CN101446062B - Reinforced concrete bridge pier column structure - Google Patents

Abstract

The invention discloses a reinforced concrete bridge pier column structure. The cross section of the bridge pier column takes the elliptical shape, and a principal stressed reinforcement is evenly distributed along an elliptical circumference; and an improvement of the structure is as follows: the bridge pier column is provided with an elliptical inner cavity, the directions of a major axis and a minor axis of the bridge pier column are respectively consistent with those of an elliptical major axis and an elliptical minor axis of the pier column profile, and the wall thickness of the bridge pier column in the elliptical major axis direction equals or is more than the wall thickness in the elliptical minor axis direction. In the structure, the bridge pier with elliptical cross section has stronger superiority in wind resistance, current rush resistance, ice slush impact resistance and the like. Furthermore, the maximum bending inertia moment of the elliptical cross section is more than that of a circular cross section under the condition of the same area, therefore, the elliptical cross section can be smaller than the circular cross section under the condition of the same stress, thus reducing the consumption of concrete.

Description

Reinforced concrete bridge pier column structure

Technical field

The present invention relates to the bridge design field, particularly relate to a kind of reinforced concrete bridge pier column structure.

Background technology

Modern bridge requires attractive in appearance and practicality is laid equal stress on, and the effect of pillarwork in bridge construction is most important, and it is not only important load-carrying members, and very big to the landscape effect influence of bridge.Existing bridge pier column often is designed to rectangle, circle and nose circle tee section, this pier stud or be that landscape effect is not good enough, or be the poor-performing of aspects such as anti-current-rush, anti-ice slush bump.

Summary of the invention

Technical problem to be solved by this invention is to solve bridge pier column anti-current-rush, the relatively poor problem of anti-ice slush collision performance.

In order to solve the problems of the technologies described above, the technical solution adopted in the present invention provides a kind of reinforced concrete bridge pier column structure, the cross section ovalize of described bridge pier column, and its stressed main reinforcement is evenly arranged along oval circumference.

As improvement of the present invention, described bridge pier column has an oval-shaped cavity, and the direction of its major axis and minor axis is consistent with oval-shaped major axis of pier stud outline and short-axis direction respectively.

Described bridge pier column is equal to or greater than wall thickness on the ellipse short shaft direction at the wall thickness on the transverse direction.

Stressed eccentric throw in described bridge pier column cross section and concrete stress expression formula satisfy:

(1) works as α ₁＜90 °, and ${\mathrm{\&alpha;}}_1\&le;{\cos }^{-1}\left(\frac{a_1{a}_2}{\sqrt{a_1^2{\mathrm{<figure-callout\; id="1"\; label="b"\; filenames="H2008101883976E00012.png"\; state="\{\{state\}\}">b</figure-callout>}}_1^2-a_2^2{\mathrm{<figure-callout\; id="1"\; label="b"\; filenames="H2008101883976E00012.png"\; state="\{\{state\}\}">b</figure-callout>}}_1^2+a_1^2{a}_2^2}}\right)$ The time

$\left\{\begin{array}{c}{e}^{\&prime;}=\frac{a_1^5{b}_1{\mathrm{<figure-callout\; id="1"\; label="W"\; filenames="H2008101883976E00012.png"\; state="\{\{state\}\}">W</figure-callout>}}_1+n\frac{\mathrm{\&mu;\&pi;}(a_1{b}_1-a_2{b}_2)}{E_p}{E}_k}{\frac{a_1{\mathrm{<figure-callout\; id="1"\; label="b"\; filenames="H2008101883976E00012.png"\; state="\{\{state\}\}">b</figure-callout>}}_1}{6}{V}_1-\mathrm{n\&mu;\&pi;}(a_1{b}_1-a_2{b}_2){\mathrm{<figure-callout\; id="1"\; label="P"\; filenames="H2008101883976E00012.png"\; state="\{\{state\}\}">P</figure-callout>}}_1}\\ {\mathrm{\&sigma;}}_h=\frac{K_{1}M}{a_1^5{b}_1{\mathrm{<figure-callout\; id="1"\; label="W"\; filenames="H2008101883976E00012.png"\; state="\{\{state\}\}">W</figure-callout>}}_1+n\frac{\mathrm{\&mu;\&pi;}(a_1{b}_1-a_2{b}_2)}{E_p}{E}_k}\end{array}\right.;$

(2) when ${\cos }^{-1}\left(\frac{a_1{a}_2}{\sqrt{a_1^2{\mathrm{<figure-callout\; id="1"\; label="b"\; filenames="H2008101883976E00012.png"\; state="\{\{state\}\}">b</figure-callout>}}_1^2-a_2^2{\mathrm{<figure-callout\; id="1"\; label="b"\; filenames="H2008101883976E00012.png"\; state="\{\{state\}\}">b</figure-callout>}}_1^2+a_1^2{a}_2^2}}\right)<{\mathrm{\&alpha;}}_1<{\cos }^{-1}\left(\frac{-a_1{a}_2}{\sqrt{a_1^2{\mathrm{<figure-callout\; id="1"\; label="b"\; filenames="H2008101883976E00012.png"\; state="\{\{state\}\}">b</figure-callout>}}_1^2-a_2^2{\mathrm{<figure-callout\; id="1"\; label="b"\; filenames="H2008101883976E00012.png"\; state="\{\{state\}\}">b</figure-callout>}}_1^2+a_1^2{a}_2^2}}\right)$ The time

$\left\{\begin{array}{c}{e}^{\&prime;}=\frac{a_1^5{b}_1{W}_1-a_2^5{b}_2{W}_2+n\frac{\mathrm{\&mu;\&pi;}(a_1{b}_1-a_2{b}_2)}{E_p}{E}_k}{\frac{a_1{\mathrm{<figure-callout\; id="1"\; label="b"\; filenames="H2008101883976E00012.png"\; state="\{\{state\}\}">b</figure-callout>}}_1}{6}{V}_1-\frac{a_2{b}_2}{6}{V}_2-\mathrm{n\&mu;\&pi;}(a_1{b}_1-a_2{b}_2){\mathrm{<figure-callout\; id="1"\; label="P"\; filenames="H2008101883976E00012.png"\; state="\{\{state\}\}">P</figure-callout>}}_1}\\ {\mathrm{\&sigma;}}_h=\frac{K_{1}M}{a_1^5{b}_1{W}_1-a_2^5{b}_2{W}_2+n\frac{\mathrm{\&mu;\&pi;}(a_1{b}_1-a_2{b}_2)}{E_p}{E}_k}\end{array}\right.;$

(3) work as α ₁＞90 °, and ${\mathrm{\&alpha;}}_1\&GreaterEqual;{\cos }^{-1}\left(\frac{{-a}_1{a}_2}{\sqrt{a_1^2{\mathrm{<figure-callout\; id="1"\; label="b"\; filenames="H2008101883976E00012.png"\; state="\{\{state\}\}">b</figure-callout>}}_1^2-a_2^2{\mathrm{<figure-callout\; id="1"\; label="b"\; filenames="H2008101883976E00012.png"\; state="\{\{state\}\}">b</figure-callout>}}_1^2+a_1^2{a}_2^2}}\right)$ The time

$\left\{\begin{array}{c}{e}^{\&prime;}=\frac{a_1^5{b}_1{W}_1-\frac{a_2^3{b}_2}{4}+n\frac{\mathrm{\&mu;\&pi;}(a_1{b}_1-a_2{b}_2)}{E_p}{E}_k}{\frac{a_1{\mathrm{<figure-callout\; id="1"\; label="b"\; filenames="H2008101883976E00012.png"\; state="\{\{state\}\}">b</figure-callout>}}_1}{6}{V}_1-\frac{\mathrm{\&pi;}(2K_1-a_1)a_{2}b}{2}-\mathrm{n\&mu;\&pi;}(a_1{b}_1-a_2{b}_2){\mathrm{<figure-callout\; id="1"\; label="P"\; filenames="H2008101883976E00012.png"\; state="\{\{state\}\}">P</figure-callout>}}_1}\\ {\mathrm{\&sigma;}}_h=\frac{K_{1}M}{a_1^5{b}_1{W}_1-\frac{a_2^3{b}_2}{8}+n\frac{\mathrm{\&mu;\&pi;}(a_1{b}_1-a_2{b}_2)}{E_p}{E}_k}\end{array}\right.;$

In the formula:

a ₁And b ₁Represent pier stud outline transverse and minor axis respectively;

a ₂And b ₂Represent wide transverse and minor axis in the pier stud respectively;

A and b represent the major axis and the minor axis of reinforcing bar oval circumference of living in the pier stud respectively;

μ represents the cross section rebar ratio;

N represents reinforcing bar modulus of elasticity and concrete;

M represents the moment of flexure value on the cross section;

a ₁The central angle of expression concrete compression cross section correspondence

θ, φ represent integration variable, value be scope be [0, a ₁]

E _p、 $E_h-E_p={\&Integral;}_0^{2\mathrm{\&pi;}}\frac{\mathrm{ab}}{\sqrt{b^2{\cos }^2\mathrm{\&phi;}+a^2{\sin }^2\mathrm{\&phi;}}}\mathrm{d\&phi;};$ $E_k={\&Integral;}_0^{2\mathrm{\&pi;}}\frac{\mathrm{ab}{\cos }^2\mathrm{\&theta;}}{{(b^2{\cos }^2\mathrm{\&theta;}+a^2{\sin }^2\mathrm{\&theta;})}^{\frac{3}{2}}}\mathrm{d\&theta;}$

${\mathrm{\&rho;}}_1\left({\mathrm{\&alpha;}}_1\right)=\frac{a_1{b}_1}{\sqrt{{\mathrm{<figure-callout\; id="1"\; label="b"\; filenames="H2008101883976E00012.png"\; state="\{\{state\}\}">b</figure-callout>}}_1^2{\cos }^2{\mathrm{\&alpha;}}_1+a_1^2{\sin }^2{\mathrm{\&alpha;}}_1}}$

2K ₁＝a ₁-ρ ₁(a ₁)cosα ₁

P ₁＝ρ ₁(α ₁)cosα ₁

${\mathrm{<figure-callout\; id="1"\; label="V"\; filenames="H2008101883976E00012.png"\; state="\{\{state\}\}">V</figure-callout>}}_1=[\frac{2{\mathrm{<figure-callout\; id="1"\; label="a"\; filenames="H2008101883976E00012.png"\; state="\{\{state\}\}">a</figure-callout>}}_1^4{\sin }^3{\mathrm{\&alpha;}}_1}{({\mathrm{<figure-callout\; id="1"\; label="b"\; filenames="H2008101883976E00012.png"\; state="\{\{state\}\}">b</figure-callout>}}_1^2{\cos }^2{\mathrm{\&alpha;}}_1+a_1^2{\sin }^2{\mathrm{\&alpha;}}_1)}-3P_1(\arctan \left(\frac{a_1\tan {\mathrm{\&alpha;}}_1}{b_1}\right)-\frac{a_1{\mathrm{<figure-callout\; id="1"\; label="b"\; filenames="H2008101883976E00012.png"\; state="\{\{state\}\}">b</figure-callout>}}_1\tan {\mathrm{\&alpha;}}_1}{({\mathrm{<figure-callout\; id="1"\; label="b"\; filenames="H2008101883976E00012.png"\; state="\{\{state\}\}">b</figure-callout>}}_1^2+a_1^2{\tan }^2{\mathrm{\&alpha;}}_1)})]$

$V_2=[\frac{2a_2^4{\sin }^3{\mathrm{\&alpha;}}_2}{(b_2^2{\cos }^2{\mathrm{\&alpha;}}_2+a_2^2{\sin }^2{\mathrm{\&alpha;}}_2)}-3P_2(\arctan \left(\frac{a_2\tan {\mathrm{\&alpha;}}_2}{b_2}\right)-\frac{a_2{b}_2\tan {\mathrm{\&alpha;}}_2}{(b_2^2+a_2^2{\tan }^2{\mathrm{\&alpha;}}_2)})]$

${\mathrm{<figure-callout\; id="1"\; label="W"\; filenames="H2008101883976E00012.png"\; state="\{\{state\}\}">W</figure-callout>}}_1=[\frac{a_1^2\arctan \left(\frac{a_1\tan {\mathrm{\&alpha;}}_1}{b_1}\right)-a_1{\mathrm{<figure-callout\; id="1"\; label="b"\; filenames="H2008101883976E00012.png"\; state="\{\{state\}\}">b</figure-callout>}}_1{\mathrm{\&alpha;}}_1}{{(a_1^2-{\mathrm{<figure-callout\; id="1"\; label="b"\; filenames="H2008101883976E00012.png"\; state="\{\{state\}\}">b</figure-callout>}}_1^2)}^2}-\frac{1}{2}(\frac{a_1{\mathrm{<figure-callout\; id="1"\; label="b"\; filenames="H2008101883976E00012.png"\; state="\{\{state\}\}">b</figure-callout>}}_1\tan {\mathrm{\&alpha;}}_1}{({\mathrm{<figure-callout\; id="1"\; label="a"\; filenames="H2008101883976E00012.png"\; state="\{\{state\}\}">a</figure-callout>}}_1^2-{\mathrm{<figure-callout\; id="1"\; label="b"\; filenames="H2008101883976E00012.png"\; state="\{\{state\}\}">b</figure-callout>}}_1^2)({\mathrm{<figure-callout\; id="1"\; label="b"\; filenames="H2008101883976E00012.png"\; state="\{\{state\}\}">b</figure-callout>}}_1^2+a_1^2{\tan }^2{\mathrm{\&alpha;}}_1)}-\frac{\arctan \left(\frac{a_1\tan {\mathrm{\&alpha;}}_1}{b_1}\right)}{(a_1^2-{\mathrm{<figure-callout\; id="1"\; label="b"\; filenames="H2008101883976E00012.png"\; state="\{\{state\}\}">b</figure-callout>}}_1^2)})]$

$-\frac{{\mathrm{<figure-callout\; id="1"\; label="P"\; filenames="H2008101883976E00012.png"\; state="\{\{state\}\}">P</figure-callout>}}_1}{3}\left[\frac{{\sin }^3{\mathrm{\&alpha;}}_1}{({\mathrm{<figure-callout\; id="1"\; label="b"\; filenames="H2008101883976E00012.png"\; state="\{\{state\}\}">b</figure-callout>}}_1^2{\cos }^2{\mathrm{\&alpha;}}_1+a_1^2{\sin }^2{\mathrm{\&alpha;}}_1)}\right]$

$W_2=[\frac{a_2^2\arctan \left(\frac{a_2\tan {\mathrm{\&alpha;}}_2}{b_2}\right)-a_2{b}_2{\mathrm{\&alpha;}}_2}{{(a_2^2-b_2^2)}^2}-\frac{1}{2}(\frac{a_2{b}_2\tan {\mathrm{\&alpha;}}_2}{(a_2^2-b_2^2)(b_2^2+a_2^2{\tan }^2{\mathrm{\&alpha;}}_2)}-\frac{\arctan \left(\frac{a_2\tan {\mathrm{\&alpha;}}_2}{b_2}\right)}{(a_2^2-b_2^2)})]$

$-\frac{P_2}{3}\left[\frac{{\sin }^3{\mathrm{\&alpha;}}_2}{(b_2^2{\cos }^2{\mathrm{\&alpha;}}_2+a_2^2{\sin }^2{\mathrm{\&alpha;}}_2)}\right]$

The present invention, the bridge pier of elliptic cross-section all has stronger superiority at aspects such as wind resistance, anti-current-rush, anti-ice slush bumps, and in addition, the maximum bending resistance moment of inertia of elliptic cross-section is bigger than circular cross-section under situation of the same area, and the ratio of the two is (a/r) ²(wherein the oval major semiaxis of a, r is a radius of a circle) therefore in that to reach under the situation of identical stress the comparable circular section of oval cross section little, thereby saved concrete amount.

Description of drawings

Fig. 1 is Reinforced Concrete Bridge pier stud schematic cross-section of the present invention and neutral axis zone division schematic diagram;

Fig. 2～Fig. 4 is a Reinforced Concrete Bridge pier stud of the present invention cross section force analysis key diagram;

Fig. 5～Fig. 6 is the E of Reinforced Concrete Bridge pier stud of the present invention _p(θ), E _k(θ) change curve.

The specific embodiment

The invention provides a kind of reinforced concrete bridge pier column structure, its cross section ovalize, and its stressed main reinforcement is evenly arranged along oval circumference.

Preferably, described bridge pier column has an oval-shaped cavity, and the direction of its major axis and minor axis is consistent with oval-shaped major axis of pier stud outline and short-axis direction respectively.

Further, described bridge pier column is equal to or greater than wall thickness on the ellipse short shaft direction at the wall thickness on the transverse direction.

Specify design philosophy of the present invention below in two kinds of situation.

One, oval ring cross section pier stud is when the large eccentric pressuring of ring section:

As shown in Figure 1, three of I, the II that the cross section position of neutral axis of bridge pier column is pressed, III divide in the zone, and neutral axis is two sides, cross section product moment to be equated and the coordinate axes chosen, and area moment is the product that selected region area and its central point arrive the distance of coordinate axes, and supposition

1. the cross section strain keeps the plane;

2. stress and strain is directly proportional;

3. do not consider concrete tensile strength;

Thus, the equilibrium of forces equation is as follows:

Can obtain by static balance condition:

$\left\{\begin{array}{c}N=N_h+N_g=N_{h1}-N_{h2}+N_g\\ M=M_h+M_g=M_{h1}-M_{h2}+M_g\end{array}\right.---\left(1\right)$

In the formula:

N represents reference axis power;

M represents calculated bending moment;

N _h, N _H1, N _H2Representative ring tee section pressure zone concrete and big or small bow-shaped cross-section makes a concerted effort respectively;

M _h, M _H1, M _H2The resultant moment of difference representative ring tee section pressure zone concrete and big or small bow-shaped cross-section;

N _g, M _gRepresent making a concerted effort and resultant moment of reinforcing bar respectively.

Two, during oval ring cross section pier stud eccentric compression,

As shown in Figure 2, in arc greatly (referring to concrete compression district area, i.e. shaded area among Fig. 2), being expressed as of oval cross section under the illustrated coordinate system:

$\frac{x^2}{a_1^2}+\frac{y^2}{{\mathrm{<figure-callout\; id="1"\; label="b"\; filenames="H2008101883976E00012.png"\; state="\{\{state\}\}">b</figure-callout>}}_1^2}=1---\left(2\right)$

Formula (2) is converted into polar expression formula, establishes x=ρ ₁Cos θ ₁Y=ρ ₁Sin θ ₁

(2) expression formula is:

${\mathrm{\&rho;}}_1\left({\mathrm{\&theta;}}_1\right)=\frac{a_1{b}_1}{\sqrt{{\mathrm{<figure-callout\; id="1"\; label="b"\; filenames="H2008101883976E00012.png"\; state="\{\{state\}\}">b</figure-callout>}}_1^2{\cos }^2{\mathrm{\&theta;}}_1+a_1^2{\sin }^2{\mathrm{\&theta;}}_1}}---\left(3\right)$

In Fig. 2 big arc the expression formula of little area be:

dA _h＝2ydx＝2ρ ₁(θ ₁)sinθ ₁·d(ρ ₁(θ ₁)cosθ ₁) (4)

(3) formula (4) formula of bringing into is obtained following expression:

${\mathrm{dA}}_h=2\frac{a_1^4{\mathrm{<figure-callout\; id="1"\; label="b"\; filenames="H2008101883976E00012.png"\; state="\{\{state\}\}">b</figure-callout>}}_1^2{\sin }^2{\mathrm{\&theta;}}_1}{{({\mathrm{<figure-callout\; id="1"\; label="b"\; filenames="H2008101883976E00012.png"\; state="\{\{state\}\}">b</figure-callout>}}_1^2{\cos }^2{\mathrm{\&theta;}}_1+a_1^2{\sin }^2{\mathrm{\&theta;}}_1)}^2}{\mathrm{d\&theta;}}_1---\left(5\right)$

According to the basic assumption of front, little area dA _hThe concrete compressive stress expression formula at place is:

${\mathrm{\&sigma;}}_{{\mathrm{h\&theta;}}_1}={\mathrm{\&sigma;}}_h\frac{{\mathrm{\&rho;}}_1\left({\mathrm{\&theta;}}_1\right)\cos {\mathrm{\&theta;}}_1-{\mathrm{<figure-callout\; id="1"\; label="P"\; filenames="H2008101883976E00012.png"\; state="\{\{state\}\}">P</figure-callout>}}_1}{2K_1}---\left(6\right)$

In the following formula:

P ₁＝ρ ₁(α ₁)cosα ₁，2K ₁＝a ₁-ρ ₁(α ₁)cosα ₁ (7)

Shown in Figure 2 big arc in, the shaded area place is concrete to make a concerted effort and the expression formula of resultant moment is:

${\mathrm{<figure-callout\; id="1"\; label="N\; h"\; filenames="H2008101883976E00012.png"\; state="\{\{state\}\}">N</figure-callout>}}_h={\&Integral;}_{A_h}{\mathrm{\&sigma;}}_{{\mathrm{h\&theta;}}_1}\&CenterDot;d{A}_h={\&Integral;}_0^{{\mathrm{\&alpha;}}_1}{\mathrm{\&sigma;}}_h\frac{{\mathrm{\&rho;}}_1\left({\mathrm{\&theta;}}_1\right)\cos {\mathrm{\&theta;}}_1-P_1}{{\mathrm{<figure-callout\; id="1"\; label="K"\; filenames="H2008101883976E00012.png"\; state="\{\{state\}\}">K</figure-callout>}}_1}\&CenterDot;\frac{a_1^4{\mathrm{<figure-callout\; id="1"\; label="b"\; filenames="H2008101883976E00012.png"\; state="\{\{state\}\}">b</figure-callout>}}_1^2{\sin }^2{\mathrm{\&theta;}}_1}{{({\mathrm{<figure-callout\; id="1"\; label="b"\; filenames="H2008101883976E00012.png"\; state="\{\{state\}\}">b</figure-callout>}}_1^2{\cos }^2{\mathrm{\&theta;}}_1+a_1^2{\sin }^2{\mathrm{\&theta;}}_1)}^2}d{\mathrm{\&theta;}}_1$

$={\mathrm{\&sigma;}}_h\frac{a_1^4{\mathrm{<figure-callout\; id="1"\; label="b"\; filenames="H2008101883976E00012.png"\; state="\{\{state\}\}">b</figure-callout>}}_1^2}{{\mathrm{<figure-callout\; id="1"\; label="K"\; filenames="H2008101883976E00012.png"\; state="\{\{state\}\}">K</figure-callout>}}_1}{\&Integral;}_0^{{\mathrm{\&alpha;}}_1}\frac{a_1{\mathrm{<figure-callout\; id="1"\; label="b"\; filenames="H2008101883976E00012.png"\; state="\{\{state\}\}">b</figure-callout>}}_1\cos {\mathrm{\&theta;}}_1{\sin }^2{\mathrm{\&theta;}}_1}{{({\mathrm{<figure-callout\; id="1"\; label="b"\; filenames="H2008101883976E00012.png"\; state="\{\{state\}\}">b</figure-callout>}}_1^2{\cos }^2{\mathrm{\&theta;}}_1+a_1^2{\sin }^2{\mathrm{\&theta;}}_1)}^{\frac{5}{2}}}-\frac{{\mathrm{<figure-callout\; id="1"\; label="P"\; filenames="H2008101883976E00012.png"\; state="\{\{state\}\}">P</figure-callout>}}_1{\sin }^2{\mathrm{\&theta;}}_1}{{({\mathrm{<figure-callout\; id="1"\; label="b"\; filenames="H2008101883976E00012.png"\; state="\{\{state\}\}">b</figure-callout>}}_1^2{\cos }^2{\mathrm{\&theta;}}_1+a_1^2{\sin }^2{\mathrm{\&theta;}}_1)}^2}d{\mathrm{\&theta;}}_1$

$={\mathrm{\&sigma;}}_h\frac{a_1{\mathrm{<figure-callout\; id="1"\; label="b"\; filenames="H2008101883976E00012.png"\; state="\{\{state\}\}">b</figure-callout>}}_1}{6K_1}{V}_1---\left(8\right)$

Wherein:

${\mathrm{<figure-callout\; id="1"\; label="V"\; filenames="H2008101883976E00012.png"\; state="\{\{state\}\}">V</figure-callout>}}_1=[\frac{2a_1^4{\sin }^3{\mathrm{\&alpha;}}_1}{{\mathrm{<figure-callout\; id="1"\; label="b"\; filenames="H2008101883976E00012.png"\; state="\{\{state\}\}">b</figure-callout>}}_1^2{\cos }^2{\mathrm{\&alpha;}}_1+a_1^2{\sin }^2{\mathrm{\&alpha;}}_1}-3P_1(\arctan \left(\frac{a_1\tan {\mathrm{\&alpha;}}_1}{b_1}\right)-\frac{a_1{\mathrm{<figure-callout\; id="1"\; label="b"\; filenames="H2008101883976E00012.png"\; state="\{\{state\}\}">b</figure-callout>}}_1\tan {\mathrm{\&alpha;}}_1}{({\mathrm{<figure-callout\; id="1"\; label="b"\; filenames="H2008101883976E00012.png"\; state="\{\{state\}\}">b</figure-callout>}}_1^2+a_1^2{\tan }^2{\mathrm{\&alpha;}}_1)})]---\left(9\right)$

${\mathrm{<figure-callout\; id="1"\; label="M\; h"\; filenames="H2008101883976E00012.png"\; state="\{\{state\}\}">M</figure-callout>}}_h={\&Integral;}_{A_h}x\&CenterDot;{\mathrm{\&sigma;}}_{{\mathrm{h\&theta;}}_1}\&CenterDot;{\mathrm{dA}}_h={\&Integral;}_0^{{\mathrm{\&alpha;}}_1}{\mathrm{\&rho;}}_1\left({\mathrm{\&theta;}}_1\right)\cos {\mathrm{\&theta;}}_1\&CenterDot;{\mathrm{\&sigma;}}_h\frac{{\mathrm{\&rho;}}_1\left({\mathrm{\&theta;}}_1\right)\cos {\mathrm{\&theta;}}_1-P_1}{{\mathrm{<figure-callout\; id="1"\; label="K"\; filenames="H2008101883976E00012.png"\; state="\{\{state\}\}">K</figure-callout>}}_1}\&CenterDot;\frac{a_1^4{\mathrm{<figure-callout\; id="1"\; label="b"\; filenames="H2008101883976E00012.png"\; state="\{\{state\}\}">b</figure-callout>}}_1^2{\sin }^2{\mathrm{\&theta;}}_1}{{({\mathrm{<figure-callout\; id="1"\; label="b"\; filenames="H2008101883976E00012.png"\; state="\{\{state\}\}">b</figure-callout>}}_1^2{\cos }^2{\mathrm{\&theta;}}_1+a_1^2{\sin }^2{\mathrm{\&theta;}}_1)}^2}{\mathrm{d\&theta;}}_1$

$={\mathrm{\&sigma;}}_h\frac{a_1^5{\mathrm{<figure-callout\; id="1"\; label="b"\; filenames="H2008101883976E00012.png"\; state="\{\{state\}\}">b</figure-callout>}}_1^3}{{\mathrm{<figure-callout\; id="1"\; label="K"\; filenames="H2008101883976E00012.png"\; state="\{\{state\}\}">K</figure-callout>}}_1}{\&Integral;}_0^{{\mathrm{\&alpha;}}_1}\frac{a_1{\mathrm{<figure-callout\; id="1"\; label="b"\; filenames="H2008101883976E00012.png"\; state="\{\{state\}\}">b</figure-callout>}}_1{\cos }^2{\mathrm{\&theta;}}_1{\sin }^2{\mathrm{\&theta;}}_1}{{({\mathrm{<figure-callout\; id="1"\; label="b"\; filenames="H2008101883976E00012.png"\; state="\{\{state\}\}">b</figure-callout>}}_1^2{\cos }^2{\mathrm{\&theta;}}_1+a_1^2{\sin }^2{\mathrm{\&theta;}}_1)}^3}-\frac{{\mathrm{<figure-callout\; id="1"\; label="P"\; filenames="H2008101883976E00012.png"\; state="\{\{state\}\}">P</figure-callout>}}_1{\sin }^2{\mathrm{\&theta;}}_1\cos {\mathrm{\&theta;}}_1}{{({\mathrm{<figure-callout\; id="1"\; label="b"\; filenames="H2008101883976E00012.png"\; state="\{\{state\}\}">b</figure-callout>}}_1^2{\cos }^2{\mathrm{\&theta;}}_1+a_1^2{\sin }^2{\mathrm{\&theta;}}_1)}^{\frac{5}{2}}}{\mathrm{d\&theta;}}_1$

$={\mathrm{\&sigma;}}_h\frac{a_1^5{b}_1}{K_1}{W}_1---\left(10\right)$

Wherein:

${\mathrm{<figure-callout\; id="1"\; label="W"\; filenames="H2008101883976E00012.png"\; state="\{\{state\}\}">W</figure-callout>}}_1=[\frac{a_1^2\arctan \left(\frac{a_1\tan {\mathrm{\&alpha;}}_1}{b_1}\right)-a_1{\mathrm{<figure-callout\; id="1"\; label="b"\; filenames="H2008101883976E00012.png"\; state="\{\{state\}\}">b</figure-callout>}}_1{\mathrm{\&alpha;}}_1}{{(a_1^2-{\mathrm{<figure-callout\; id="1"\; label="b"\; filenames="H2008101883976E00012.png"\; state="\{\{state\}\}">b</figure-callout>}}_1^2)}^2}-\frac{1}{2}(\frac{a_1{\mathrm{<figure-callout\; id="1"\; label="b"\; filenames="H2008101883976E00012.png"\; state="\{\{state\}\}">b</figure-callout>}}_1\tan {\mathrm{\&alpha;}}_1}{(a_1^2-{\mathrm{<figure-callout\; id="1"\; label="b"\; filenames="H2008101883976E00012.png"\; state="\{\{state\}\}">b</figure-callout>}}_1^2)({\mathrm{<figure-callout\; id="1"\; label="b"\; filenames="H2008101883976E00012.png"\; state="\{\{state\}\}">b</figure-callout>}}_1^2+a_1^2{\tan }^2{\mathrm{\&alpha;}}_1)}-\frac{\arctan \left(\frac{a_1\tan {\mathrm{\&alpha;}}_1}{b_1}\right)}{(a_1^2-{\mathrm{<figure-callout\; id="1"\; label="b"\; filenames="H2008101883976E00012.png"\; state="\{\{state\}\}">b</figure-callout>}}_1^2)})]$

$-\frac{{\mathrm{<figure-callout\; id="1"\; label="P"\; filenames="H2008101883976E00012.png"\; state="\{\{state\}\}">P</figure-callout>}}_1}{3}\left[\frac{{\sin }^3{\mathrm{\&alpha;}}_1}{({\mathrm{<figure-callout\; id="1"\; label="b"\; filenames="H2008101883976E00012.png"\; state="\{\{state\}\}">b</figure-callout>}}_1^2{\cos }^2{\mathrm{\&alpha;}}_1+a_1^2{\sin }^2{\mathrm{\&alpha;}}_1)}\right]---\left(11\right)$

In like manner for little bow-shaped cross-section as shown in Figure 3, the equation of establishing the innermost layer ellipse is:

${\mathrm{\&rho;}}_2\left({\mathrm{\&theta;}}_2\right)=\frac{a_2{b}_2}{\sqrt{b_2^2{\cos }^2{\mathrm{\&theta;}}_2+a_2^2{\sin }^2{\mathrm{\&theta;}}_2}}---\left(12\right)$

Can obtain the conclusion similar to the front:

$\left\{\begin{array}{c}{N}_{h2}={\mathrm{\&sigma;}}_h\frac{a_2{b}_2}{6K_1}{V}_2\\ M_{h2}={\mathrm{\&sigma;}}_h\frac{a_2^5{b}_2}{K_1}{W}_2\end{array}\right.---\left(13\right)$

Wherein

$\left\{\begin{array}{c}{P}_2={\mathrm{\&rho;}}_2\left({\mathrm{\&alpha;}}_2\right)\cos {\mathrm{\&alpha;}}_2\\ V_2=[\frac{2a_2^4{\sin }^3{\mathrm{\&alpha;}}_2}{(b_2^2{\cos }^2{\mathrm{\&alpha;}}_2+a_2^2{\sin }^2{\mathrm{\&alpha;}}_2)}-3P_2(\arctan \left(\frac{a_2\tan {\mathrm{\&alpha;}}_2}{b_2}\right)-\frac{a_2{b}_2\tan {\mathrm{\&alpha;}}_2}{(b_2^2+a_2^2{\tan }^2{\mathrm{\&alpha;}}_2)})]\\ W_2=[\frac{a_2^2\arctan \left(\frac{a_2\tan {\mathrm{\&alpha;}}_2}{b_2}\right)-a_2{b}_2{\mathrm{\&alpha;}}_2}{{(a_2^2-b_2^2)}^2}-\frac{1}{2}(\frac{a_2{b}_2\tan {\mathrm{\&alpha;}}_2}{(a_2^2-b_2^2)(b_2^2+a_2^2{\tan }^2{\mathrm{\&alpha;}}_2)}-\frac{\arctan \left(\frac{a_2\tan {\mathrm{\&alpha;}}_2}{b_2}\right)}{(a_2^2-b_2^2)})]\\ -\frac{P_2}{3}\left[\frac{{\sin }^3{\mathrm{\&alpha;}}_2}{(b_2^2{\cos }^2{\mathrm{\&alpha;}}_2+a_2^2{\sin }^2{\mathrm{\&alpha;}}_2)}\right]\end{array}\right.---\left(14\right)$

The reinforcement ratio of supposing reinforcing bar is μ, and the oval expression formula that reinforcing bar surrounds is:

${\mathrm{\&rho;}}_g\left(\mathrm{\&phi;}\right)=\frac{\mathrm{ab}}{\sqrt{b^2{\cos }^2\mathrm{\&phi;}+a^2{\sin }^2\mathrm{\&phi;}}},$ Its thickness is t,

As shown in Figure 2, then have following equation to set up for the area of reinforcement:

tE _p＝μπ(a ₁b ₁-a ₂b ₂) (15)

E wherein _pExpression formula for integration: $E_p={\&Integral;}_0^{2\mathrm{\&pi;}}\frac{\mathrm{ab}}{\sqrt{b^2{\cos }^2\mathrm{\&phi;}+a^2{\sin }^2\mathrm{\&phi;}}}\mathrm{d\&phi;},$ Its result is the ellptic integral function, can consult relevant mathematics formulary and draw.

Obtain by (15) formula:

$t=\frac{\mathrm{\&mu;\&pi;}(a_1{b}_1-a_2{b}_2)}{E_p}---\left(16\right)$

On the elliptical ring that reinforcing bar converts, get little area dA _g, then

${\mathrm{dA}}_g=\mathrm{t\&rho;}\left(\mathrm{\&phi;}\right)\mathrm{d\&phi;}=\frac{\mathrm{\&mu;\&pi;}(a_1{b}_1-a_2{b}_2)}{E_p}\frac{\mathrm{ab}}{\sqrt{b^2{\cos }^2\mathrm{\&phi;}+a^2{\sin }^2\mathrm{\&phi;}}}\mathrm{d\&phi;}---\left(17\right)$

Relational expression by stress ratio obtains:

${\mathrm{\&sigma;}}_{\mathrm{g\&theta;}}=n\frac{{\mathrm{\&sigma;}}_h}{2K_1}({\mathrm{\&rho;}}_g\left(\mathrm{\&phi;}\right)\cos \mathrm{\&phi;}-P_1)---\left(18\right)$

In the formula, n is a reinforcing bar and concrete bullet mould ratio, and a and b represent the major axis and the minor axis of reinforcing bar oval circumference of living in the pier stud respectively.

By (17), (18) formula obtain reinforcing bar make a concerted effort with the expression formula of resultant moment be:

$N_g={\&Integral;}_{A_g}{\mathrm{\&sigma;}}_g\&CenterDot;{\mathrm{dA}}_g=n\frac{\mathrm{\&mu;\&pi;}(a_1{b}_1-a_2{b}_2)\mathrm{ab}{\mathrm{\&sigma;}}_h}{{\mathrm{<figure-callout\; id="1"\; label="E\; p\; K"\; filenames="H2008101883976E00012.png"\; state="\{\{state\}\}">E</figure-callout>}}_p{K}_{}{}_1}{\&Integral;}_0^{\mathrm{\&pi;}}(\frac{\mathrm{ab}\cos \mathrm{\&phi;}}{b^2{\cos }^2\mathrm{\&phi;}+a^2{\sin }^2\mathrm{\&phi;}}-\frac{P_1}{\sqrt{b^2{\cos }^2\mathrm{\&phi;}+a^2{\sin }^2\mathrm{\&phi;}}})\mathrm{d\&phi;}$

$=-n\frac{\mathrm{\&mu;\&pi;}(a_1{b}_1-a_2{b}_2){\mathrm{<figure-callout\; id="1"\; label="P"\; filenames="H2008101883976E00012.png"\; state="\{\{state\}\}">P</figure-callout>}}_1{\mathrm{\&sigma;}}_h}{K_1}---\left(19\right)$

Work as can be seen by (19) formula $\mathrm{\&phi;}<\frac{\mathrm{\&pi;}}{2},$ The tensile stress of making a concerted effort to be of reinforcing bar.

$M_g={\&Integral;}_{A_g}\mathrm{\&rho;}\left(\mathrm{\&theta;}\right)\cos \left(\mathrm{\&theta;}\right)\&CenterDot;{\mathrm{\&sigma;}}_g\&CenterDot;{\mathrm{dA}}_g=n\frac{\mathrm{\&mu;\&pi;}(a_1{b}_1-a_2{b}_2)a^2{b}^2{\mathrm{\&sigma;}}_h}{2{\mathrm{<figure-callout\; id="1"\; label="E\; p\; K"\; filenames="H2008101883976E00012.png"\; state="\{\{state\}\}">E</figure-callout>}}_p{K}_{}{}_1}{\&Integral;}_0^{2\mathrm{\&pi;}}(\frac{\mathrm{ab}{\cos }^2\mathrm{\&theta;}}{{(b^2{\cos }^2\mathrm{\&theta;}+a^2{\sin }^2\mathrm{\&theta;})}^{\frac{3}{2}}}-\frac{{\mathrm{<figure-callout\; id="1"\; label="P"\; filenames="H2008101883976E00012.png"\; state="\{\{state\}\}">P</figure-callout>}}_1\cos \mathrm{\&theta;}}{b^2{\cos }^2\mathrm{\&theta;}+a^2{\sin }^2\mathrm{\&theta;}})\mathrm{d\&theta;}$

$=n\frac{\mathrm{\&mu;\&pi;}(a_1{b}_1-a_2{b}_2){\mathrm{\&sigma;}}_h}{E_p{K}_1}{E}_k---\left(20\right)$

(20) E in the formula _kExpression formula for integration: $E_k={\&Integral;}_0^{2\mathrm{\&pi;}}\frac{\mathrm{ab}{\cos }^2\mathrm{\&theta;}}{{(b^2{\cos }^2\mathrm{\&theta;}+a^2{\sin }^2\mathrm{\&theta;})}^{\frac{3}{2}}}\mathrm{d\&theta;},$ Its result is the ellptic integral function, can consult relevant mathematics formulary.

Know when neutral axis during by Fig. 2, i.e. θ at I ₁＜90 °, and ${\mathrm{\&theta;}}_1\&le;{\cos }^{-1}\left(\frac{a_1{a}_2}{\sqrt{a_1^2{\mathrm{<figure-callout\; id="1"\; label="b"\; filenames="H2008101883976E00012.png"\; state="\{\{state\}\}">b</figure-callout>}}_1^2-a_2^2{\mathrm{<figure-callout\; id="1"\; label="b"\; filenames="H2008101883976E00012.png"\; state="\{\{state\}\}">b</figure-callout>}}_1^2+a_1^2{a}_2^2}}\right)$ The time,

$N_{h_2}=M_{h2}=0$

Can obtain

$\left\{\begin{array}{c}N={\mathrm{<figure-callout\; id="1"\; label="N\; h"\; filenames="H2008101883976E00012.png"\; state="\{\{state\}\}">N</figure-callout>}}_h+N_g={\mathrm{\&sigma;}}_h\frac{a_1{\mathrm{<figure-callout\; id="1"\; label="b"\; filenames="H2008101883976E00012.png"\; state="\{\{state\}\}">b</figure-callout>}}_1}{6K_1}{V}_1-n\frac{\mathrm{\&mu;\&pi;}(a_1{b}_1-a_2{b}_2){\mathrm{<figure-callout\; id="1"\; label="P"\; filenames="H2008101883976E00012.png"\; state="\{\{state\}\}">P</figure-callout>}}_1{\mathrm{\&sigma;}}_h}{K_1}\\ M={\mathrm{<figure-callout\; id="1"\; label="M\; h"\; filenames="H2008101883976E00012.png"\; state="\{\{state\}\}">M</figure-callout>}}_h+M_g={\mathrm{\&sigma;}}_h\frac{a_1^5{b}_1}{K_1}{\mathrm{<figure-callout\; id="1"\; label="W"\; filenames="H2008101883976E00012.png"\; state="\{\{state\}\}">W</figure-callout>}}_1+n\frac{\mathrm{\&mu;\&pi;}(a_1{b}_1-a_2{b}_2){\mathrm{\&sigma;}}_h}{E_p{K}_1}{E}_k\end{array}\right.---\left(21\right)$

Obtain by (21) formula:

$\left\{\begin{array}{c}{e}^{\&prime;}=\frac{a_1^5{b}_1{\mathrm{<figure-callout\; id="1"\; label="W"\; filenames="H2008101883976E00012.png"\; state="\{\{state\}\}">W</figure-callout>}}_1+n\frac{\mathrm{\&mu;\&pi;}(a_1{b}_1-a_2{b}_2)}{E_p}{E}_k}{\frac{a_1{\mathrm{<figure-callout\; id="1"\; label="b"\; filenames="H2008101883976E00012.png"\; state="\{\{state\}\}">b</figure-callout>}}_1}{6}{V}_1-\mathrm{n\&mu;\&pi;}(a_1{b}_1-a_2{b}_2){\mathrm{<figure-callout\; id="1"\; label="P"\; filenames="H2008101883976E00012.png"\; state="\{\{state\}\}">P</figure-callout>}}_1}\\ {\mathrm{\&sigma;}}_h=\frac{K_{1}M}{a_1^5{b}_1{\mathrm{<figure-callout\; id="1"\; label="W"\; filenames="H2008101883976E00012.png"\; state="\{\{state\}\}">W</figure-callout>}}_1+n\frac{\mathrm{\&mu;\&pi;}(a_1{b}_1-a_2{b}_2)}{E_p}{E}_k}\end{array}\right.---\left(22\right)$

When neutral axis during at II, promptly ${\cos }^{-1}\left(\frac{a_1{a}_2}{\sqrt{a_1^2{\mathrm{<figure-callout\; id="1"\; label="b"\; filenames="H2008101883976E00012.png"\; state="\{\{state\}\}">b</figure-callout>}}_1^2-a_2^2{\mathrm{<figure-callout\; id="1"\; label="b"\; filenames="H2008101883976E00012.png"\; state="\{\{state\}\}">b</figure-callout>}}_1^2+a_1^2{a}_2^2}}\right)<{\mathrm{\&theta;}}_1<{\cos }^{-1}\left(\frac{-a_1{a}_2}{\sqrt{a_1^2{\mathrm{<figure-callout\; id="1"\; label="b"\; filenames="H2008101883976E00012.png"\; state="\{\{state\}\}">b</figure-callout>}}_1^2-a_2^2{\mathrm{<figure-callout\; id="1"\; label="b"\; filenames="H2008101883976E00012.png"\; state="\{\{state\}\}">b</figure-callout>}}_1^2+a_1^2{a}_2^2}}\right)$ The time, can obtain:

$\left\{\begin{array}{c}N=N_{h1}-N_{h2}+N_g={\mathrm{\&sigma;}}_h\frac{a_1{\mathrm{<figure-callout\; id="1"\; label="b"\; filenames="H2008101883976E00012.png"\; state="\{\{state\}\}">b</figure-callout>}}_1}{6K_1}{V}_1-{\mathrm{\&sigma;}}_h\frac{a_2{b}_2}{6K_1}{V}_2-n\frac{\mathrm{\&mu;\&pi;}(a_1{b}_1-a_2{b}_2){\mathrm{<figure-callout\; id="1"\; label="P"\; filenames="H2008101883976E00012.png"\; state="\{\{state\}\}">P</figure-callout>}}_1{\mathrm{\&sigma;}}_h}{K_1}\\ M=M_{h1}-M_{h2}+M_g={\mathrm{\&sigma;}}_h\frac{a_1^5{b}_1}{K_1}{W}_1-{\mathrm{\&sigma;}}_h\frac{a_2^5{b}_2}{K_1}{\mathrm{<figure-callout\; id="1"\; label="W"\; filenames="H2008101883976E00012.png"\; state="\{\{state\}\}">W</figure-callout>}}_1+n\frac{\mathrm{\&mu;\&pi;}(a_1{b}_1-a_2{b}_2){\mathrm{\&sigma;}}_h}{E_p{K}_1}{E}_k\end{array}\right.---\left(23\right)$

$\left\{\begin{array}{c}{e}^{\&prime;}=\frac{a_1^5{b}_1{W}_1-a_2^5{b}_2{W}_2+n\frac{\mathrm{\&mu;\&pi;}(a_1{b}_1-a_2{b}_2)}{E_p}{E}_k}{\frac{a_1{\mathrm{<figure-callout\; id="1"\; label="b"\; filenames="H2008101883976E00012.png"\; state="\{\{state\}\}">b</figure-callout>}}_1}{6}{V}_1-\frac{a_2{b}_2}{6}{V}_2-\mathrm{n\&mu;\&pi;}(a_1{b}_1-a_2{b}_2){\mathrm{<figure-callout\; id="1"\; label="P"\; filenames="H2008101883976E00012.png"\; state="\{\{state\}\}">P</figure-callout>}}_1}\\ {\mathrm{\&sigma;}}_h=\frac{K_{1}M}{a_1^5{b}_1{W}_1-a_2^5{b}_2{W}_2+n\frac{\mathrm{\&mu;\&pi;}(a_1{b}_1-a_2{b}_2)}{E_p}{E}_k}\end{array}\right.---\left(24\right)$

When neutral axis is positioned at area I II, i.e. θ ₁＞90 °, and ${\mathrm{\&theta;}}_1\&GreaterEqual;{\cos }^{-1}\left(\frac{{-a}_1{a}_2}{\sqrt{a_1^2{\mathrm{<figure-callout\; id="1"\; label="b"\; filenames="H2008101883976E00012.png"\; state="\{\{state\}\}">b</figure-callout>}}_1^2-a_2^2{\mathrm{<figure-callout\; id="1"\; label="b"\; filenames="H2008101883976E00012.png"\; state="\{\{state\}\}">b</figure-callout>}}_1^2+a_1^2{a}_2^2}}\right)$ The time, in innermost layer ellipse shown in Figure 4, get little area dA, the stress that the compressive stress at this little area place can obtain the x place by stress relation is:

${\mathrm{\&sigma;}}_{\mathrm{hx}}=\frac{{\mathrm{\&sigma;}}_h}{2K_1}(2{\mathrm{<figure-callout\; id="1"\; label="K"\; filenames="H2008101883976E00012.png"\; state="\{\{state\}\}">K</figure-callout>}}_1+x-a_1)---\left(25\right)$

Therefore obtain:

$\left\{\begin{array}{c}{N}_{h2}=\underset{A_{h2}}{\&Integral;}{\mathrm{\&sigma;}}_{\mathrm{hx}}\&CenterDot;{\mathrm{dA}}_{h2}=\underset{A_h}{\&Integral;}\frac{{\mathrm{\&sigma;}}_h}{2K_1}({\mathrm{<figure-callout\; id="1"\; label="K"\; filenames="H2008101883976E00012.png"\; state="\{\{state\}\}">K</figure-callout>}}_1+x-a_1){\mathrm{dA}}_{h2}=\frac{{\mathrm{\&pi;\&sigma;}}_h}{2K_1}(2K_1-a_1)a_2{b}_2\\ M_{h2}=\underset{A_{h2}}{\&Integral;}x\&CenterDot;{\mathrm{\&sigma;}}_{h2}\&CenterDot;{\mathrm{dA}}_{h2}=\underset{A_{h2}}{\&Integral;}x\&CenterDot;\frac{{\mathrm{\&sigma;}}_h}{2K_1}(2{\mathrm{<figure-callout\; id="1"\; label="K"\; filenames="H2008101883976E00012.png"\; state="\{\{state\}\}">K</figure-callout>}}_1+x-a_1)\&CenterDot;{\mathrm{dA}}_{h2}=\frac{{\mathrm{\&sigma;}}_h}{8{\mathrm{<figure-callout\; id="1"\; label="K"\; filenames="H2008101883976E00012.png"\; state="\{\{state\}\}">K</figure-callout>}}_1}\&CenterDot;a_2^3{b}_2\end{array}\right.---\left(26\right)$

Because $\underset{A_{h2}}{\&Integral;}x\&CenterDot;{\mathrm{dA}}_{h2}=0$ (A _H2Area moment to the oval cross section center line

Convolution (8), (10), (19), (20), (26) obtain neutral axis when the III district

$\left\{\begin{array}{c}N=N_{h1}-N_{h2}+N_g={\mathrm{\&sigma;}}_h\frac{a_1{\mathrm{<figure-callout\; id="1"\; label="b"\; filenames="H2008101883976E00012.png"\; state="\{\{state\}\}">b</figure-callout>}}_1}{6K_1}{V}_1-\frac{{\mathrm{\&pi;\&sigma;}}_h}{2K_1}(2K_1-a_1)a_2{b}_2-n\frac{\mathrm{\&mu;\&pi;}(a_1{b}_1-a_2{b}_2){\mathrm{<figure-callout\; id="1"\; label="P"\; filenames="H2008101883976E00012.png"\; state="\{\{state\}\}">P</figure-callout>}}_1{\mathrm{\&sigma;}}_h}{K_1}\\ M=M_{h1}-M_{h2}+M_g={\mathrm{\&sigma;}}_h\frac{a_1^5{b}_1}{K_1}{W}_1-\frac{{\mathrm{\&sigma;}}_h}{8{\mathrm{<figure-callout\; id="1"\; label="K"\; filenames="H2008101883976E00012.png"\; state="\{\{state\}\}">K</figure-callout>}}_1}\&CenterDot;a_2^3{b}_2+n\frac{\mathrm{\&mu;\&pi;}(a_1{b}_1-a_2{b}_2){\mathrm{\&sigma;}}_h}{E_p{K}_1}{E}_k\end{array}\right.---\left(27\right)$

$\left\{\begin{array}{c}{e}^{\&prime;}=\frac{a_1^5{b}_1{W}_1-\frac{a_2^3{b}_2}{4}+n\frac{\mathrm{\&mu;\&pi;}(a_1{b}_1-a_2{b}_2)}{E_p}{E}_k}{\frac{a_1{\mathrm{<figure-callout\; id="1"\; label="b"\; filenames="H2008101883976E00012.png"\; state="\{\{state\}\}">b</figure-callout>}}_1}{6}{V}_1-\frac{{\mathrm{\&pi;}(2K_1-a_1)a}_{2}b}{2}-\mathrm{n\&mu;\&pi;}(a_1{b}_1-a_2{b}_2){\mathrm{<figure-callout\; id="1"\; label="P"\; filenames="H2008101883976E00012.png"\; state="\{\{state\}\}">P</figure-callout>}}_1}\\ {\mathrm{\&sigma;}}_h=\frac{K_{1}M}{a_1^5{b}_1{W}_1-\frac{a_2^3{b}_2}{8}+n\frac{\mathrm{\&mu;\&pi;}(a_1{b}_1-a_2{b}_2)}{E_p}{E}_k}\end{array}\right.---\left(28\right)$

Fig. 5～Fig. 6 is the E of Reinforced Concrete Bridge pier stud of the present invention _p(θ), E _k(θ) change curve.

The bridge pier of elliptic cross-section all has stronger superiority at aspects such as wind resistance, anti-current-rush, anti-ice slush bumps, by the above derivation of equation as can be seen, the maximum bending resistance moment of inertia of elliptic cross-section is bigger than circular cross-section under situation of the same area, therefore in that to reach under the situation of identical stress the comparable circular section of oval cross section little, thereby saved concrete amount.The expression formula of the oval ring cross section pier stud that the present invention provides concrete and reinforcement stresses under the large eccentric pressuring situation, this expression formula is to solid section (a ₂=b ₂=0) same being suitable for.Derivation by above obtains oval pier stud concrete on three different pressure zones, the stress expression formula of reinforcing bar is different, in the actual calculation process, should determine the scope of pressure zone by tentative calculation according to known parameters and eccentric throw, select different stress calculation formula.

The present invention is not limited to above-mentioned preferred forms, and anyone should learn the structural change of making under enlightenment of the present invention, and every have identical or close technical scheme with the present invention, all falls within protection scope of the present invention.

Claims (2)

1. Gang Zhu concrete bridge pier column structure, it is characterized in that the cross section ovalize of described bridge pier column, and have an oval-shaped cavity that the direction of its major axis and minor axis is consistent with oval-shaped major axis of pier stud outline and short-axis direction respectively, stressed main reinforcement is evenly arranged along oval circumference

Stressed eccentric throw in bridge pier column cross section and concrete stress expression formula satisfy:

(1) works as α ₁＜90 °, and ${\mathrm{\&alpha;}}_1\&le;{\cos }^{-1}\left(\frac{a_1{a}_2}{\sqrt{a_1^2{b}_1^2-a_2^2{b}_1^2+a_1^2{a}_2^2}}\right)$ The time

$\left\{\begin{array}{c}{e}^{\&prime;}=\frac{a_1^5{b}_1{W}_1+n\frac{\mathrm{\&mu;\&pi;}(a_1{b}_1-a_2{b}_2)a^2{b}^2}{2E_p}{E}_k}{\frac{a_1{b}_1}{6}{V}_1-\frac{\mathrm{n\&mu;\&pi;}(a_1{b}_1-a_2{b}_2)P_1}{2}}\\ {\mathrm{\&sigma;}}_h=\frac{K_{1}M}{a_1^5{b}_1{W}_1+n\frac{\mathrm{\&mu;\&pi;}(a_1{b}_1-a_2{b}_2)}{2E_p}{E}_k}\end{array}\right.;$

(2) when ${\cos }^{-1}\left(\frac{a_1{a}_2}{\sqrt{a_1^2{b}_1^2-a_2^2{b}_1^2+a_1^2{a}_2^2}}\right)<{\mathrm{\&alpha;}}_1<{\cos }^{-1}\left(\frac{-a_1{a}_2}{\sqrt{a_1^2{b}_1^2-a_2^2{b}_1^2+a_1^2{a}_2^2}}\right)$ The time

$\left\{\begin{array}{c}{e}^{\&prime;}=\frac{a_1^5{b}_1{W}_1-a_2^5{b}_2{W}_2+n\frac{\mathrm{\&mu;\&pi;}(a_1{b}_1-a_2{b}_2)a^2{b}^2}{2E_p}{E}_k}{\frac{a_1{b}_1}{6}{V}_1-\frac{a_2{b}_2}{6}{V}_2-\mathrm{n\&mu;\&pi;}(a_1{b}_1-a_2{b}_2)P_1}\\ {\mathrm{\&sigma;}}_h=\frac{K_{1}M}{a_1^5{b}_1{W}_1-a_2^5{b}_2{W}_2+n\frac{\mathrm{\&mu;\&pi;}(a_1{b}_1-a_2{b}_2)a^2{b}^2}{2E_p}{E}_k}\end{array}\right.;$

(3) work as α ₁＞90 °, and ${\mathrm{\&alpha;}}_1\&GreaterEqual;{\cos }^{-1}\left(\frac{-a_1{a}_2}{\sqrt{a_1^2{b}_1^2-a_2^2{b}_1^2+a_1^2{a}_2^2}}\right)$ The time

$\left\{\begin{array}{c}{e}^{\&prime;}=\frac{a_1^5{b}_1{W}_1-\frac{\mathrm{\&pi;}{a}_2^3{b}_2}{8}+n\frac{\mathrm{\&mu;\&pi;}(a_1{b}_1-a_2{b}_2)a^2{b}^2}{2E_p}{E}_k}{\frac{a_1{b}_1}{6}{V}_1-\frac{\mathrm{\&pi;}(2K_1-a_1)a_2{b}_2}{2}-\mathrm{n\&mu;\&pi;}(a_1{b}_1-a_2{b}_2)P_1}\\ {\mathrm{\&sigma;}}_h=\frac{K_{1}M}{a_1^5{b}_1{W}_1-\frac{\mathrm{\&pi;}{a}_2^3{b}_2}{8}+n\frac{\mathrm{\&mu;\&pi;}(a_1{b}_1-a_2{b}_2)a^2{b}^2}{2E_p}{E}_k}\end{array}\right.;$

In the formula:

a ₁And b ₁Represent pier stud outline transverse and minor axis respectively;

a ₂And b ₂Represent wide transverse and minor axis in the pier stud respectively;

A and b represent the major axis and the minor axis of the oval circumference in reinforcing bar place in the pier stud respectively;

μ represents the cross section rebar ratio;

N represents reinforcing bar modulus of elasticity and concrete;

M represents the moment of flexure value on the cross section;

α ₁The central angle of expression concrete compression cross section correspondence

θ, φ represent integration variable, value be scope be [0, α ₁]

$E_p,E_k-E_p={\&Integral;}_0^{2\mathrm{\&pi;}}\frac{\mathrm{ab}}{\sqrt{b^2{\cos }^2\mathrm{\&phi;}+a^2{\sin }^2\mathrm{\&phi;}}}\mathrm{d\&phi;};$ $E_k={\&Integral;}_0^{2\mathrm{\&pi;}}\frac{\mathrm{ab}{\cos }^2\mathrm{\&theta;}}{{(b^2{\cos }^2\mathrm{\&theta;}+a^2{\sin }^2\mathrm{\&theta;})}^{\frac{3}{2}}}\mathrm{d\&theta;}$

${\mathrm{\&rho;}}_1\left({\mathrm{\&theta;}}_1\right)=\frac{a_1{b}_1}{\sqrt{b_1^2{\cos }^2{\mathrm{\&theta;}}_1+a_1^2{\sin }^2{\mathrm{\&theta;}}_1}}$

2K ₁＝a ₁-ρ ₁(α ₁)cosα ₁

P ₁＝ρ ₁(α ₁)cosα ₁

$V_1=[\frac{2a_1^4{\sin }^3{\mathrm{\&alpha;}}_1}{{(b_1^2{\cos }^2{\mathrm{\&alpha;}}_1+a_1^2{\sin }^2{\mathrm{\&alpha;}}_1)}^{\frac{3}{2}}}-3P_1(\arctan \left(\frac{a_1\tan {\mathrm{\&alpha;}}_1}{b_1}\right)-\frac{a_1{b}_1\tan {\mathrm{\&alpha;}}_1}{(b_1^2+a_1^2{\tan }^2{\mathrm{\&alpha;}}_1)})]$

$V_2=[\frac{2a_2^4{\sin }^3{\mathrm{\&alpha;}}_2}{{(b_2^2{\cos }^2{\mathrm{\&alpha;}}_2+a_2^2{\sin }^2{\mathrm{\&alpha;}}_2)}^{\frac{3}{2}}}-3P_2(\arctan \left(\frac{a_2\tan {\mathrm{\&alpha;}}_2}{b_2}\right)-\frac{a_2{b}_2\tan {\mathrm{\&alpha;}}_2}{(b_2^2+a_2^2{\tan }^2{\mathrm{\&alpha;}}_2)})]$

$W_1=[\frac{\arctan \left(\frac{a_1\tan {\mathrm{\&alpha;}}_1}{b_1}\right)}{8{a_1}^2{b}_1}+\frac{\tan {\mathrm{\&alpha;}}_1(a_1^2{\tan }^2{\mathrm{\&alpha;}}_1-b_1^2)}{8a_1{(b_1^2+a_1^2{\tan }^2{a}_1)}^2}]-\frac{P_1}{3}\left[\frac{{\sin }^3{\mathrm{\&alpha;}}_1}{{(b_1^2{\cos }^2{\mathrm{\&alpha;}}_1+a_1^2{\sin }^2{\mathrm{\&alpha;}}_1)}^{\frac{3}{2}}}\right]$

$W_2=[\frac{\arctan \left(\frac{a_2\tan {\mathrm{\&alpha;}}_2}{b_2}\right)}{8{a_2}^2{b}_2}+\frac{\tan {\mathrm{\&alpha;}}_2(a_2^2{\tan }^2{\mathrm{\&alpha;}}_2-b_2^2)}{8a_2{(b_2^2+a_2^2{\tan }^2{a}_2)}^2}]-\frac{P_2}{3}\left[\frac{{\sin }^3{\mathrm{\&alpha;}}_2}{{(b_2^2{\cos }^2{\mathrm{\&alpha;}}_2+a_2^2{\sin }^2{\mathrm{\&alpha;}}_2)}^{\frac{3}{2}}}\right].$

2. Gang Zhu concrete bridge pier column structure as claimed in claim 1 is characterized in that described bridge pier column is equal to or greater than wall thickness on the ellipse short shaft direction at the wall thickness on the transverse direction.

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