\frac{\left({\phi }_1^{u-k}-{\phi }_2^{u-k+1}\right)\left(1+\sqrt{3}\right)}{{\phi }_1^u-{\phi }_2^u},$ wherein, k=1, 2, . . . , u; according to the symmetry, the temperature-dependent mid-span vertical displacement ΔS.sub.2u+1−k of the main girder of the (2u+1−k)-th span is:
ΔS.sub.2u+1−k=ΔS.sub.k, wherein, k=1,2, . . . ,u.

4. The method according to claim 1, wherein when the plurality of spans of the girder bridge each have an identical length: the temperature-dependent mid-span vertical displacement of the main girder of an outermost side span has a largest change magnitude caused by the temperature difference between the top surface and the bottom surface of the main girder; when the total number n of the plurality of spans increases, the temperature-dependent mid-span vertical displacement of the main girder of the outermost side span approaches a limit value ΔS.sub.lim.sup.max: ${\Delta }_{\lim }^{\max }=-\frac{\left(\sqrt{3}-1\right)l_0^2.Math.\alpha .Math.\Delta T}{16h};$ the temperature-dependent mid-span vertical displacement of the main girder of a middlemost span has a smallest change magnitude caused by the temperature difference between the top surface and the bottom surface of the main girder; and when the total number n of the plurality of spans increases, the temperature-dependent mid-span vertical displacement of the main girder of the middlemost span approaches 0.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

(1) FIG. 1 is an analytical model for a girder bridge with odd-numbered spans according to an example of the disclosure.

(2) FIG. 2 is an analytical model for a girder bridge with even-numbered spans according to an example of the disclosure.

DETAILED DESCRIPTION OF THE EMBODIMENTS

(3) In order to make the to-be-solved technical problems, technical solutions and advantages of the disclosure clearer, the disclosure is described in detail below with reference to the accompanying drawings and specific examples.

(4) The disclosure provides a method for calculating a temperature-dependent mid-span vertical displacement of a girder bridge.

(5) The method includes the following steps:

(6) (1) Set a joint rotation of a main girder at each support as an unknown quantity, and establish an equation according to a bending moment equilibrium condition at the joint:

(7) when the total number of the spans of the girder bridge is odd:

(8) $\{\begin{array}{cc}4i_0.Math.z_k+2i_0.Math.z_{k+1}+M_T=0\mathrm{when}k=1& \left(1-1\right)\\ 2i_0.Math.z_{k-1}+8i_0.Math.z_k+2i_0.Math.z_{k+1}=0\mathrm{when}k=2,3,\mathrm{Lu}-1& \left(1-2\right)\\ 2i_0.Math.z_{k-1}+\left(4i_0+2i_1\right).Math.z_k=0\mathrm{when}k=u& \left(1-3\right)\end{array}$

(9) when the total number of the spans of the girder bridge is even:

(10) $\{\begin{array}{cc}4i_0.Math.z_k+2i_0.Math.z_{k+1}+M_T=0\mathrm{when}k=1& \left(1-1\right)\\ 2i_0.Math.z_{k-1}+8i_0.Math.z_k+2i_0.Math.z_{k+1}=0\mathrm{when}k=2,3,\mathrm{Lu}-1& \left(1-2\right)\\ 2i_0.Math.z_{k-1}+\left(4i_0+4i_1\right).Math.z_k=0\mathrm{when}k=u& \left(1-3\right)\end{array}$

(11) where, z.sub.k, z.sub.k+1, and z.sub.k−1 are unknowns of joint rotations, with clockwise rotation as positive; k is a subscript of the variable, k=1, 2, . . . , u; u is the number of the unknowns, u=┌n/2┐, which is the smallest integer not less than n/2; n is the total number of the spans of the girder bridge.

(12) When the total number n of the spans is an odd number, a u-th span in the middlemost is designated as a main span, with a length of l.sub.1, and the rest spans are side spans, with a length of l.sub.0; when the total number of the spans n is an even number, a u-th span and a (u+1)-th span in the middlemost are designated as main spans, with a length of l.sub.1, and the rest spans are side spans, with a length of l.sub.0.

(13) i.sub.0 and i.sub.1 are relative flexural stiffness of main girders of the side span and the main span respectively, defined as i.sub.0=(E.Math.I)/l.sub.0 and i.sub.1=(E.Math.I)/l.sub.1, where E is an elastic modulus of a main girder material, and I is an area moment of inertia of a main girder section.

(14) M.sub.T is a fixed-end bending moment of a sing-span girder fixed at both ends, caused by a temperature difference between top and bottom surfaces of the main girder, with the bottom of the main girder being stretched as positive, M.sub.T=(α.Math.ΔT.Math.E.Math.I)/h, where α is a linear expansion coefficient of the main girder material; ΔT is the temperature difference between the top and bottom surfaces of the main girder, with a top surface temperature higher than a bottom surface temperature as positive; h is a section height.

(15) (2) Introduce a sequence to establish a quantitative relationship between the unknown joint rotations:

(16) A sequence {a.sub.k} is introduced, which has a general term of:

(17) $a_{\kappa }=\frac{\lambda .Math.{\phi }_1^{k-1}+{\phi }_2^{k-1}}{\lambda .Math.{\phi }_1^k+{\phi }_2^k}$

(18) where, as constants, φ.sub.1=2+√{square root over (3)}, φ.sub.2=2−√{square root over (3)}; λ is also a constant, which is equal to λ=λ.sub.1=(√{square root over (3)}+ξ)/(√{square root over (3)}−ξ) for a girder bridge with an odd number of spans, and is equal to λ=λ.sub.2=(√{square root over (3)}+2ξ)/(√{square root over (3)}−2ξ) for a girder bridge with an even number of spans; ξ is a side-to-main span ratio, that is, ξ=l.sub.0/l.sub.1; k=1, 2, . . . , u−1 represents a subscript of the sequence {a.sub.k}.

(19) With the help of the sequence {a.sub.k}, the relationship between z.sub.k and z.sub.k−1 is expressed as follows:
z.sub.k=a.sub.u−k+1.Math.z.sub.k−1
where, the subscript k=2, 3, . . . , u.

(20) (3) Substitute the relationship between the variables in step (2) into the equation in step (1), to obtain an analytical formula for each joint rotation:

(21) $z_k={\left(-1\right)}^k\frac{\alpha .Math.\Delta T.Math.l_0}{2\sqrt{3}.Math.h}.Math.\frac{\lambda .Math.{\phi }_1^{u-k}+{\phi }_2^{u-k}}{\lambda .Math.{\phi }_1^{u-1}-{\phi }_2^{u-1}}$

(22) where, k=1, 2, . . . , u; for a girder bridge with an odd number of spans, λ takes λ.sub.1, and according to symmetry, z.sub.2u+1−k=−z.sub.k; for a girder bridge with an even number of spans, λ takes λ.sub.2, and according to symmetry, z.sub.2u+2−k=−z.sub.k, z.sub.u+1=0.

(23) (4) Use a result of step (3) to obtain an analytical formula for the bending moment of each joint by a superposition principle.

(24) If a bending moment that makes the girder bottom stretched is assumed as a positive bending moment, when the total number of the spans of the girder bridge is an odd number, a bending moment M.sub.k at a k-th joint is:

(25) 0$M_k=\left[1+{\left(-1\right)}^k.Math.\frac{{\lambda }_1.Math.{\phi }_1^{u-k}-{\phi }_2^{u-k}}{{\lambda }_1.Math.{\phi }_1^{u-1}-{\phi }_2^{u-1}}\right].Math.\frac{\alpha \mathrm{EI}.Math.\Delta T}{h},$
where, k=1, 2, . . . , u−1;

(26) a bending moment M.sub.u at a u-th joint is:

(27) $M_u=\left[1+{\left(-1\right)}^u\frac{\xi }{\sqrt{3}}.Math.\frac{{\lambda }_1+1}{{\lambda }_1.Math.{\phi }_1^{u-1}-{\phi }_2^{u-1}}\right].Math.\frac{\alpha \mathrm{EI}.Math.\Delta T}{h}$

(28) according to symmetry, a bending moment at a (2u+1−k)-th joint is equal to the bending moment at the k-th joint, namely
M.sub.2u+1−k=M.sub.k where, k=1,2, . . . ,u.

(29) When the total number of the spans of the girder bridge is an even number, the bending moment M.sub.k at the k-th joint is:

(30) $M_k=\left[1+{\left(-1\right)}^k.Math.\frac{{\lambda }_2.Math.{\phi }_1^{u-k}-{\phi }_2^{u-k}}{{\lambda }_2.Math.{\phi }_1^{u-1}-{\phi }_2^{u-1}}\right].Math.\frac{\alpha \mathrm{EI}.Math.\Delta T}{h},$
where, k=1, 2, . . . , u;

(31) a bending moment M.sub.u+1 at a (u+1)-th joint is:

(32) $M_{u+1}=\left[1+{\left(-1\right)}^{u+1}\frac{\xi }{\sqrt{3}}.Math.\frac{{\lambda }_2+1}{{\lambda }_2.Math.{\phi }_1^{u-1}-{\phi }_2^{u-1}}\right].Math.\frac{\alpha \mathrm{EI}.Math.\Delta T}{h}$

(33) according to symmetry, the bending moment at a (2u+2−k)-th joint is equal to the bending moment at the k-th joint, namely
M.sub.2u+2−k=M.sub.k, where, k=1,2, . . . ,u.

(34) (5) Use a result of step (4) to obtain an analytical formula for the mid-span vertical displacement of each span girder by a principle of virtual work.

(35) If it is assumed that the mid-span downward movement of the main girder corresponds to a positive displacement, when the total number of the spans of the girder bridge is an odd number, the mid-span vertical displacement ΔS.sub.k of the k-th span girder is:

(36) ${\Delta }_k={\left(-1\right)}^k\frac{l_0^2.Math.\alpha .Math.\Delta T}{16h}.Math.\frac{\left({\lambda }_1.Math.{\phi }_1^{u-k-1}+{\phi }_2^{u-k}\right)\left(1+\sqrt{3}\right)}{{\lambda }_1.Math.{\phi }_1^{u-1}-{\phi }_2^{u-1}},$
where, k=1, 2, . . . , u−1;

(37) the mid-span vertical displacement ΔS.sub.u of the u-th span girder is:

(38) ${\Delta }_u={\left(-1\right)}^u\frac{l_0^2.Math.\alpha .Math.\Delta T}{8\sqrt{3}.Math.\xi .Math.h}.Math.\frac{{\lambda }_1+1}{{\lambda }_1.Math.{\phi }_1^{u-1}-{\phi }_2^{u-1}}$

(39) according to symmetry, the mid-span vertical displacement ΔS.sub.2u−k of the (2u−k)-th span girder is:
ΔS.sub.2u−k=ΔS.sub.k, where, k=1,2, . . . ,u−1.

(40) When the total number of the spans of the girder bridge is an even number, the mid-span vertical displacement ΔS.sub.k of the k-th span girder is:

(41) ${\Delta }_k={\left(-1\right)}^k\frac{l_0^2.Math.\alpha .Math.\Delta T}{16h}.Math.\frac{\left({\lambda }_2.Math.{\phi }_1^{u-k-1}+{\phi }_2^{u-k}\right)\left(1+\sqrt{3}\right)}{{\lambda }_2.Math.{\phi }_1^{u-1}-{\phi }_2^{u-1}},$
where, k=1, 2, . . . , u−1;

(42) the mid-span vertical displacement ΔS.sub.u of the u-th span girder is:

(43) ${\Delta }_u={\left(-1\right)}^u\frac{l_0^2.Math.\alpha .Math.\Delta T}{16\sqrt{3}.Math.\xi .Math.h}.Math.\frac{{\lambda }_2+1}{{\lambda }_2.Math.{\phi }_1^{u-1}-{\phi }_2^{u-1}}$

(44) according to symmetry, the mid-span vertical displacement ΔS.sub.2u+1+k of the (2u+1−k)-th span girder is:
ΔS.sub.2u+1−k=ΔS.sub.k, where, k=1,2, . . . ,u.

(45) In a specific application, in step (1), the joint rotation of the main girder at each support is set as an unknown quantity to establish a bending moment equilibrium equation at the joint, specifically:

(46) In an analysis model for a girder bridge with odd-numbered spans in FIG. 1, the total number of spans is n=2.Math.u−1, where an integer u≥1, u=┌n/2┐ (i.e., the smallest integer not less than n/2). The u-th span is designated as the main span, with a length of l.sub.1, and the rest spans are side spans, with a length of l.sub.0, i.sub.0 and i.sub.1 are the relative flexural stiffness of the main girders of the side span and the main span respectively, i.sub.0=(E.Math.I)/l.sub.0, i.sub.1=(E.Math.I)/l.sub.1. E is an elastic modulus of a main girder material, and I is an area moment of inertia of a main girder section. h is a section depth of the main girder, ΔT is a temperature difference between the top and bottom surfaces of the main girder, and α is a linear expansion coefficient of the main girder material. If a side-to-main span ratio ξ=l.sub.0/l.sub.1 is introduced, i.sub.1=i.sub.0ξ.

(47) The bending moment equilibrium equation of the main girder at each joint can be written according to the slope-deflection equation in structural mechanics. At joint 1, the bending moment equilibrium equation is as follows:
4i.sub.0.Math.z.sub.1+2i.sub.0.Math.z.sub.2+M.sub.T=0  (1)

(48) In the equation, M.sub.T is a fixed-end bending moment of a single-span girder fixed at both ends, caused by the temperature difference between the top and the bottom surfaces. If the bending moment which makes the girder bottom stretched is considered positive, by referring to the ready-made tables in textbooks, the fixed-end bending moment is

(49) $\begin{array}{cc}{M}_T=\frac{\alpha .Math.\Delta T}{h}\mathrm{EI}& \left(2\right)\end{array}$

(50) At joints k=2, 3, . . . , u−1, the bending moment equilibrium equation is as follows:
2i.sub.0.Math.z.sub.k−1+8i.sub.0.Math.z.sub.k+2i.sub.0.Math.z.sub.k+1=0  (3)

(51) At joint u, the bending moment equilibrium equation is as follows:
2i.sub.0.Math.z.sub.u−1+(4i.sub.0+4i.sub.1).Math.z.sub.u+2i.sub.1.Math.z.sub.u+1=0  (4)

(52) Because the structure and temperature change are symmetrical about the centerline of the girder, z.sub.k=−z.sub.2u+1−k, where k=1, 2, . . . , u, so Eq. (4) can be written as:
2i.sub.0.Math.z.sub.u−1+(4i.sub.0+2i.sub.1).Math.z.sub.u=0  (5)

(53) Meanwhile, the bending moment equilibrium equations at joints k=u+1, u+2, . . . , 2u are equivalent to Eqs. (1), (3) and (5). Therefore, the number of independent unknown joint rotations is u.

(54) Similarly, in an analysis model for a girder bridge with even-numbered spans in FIG. 2, the total number of spans is n=2.Math.u, where the integer u≥1. The u-th span and the (u+1)-th span in the middlemost are designated as main spans, with a length of l.sub.1, and the rest spans are side spans, with a length of l.sub.0. According to the slope-deflection equation in structural mechanics, the bending moment equilibrium equation of the main girder at joint 1 is the same as Eq. (1), the bending moment equilibrium equation of the main girder at joints k=2, 3, . . . , u−1 is the same as Eq. (3), and the bending moment equilibrium equation of the main girder at joint u is:
2i.sub.0.Math.z.sub.u−1+(4i.sub.0+4i.sub.1).Math.z.sub.u+2i.sub.1.Math.z.sub.u+1=0  (6)

(55) Because the structure and temperature change are symmetrical about the centerline of the girder, z.sub.k=−z.sub.2u+2−k, where k=1, 2, . . . , u, and z.sub.n+1=0. Thus, Eq. (6) can be written as:
2i.sub.0.Math.z.sub.u−1+(4i.sub.0+4i.sub.1).Math.z.sub.u=0  (7)

(56) Meanwhile, the bending moment equilibrium equation at joint k=u+1 is reduced to an identity, and the bending moment equilibrium equation at joints k=u+2, u+3, . . . , 2u+1 are equivalent to those written for the joints k=1, 2, . . . , u. Therefore, the number of independent unknown joint rotations is still u.

(57) Among the above derivation, when u=1, only Eq. (1) is valid, and the girder bridge has only the main span and no side spans, so it is assumed that l.sub.0=l.sub.1. When u=2, only Eqs. (1) and (5) or Eqs. (1) and (7) are valid. When u≥3, the above equations are all valid.

(58) In step (2), a sequence is introduced to establish a quantitative relationship between the unknown joint rotations, specifically:

(59) For the girder bridge with odd-numbered spans, it is derived from Eq. (5) that

(60) $\begin{array}{cc}{z}_u=-\frac{2i_0}{4i_0+2i_1}{z}_{u-1}=-\frac{1}{2+\xi }{z}_{u-1}& \left(8\right)\end{array}$

(61) By introducing a.sub.1=1/(2+ξ) and substituting z.sub.u=−a.sub.1.Math.z.sub.u−1 into Eq. (3), the relationship between z.sub.k and z.sub.k−1 (k=u−1, u−2, . . . , 2) can be rewritten as:
z.sub.k=−a.sub.u−k+1.Math.z.sub.k−1  (9)

(62) In the equation, the sequence a.sub.k (k=2, 3, . . . , u−1) satisfies:

(63) 0$\begin{array}{cc}{a}_k=\frac{1}{4-a_{k-1}}& \left(10\right)\end{array}$

(64) Subtracting a constant φ from both sides of Eq. (10) yields

(65) $\begin{array}{cc}{a}_k-\phi =\frac{1}{4-a_{k-1}}-\phi & \left(11\right)\end{array}$

(66) Through an identical transformation, the right hand side of Eq. (11) is equivalent to

(67) $\begin{array}{cc}\frac{1}{4-a_{k-1}}-\phi =\frac{1-\phi \left(4-a_{k-1}\right)}{4-a_{k-1}}=\frac{\phi a_{k-1}+1-4\phi }{4-a_{k-1}}=\frac{\phi \left(a_{k-1}-\phi \right)+1-4\phi +{\phi }^2}{4-a_{k-1}}& \left(12\right)\end{array}$

(68) Let 1−4φ+φ.sup.2=0 in the numerator, and solutions to this quadratic equation are φ.sub.1=2+√{square root over (3)} and φ.sub.2=2−√{square root over (3)}. According to Eq. (11), the following two equations are obtained:

(69) $\begin{array}{cc}{a}_k-{\phi }_1=\frac{{\phi }_1\left(a_{k-1}-{\phi }_1\right)}{4-a_{k-1}}& \left(13\right)\end{array}$$\begin{array}{cc}{a}_k-{\phi }_2=\frac{{\phi }_2\left(a_{k-1}-{\phi }_2\right)}{4-a_{k-1}}& \left(14\right)\end{array}$

(70) Dividing Eq. (13) by Eq. (14) leads to:

(71) $\begin{array}{cc}\frac{a_k-{\phi }_1}{a_k-{\phi }_2}=\frac{{\phi }_1}{{\phi }_2}.Math.\frac{a_{k-1}-{\phi }_1}{a_{k-1}-{\phi }_2}& \left(15\right)\end{array}$

(72) Therefore, the sequence {(a.sub.k−φ.sub.1)/(a.sub.k−φ.sub.2)} is a geometric sequence with φ.sub.1/φ.sub.2 as a common ratio and (a.sub.1−φ.sub.1)/(a.sub.1−φ.sub.2) as the first term, thus:

(73) $\begin{array}{cc}\frac{a_k-{\phi }_1}{a_k-{\phi }_2}={\left(\frac{{\phi }_1}{{\phi }_2}\right)}^{k-1}.Math.\frac{a_1-{\phi }_1}{a_1-{\phi }_2}& \left(16\right)\end{array}$

(74) As a.sub.1=1/(2+ξ), the expression of a.sub.k (k=1, 2, . . . , u−1) is obtained from Eq. (16):

(75) $\begin{array}{cc}{a}_k=\frac{\lambda .Math.{\phi }_1^{k-1}+{\phi }_2^{k-1}}{\lambda .Math.{\phi }_1^k+{\phi }_2^k}& \left(17\right)\end{array}$

(76) where the constants λ=λ.sub.1=(√{square root over (3)}+ξ)/(√{square root over (3)}−ξ), φ.sub.1=2+√{square root over (3)}, and φ.sub.2=2−√{square root over (3)}.

(77) For the girder bridge with even-numbered spans, it is derived from Eq. (7) that

(78) $\begin{array}{cc}{z}_u=-\frac{2i_0}{4i_0+4i_1}{z}_{u-1}=-\frac{1}{2+2\xi }{z}_{u-1}& \left(18\right)\end{array}$

(79) By introducing a.sub.1=1/(2+2ξ) and following the same derivation process as that of the girder bridge with odd-numbered spans, Eq. (9) still applies to the relationship between z.sub.k and z.sub.k−1. With the replacement of λ=λ.sub.1 with λ=λ.sub.2=(√{square root over (3)}+2ξ)/(√{square root over (3)}−2ξ), the expression of a.sub.k (k=1, 2, . . . , u−1), namely Eq. (17) is also valid.

(80) In step (3), the relationship between the variables in step (2) is substituted into the equations in step (1), to obtain an analytical formula of each joint rotation. Specifically:

(81) The general term of a.sub.k in Eq. (17) produces:

(82) $\begin{array}{cc}{a}_{u-1}=\frac{\lambda .Math.{\phi }_1^{u-2}+{\phi }_2^{u-2}}{\lambda .Math.{\phi }_1^{u-1}+{\phi }_2^{u-1}}& \left(19\right)\end{array}$

(83) By substituting z.sub.2=−a.sub.u−1.Math.z.sub.1 into Eq. (1), z.sub.1 is solved as:

(84) $\begin{array}{cc}{z}_1=-\frac{M_T}{2\sqrt{3}.Math.i_0}.Math.\frac{\lambda .Math.{\phi }_1^{u-1}+{\phi }_2^{u-1}}{\lambda .Math.{\phi }_1^{u-1}-{\phi }_2^{u-1}}& \left(20\right)\end{array}$

(85) The rest unknowns are further solved according to Eq. (9):

(86) 0$\begin{array}{cc}{z}_k={\left(-1\right)}^k\frac{M_T}{2\sqrt{3}.Math.i_0}.Math.\frac{\lambda .Math.{\phi }_1^{u-k}+{\phi }_2^{u-k}}{\lambda .Math.{\phi }_1^{u-1}-{\phi }_2^{u-1}}& \left(21\right)\end{array}$

(87) Eq. (21) is applicable for k=1, 2, . . . , u. For the girder bridge with odd-numbered spans, λ takes λ.sub.1, and according to symmetry, z.sub.2u+1−k=−z.sub.k. For the girder bridge with even-numbered spans, λ takes λ.sub.2 and according to symmetry, z.sub.2u+2−k=−z.sub.k and z.sub.u+1=0.

(88) In step (4), a result of step (3) is used to obtain an analytical formula for the bending moment at each joint by a superposition principle. Specifically:

(89) The bending moment that makes the girder bottom stretched is assumed as a positive bending moment. For the girder bridge with odd-numbered spans, the bending moment M.sub.k at the k-th joint (k=1, 2, . . . , u−1) is
M.sub.k=4i.sub.0.Math.z.sub.k+2i.sub.0.Math.z.sub.k+1+M.sub.T  (22)

(90) Substituting Eqs. (2) and (21) into Eq. (22) and simplifying lead to

(91) $\begin{array}{cc}{M}_k=\left[1+{\left(-1\right)}^k.Math.\frac{{\lambda }_1.Math.{\phi }_1^{u-k}-{\phi }_2^{u-k}}{{\lambda }_1.Math.{\phi }_1^{u-1}-{\phi }_2^{u-1}}\right].Math.\frac{\alpha \mathrm{EI}.Math.\Delta T}{h}& \left(23\right)\end{array}$

(92) Because of symmetry (z.sub.u+1=−z.sub.u), the bending moment at the u-th joint is:

(93) $\begin{array}{cc}{M}_u=4i_1.Math.z_u+2i_1.Math.z_{u+1}+M_T=2i_1.Math.z_u+M_T=\left[1+{\left(-1\right)}^u\frac{\xi }{\sqrt{3}}.Math.\frac{{\lambda }_1+1}{{\lambda }_1.Math.{\phi }_1^{u-1}-{\phi }_2^{u-1}}\right].Math.\frac{\alpha \mathrm{EI}.Math.\Delta T}{h}& \left(24\right)\end{array}$

(94) Because of symmetry, the bending moment at the (2u+1−k)-th joint (k=1, 2, . . . , u) is:
M.sub.2u+1−k=M.sub.k  (25)

(95) For the girder bridge with even-numbered spans, the bending moment M.sub.k at the k-th joint (k=1, 2, . . . , u) can be calculated by Eq. (22). However, different from the case of the odd-numbered-span bridge, the expression of z.sub.k uses λ.sub.2 instead of λ.sub.1. Thus, for the girder bridge with even-numbered spans, the bending moment M.sub.k at the k-th joint (k=1, 2, . . . , u) is:

(96) $\begin{array}{cc}{M}_k=\left[1+{\left(-1\right)}^k.Math.\frac{{\lambda }_2.Math.{\phi }_1^{u-k}-{\phi }_2^{u-k}}{{\lambda }_2.Math.{\phi }_1^{u-1}-{\phi }_2^{u-1}}\right].Math.\frac{\alpha \mathrm{EI}.Math.\Delta T}{h}& \left(26\right)\end{array}$

(97) Because z.sub.u+1=0, the bending moment M.sub.u+1 at the (u+1)-th joint is:

(98) $\begin{array}{cc}{M}_u=-2i_1.Math.z_u-4i_1.Math.z_{u+1}+M_T=-2i_1.Math.z_u+M_T=\left[1+{\left(-1\right)}^u\frac{\xi }{\sqrt{3}}.Math.\frac{{\lambda }_1+1}{{\lambda }_1.Math.{\phi }_1^{u-1}-{\phi }_2^{u-1}}\right].Math.\frac{\alpha \mathrm{EI}.Math.\Delta T}{h}& \left(27\right)\end{array}$

(99) Because of symmetry, the bending moment at the (2u+2−k)-th joint (k=1, 2, . . . , u) is:
M.sub.2u+2−k=M.sub.k  (28)

(100) In step (5), a result of step (4) is used to obtain an analytical formula for the mid-span vertical displacement of each span girder by a principle of virtual work. Specifically:

(101) The mid-span downward movement of the main girder is assumed to be a positive displacement. When the total number of the spans of the girder bridge is an odd number, according to the principle of virtual work combined with a method of diagram multiplication, the mid-span vertical displacement of the k-th span girder (k=1, 2, . . . , u−1) is:

(102) $\begin{array}{cc}{\Delta }_k=\frac{1}{\mathrm{EI}}.Math.\frac{l_0}{4}.Math.\frac{l_0}{2}.Math.\frac{M_k^L+M_k^R}{2}-\frac{\alpha .Math.\Delta T}{h}\frac{l_0}{2}.Math.\frac{l_0}{4}={\left(-1\right)}^k\frac{l_0^2.Math.\alpha .Math.\Delta T}{16h}.Math.\frac{\left({\lambda }_1.Math.{\phi }_1^{u-k-1}+{\phi }_2^{u-k}\right)\left(1+\sqrt{3}\right)}{{\lambda }_1.Math.{\phi }_1^{u-1}-{\phi }_2^{u-1}}& \left(29\right)\end{array}$

(103) Similarly, the mid-span vertical displacement of the u-th span girder is:

(104) $\begin{array}{cc}{\Delta }_u=\frac{1}{\mathrm{EI}}.Math.\frac{l_1}{4}.Math.\frac{l_1}{2}.Math.\frac{M_u^L+M_u^R}{2}-\frac{\alpha .Math.\Delta T}{h}\frac{l_1}{2}.Math.\frac{l_1}{4}={\left(-1\right)}^u\frac{l_0^2.Math.\alpha .Math.\Delta T}{8\sqrt{3}.Math.\xi .Math.h}.Math.\frac{{\lambda }_1+1}{{\lambda }_1.Math.{\phi }_1^{u-1}-{\phi }_2^{u-1}}& \left(30\right)\end{array}$

(105) According to symmetry, the mid-span vertical displacement of the (2u−k)-th span girder (k=1, 2, . . . , u−1) is:
ΔS.sub.2u−k=ΔS.sub.k  (31)

(106) When the total number of the spans of the girder bridge is an even number, the mid-span vertical displacement of the k-th span girder (k=1, 2, . . . , u−1) is:

(107) $\begin{array}{cc}{\Delta }_k=\frac{1}{\mathrm{EI}}.Math.\frac{l_0}{4}.Math.\frac{l_0}{2}.Math.\frac{M_k^L+M_k^R}{2}-\frac{\alpha .Math.\Delta T}{h}\frac{l_0}{2}.Math.\frac{l_0}{4}={\left(-1\right)}^k\frac{l_0^2.Math.\alpha .Math.\Delta T}{16h}.Math.\frac{\left({\lambda }_2.Math.{\phi }_1^{u-k-1}+{\phi }_2^{u-k}\right)\left(1+\sqrt{3}\right)}{{\lambda }_2.Math.{\phi }_1^{u-1}-{\phi }_2^{u-1}}& \left(32\right)\end{array}$

(108) Similarly, the mid-span vertical displacement of the u-th span girder is:

(109) $\begin{array}{cc}{\Delta }_u=\frac{1}{\mathrm{EI}}.Math.\frac{l_1}{4}.Math.\frac{l_1}{2}.Math.\frac{M_u^L+M_u^R}{2}-\frac{\alpha .Math.\Delta T}{h}\frac{l_1}{2}.Math.\frac{l_1}{4}={\left(-1\right)}^u\frac{l_0^2.Math.\alpha .Math.\Delta T}{16\sqrt{3}.Math.\xi .Math.h}.Math.\frac{{\lambda }_2+1}{{\lambda }_2.Math.{\phi }_1^{u-1}-{\phi }_2^{u-1}}& \left(33\right)\end{array}$

(110) According to symmetry, the mid-span vertical displacement of the (2u+1−k)-th span girder (k=1, 2, . . . , u) is:
ΔS.sub.2u+1−k=ΔS.sub.k  (34)

(111) When the length of each span of the girder bridge is identical, ξ=1. For the girder bridge with odd-numbered spans, λ.sub.1=(√{square root over (3)}+1)/(√{square root over (3)}−1), and Eqs. (29) and (30) for calculating the mid-span vertical displacement of each span girder can be unified as:

(112) $\begin{array}{cc}{\Delta }_k={\left(-1\right)}^k\frac{l_0^2.Math.\alpha .Math.\Delta T}{16h}.Math.\frac{\left({\phi }_1^{u-k}+{\phi }_2^{u-k}\right)\left(1+\sqrt{3}\right)}{{\phi }_1^u-{\phi }_2^{u-1}}& \left(35\right)\end{array}$

(113) where k=1, 2, . . . , u. According to symmetry, the mid-span vertical displacement of the (2u−k)-th span girder (k=1, 2, . . . , u−1) is calculated by Eq. (31).

(114) When the length of each span of the girder bridge is equal, for the girder bridge with even-numbered spans, λ.sub.2=(√{square root over (3)}+2)/(√{square root over (3)}−2), and Eqs. (32) and (33) for calculating the mid-span vertical displacement of each span girder can be unified as:

(115) 0$\begin{array}{cc}{\Delta }_k={\left(-1\right)}^k\frac{l_0^2.Math.\alpha .Math.\Delta T}{16h}.Math.\frac{\left({\phi }_1^{u-k}+{\phi }_2^{u-k+1}\right)\left(1+\sqrt{3}\right)}{{\phi }_1^u-{\phi }_2^u}& \left(35\right)\end{array}$

(116) where k=1, 2, . . . , u. According to symmetry, the mid-span vertical displacement of the 2u+1−k-th span girder (k=1, 2, . . . , u) is calculated by Eq. (34).

(117) According to Eq. (35), for the odd-numbered-span girder bridge with a given u, the mid-span vertical displacement magnitude |ΔS.sub.k| of each span girder is proportional to φ.sub.1.sup.u−k+φ.sub.2.sup.u−k. As φ.sub.2=1/φ.sub.1, two functions ƒ.sub.1(x)=x+1/x and g.sub.1(x)=φ.sub.1.sup.u−x are introduced to define a composite function, which leads to ƒ.sub.1(g.sub.1(k))=φ.sub.1.sup.u−k+φ.sub.2.sup.u−k. For k=1, 2, . . . , u, g.sub.1(k) decreases with the increase of k and g.sub.1(k)≥1. On the other hand, the derivative ƒ′.sub.1(x)=1−1/x.sup.2 implies that ƒ.sub.1(x) is a monotonically increasing function in the region of x≥1. Therefore, ƒ.sub.1(g.sub.1(k)) decreases monotonically with the increase of k, that is, the outermost span girder (k=1) has the largest magnitude of the mid-span vertical displacement while the middlemost span girder (k=u) has the smallest magnitude.

(118) Similarly, for the even-numbered-span girder bridge with a given u, according to Eq. (36), the mid-span vertical displacement magnitude |ΔS.sub.k| of each span girder is proportional to φ.sub.1.sup.u−k−φ.sub.2.sup.u−k+1. A new function ƒ.sub.2(x)=x−φ.sub.2/x is introduced and composed with g.sub.1(x)=φ.sub.1.sup.u−x, which leads to ƒ.sub.2(g.sub.1(k))=φ.sub.1.sup.u−k−φ.sub.2.sup.u−k+1. For k=1, 2, . . . , u, g.sub.1(k) decreases with the increase of k and g.sub.1(k)≥1. Meanwhile, the derivative ƒ′.sub.2(x)=1+φ.sub.2/x.sup.2>0, so ƒ.sub.2 (g.sub.1(k)) decreases monotonically with the increase of k. Therefore, for the girder bridge with even-numbered spans, the mid-span vertical displacement magnitude of the outermost span girder (k=1) is the largest, while those of the middlemost span girders (k=u and k=u+1) are the smallest.

(119) In summary, for the girder bridges with either odd- or even-numbered spans, the outermost and the middlemost span girders respectively have the largest and smallest magnitudes of the mid-span vertical displacement induced by the temperature difference between the top and bottom surfaces of the girder.

(120) For the main girder in the span of k=1, when u approaches infinity, Eqs. (35) and (36) both converge to a constant ΔS.sub.lim.sup.max:

(121) $\begin{array}{cc}{\Delta }_{\lim }^{\max }=-\frac{\left(\sqrt{3}-1\right)l_0^2.Math.\alpha .Math.\Delta T}{16h}& \left(37\right)\end{array}$

(122) Therefore, the mid-span vertical displacement of the outermost span girder in a girder bridge has a limit value as the number of bridge spans increases, namely ΔS.sub.lim.sup.max:

(123) Similarly, for the main girder in the span of k=u, when u approaches infinity, Eqs. (35) and (36) both converge to 0. Therefore, the mid-span vertical displacement of the middlemost span girder in a girder bridge also has a limit value as the number of bridge spans increases, namely 0.

(124) The disclosure is described in detail below with reference to the specific examples.

Example 1

(125) A 6-span continuous girder bridge has equal spans of 110 m. The prismatic girder is made of steel, and has a depth of 4.5 m. The temperature difference between the top and bottom surfaces of the main girder due to direct sunlight is 20° C. The mid-span vertical displacement of the 1.sup.st and 3.sup.rd spans needs to be determined.

(126) As the number of spans is n=6, the abovementioned girder bridge has an even number of spans, and the number of independent unknown joint rotations is u=3. The main and side spans have identical length of l.sub.0=l.sub.1=110 m; the linear expansion coefficient of the main girder is α=1.2×10.sup.−5° C..sup.−1; the girder depth is h=4.5 m; and the temperature difference between the top and bottom surfaces of the main girder is ΔT=20° C. Substituting the above parameters into Eq. (36) leads to ΔS.sub.1=−0.0295 m for k=1, and ΔS.sub.3=−0.0016 m for k=3. It can be seen that the mid-span vertical displacement magnitude of the 1st span girder is greater than that of the 3.sup.rd span girder, and the negative sign before the displacement value means that the mid-span deck moves upward.

Example 2

(127) A girder bridge has n spans, each span being 110 m in length. The prismatic girder is made of steel, and has a depth of 4.5 m. The temperature difference between the top and bottom surfaces of the main girder is 20° C. For n=1, 2, . . . , 10, the mid-span vertical displacement of the outermost span girder needs to be calculated.

(128) It can be seen from the above that l.sub.0=l.sub.1=110 m, α=1.2×10.sup.−5° C..sup.−1, h=4.5 m, ΔT=20° C., and k=1. According to the parity of the span number n, these parameters are substituted into Eqs. (35) or (36) with u=┌n/2┐ (i.e., the smallest integer not less than n/2). As a result, the mid-span vertical displacement of the outermost span girder corresponding to different total span numbers is obtained, that is, −0.0807 m, −0.0202 m, −0.0323 m, −0.0288 m, −0.0297 m, −0.0295 m, −0.0295 m, −0.0295 m, −0.0295 m, and −0.0295 m, which are listed in the order of increasing n. From n=6, the mid-span vertical displacement of the outermost span girder is close to −0.0295 m, which is the limit value calculated by Eq. (37). This example clearly indicates that when the continuous girder bridge has a large number of spans, the girder mid-span vertical displacement caused by the temperature difference between the top and the bottom surfaces of the main girder is small, and is thus difficult to be accurately measured by the commonly used equipment.

(129) The above described are preferred implementations of the disclosure. It should be noted that a person of ordinary skill in the art may further make several improvements and modifications without departing from the principle of the disclosure, but such improvements and modifications should also be deemed as falling within the protection scope of the disclosure.