CN105040569A - Simply supported steel box girder prestressing cable shape optimizing method - Google Patents

Abstract

The invention discloses a simply supported steel box girder prestressing cable shape optimizing method. By introducing a thin-walled box girder theory, the method is more accurate in mechanical analysis; compared with a traditional prestressing cable shape, the dual effects of obtaining pre-camber and exerting prestress can be well achieved. Due to the arrangement of optimized prestressing cables, a box girder is in a better mechanical state, and therefore bad damage such as girder box cracks and local instability can be easily avoided, and the durability of the structure is improved. Meanwhile, the method is clear in mechanical concept, simple in calculation and wide in application prospect and practical value. By means of the method, the mechanical performance of the box girder can be optimized, and the steel material consumption can be further saved.

Description

The method for optimizing that a kind of simply supported steel box girder presstressed reinforcing steel is linear

Technical field

The present invention relates to the method for optimizing that a kind of simply supported steel box girder presstressed reinforcing steel is linear.

Background technology

Prestressing technique is widely used in modern structure engineering, particularly bridge engineering, the combining structure that steel box girder with prestressed tendons mainly arranges external curved prestressing tendon and formed in steel case, for steel box-girder, the introducing of presstressed reinforcing steel greatly improves the mechanical characteristic of structure, improve the span ability of bridge, steel about 30% can be saved compared with normal steel structure, as the construction of Shenyang City's Second Ring Road viaduct and the reparation etc. of Harbin City workers, peasants and soldiers bridge all have employed steel box girder with prestressed tendons.The layout of presstressed reinforcing steel is mainly prestressing with bond muscle and external prestressing steels, and prestressing with bond muscle is mainly used in concrete-bridge, and external prestressing steels then great majority is applied to Bridge repair and reinforcement and steel box girder with prestressed tendons.

In steel box-girder, the presstressed reinforcing steel of employing is linear varied, as linear, fold-line-shaped and shaped form etc., and the still shaped form be wherein most widely used.But, according to existing document, the selection linear for presstressed reinforcing steel does not carry out systematic research, and thus the effect of presstressed reinforcing steel fails to give full play to, but present stage is due to the extensive application of steel box girder with prestressed tendons, so the research of this aspect and invention have more necessity.But, owing to affecting by Shear Lag Effect, such structure mechanics analysis is more complicated, compared with concrete box girders, the Liang Bi of steel box girder with prestressed tendons is thinner, so more careful mechanical analysis is not carried out to it, the Local Cracking of this class formation, local buckling may be caused, or more serious bridge defect.

Summary of the invention

The present invention seeks to: provide a kind of mechanical analysis accurate, and the double action that it obtains camber and Shi Hanzhang can be played very well, steel box-girder is made to be in better mechanical state, be conducive to avoiding the bad diseases such as beam body cracking, local buckling, and then the method for optimizing that the simply supported steel box girder presstressed reinforcing steel improving such structure durability is linear.

Technical scheme of the present invention is: the method for optimizing that a kind of simply supported steel box girder presstressed reinforcing steel is linear, comprises the following steps:

Step 1): introduce 3 generalized displacements when analyzing steel box-girder vertical bending, namely w (x) is case beam vertical deflection, the maximum lonitudinal warping displacement difference function that u (x) is case beam wing plate, θ (x) is for box-girder is about the vertical corner of z-axis, and x is girder span direction, then the length travel of box-girder wing plate can be expressed as:

$U(x,z)=\±h\[\mathrm{\θ}\left(x\right)+(1-\frac{z^2}{b^2})u\left(x\right)\]---\left(1\right)$

In formula: b is the half of case beam clear span;

H is the vertical y coordinate in box-girder cross section;

Step 2): (1) web potential energy of deformation

$V_1=\frac{1}{2}{\mathrm{EI}}_w{\∫}_0^l{\left({\mathrm{\θ}}^{\′}\right)}^{2}dx---\left(2\right)$

(2) wing plate potential energy of deformation

$V_2=\frac{1}{2}{\mathrm{EI}}_1{\∫}_0^l\[{\left({\mathrm{\θ}}^{\′}\right)}^2+\frac{4}{3}{\mathrm{\θ}}^{\′}{u}^{\′}+\frac{8}{15}{\left(u^{\′}\right)}^2\]dx+\frac{1}{2}{\∫}_0^l\frac{4{\mathrm{GI}}_1}{3b^2}{u}^{2}dx---\left(3\right)$

(3) hophornbeam pungent Ke's shear strain potential energy

$V_3=\frac{1}{2}{\∫}_0^{l}kGA{(\mathrm{\θ}-w^{\′})}^{2}dx---\left(4\right)$

(4) load potential energy

$V_p=-{\∫}_0^{l}q\left(x\right)w\left(x\right)dx-\[Q\left(x\right)w\left(x\right)+M_1\left(x\right)u\left(x\right)+M\left(x\right)\mathrm{\θ}\left(x\right)\]{|}_0^l---\left(5\right)$

So total potential energy of structural system is:

V＝V ₁+V ₂+V ₃+V _p(6)

In formula: M ₁(x) for case beam wing plate Girder with Shear Lag Effect produce about z-axis moment of flexure; M (x) for beam section end produce vertical corner θ (x) time about z-axis moment of flexure; Q (x), q (x) are distributed force vertical on beam section end vertical shear and case beam; E, G are Young's modulus of elasticity and the coefficient of rigidity of material; K is cross section shape coefficient; I _w, I ₁for box girder web and wing plate are about the moment of inertia of z-axis; A is box section area.

Step 3): according to variation principle obtaining its differential equation is:

${\mathrm{EI\θ}}^{\′\′}+\frac{2}{3}{\mathrm{EI}}_1{u}^{\′\′}-kGA(\mathrm{\θ}-w^{\′})=0---\left(7\right)$

$\frac{2}{3}{\mathrm{EI}}_1{\mathrm{\θ}}^{\′\′}+\frac{8}{15}{\mathrm{EI}}_1{u}^{\′\′}-\frac{4{\mathrm{GI}}_1}{3b^2}u=0---\left(8\right)$

kGA(θ'-w”)-q(x)＝0(9)

Its fringe conditions is:

$\[{\mathrm{EI\θ}}^{\′}+\frac{2}{3}{\mathrm{EI}}_1{u}^{\′}-M\]{|}_0^l\mathrm{\δ}\mathrm{\θ}=0---\left(10\right)$

$\[\frac{2}{3}{\mathrm{EI}}_1{\mathrm{\θ}}^{\′}+\frac{8}{15}{\mathrm{EI}}_1{u}^{\′}-M_1\]{|}_0^l\mathrm{\δ}u=0---\left(11\right)$

$\[kGA(\mathrm{\θ}-w^{\′})+Q\left(x\right)\]{|}_0^l\mathrm{\δ}w=0---\left(12\right)$

Step 4): by equation (7), between (8) and (9) arrangement conversion, trying to achieve new Solutions of Ordinary Differential Equations is:

$u^{\left(3\right)}+\frac{2G}{{\mathrm{Emb}}^2}u-\frac{q}{E\mathrm{Im}}=0---\left(13\right)$

Wherein: i=I _w+ I ₁, and u ⁽³⁾for 3 differentiate formulas of u;

Then its characteristic equation solution is:

γ _1,2＝±(α ₁+β ₁i)；γ ₃＝0。

Finally, the solution of u (x), θ (x) and w (x) is respectively:

$u\left(x\right)=c_{1}sh({\mathrm{\α}}_1+{\mathrm{\β}}_{1}i)x+c_{2}ch({\mathrm{\α}}_1+{\mathrm{\β}}_{1}i)x+c_3+\frac{b^2}{2GI}qx---\left(14\right)$

$\mathrm{\θ}\left(x\right)=c_1{B}_{1}sh({\mathrm{\α}}_1+{\mathrm{\β}}_{1}i)x+c_2{B}_{1}ch({\mathrm{\α}}_1+{\mathrm{\β}}_{1}i)x+c_3\frac{m_2}{2}{x}^2+c_{4}x+c_5+\frac{q}{6EI}{x}^3---\left(15\right)$

$\begin{array}{c}w\left(x\right)=c_1-\frac{B_1}{{\mathrm{\α}}_1+{\mathrm{\β}}_{1}i}ch({\mathrm{\α}}_1+{\mathrm{\β}}_{1}i)x+c_2\frac{B_1}{{\mathrm{\α}}_1+{\mathrm{\β}}_{1}i}sh({\mathrm{\α}}_1+{\mathrm{\β}}_{1}i)x+c_3(\frac{m_2}{6}{x}^3\\ -\frac{m_{2}EI}{kGA}x)+\frac{c_4}{2}{x}^2+c_{5}x+c_6+\frac{q}{24EI}{x}^4-\frac{q}{2kGA}{x}^2\end{array}---\left(16\right)$

Wherein: $m_1=-\frac{4}{5};m_2=\frac{2G}{{\mathrm{Eb}}^2};B_1=\frac{m_2+m_1{({\mathrm{\α}}_1+{\mathrm{\β}}_{1}i)}^2}{{({\mathrm{\α}}_1+{\mathrm{\β}}_{1}i)}^2};$ And c ₁; c ₂; c ₃; c ₄; c ₅; c ₆for constant coefficient;

Step 5) suppose that the curvilinear equation of presstressed reinforcing steel is: wherein l is steel box-girder span, and f is the sag of presstressed reinforcing steel span centre, and h is the distance that presstressed reinforcing steel departs from neutral axis, and q is the uniform vertical load of pre-applied force equivalence;

Step 6) according to specific border condition, by equation w (x), θ (x), u (x) or its derivative formula substitute into fringe conditions, first calculate undetermined constant c ₁; c ₂; c ₃; c ₄; c ₅; c ₆, then calculate w (x), the size of steel box-girder camber can be obtained, the size of lower wing plate prestressing can be calculated, that is: simultaneously after considering, filter out simply supported steel box girder optimum prestress muscle (rope) and arrange linear.

2. the method for optimizing that simply supported steel box girder presstressed reinforcing steel according to claim 1 is linear, is characterized in that, step 6) in filter out simply supported steel box girder optimum prestress muscle and arrange that linear concrete grammar is as follows: first according to the curvilinear equation of presstressed reinforcing steel h is divided into some deciles as circulation item, then under specific border condition and certain prestressing force condition, forms matrix equation according to fringe conditions, calculate undetermined constant c based on this ₁; c ₂; c ₃; c ₄; c ₅; c ₆, and then constant term is inputted w (x), θ ' (x), u'(x) and camber and wing plate prestress value can be obtained, finally according to camber size, take into account flange plate stress value and filter out the linear equation of presstressed reinforcing steel.

Advantage of the present invention is:

1. the present invention introduces Thin-walled Box Girder theory, its mechanical analysis is more accurate, compare with traditional presstressed reinforcing steel is linear, the double utility of its acquisition camber of fine performance and Shi Hanzhang, presstressed reinforcing steel after optimization is arranged and is made steel box-girder be in better mechanical state, be conducive to avoiding the bad diseases such as beam body cracking, local buckling, and then improve the durability of this class formation;

2. mechanical concept of the present invention clear, calculate simple, there is broad application prospect and practical value, not only can optimize the mechanical property of steel box-girder, and can saving steel consumption further.Meanwhile, the maintenance and reinforcement for concrete box girder has directive significance (adopting external prestressing steels (rope)).

Accompanying drawing explanation

Below in conjunction with drawings and Examples, the invention will be further described:

Fig. 1 is the linear schematic diagram of steel box-girder presstressed reinforcing steel of the present invention;

Fig. 2 is steel box-girder schematic cross-section of the present invention (O is the case beam centre of form);

Fig. 3 is the equivalent load schematic diagram of presstressed reinforcing steel of the present invention;

Fig. 4 is steel box-girder cross section of the present invention cloth muscle schematic diagram;

Fig. 5 is steel box-girder span centre cross section cloth muscle schematic diagram in embodiment 1;

Detailed description of the invention

Embodiment 1: as shown in Fig. 2, Fig. 5, geometric parameter and the material parameter of steel box-girder are respectively: t ₁=t ₂=t ₃=3mm; t ₄=4mm; B=0.25m; α b=0.25m; H=0.4m; L=5m; E=2.01 × 10 ⁵mpa; G=7.9 × 10 ⁴mpa, and the pre-applied force of every root presstressed reinforcing steel is 50KN, wherein E ₁; F ₁; G ₁and E ₂; F ₂; G ₂be respectively pressure detection point or calculation level.Finally according to the pre-applied force applied, and different reinforcing plan can obtain different cambers and different flange plate stress distribution forms, and the layout that can obtain optimum prestress muscle is accordingly linear.

The main purpose that presstressed reinforcing steel is arranged makes bridge construction obtain appropriate camber and prestressing, and then make structure have good mechanical property, steel box-girder is thin wall construction, there is unique mechanical characteristic, due to the impact by Shear Lag Effect, on it, the distribution of lower wing plate normal stress is uneven, thus under prestressing force effect, just likely causes upper flange and web junction (as E ₁; G ₁near point) ftracture, and lower wing plate and web junction are (as E ₂; G ₂near point) there is local buckling.

Through circulation tentative calculation, comprehensive every factor, this experiment beam selects h to be 3cm, and now whole steel box-girder section is all subject to compressive stress impact, and camber is enough large, and thus the screening of presstressed reinforcing steel is linear should be but tradition is selected to should be h=8cm, and its linear equations is now steel box-girder upper flange still has larger tensile stress, thus fails to reach good prestressing force effect.

Table 1 simply supported steel box girder camber and wing plate normal stress sample calculation (wherein 3 groups of data)

Note: negative stress is compressive stress, normal stress is tensile stress.

Claims (2)

1. the method for optimizing that simply supported steel box girder presstressed reinforcing steel is linear, is characterized in that, comprises the following steps:

Step 1): introduce 3 generalized displacements when analyzing steel box-girder vertical bending, namely w (x) is case beam vertical deflection, the maximum lonitudinal warping displacement difference function that u (x) is case beam wing plate, θ (x) is for box-girder is about the vertical corner of z-axis, and x is girder span direction, then the length travel of box-girder wing plate can be expressed as:

$U(x,z)=\±h\[\mathrm{\θ}\left(x\right)+(1-\frac{z^2}{b^2})u\left(x\right)\]---\left(1\right)$

In formula: b is the half of case beam clear span;

H is the vertical y coordinate in box-girder cross section;

Step 2): (1) web potential energy of deformation

$V_1=\frac{1}{2}{\mathrm{EI}}_w{\∫}_0^l{\left({\mathrm{\θ}}^{\′}\right)}^{2}dx---\left(2\right)$

(2) wing plate potential energy of deformation

$V_2=\frac{1}{2}{\mathrm{EI}}_1{\∫}_0^l\[{\left({\mathrm{\θ}}^{\′}\right)}^2+\frac{4}{3}{\mathrm{\θ}}^{\′}{u}^{\′}+\frac{8}{15}{\left(u^{\′}\right)}^2\]dx+\frac{1}{2}{\∫}_0^l\frac{4{\mathrm{GI}}_1}{3b^2}{u}^{2}dx---\left(3\right)$

(3) hophornbeam pungent Ke's shear strain potential energy

$V_3=\frac{1}{2}{\∫}_0^{l}kGA{(\mathrm{\θ}-w^{\′})}^{2}dx---\left(4\right)$

(4) load potential energy

$V_p=-{\∫}_0^{l}q\left(x\right)w\left(x\right)dx-\[Q\left(x\right)w\left(x\right)+M_1\left(x\right)u\left(x\right)+M\left(x\right)\mathrm{\θ}\left(x\right)\]{|}_0^l---\left(5\right)$

So total potential energy of structural system is:

V＝V ₁+V ₂+V ₃+V _p(6)

In formula: M ₁(x) for case beam wing plate Girder with Shear Lag Effect produce about z-axis moment of flexure; M (x) for beam section end produce vertical corner θ (x) time about z-axis moment of flexure; Q (x), q (x) are distributed force vertical on beam section end vertical shear and case beam; E, G are Young's modulus of elasticity and the coefficient of rigidity of material; K is cross section shape coefficient; I _w, I ₁for box girder web and wing plate are about the moment of inertia of z-axis; A is box section area.

Step 3): according to variation principle obtaining its differential equation is:

${\mathrm{EI\θ}}^{\′\′}+\frac{2}{3}{\mathrm{EI}}_1{u}^{\′\′}-kGA(\mathrm{\θ}-w^{\′})=0---\left(7\right)$

$\frac{2}{3}{\mathrm{EI}}_1{\mathrm{\θ}}^{\′\′}+\frac{8}{15}{\mathrm{EI}}_1{u}^{\′\′}-\frac{4{\mathrm{GI}}_1}{3b^2}u=0---\left(8\right)$

kGA(θ′-w″)-q(x)＝0(9)

Its fringe conditions is:

$\[{\mathrm{EI\θ}}^{\′}+\frac{2}{3}{\mathrm{EI}}_1{u}^{\′}-M\]{|}_0^l\mathrm{\δ}\mathrm{\θ}=0---\left(10\right)$

$\[\frac{2}{3}{\mathrm{EI}}_1{\mathrm{\θ}}^{\′}+\frac{8}{15}{\mathrm{EI}}_1{u}^{\′}-M_1\]{|}_0^l\mathrm{\δ}u=0---\left(11\right)$

$\[kGA(\mathrm{\θ}-w^{\′})+Q\left(x\right)\]{|}_0^l\mathrm{\δ}w=0---\left(12\right)$

Step 4): by equation (7), between (8) and (9) arrangement conversion, trying to achieve new Solutions of Ordinary Differential Equations is:

$u^{\left(3\right)}+\frac{2G}{{\mathrm{Emb}}^2}u-\frac{q}{E\mathrm{Im}}=0---\left(13\right)$

Wherein: i=I _w+ I ₁, and u ⁽³⁾for 3 differentiate formulas of u;

Then its characteristic equation solution is:

γ _1,2＝±(α ₁+β ₁i)；γ ₃＝0。

Finally, the solution of u (x), θ (x) and w (x) is respectively:

$u\left(x\right)=c_{1}sh({\mathrm{\α}}_1+{\mathrm{\β}}_{1}i)x+c_{2}ch({\mathrm{\α}}_1+{\mathrm{\β}}_{1}i)x+c_3+\frac{b^2}{2GI}qx---\left(14\right)$

$\mathrm{\θ}\left(x\right)=c_1{B}_{1}sh({\mathrm{\α}}_1+{\mathrm{\β}}_{1}i)x+c_2{B}_{1}ch({\mathrm{\α}}_1+{\mathrm{\β}}_{1}i)x+c_3\frac{m_2}{2}{x}^2+c_{4}x+c_5+\frac{q}{6EI}{x}^3---\left(15\right)$

$\begin{array}{c}w\left(x\right)=c_1-\frac{B_1}{{\mathrm{\α}}_1+{\mathrm{\β}}_{1}i}ch({\mathrm{\α}}_1+{\mathrm{\β}}_{1}i)x+c_2\frac{B_1}{{\mathrm{\α}}_1+{\mathrm{\β}}_{1}i}sh({\mathrm{\α}}_1+{\mathrm{\β}}_{1}i)x+c_3(\frac{m_2}{6}{x}^3\\ -\frac{m_{2}EI}{kGA}x)+\frac{c_4}{2}{x}^2+c_{5}x+c_6+\frac{q}{24EI}{x}^4-\frac{q}{2kGA}{x}^2\end{array}---\left(16\right)$

Wherein: $m_1=-\frac{4}{5};$ $m_2=\frac{2G}{{\mathrm{Eb}}^2};$ $B_1=\frac{m_2+m_1{({\mathrm{\α}}_1+{\mathrm{\β}}_{1}i)}^2}{{({\mathrm{\α}}_1+{\mathrm{\β}}_{1}i)}^2};$ And c ₁; c ₂; c ₃; c ₄; c ₅; c ₆for constant coefficient;

Step 5) suppose that the curvilinear equation of presstressed reinforcing steel is: wherein l is steel box-girder span, and f is the sag of presstressed reinforcing steel span centre, and h is the distance that presstressed reinforcing steel departs from neutral axis, and q is the uniform vertical load of pre-applied force equivalence;

Step 6) according to specific border condition, by equation w (x), θ (x), u (x) or its derivative formula substitute into fringe conditions, first calculate undetermined constant c ₁; c ₂; c ₃; c ₄; c ₅; c ₆, then calculate w (x), the size of steel box-girder camber can be obtained, the size of lower wing plate prestressing can be calculated, that is: simultaneously after considering, filter out simply supported steel box girder optimum prestress muscle (rope) and arrange linear.

2. the method for optimizing that simply supported steel box girder presstressed reinforcing steel according to claim 1 is linear, is characterized in that, step 6) in filter out simply supported steel box girder optimum prestress muscle and arrange that linear concrete grammar is as follows: first according to the curvilinear equation of presstressed reinforcing steel h is divided into some deciles as circulation item, then under specific border condition and certain prestressing force condition, forms matrix equation according to fringe conditions, calculate undetermined constant c based on this ₁; c ₂; c ₃; c ₄; c ₅; c ₆, and then constant term is inputted w (x), θ ' (x), u ' (x) can obtain camber and wing plate prestress value, finally according to camber size, takes into account flange plate stress value and filters out the linear equation of presstressed reinforcing steel.