# In-Plane Instability of Parabolic Arches under Uniformly Distributed Vertical Load Coupled with Temperature Gradient Field

Authors: Zhenyu Pan; Deyuan Deng; Changsheng Feng; Hanwen Lu; Zhou Chen; Jinman Zhou; Angfeng Jiang
Journal: Advances in Civil Engineering
 (2022)

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## Abstract

In civil engineering, arches, such as steel arch roofs and arch bridges, are often subjected to linear temperature gradient field. It is known that the in-plane instability of parabolic arches is caused by the significant axial force. The arch under the linear temperature gradient field produces complex axial force, and so the instability of arches would be affected by temperature gradient field significantly. However, the analytical solutions of in-plane instability of parabolic arches being subjected to the uniformly distributed vertical load and the temperature gradient field are not solved in the opening literature. In this paper, in-plane instability of a fixed steel parabolic arch under linear temperature gradient field and vertical uniform load is analyzed theoretically. Firstly, the cross-sectional effective centroid and effective stiffness of the cross section for arches under the linear temperature gradient field are derived. Secondly, the preinstability internal force analysis of the parabolic arch under the linear temperature gradient field and the vertical uniform load is carried out based on the force methods. Novel theoretical solutions for in-plane instability load for fixed steel parabolic arches under the linear temperature gradient field and the vertical uniform load are obtained. It is found that the gradient temperature, slenderness, and rise-span ratio have important influences on the critical in-plane instability load of the shallow parabolic arch, while there is no significant effect on the deep parabolic arch.


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## Body

## 1. Introduction

Long-span steel arch structures are widely applied to engineering, for instance, long-span steel roofs of the terminal, long-span steel arch bridge, etc. For large-span steel roofs, the temperature inside and outside of the roof is different due to sun exposure. In summer, the temperature inside the structure is lower than that outside the structure, while in winter, the temperature inside the structure is higher than that outside the structure. The inconsistence of internal and external temperature will produce linear gradient temperature field and internal force to the structure, and the linear gradient temperature field and internal force affect the bearing capacity of the structure. Except for the linear gradient temperature field caused by solar irradiation, the fire inside the structure will cause the internal temperature of the structure to be higher than the external temperature and then generates the linear gradient temperature field. Hence, the study of the steel arch structure under linear temperature gradient field has a profound significance on the fire resistance design of the steel arch structure.Many scholars have conducted a series of research studies on the buckling of steel arch structures. Pi and Trahair. [1] studied the inelastic lateral buckling strength and design of steel arches under general loading using an advanced nonlinear inelastic finite element method of analysis. Mallon et al. [2] researched the influence of the initial curvature of thin shallow arches on the dynamic pulse buckling load. Moon et al. [3] investigated the geometrically nonlinear behavior of pin-ended shallow parabolic steel arches subjected to a vertically distributed load for assessing the buckling load. Pi and Bradford. [4] researched the dynamic in-plane buckling of a shallow pin-ended circular arch under a central radial load that is applied suddenly with infinite duration. Bradford et al. [5] researched the prebuckling behavior of a pin-ended circular arch under a uniform radial load. Han et al. [6] researched the in-plane nonlinear behavior and stability of shallow circular arches with elastic horizontal supports under a uniform radial load by the principle of virtual work. Zeng et al. [7] studied the stability analysis of elastic restrained arc steel arch under concentrated load. Kang et al. [8] conducted a dynamic response analysis of an elastically supported arc steel arch structure under blast impact loading. Yan et al. [9] researched a nonuniform shallow arch characterized by three constant stiffness regions under a central concentrated load. Li et al. [10] established an analytical solution to predict the buckling load of the thin-walled arch under a point load at midspan position. Fan et al. [11] carried out an analytical study and a numerical simulation on the nonlinear in-plane buckling behavior of the shallow parabolic steel arches with tension cables and pin joints. Pi and Bradford [12] researched the nonlinear thermoelastic buckling behavior of articulated shallow arc steel arches under the impact of linear temperature gradient field. Bouras and Vrcelj. [13] carried out the nonlinear elastic prebuckling and in-plane buckling analysis of the circular shallow arches under a uniformly distributed load and time-varying uniform temperature field. Song et al. [14] analyzed an in-plane jump buckling and bending behavior for arc steel arches subjected to fire. Asgari et al. [15] theoretically researched the nonlinear thermoelastic behavior of pin-ended functionally graded material (FGM) circular shallow arches. Li and Zheng, [16] investigated the buckling of confined thin-walled functionally graded material (FGM) arch subjected to external pressure. Tang et al. [17] researched the in-plane asymmetric buckling of the heated functionally graded material (FGM) circular arches under uniform pressure fields. Li et al. [18] researched the analytical process of the functionally graded porous (FGP) arch structure in an elevated thermal field. Cai et al. [19] researched the in-plane stability of rotationally restrained parabolic shallow steel arches under a vertical uniform load and temperature changes below 100°C and used the virtual work principle method to establish the nonlinear equilibrium and buckling equations. Pi and Bradford [20] researched nonlinear in-plane buckling of circular arches being subjected to uniform radial and thermal loading. However, only a few scholars have researched the parabolic steel arches under the temperature field, especially the effect of temperature gradient field. Parabolic arches are widely used in practical engineering. The internal force analysis and stability analysis of parabolic arches are important parts of arch design, construction and maintenance, etc. However, the analytical solutions of in-plane instability of parabolic arches being subjected to uniformly distributed vertical load and temperature gradient field are not solved in the opening.Hence, this paper derives the cross-sectional effective centroid and effective stiffness for parabolic arches under the linear temperature gradient field and obtains the axial and bending actions of parabolic arches under linear temperature gradient field coupled with vertical uniform load. In addition, the analytical solution of the critical load for in-plane instability of the parabolic arches under temperature gradient field coupled with vertical uniform load is also obtained, and it is verified by the numerical simulations software ANSYS.


## 2. Analysis of Geometrical and Material Properties of Parabolic Arch

### 2.1. Basic Analysis

#### 2.1.1. Mechanical Model

The fixed parabolic steel arch under linear gradient temperature field coupled with vertical uniformly distributed load is considered as the study object, which is plotted in Figure1.Figure 1
Mechanical diagram of parabolic arch. (a) Fixed arch. (b) Cross section.
(a)(b)o∗x∗y∗z∗ denotes the initial coordinates of the parabolic steel arch, o∗ is situated in the cross-sectional geometric center of vault, and o∗x∗, o∗y∗, and o∗z∗ are the horizontal coordinate axis, the vertical coordinate axis, and the lateral coordinate axis of the initial coordinates, respectively. o∗s∗ is the geometric centroid axis of the initial coordinates. Figure 1(a) shows that f, L, l, and S are the rise, span, half-span, and length of the parabolic steel arch, respectively. Figure 1(b) shows that T1 and T2 are top and bottom cross-sectional temperatures, respectively. h represents the cross-sectional height.In addition, the equation of geometric coordinates of the parabolic steel arch can be defined based on Figure1 as(1)y∗=a1l2x12,where a1 represents the shape factor of the parabolic steel arch, which can be given by(2)a1=2fl.x1 represents nondimensional coordinate of the o∗x∗ axis, which is denoted as x1=x∗/l, and so the arc differential ds of the parabolic arch can be calculated by(3)ds=l1+a1x12dx1.


#### 2.1.2. Basic Hypothesis

The in-plane instability analysis for parabolic steel arches under the linear temperature gradient field studied in this paper satisfies the following hypothesis.(1)
The environment temperature is20°C.(2)
Cross-sectional temperature, parabolic arch deformation, temperature dilatation factora, and Poisson’s ratio v are independent of time.(3)
The temperatureT of the cross section of the arch varies uniformly along the o∗s∗ axis and the o∗x∗ axis and linearly along the o∗y∗ axis, as shown in Figure 2, where T1<T2.Figure 2
Linear gradient temperature field.Hence, the temperature at any point on the cross section of parabolic arch can be calculated by(4)Ty=Ta+y∗ΔTgh,where Ta represents the cross-sectional average temperature of the parabolic arch, which can be obtained by(5)Ta=T1+T22,and ΔTg represents temperature difference value between top and bottom of the cross section, which can be calculated by(6)ΔTg=T2−T1.


#### 2.1.3. Modulus of Elasticity

In this paper, steel is chosen as the material for the arch. The modulus of elasticity of steelET=ξT⋅E0, where E0 is the modulus of elasticity of Q235 steel at temperature 20°C, and ξT is the temperature affection factor, which can be given by (7)ξT=7T−47806T−4760,0°C<T≤600°C.Figure3 shows the influence of temperature on steel elastic modulus. As shown in the picture, steel elastic modulus decreases with the an increase of material temperature, and the value decreases more remarkably with the augment of temperature.Figure 3
Relationship between modulus of elasticity and temperature.


#### 2.1.4. Effective Stiffness and Effective Centroid

The I section is taken as the cross section of parabolic steel arch studied in this paper, and when the parabolic steel arch is under the linear gradient temperature field, the elastic modulus changes along theo∗y∗ axis, and the vertical coordinate of effective centroid also changes. Therefore, the effective coordinate system oxyz can be determined by the location of the effective centroid, which is shown in Figure 4.Figure 4
Microelement of I-section arch structure.ox, oy, and oz denote the effective horizontal axis, effective vertical axis, and effective lateral axis, respectively. os is the effective centroid axis of the effective coordinate system. By substituting (4) into (7), the elastic modulus along the o∗y∗ axis is given by(8)ETy∗=E07Ta−4780h+7ΔTgy∗6Ta−4760h+6ΔTgy∗.Accordingly, the vertical coordinate of effective centroid of theo∗y∗ axis is given by(9)yc∗=−yc=−−∫AETy∗y∗dA∫AETy∗dA=∫AETy∗y∗dA∫AETy∗dA,where yc is the vertical coordinate of effective centroid of the oy axis.The effect of linear gradient temperature field on the effective center of parabolic arch cross section is shown in Figure5, and it can be seen that the effective center of shape is shifted to the side with lower temperature under linear gradient temperature field, and the effect of gradient temperature on the I section is greater than that on the rectangular section.Figure 5
Influence of the temperature difference with the vertical coordinate of effective centroid.In addition,To is the temperature of the effective centroid, which can be given by(10)To=Ta+ΔTgyc∗h.By replacing the vertical coordinatey∗ with y∗=y−yc, the elastic modulus along the oy axis can be given by(11)ETy=E07Ta−4780h+7ΔTgy−yc6Ta−4760h+6ΔTgy−yc.For ensuring that the analysis of the internal forces of the parabolic arch under gradient temperature is precise, the effective stiffnessesEI¯ and EA¯ of the arch section are derived, which can be given separately by(12)EI¯=∫AETyy2dA,=2320Tsh2E027ΔTg212twh+tfb−tw+7E0twh372ΔTg+7E0tf12ΔTgh2−2tfh+43tf2b−tw−yc2EA¯+1160h3Ts2E081ΔTg3b−twΞ1+bΞ2,EA¯=∫AETydA=1160hE09ΔTgb−twΞ1+bΞ2+7E0312twh+tfb−tw,where tf represents the thickness of the flange plate of the I section, tw represents the thickness of the web of the I section, and b,h, and A represent the width, height, and area of the I section, separately. Beyond that, the parameters Ts, Ξ12 can be mathematically expressed as(13)Ts=2380−3Ta,Ξ1=ln2Tsh+3h−2tfΔTg2Tsh−3h−2tfΔTg,Ξ2=ln2Ts−3ΔTg2Ts+3ΔTg.


## 2.1. Basic Analysis

### 2.1.1. Mechanical Model

The fixed parabolic steel arch under linear gradient temperature field coupled with vertical uniformly distributed load is considered as the study object, which is plotted in Figure1.Figure 1
Mechanical diagram of parabolic arch. (a) Fixed arch. (b) Cross section.
(a)(b)o∗x∗y∗z∗ denotes the initial coordinates of the parabolic steel arch, o∗ is situated in the cross-sectional geometric center of vault, and o∗x∗, o∗y∗, and o∗z∗ are the horizontal coordinate axis, the vertical coordinate axis, and the lateral coordinate axis of the initial coordinates, respectively. o∗s∗ is the geometric centroid axis of the initial coordinates. Figure 1(a) shows that f, L, l, and S are the rise, span, half-span, and length of the parabolic steel arch, respectively. Figure 1(b) shows that T1 and T2 are top and bottom cross-sectional temperatures, respectively. h represents the cross-sectional height.In addition, the equation of geometric coordinates of the parabolic steel arch can be defined based on Figure1 as(1)y∗=a1l2x12,where a1 represents the shape factor of the parabolic steel arch, which can be given by(2)a1=2fl.x1 represents nondimensional coordinate of the o∗x∗ axis, which is denoted as x1=x∗/l, and so the arc differential ds of the parabolic arch can be calculated by(3)ds=l1+a1x12dx1.


### 2.1.2. Basic Hypothesis

The in-plane instability analysis for parabolic steel arches under the linear temperature gradient field studied in this paper satisfies the following hypothesis.(1)
The environment temperature is20°C.(2)
Cross-sectional temperature, parabolic arch deformation, temperature dilatation factora, and Poisson’s ratio v are independent of time.(3)
The temperatureT of the cross section of the arch varies uniformly along the o∗s∗ axis and the o∗x∗ axis and linearly along the o∗y∗ axis, as shown in Figure 2, where T1<T2.Figure 2
Linear gradient temperature field.Hence, the temperature at any point on the cross section of parabolic arch can be calculated by(4)Ty=Ta+y∗ΔTgh,where Ta represents the cross-sectional average temperature of the parabolic arch, which can be obtained by(5)Ta=T1+T22,and ΔTg represents temperature difference value between top and bottom of the cross section, which can be calculated by(6)ΔTg=T2−T1.


### 2.1.3. Modulus of Elasticity

In this paper, steel is chosen as the material for the arch. The modulus of elasticity of steelET=ξT⋅E0, where E0 is the modulus of elasticity of Q235 steel at temperature 20°C, and ξT is the temperature affection factor, which can be given by (7)ξT=7T−47806T−4760,0°C<T≤600°C.Figure3 shows the influence of temperature on steel elastic modulus. As shown in the picture, steel elastic modulus decreases with the an increase of material temperature, and the value decreases more remarkably with the augment of temperature.Figure 3
Relationship between modulus of elasticity and temperature.


### 2.1.4. Effective Stiffness and Effective Centroid

The I section is taken as the cross section of parabolic steel arch studied in this paper, and when the parabolic steel arch is under the linear gradient temperature field, the elastic modulus changes along theo∗y∗ axis, and the vertical coordinate of effective centroid also changes. Therefore, the effective coordinate system oxyz can be determined by the location of the effective centroid, which is shown in Figure 4.Figure 4
Microelement of I-section arch structure.ox, oy, and oz denote the effective horizontal axis, effective vertical axis, and effective lateral axis, respectively. os is the effective centroid axis of the effective coordinate system. By substituting (4) into (7), the elastic modulus along the o∗y∗ axis is given by(8)ETy∗=E07Ta−4780h+7ΔTgy∗6Ta−4760h+6ΔTgy∗.Accordingly, the vertical coordinate of effective centroid of theo∗y∗ axis is given by(9)yc∗=−yc=−−∫AETy∗y∗dA∫AETy∗dA=∫AETy∗y∗dA∫AETy∗dA,where yc is the vertical coordinate of effective centroid of the oy axis.The effect of linear gradient temperature field on the effective center of parabolic arch cross section is shown in Figure5, and it can be seen that the effective center of shape is shifted to the side with lower temperature under linear gradient temperature field, and the effect of gradient temperature on the I section is greater than that on the rectangular section.Figure 5
Influence of the temperature difference with the vertical coordinate of effective centroid.In addition,To is the temperature of the effective centroid, which can be given by(10)To=Ta+ΔTgyc∗h.By replacing the vertical coordinatey∗ with y∗=y−yc, the elastic modulus along the oy axis can be given by(11)ETy=E07Ta−4780h+7ΔTgy−yc6Ta−4760h+6ΔTgy−yc.For ensuring that the analysis of the internal forces of the parabolic arch under gradient temperature is precise, the effective stiffnessesEI¯ and EA¯ of the arch section are derived, which can be given separately by(12)EI¯=∫AETyy2dA,=2320Tsh2E027ΔTg212twh+tfb−tw+7E0twh372ΔTg+7E0tf12ΔTgh2−2tfh+43tf2b−tw−yc2EA¯+1160h3Ts2E081ΔTg3b−twΞ1+bΞ2,EA¯=∫AETydA=1160hE09ΔTgb−twΞ1+bΞ2+7E0312twh+tfb−tw,where tf represents the thickness of the flange plate of the I section, tw represents the thickness of the web of the I section, and b,h, and A represent the width, height, and area of the I section, separately. Beyond that, the parameters Ts, Ξ12 can be mathematically expressed as(13)Ts=2380−3Ta,Ξ1=ln2Tsh+3h−2tfΔTg2Tsh−3h−2tfΔTg,Ξ2=ln2Ts−3ΔTg2Ts+3ΔTg.


## 2.1.1. Mechanical Model

The fixed parabolic steel arch under linear gradient temperature field coupled with vertical uniformly distributed load is considered as the study object, which is plotted in Figure1.Figure 1
Mechanical diagram of parabolic arch. (a) Fixed arch. (b) Cross section.
(a)(b)o∗x∗y∗z∗ denotes the initial coordinates of the parabolic steel arch, o∗ is situated in the cross-sectional geometric center of vault, and o∗x∗, o∗y∗, and o∗z∗ are the horizontal coordinate axis, the vertical coordinate axis, and the lateral coordinate axis of the initial coordinates, respectively. o∗s∗ is the geometric centroid axis of the initial coordinates. Figure 1(a) shows that f, L, l, and S are the rise, span, half-span, and length of the parabolic steel arch, respectively. Figure 1(b) shows that T1 and T2 are top and bottom cross-sectional temperatures, respectively. h represents the cross-sectional height.In addition, the equation of geometric coordinates of the parabolic steel arch can be defined based on Figure1 as(1)y∗=a1l2x12,where a1 represents the shape factor of the parabolic steel arch, which can be given by(2)a1=2fl.x1 represents nondimensional coordinate of the o∗x∗ axis, which is denoted as x1=x∗/l, and so the arc differential ds of the parabolic arch can be calculated by(3)ds=l1+a1x12dx1.


## 2.1.2. Basic Hypothesis

The in-plane instability analysis for parabolic steel arches under the linear temperature gradient field studied in this paper satisfies the following hypothesis.(1)
The environment temperature is20°C.(2)
Cross-sectional temperature, parabolic arch deformation, temperature dilatation factora, and Poisson’s ratio v are independent of time.(3)
The temperatureT of the cross section of the arch varies uniformly along the o∗s∗ axis and the o∗x∗ axis and linearly along the o∗y∗ axis, as shown in Figure 2, where T1<T2.Figure 2
Linear gradient temperature field.Hence, the temperature at any point on the cross section of parabolic arch can be calculated by(4)Ty=Ta+y∗ΔTgh,where Ta represents the cross-sectional average temperature of the parabolic arch, which can be obtained by(5)Ta=T1+T22,and ΔTg represents temperature difference value between top and bottom of the cross section, which can be calculated by(6)ΔTg=T2−T1.


## 2.1.3. Modulus of Elasticity

In this paper, steel is chosen as the material for the arch. The modulus of elasticity of steelET=ξT⋅E0, where E0 is the modulus of elasticity of Q235 steel at temperature 20°C, and ξT is the temperature affection factor, which can be given by (7)ξT=7T−47806T−4760,0°C<T≤600°C.Figure3 shows the influence of temperature on steel elastic modulus. As shown in the picture, steel elastic modulus decreases with the an increase of material temperature, and the value decreases more remarkably with the augment of temperature.Figure 3
Relationship between modulus of elasticity and temperature.


## 2.1.4. Effective Stiffness and Effective Centroid

The I section is taken as the cross section of parabolic steel arch studied in this paper, and when the parabolic steel arch is under the linear gradient temperature field, the elastic modulus changes along theo∗y∗ axis, and the vertical coordinate of effective centroid also changes. Therefore, the effective coordinate system oxyz can be determined by the location of the effective centroid, which is shown in Figure 4.Figure 4
Microelement of I-section arch structure.ox, oy, and oz denote the effective horizontal axis, effective vertical axis, and effective lateral axis, respectively. os is the effective centroid axis of the effective coordinate system. By substituting (4) into (7), the elastic modulus along the o∗y∗ axis is given by(8)ETy∗=E07Ta−4780h+7ΔTgy∗6Ta−4760h+6ΔTgy∗.Accordingly, the vertical coordinate of effective centroid of theo∗y∗ axis is given by(9)yc∗=−yc=−−∫AETy∗y∗dA∫AETy∗dA=∫AETy∗y∗dA∫AETy∗dA,where yc is the vertical coordinate of effective centroid of the oy axis.The effect of linear gradient temperature field on the effective center of parabolic arch cross section is shown in Figure5, and it can be seen that the effective center of shape is shifted to the side with lower temperature under linear gradient temperature field, and the effect of gradient temperature on the I section is greater than that on the rectangular section.Figure 5
Influence of the temperature difference with the vertical coordinate of effective centroid.In addition,To is the temperature of the effective centroid, which can be given by(10)To=Ta+ΔTgyc∗h.By replacing the vertical coordinatey∗ with y∗=y−yc, the elastic modulus along the oy axis can be given by(11)ETy=E07Ta−4780h+7ΔTgy−yc6Ta−4760h+6ΔTgy−yc.For ensuring that the analysis of the internal forces of the parabolic arch under gradient temperature is precise, the effective stiffnessesEI¯ and EA¯ of the arch section are derived, which can be given separately by(12)EI¯=∫AETyy2dA,=2320Tsh2E027ΔTg212twh+tfb−tw+7E0twh372ΔTg+7E0tf12ΔTgh2−2tfh+43tf2b−tw−yc2EA¯+1160h3Ts2E081ΔTg3b−twΞ1+bΞ2,EA¯=∫AETydA=1160hE09ΔTgb−twΞ1+bΞ2+7E0312twh+tfb−tw,where tf represents the thickness of the flange plate of the I section, tw represents the thickness of the web of the I section, and b,h, and A represent the width, height, and area of the I section, separately. Beyond that, the parameters Ts, Ξ12 can be mathematically expressed as(13)Ts=2380−3Ta,Ξ1=ln2Tsh+3h−2tfΔTg2Tsh−3h−2tfΔTg,Ξ2=ln2Ts−3ΔTg2Ts+3ΔTg.


## 3. Results and Discussion

### 3.1. Internal Force Analysis

As the parabolic steel arches are linear elastic, their strain energy in the preinstability state under linear temperature gradient field coupled with vertical uniformly distributed loadq can be given by(14)Uss0=12∫−S/2S/2∫AETεxx02dAds−∫−S/2S/2∫AETαΔTo−yΔTghεxx0dAds,with ΔTo=To−20°C, where A is the cross-sectional area, ET, α, and εxx0 are the modulus of elasticity, the thermal coefficient of the steel, and the linear normal strain, respectively. εxx0 is given by(15)εxx0=εm0+εb0=ds1−dsds+y1ρ1−1ρ,where ds1 is the arc differential after deformation and ρ and ρ1 are radii of curvature of the parabolic steel arches before and after deformation, respectively. In addition, the axial force N and bending moment M can be expressed as(16)N=−∫AETεxx0dA=−∫AETyds1−dsds+y1ρ1−1ρdA=−EA¯εm0,(17)M=−∫AETyεxx0dA=−∫AETyyds1−dsds+y1ρ1−1ρdA=−EI¯εb0.Accordingly, by substituting (16) and (17) into (14), the term of the strain energy can be simplified as(18)∫AETεxx02dA=∫AETyds1−dsds+y1ρ1−1ρ2dA=EA¯εm02+EI¯εb02=N2EA¯+M2EI¯.By substituting equations (3), (16), (17), and (18) into equation (14), the strain energy can be further simplified as(19)Uss0=12∫−11N2EA¯+M2EI¯lV1dx1−αΔT0∫−11NlV1dx1+αΔTgh∫−11MlV1dx1,where(20)V1=1+a1x12.The accurate solutions of the preinstability axial forceN and bending moment M of a parabolic steel arch are essential for the in-plane instability of the arch. However, no accurate solutions of internal forces for arches being subjected to linear temperature gradient filed and vertical uniformly distributed load q can be obtained in the opening literature. The internal force of the arch can be solved by the force method which is based on Castigliano’s theorems, and when an arch is under linear temperature gradient field coupled with vertical uniformly distributed load q, the internal forces for the arch can be herein calculated by the force method in this paper.Based on the force method, the parabolic steel arch is cut into two pieces at the top, which is shown in Figure6. These two parts are the statically determinate structures with unknown additional axial forces Nc, bending moment Mc, and shear force Qyc. However, according to the principle of structural symmetry, the unknown additional shear force Qyc is equal to zero.Figure 6
Parabolic arch force method diagram.In addition, to make the cut arch equivalent to the original arch, the relative axial displacementΔNC and the relative rotation ΔMC at the cutting location should be equal to zero. Based on Castigliano’s theorems, the relative axial displacement ΔNC and the relative rotation ΔMC can be, respectively, given by(21)ΔNC=∂Uss0∂Nc=0,(22)ΔMC=∂Uss0∂Mc=0.Substituting (19) into (21) and (22), respectively, the corresponding force method equation can be obtained as(23)∫−11NEA¯∂N∂Nc+MEI¯∂M∂NclV1dx1−αΔT0∫−11∂N∂NclV1dx1+αΔTgh∫−11∂M∂NclV1dx1=0,(24)∫−11NEA¯∂N∂Mc+MEI¯∂M∂MclV1dx1−αΔT0∫−11∂N∂MclV1dx1+αΔTgh∫−11∂M∂MclV1dx1=0.Because the segmentations of two part of the parabolic steel arch are the statically determinate structures, based on the principle of force equilibrium, the axial forceN and bending moment M can be solved, respectively, as(25)N=qla1x12−Nca12x12+1,(26)M=Mc−qlx122−Nca1lx1.The solutions of the internal force obtained from (25) and (26) have two unknown values (Nc and Mc); to solve them, substituting (25) and (26) into (23) and (24), Nc and Mc can be obtained as(27)Nc=Γ1EA¯αΔTo−Γ2qlΨ,(28)Mc=3lΓ4ql−Γ3EA¯αΔToΨΞ−2a1a12+1−EI¯αΔTgh,where(29)Γ1=768rx2a14Ξ1−2a1a12+1,Γ2=3Ξ264a12rx2+l2−4a18a14+8a12+3Ξl2a12+1+4a12a12+14a14−8a12−9l2−192rx2a12,Γ3=32rx2a12Ξ2+4a13Ξa12+1−4a122a14−3a12−1Γ4=8rx24a124a12+1a12+1Ξ−Ξ3−2a1a12+12a12−1Ξ2−8a132a12+1a12+13,Ψ=3Ξ2a1128a12rx2+l2−4a12a12+18a14+8a12+3l2+192rx2a12Ξ+4a13l24a14−8a12−9a12+1,Ξ=lna12+1+a1−lna12+1−a1.Hence, the accurate solutions of axial forceN and bending moment M can be obtained by substituting (27) and (28) into (25) and (26).In order to expound the effect of the linear temperature gradient field onN and M in (23) and (24), the changes of Nc and Mc with rise-span ratio f/L for parabolic arches having temperature difference between the top and base of the cross section ΔTg (i.e., ΔTg=50°C,100°C,150°C) are shown in Figures 7(a) and 7(b), respectively, where Nc and Mc are the central axial and bending actions, and Ta=200°C, q=2×105kN/m, and L/γx=30 with γx being the radius of gyration of the arch. Beyond that, the changes of Nc and Mc with rise-span ratio f/L for parabolic arches having different average temperatures of the cross section Ta (i.e., Ta=150°C,200°C,250°C) are plotted in Figures 7(c) and 7(d), respectively, where ΔTg=100°C, q=2×105kN/m, and L/γx=30.Figure 7
Influences of the temperature gradient on internal forces. (a) Rise-span ratiof/L vs. axial forces Nc with different ΔTg. (b) Rise-span ratio f/L vs. bending moment Mc with different ΔTg. (c) Rise-span ratio f/L vs. axial forces Nc with different Ta. (d) Rise-span ratio f/L vs. bending moment Mc with different Ta.
(a)(b)(c)(d)According to Figure7(a), Nc almost does not change for parabolic arches having different ΔTg, which is consistent with (27) without ΔTg. Figure 7(a) also reveals that Nc augments as the rise-span ratio f/L augments at the beginning; after that, it decreases as the rise-span ratio f/L augments in case the rise-span ratio of arches reaches to a certain value. Figure 7(b) shows that Mc decreases as the temperature difference between the top and bottom of the arch section ΔTg augments. Figure 7(b) also shows that Mc decreases as the rise-span ratio f/L augments initially; after that, it augments as the rise-span ratio f/L augments in case the rise-span ratio of arches reaches to a certain value. Figure 7(c) shows that Nc augments as the average temperature of the arch section Ta augments. Figure 7(d) shows that Mc decreases as the average temperature of the arch section Ta augments.To demonstrate the influences of the uniformly distributed vertical load onN and M given by (23) and (24), the changes of Nc and Mc with rise-span ratio f/L for parabolic arches having different uniformly distributed vertical loads q (i.e., q=105kN/m,2×105kN/m,3×105kN/m) are plotted in Figures 8(a) and 8(b), respectively, where ΔTg=100°C, Ta=200°C, and L/γx=30.Figure 8
Influences of the uniformly distributed vertical load on internal forces. (a) Axial forces. (b) Bending moments.
(a)(b)According to Figure8, the central axial and bending actions Nc and Mc augment as the uniformly distributed vertical load q augments.Figures9 and 10 show the variation of the dimensionless internal force with the rise-span ratio for parabolic arch under the linear gradient temperature field and the vertical uniformly distributed load. Figure 9(a) shows that the central axial force Nc augments with the augments of the slenderness ratio L/rx when temperature difference ΔTg=0°C and the average temperature Ta=20°C. The central axial force Nc augments as the rise-span ratio f/L augments when the rise-span ratio f/L is less than 0.1 and then decreases as the rise-span ratio f/L augments when the rise-span ratio f/L is bigger than 0.1. Figure 9(b) shows that the central axial force Nc decreases with the augments of the slenderness ratio L/rx when temperature difference ΔTg=100°C and the average temperature Ta=200°C. The central axial force Nc decreases as the rise-span ratio f/L augments.Figure 9
Relationship between rise-span ratio and central force. (a) Only considering the vertical uniformly distributed loadq. (b) Only considering the gradient temperature field.
(a)(b)Figure 10
Relationship between rise-span ratio and central bending moment. (a) Only considering the vertical uniformly distributed loadq. (b) Only considering the gradient temperature field.
(a)(b)Figure10(a) shows that the central bending moment Mc augments with the augments of the slenderness ratio L/rx when temperature difference ΔTg=0°C and the average temperature Ta=20°C. The central bending moment Mc decreases as the rise-span ratio f/L augments. Figure 10(b) shows that the central bending moment Mc decreases with the augments of the slenderness ratio L/rx when temperature difference ΔTg=100°C, the average temperature Ta=200°C, and the rise-span ratio f/L is approximately less than 0.1. The central bending moment Mc decreases as the rise-span ratio f/L augments, while the central bending moment Mc augments with the augments of the slenderness ratio L/rx when the rise-span ratio f/L is approximately bigger than 0.1. The central bending moment Mc augments as the rise-span ratio augments f/L.


### 3.2. In-Plane Instability Analysis

According to the paper of Pi and Bradford [20], the in-plane instability critical load of parabolic arches boundary load can be compared with the relevant equation of circular arc arch. According to paper of Song et al. [14], the critical axial force of a solidly connected circular arch at both ends can be calculated by the following equation:(30)Ncr=EI¯ηπ2S/22−EI¯R2,where the parameter ηπ can be expressed as(31)ηπ=1.403π.For a parabolic arch, the focal collimation distancep can be calculated by the following equation:(32)p=L28f.In addition, the radius of curvature of the parabolic arch is given by(33)R=1+y′22/3y″=p1+xp22/3≈p.For the sake of solving the critical load of instability of the parabolic arch with fixed ends, the axial force term of (30) can be replaced by the axial force at the top of the arch in the analytical solution of the internal force in the previous section [19].(34)Ncr=Γ2ql−Γ1EA¯αΔT0Ψ.Substitute (33) and (34) into (30) and obtain(35)qcr=ΨEA¯ηπ2/S/22−EI¯/p2+Γ1EA¯αΔT0Γ2l.Equation (35) is the critical in-plane instability load of a fixed parabolic steel arch under the gradient temperature coupled with vertical uniform load.The change regular of dimensionless instability critical load with rise-span ratio for parabolic arches having different gradient temperature fields is shown in Figure11, where NE2 is the second mode flexural instability load of an axially compressive fixed column having the same length of the fixed parabolic steel arch, which can be expressed as (36)NE2=1.4303π2E0IxS/22.Figure 11
The relationship between the critical load and rise-span ratio for arches having differentTb.According to Figure11, when the rise-span ratio is less than 0.15, the dimensionless critical in-plane instability load of parabolic arch is obviously influenced by the temperature gradient field and decreases with the augment of gradient temperature difference. However, when the rise-span ratio is larger than 0.15, the dimensionless critical in-plane instability load of parabolic arch is slightly influenced by the gradient temperature field. In addition, the dimensionless critical in-plane instability load of parabolic arches decreases with the increasing rise-span ratio.The variation of the dimensionless instability critical load with the rise-span ratio for the parabolic arch under the linear gradient temperature field is shown in Figure12. As seen from Figure 12, the dimensionless critical in-plane instability load of parabolic arch decreases with the augment of slenderness.Figure 12
The relationship between the critical load and rise-span ratio for arches having differentS/rx.


### 3.3. Finite Element Validation Analysis

Finite element software ANSYS was employed to verify the accuracy of the above theoretical research. A biaxially symmetric I-beam section of the arch has a heighth  = 250 mm, a width b  = 150 mm, a span length L  = 5000 mm, a flange thickness tf  = 10 mm, and a web thickness tw  = 6 mm, and the material properties of Q235 steel were selected for modeling, which can be shown in Figure 13.Figure 13
Finite element model of parabolic arch under the linear gradient temperature.The beam188 element was chosen to build the parabolic arch mode. According to the help file of the beam188 element of ANSYS, the beam188 element can be used to specify temperature gradients that vary linearly both over the cross section and along the length of the element. The model is built and solved numerically for critical loads and then compared and analyzed with the theoretical solution.The comparison between the theoretical solution and the finite element results for the dimensionless critical in-plane instability load of the parabolic arch under linear gradient temperature field coupled with vertical uniform load can be seen from Figure14. According to Figure 14, the theoretical solutions agree well with the finite element consequence data, indicating that (35) can accurately predict the instability critical load of the fixed parabolic steel arch under the gradient temperature field coupled with the vertical uniform load.Figure 14
Finite element verification of dimensionless critical buckling loads for parabolic arches.


## 3.1. Internal Force Analysis

As the parabolic steel arches are linear elastic, their strain energy in the preinstability state under linear temperature gradient field coupled with vertical uniformly distributed loadq can be given by(14)Uss0=12∫−S/2S/2∫AETεxx02dAds−∫−S/2S/2∫AETαΔTo−yΔTghεxx0dAds,with ΔTo=To−20°C, where A is the cross-sectional area, ET, α, and εxx0 are the modulus of elasticity, the thermal coefficient of the steel, and the linear normal strain, respectively. εxx0 is given by(15)εxx0=εm0+εb0=ds1−dsds+y1ρ1−1ρ,where ds1 is the arc differential after deformation and ρ and ρ1 are radii of curvature of the parabolic steel arches before and after deformation, respectively. In addition, the axial force N and bending moment M can be expressed as(16)N=−∫AETεxx0dA=−∫AETyds1−dsds+y1ρ1−1ρdA=−EA¯εm0,(17)M=−∫AETyεxx0dA=−∫AETyyds1−dsds+y1ρ1−1ρdA=−EI¯εb0.Accordingly, by substituting (16) and (17) into (14), the term of the strain energy can be simplified as(18)∫AETεxx02dA=∫AETyds1−dsds+y1ρ1−1ρ2dA=EA¯εm02+EI¯εb02=N2EA¯+M2EI¯.By substituting equations (3), (16), (17), and (18) into equation (14), the strain energy can be further simplified as(19)Uss0=12∫−11N2EA¯+M2EI¯lV1dx1−αΔT0∫−11NlV1dx1+αΔTgh∫−11MlV1dx1,where(20)V1=1+a1x12.The accurate solutions of the preinstability axial forceN and bending moment M of a parabolic steel arch are essential for the in-plane instability of the arch. However, no accurate solutions of internal forces for arches being subjected to linear temperature gradient filed and vertical uniformly distributed load q can be obtained in the opening literature. The internal force of the arch can be solved by the force method which is based on Castigliano’s theorems, and when an arch is under linear temperature gradient field coupled with vertical uniformly distributed load q, the internal forces for the arch can be herein calculated by the force method in this paper.Based on the force method, the parabolic steel arch is cut into two pieces at the top, which is shown in Figure6. These two parts are the statically determinate structures with unknown additional axial forces Nc, bending moment Mc, and shear force Qyc. However, according to the principle of structural symmetry, the unknown additional shear force Qyc is equal to zero.Figure 6
Parabolic arch force method diagram.In addition, to make the cut arch equivalent to the original arch, the relative axial displacementΔNC and the relative rotation ΔMC at the cutting location should be equal to zero. Based on Castigliano’s theorems, the relative axial displacement ΔNC and the relative rotation ΔMC can be, respectively, given by(21)ΔNC=∂Uss0∂Nc=0,(22)ΔMC=∂Uss0∂Mc=0.Substituting (19) into (21) and (22), respectively, the corresponding force method equation can be obtained as(23)∫−11NEA¯∂N∂Nc+MEI¯∂M∂NclV1dx1−αΔT0∫−11∂N∂NclV1dx1+αΔTgh∫−11∂M∂NclV1dx1=0,(24)∫−11NEA¯∂N∂Mc+MEI¯∂M∂MclV1dx1−αΔT0∫−11∂N∂MclV1dx1+αΔTgh∫−11∂M∂MclV1dx1=0.Because the segmentations of two part of the parabolic steel arch are the statically determinate structures, based on the principle of force equilibrium, the axial forceN and bending moment M can be solved, respectively, as(25)N=qla1x12−Nca12x12+1,(26)M=Mc−qlx122−Nca1lx1.The solutions of the internal force obtained from (25) and (26) have two unknown values (Nc and Mc); to solve them, substituting (25) and (26) into (23) and (24), Nc and Mc can be obtained as(27)Nc=Γ1EA¯αΔTo−Γ2qlΨ,(28)Mc=3lΓ4ql−Γ3EA¯αΔToΨΞ−2a1a12+1−EI¯αΔTgh,where(29)Γ1=768rx2a14Ξ1−2a1a12+1,Γ2=3Ξ264a12rx2+l2−4a18a14+8a12+3Ξl2a12+1+4a12a12+14a14−8a12−9l2−192rx2a12,Γ3=32rx2a12Ξ2+4a13Ξa12+1−4a122a14−3a12−1Γ4=8rx24a124a12+1a12+1Ξ−Ξ3−2a1a12+12a12−1Ξ2−8a132a12+1a12+13,Ψ=3Ξ2a1128a12rx2+l2−4a12a12+18a14+8a12+3l2+192rx2a12Ξ+4a13l24a14−8a12−9a12+1,Ξ=lna12+1+a1−lna12+1−a1.Hence, the accurate solutions of axial forceN and bending moment M can be obtained by substituting (27) and (28) into (25) and (26).In order to expound the effect of the linear temperature gradient field onN and M in (23) and (24), the changes of Nc and Mc with rise-span ratio f/L for parabolic arches having temperature difference between the top and base of the cross section ΔTg (i.e., ΔTg=50°C,100°C,150°C) are shown in Figures 7(a) and 7(b), respectively, where Nc and Mc are the central axial and bending actions, and Ta=200°C, q=2×105kN/m, and L/γx=30 with γx being the radius of gyration of the arch. Beyond that, the changes of Nc and Mc with rise-span ratio f/L for parabolic arches having different average temperatures of the cross section Ta (i.e., Ta=150°C,200°C,250°C) are plotted in Figures 7(c) and 7(d), respectively, where ΔTg=100°C, q=2×105kN/m, and L/γx=30.Figure 7
Influences of the temperature gradient on internal forces. (a) Rise-span ratiof/L vs. axial forces Nc with different ΔTg. (b) Rise-span ratio f/L vs. bending moment Mc with different ΔTg. (c) Rise-span ratio f/L vs. axial forces Nc with different Ta. (d) Rise-span ratio f/L vs. bending moment Mc with different Ta.
(a)(b)(c)(d)According to Figure7(a), Nc almost does not change for parabolic arches having different ΔTg, which is consistent with (27) without ΔTg. Figure 7(a) also reveals that Nc augments as the rise-span ratio f/L augments at the beginning; after that, it decreases as the rise-span ratio f/L augments in case the rise-span ratio of arches reaches to a certain value. Figure 7(b) shows that Mc decreases as the temperature difference between the top and bottom of the arch section ΔTg augments. Figure 7(b) also shows that Mc decreases as the rise-span ratio f/L augments initially; after that, it augments as the rise-span ratio f/L augments in case the rise-span ratio of arches reaches to a certain value. Figure 7(c) shows that Nc augments as the average temperature of the arch section Ta augments. Figure 7(d) shows that Mc decreases as the average temperature of the arch section Ta augments.To demonstrate the influences of the uniformly distributed vertical load onN and M given by (23) and (24), the changes of Nc and Mc with rise-span ratio f/L for parabolic arches having different uniformly distributed vertical loads q (i.e., q=105kN/m,2×105kN/m,3×105kN/m) are plotted in Figures 8(a) and 8(b), respectively, where ΔTg=100°C, Ta=200°C, and L/γx=30.Figure 8
Influences of the uniformly distributed vertical load on internal forces. (a) Axial forces. (b) Bending moments.
(a)(b)According to Figure8, the central axial and bending actions Nc and Mc augment as the uniformly distributed vertical load q augments.Figures9 and 10 show the variation of the dimensionless internal force with the rise-span ratio for parabolic arch under the linear gradient temperature field and the vertical uniformly distributed load. Figure 9(a) shows that the central axial force Nc augments with the augments of the slenderness ratio L/rx when temperature difference ΔTg=0°C and the average temperature Ta=20°C. The central axial force Nc augments as the rise-span ratio f/L augments when the rise-span ratio f/L is less than 0.1 and then decreases as the rise-span ratio f/L augments when the rise-span ratio f/L is bigger than 0.1. Figure 9(b) shows that the central axial force Nc decreases with the augments of the slenderness ratio L/rx when temperature difference ΔTg=100°C and the average temperature Ta=200°C. The central axial force Nc decreases as the rise-span ratio f/L augments.Figure 9
Relationship between rise-span ratio and central force. (a) Only considering the vertical uniformly distributed loadq. (b) Only considering the gradient temperature field.
(a)(b)Figure 10
Relationship between rise-span ratio and central bending moment. (a) Only considering the vertical uniformly distributed loadq. (b) Only considering the gradient temperature field.
(a)(b)Figure10(a) shows that the central bending moment Mc augments with the augments of the slenderness ratio L/rx when temperature difference ΔTg=0°C and the average temperature Ta=20°C. The central bending moment Mc decreases as the rise-span ratio f/L augments. Figure 10(b) shows that the central bending moment Mc decreases with the augments of the slenderness ratio L/rx when temperature difference ΔTg=100°C, the average temperature Ta=200°C, and the rise-span ratio f/L is approximately less than 0.1. The central bending moment Mc decreases as the rise-span ratio f/L augments, while the central bending moment Mc augments with the augments of the slenderness ratio L/rx when the rise-span ratio f/L is approximately bigger than 0.1. The central bending moment Mc augments as the rise-span ratio augments f/L.


## 3.2. In-Plane Instability Analysis

According to the paper of Pi and Bradford [20], the in-plane instability critical load of parabolic arches boundary load can be compared with the relevant equation of circular arc arch. According to paper of Song et al. [14], the critical axial force of a solidly connected circular arch at both ends can be calculated by the following equation:(30)Ncr=EI¯ηπ2S/22−EI¯R2,where the parameter ηπ can be expressed as(31)ηπ=1.403π.For a parabolic arch, the focal collimation distancep can be calculated by the following equation:(32)p=L28f.In addition, the radius of curvature of the parabolic arch is given by(33)R=1+y′22/3y″=p1+xp22/3≈p.For the sake of solving the critical load of instability of the parabolic arch with fixed ends, the axial force term of (30) can be replaced by the axial force at the top of the arch in the analytical solution of the internal force in the previous section [19].(34)Ncr=Γ2ql−Γ1EA¯αΔT0Ψ.Substitute (33) and (34) into (30) and obtain(35)qcr=ΨEA¯ηπ2/S/22−EI¯/p2+Γ1EA¯αΔT0Γ2l.Equation (35) is the critical in-plane instability load of a fixed parabolic steel arch under the gradient temperature coupled with vertical uniform load.The change regular of dimensionless instability critical load with rise-span ratio for parabolic arches having different gradient temperature fields is shown in Figure11, where NE2 is the second mode flexural instability load of an axially compressive fixed column having the same length of the fixed parabolic steel arch, which can be expressed as (36)NE2=1.4303π2E0IxS/22.Figure 11
The relationship between the critical load and rise-span ratio for arches having differentTb.According to Figure11, when the rise-span ratio is less than 0.15, the dimensionless critical in-plane instability load of parabolic arch is obviously influenced by the temperature gradient field and decreases with the augment of gradient temperature difference. However, when the rise-span ratio is larger than 0.15, the dimensionless critical in-plane instability load of parabolic arch is slightly influenced by the gradient temperature field. In addition, the dimensionless critical in-plane instability load of parabolic arches decreases with the increasing rise-span ratio.The variation of the dimensionless instability critical load with the rise-span ratio for the parabolic arch under the linear gradient temperature field is shown in Figure12. As seen from Figure 12, the dimensionless critical in-plane instability load of parabolic arch decreases with the augment of slenderness.Figure 12
The relationship between the critical load and rise-span ratio for arches having differentS/rx.


## 3.3. Finite Element Validation Analysis

Finite element software ANSYS was employed to verify the accuracy of the above theoretical research. A biaxially symmetric I-beam section of the arch has a heighth  = 250 mm, a width b  = 150 mm, a span length L  = 5000 mm, a flange thickness tf  = 10 mm, and a web thickness tw  = 6 mm, and the material properties of Q235 steel were selected for modeling, which can be shown in Figure 13.Figure 13
Finite element model of parabolic arch under the linear gradient temperature.The beam188 element was chosen to build the parabolic arch mode. According to the help file of the beam188 element of ANSYS, the beam188 element can be used to specify temperature gradients that vary linearly both over the cross section and along the length of the element. The model is built and solved numerically for critical loads and then compared and analyzed with the theoretical solution.The comparison between the theoretical solution and the finite element results for the dimensionless critical in-plane instability load of the parabolic arch under linear gradient temperature field coupled with vertical uniform load can be seen from Figure14. According to Figure 14, the theoretical solutions agree well with the finite element consequence data, indicating that (35) can accurately predict the instability critical load of the fixed parabolic steel arch under the gradient temperature field coupled with the vertical uniform load.Figure 14
Finite element verification of dimensionless critical buckling loads for parabolic arches.


## 4. Conclusion

This paper has presented a theoretical study on the internal forces and critical in-plane instability loads for parabolic arches with I section under the linear gradient temperature field and vertical uniform load. The cross-sectional effective centroid and effective stiffness for parabolic arches under the linear temperature gradient field have been derived. The axial and bending actions of the parabolic arch under linear gradient temperature field coupled with vertical uniform load have also been obtained. In addition, the precise analytical solutions of the critical load for in-plane instability of the parabolic arch under gradient temperature field coupled with vertical uniform load have been obtained, and these solutions have been verified by numerical simulations of ANSYS. The conclusions of this article can be summarized as follows:(1)
For parabolic arches under the linear gradient temperature field and vertical uniform load, the central axial force and the central bending moment decrease with the augment of slenderness.(2)
The central axial force of the arch first augments with the augment of the rise-span ratio and then decreases with the augment of the rise-span ratio while the rise-span ratio reaches a certain value. The central bending moment decreases with the augment of the rise-span ratio.(3)
When the rise-span ratio is less than 0.15, the dimensionless critical in-plane instability load of parabolic arch is obviously influenced by the temperature gradient field and decreases with the augment of gradient temperature difference; however, when the rise-span ratio is larger than 0.15, the dimensionless critical in-plane instability load of parabolic arch is slightly influenced by the gradient temperature field.(4)
The dimensionless critical in-plane instability load of parabolic arches decreases with the increasing rise-span ratio.(5)
The dimensionless critical in-plane instability load of parabolic arch decreases with the augment of slenderness.



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*Source: 8973013-2022-02-22.xml*