# Earthquake Response Analysis of Tall Reinforced Concrete Chimneys considering Eccentricity

Authors: Yingchun Jiang; Tielin Liu; Yikui Bai
Journal: Shock and Vibration
 (2020)

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## Abstract

A numerical algorithm is presented to analyze earthquake response of tall reinforced concrete (RC) chimneys based on stick multidegree-of-freedom models. The algorithm considers the eccentricity phenomena between spatial discrete nodes and corresponding centroids of investigated lumps. The spatial discrete segments of the chimney are used to construct the investigated lumps. The equations of dynamic equilibrium of the investigated lumps are derived, and the numerical calculation procedure is implemented. Phenomena of eccentricity are studied for 150 m and 210 m RC chimneys. Seismic stresses and effects of vertical ground motion for the two chimneys are also studied. Numerical results show that the tensile and compressive stresses of the seismic control cross sections of the chimneys may increase under the actions of several specific earthquake waves by considering existing eccentricities. The effect of eccentricity on the earthquake responses of tall RC chimney should be considered, and stresses caused by vertical ground motion should not be neglected to obtain accurate earthquake response of chimneys.


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## Body

## 1. Introduction

### 1.1. Literature Review

Many tall reinforced concrete (RC) chimneys have been built in places such as coal-fired power stations and refinery because of the requirements of air pollution control standards. Slender tall chimneys are sensitive to earthquake. Strong earthquake threatens the safety of the chimney and other buildings around it because of the nonuniformity of the diameter along the height of the chimney and the variation in dead weights at different heights. Research and calculations of earthquake response are necessary for the designs of tall RC chimneys under the action of earthquakes.Destruction and collapse of tall chimneys bring huge economic losses. Research on seismic damage of chimneys indicates that the destructive effect of vertical earthquake on chimneys should be considered [1]. Liu and He [2] analyzed the seismic response of a chimney under the action of combined horizontal and vertical ground motions. They concluded that vertical stress cannot be neglected completely although horizontal seismic stress is the controlling part. Wang and Liao [3] developed a calculation technique for analyzing the vertical earthquake response of a chimney based on multiple reflections of traveling waves in the chimney. Liu et al. [4] conducted earthquake transient response of multidegree-of-freedom (MDOF) systems by the model superposition method. Ding et al. [5] conducted a vertical impact to the bottoms of chimney models. The test results showed that only one circle crack appeared in the chimney models, and the crack had relations with the ratio of input wave frequency to the natural frequency of the chimney model. Chen et al. [6] used a shaking table to study the fracture phenomenon of a chimney under the action of vertical earthquake. They found that vertical seismic action should be considered in the structural design of chimneys.A 115 m high RC chimney collapsed on August 17, 1999, during the Izmit earthquake in Turkey [7–10]. Huang et al. [7, 8] investigated a collapsed chimney with methods of linear dynamic response spectrum and nonlinear static capacity spectrum. Akinci [9] used shell elements to analyze a collapsed chimney. The analysis results indicated that low ductility and inadequate capacity due to a large duct were the two significant factors for the collapsed chimney. Huang and Gould [10] further proposed a new three-dimensional (3-D) pushover analysis method to study the dynamic response of the stack due to an earthquake motion recorded at a nearby site. Lopes et al. [11] constructed a realistic numerical model to evaluate the seismic vulnerability of a chimney and simulated its damaged state. Minghini et al. [12] analyzed a brickwork chimney and a shear failure mechanism of the upper part of the chimney damaged by the 2012 Emilia earthquake. Zhou et al. selected an existing 240 m tall RC chimney as the research object and analyzed the seismic vulnerability curve of the chimney by considering structural damage [13], structural nonlinearity [14], and multidimensional earthquake [15] and evaluated and analyzed the vulnerability of the chimney structure.The modal frequency and mode shape are the main dynamic characteristics of chimneys. Chen et al. [16] used the Adomian decomposition method to decompose dimensionless modal shape and calculated the dynamic characteristics of a high-rise chimney. Sancibrian et al. [17] used the continuum structure system to analyze the dynamic characteristics of a chimney, including natural frequencies and mode shapes. Tatara and Ratajewicz [18] evaluated the dynamic characteristics of a typical RC industrial chimney that belongs to a heating plant and determined the lowest vibration frequency of the structure.Soil-structure interaction (SSI) has a great influence on the seismic performance of tall chimney structures [19, 20]. Halabian and Kabiri [21] studied the effect of foundation stiffness of different types of RC chimneys on the ductility reduction coefficient. A series of inelastic and elastic time history analyses up to the collapse status for a large variety of RC chimneys subjected to five scaled artificial records were conducted. Jisha et al. [22] analyzed 3-D SSI of tall RC chimneys with annular raft foundation and evaluated the effect of SSI by changing the stiffness of supporting soil and foundation [23]. Zhou et al. [24, 25] studied the effect of SSI on the seismic collapse resistance of chimney structure for a 240 m high RC chimney.Time period is an important approach of evaluating the structural response of a structure subjected to large-scale earthquake vibrations. The seismic response of chimney structures uses time history analysis considering earthquake, which is a long-duration earthquake impulse [19]. Sarkar et al. [26] studied the effects of different base conditions on the design value of shear force and bending moment of different sections of a 275 m tall RCC multiflue chimney by using a response spectrum method. The spectrum compatibility time history of specific engineering site was considered, and the time history dynamic analysis of the chimney structure was conducted in detail. Wilson [27] studied the inelastic response of 10 RC chimneys ranging from 115 m to 301 m subject to earthquake excitation. Liu et al. [28] studied the structural dynamic performance and elastic-plastic seismic response of a 240 m high special-shaped RC chimney. The weak position of the chimney was discussed by analyzing its interlayer displacement angle.


### 1.2. Motivation

When the stick MDOF structural model is used to study the dynamic response of tall chimney structures, the chimney structure is usually discretized into beam elements of a certain length. Thin beam elements are good, but they are not sufficiently thin in practical analysis. The cross section area of a chimney structure is variable at different heights. Therefore, even if the discrete length of space is equal, the vertical eccentricity of mass will occur because the mass of each element will be uneven. The vertical eccentricity of mass must be considered to improve the accuracy and validity of dynamic response analysis.A numerical algorithm is presented in this study to analyze the earthquake response of tall RC chimneys. The chimney is spatially discretized into several segments of finite length, and the investigated lumps are then constructed by using these segments. The calculating formulae of axial force, shear force, and bending moment acting on the median cross section of each segment are derived. These interior forces are associated with end displacement and angle. Not only the inertial forces of the investigated lumps are considered but also the rotary inertias of investigated lumps. Existing eccentricity phenomena between spatial discrete nodes and corresponding centroids of investigated lumps are considered. Dynamic equations and relations are applied alternately in the time domain to calculate the seismic response of the chimney. Two RC chimneys of 150 m and 210 m height are studied. The numerical results show that the effects of such eccentricities on earthquake response should not be neglected for tall chimneys, as well as the vertical ground motion.


## 1.1. Literature Review

Many tall reinforced concrete (RC) chimneys have been built in places such as coal-fired power stations and refinery because of the requirements of air pollution control standards. Slender tall chimneys are sensitive to earthquake. Strong earthquake threatens the safety of the chimney and other buildings around it because of the nonuniformity of the diameter along the height of the chimney and the variation in dead weights at different heights. Research and calculations of earthquake response are necessary for the designs of tall RC chimneys under the action of earthquakes.Destruction and collapse of tall chimneys bring huge economic losses. Research on seismic damage of chimneys indicates that the destructive effect of vertical earthquake on chimneys should be considered [1]. Liu and He [2] analyzed the seismic response of a chimney under the action of combined horizontal and vertical ground motions. They concluded that vertical stress cannot be neglected completely although horizontal seismic stress is the controlling part. Wang and Liao [3] developed a calculation technique for analyzing the vertical earthquake response of a chimney based on multiple reflections of traveling waves in the chimney. Liu et al. [4] conducted earthquake transient response of multidegree-of-freedom (MDOF) systems by the model superposition method. Ding et al. [5] conducted a vertical impact to the bottoms of chimney models. The test results showed that only one circle crack appeared in the chimney models, and the crack had relations with the ratio of input wave frequency to the natural frequency of the chimney model. Chen et al. [6] used a shaking table to study the fracture phenomenon of a chimney under the action of vertical earthquake. They found that vertical seismic action should be considered in the structural design of chimneys.A 115 m high RC chimney collapsed on August 17, 1999, during the Izmit earthquake in Turkey [7–10]. Huang et al. [7, 8] investigated a collapsed chimney with methods of linear dynamic response spectrum and nonlinear static capacity spectrum. Akinci [9] used shell elements to analyze a collapsed chimney. The analysis results indicated that low ductility and inadequate capacity due to a large duct were the two significant factors for the collapsed chimney. Huang and Gould [10] further proposed a new three-dimensional (3-D) pushover analysis method to study the dynamic response of the stack due to an earthquake motion recorded at a nearby site. Lopes et al. [11] constructed a realistic numerical model to evaluate the seismic vulnerability of a chimney and simulated its damaged state. Minghini et al. [12] analyzed a brickwork chimney and a shear failure mechanism of the upper part of the chimney damaged by the 2012 Emilia earthquake. Zhou et al. selected an existing 240 m tall RC chimney as the research object and analyzed the seismic vulnerability curve of the chimney by considering structural damage [13], structural nonlinearity [14], and multidimensional earthquake [15] and evaluated and analyzed the vulnerability of the chimney structure.The modal frequency and mode shape are the main dynamic characteristics of chimneys. Chen et al. [16] used the Adomian decomposition method to decompose dimensionless modal shape and calculated the dynamic characteristics of a high-rise chimney. Sancibrian et al. [17] used the continuum structure system to analyze the dynamic characteristics of a chimney, including natural frequencies and mode shapes. Tatara and Ratajewicz [18] evaluated the dynamic characteristics of a typical RC industrial chimney that belongs to a heating plant and determined the lowest vibration frequency of the structure.Soil-structure interaction (SSI) has a great influence on the seismic performance of tall chimney structures [19, 20]. Halabian and Kabiri [21] studied the effect of foundation stiffness of different types of RC chimneys on the ductility reduction coefficient. A series of inelastic and elastic time history analyses up to the collapse status for a large variety of RC chimneys subjected to five scaled artificial records were conducted. Jisha et al. [22] analyzed 3-D SSI of tall RC chimneys with annular raft foundation and evaluated the effect of SSI by changing the stiffness of supporting soil and foundation [23]. Zhou et al. [24, 25] studied the effect of SSI on the seismic collapse resistance of chimney structure for a 240 m high RC chimney.Time period is an important approach of evaluating the structural response of a structure subjected to large-scale earthquake vibrations. The seismic response of chimney structures uses time history analysis considering earthquake, which is a long-duration earthquake impulse [19]. Sarkar et al. [26] studied the effects of different base conditions on the design value of shear force and bending moment of different sections of a 275 m tall RCC multiflue chimney by using a response spectrum method. The spectrum compatibility time history of specific engineering site was considered, and the time history dynamic analysis of the chimney structure was conducted in detail. Wilson [27] studied the inelastic response of 10 RC chimneys ranging from 115 m to 301 m subject to earthquake excitation. Liu et al. [28] studied the structural dynamic performance and elastic-plastic seismic response of a 240 m high special-shaped RC chimney. The weak position of the chimney was discussed by analyzing its interlayer displacement angle.


## 1.2. Motivation

When the stick MDOF structural model is used to study the dynamic response of tall chimney structures, the chimney structure is usually discretized into beam elements of a certain length. Thin beam elements are good, but they are not sufficiently thin in practical analysis. The cross section area of a chimney structure is variable at different heights. Therefore, even if the discrete length of space is equal, the vertical eccentricity of mass will occur because the mass of each element will be uneven. The vertical eccentricity of mass must be considered to improve the accuracy and validity of dynamic response analysis.A numerical algorithm is presented in this study to analyze the earthquake response of tall RC chimneys. The chimney is spatially discretized into several segments of finite length, and the investigated lumps are then constructed by using these segments. The calculating formulae of axial force, shear force, and bending moment acting on the median cross section of each segment are derived. These interior forces are associated with end displacement and angle. Not only the inertial forces of the investigated lumps are considered but also the rotary inertias of investigated lumps. Existing eccentricity phenomena between spatial discrete nodes and corresponding centroids of investigated lumps are considered. Dynamic equations and relations are applied alternately in the time domain to calculate the seismic response of the chimney. Two RC chimneys of 150 m and 210 m height are studied. The numerical results show that the effects of such eccentricities on earthquake response should not be neglected for tall chimneys, as well as the vertical ground motion.


## 2. Numerical Algorithm

### 2.1. Governing Equations of Investigated Lumps

A chimney with variable cross section can be discretized into many segments of different cross section areas along the vertical direction in practical calculation based on the stick MDOF model. These segments are used to construct the investigated lumps. Figure1 shows a typical investigated lump i (the nodal numbering i is also used to describe the investigated lump) consisting of half the upper segment and half the lower segment around the discrete node i. Sign C is the centroid of the investigated lump. The centroid of the investigated lump is commonly not at the same point of the corresponding discrete node i for the chimney with variable cross-sectional area. Here, the distance between the discrete node and the centroid of the investigated lump within the same investigated lump is called eccentricity.Figure 1
Sketch of deformation of investigated lumpi.The vertical equilibrium equation for the centroid of investigated lumpi is(1)miw¨ci=Nl−1−Nl−mig,where mi is the mass of investigated lump i, w¨ci is the vertical acceleration of centroid C, and Nl−1 and Nl are the axial forces acting on investigated lump i.The transverse equilibrium equation for the centroid of investigated lumpi is(2)miu¨ci=Vl−Vl−1,where u¨ci is the horizontal acceleration of the centroid of investigated lump i and Vl and Vl−1 are the shear forces acting on investigated lump i.On the basis of mechanics theory, the rotational equilibrium equation should be established about the horizontal axis (y-axis) through centroid C of investigated lump i, that is,(3)Jciθ¨i=Ml−Ml−1+Vlhl2+zci+Vl−1hl−12−zci+Nlui+1−ui2+zciui−ui−1hl−1+Nl−112−zcihl−1ui−ui−1,where Jci is the moment of inertia of investigated lump i about the y-axis through its centroid; θ¨i and zci are the angular acceleration and the eccentricity of investigated lump i, respectively; Ml and hl are the bending moments and the length of segment l, respectively; and ui is the transverse displacement of node i.Figure2 shows that the following relation holds:(4)u¨i=u¨ci+zciθ¨i,where u¨i is the horizontal acceleration of node i. In Figure 2, w¨i is the vertical acceleration of node i.Figure 2
Accelerations of nodei and centroid C of the investigated lump.


### 2.2. Forces and Bending Moment on the Central Cross Section of the Segment

Figure3 shows the free-body diagrams of the upper and lower halves of segment l by an artificial cut at the central section of segment l. In Figure 3, Ni′ and Ni+1′ are the axial compressive forces at the bottom and top ends of segment l, respectively, because of its vertical deformation; Vi′ and Vi+1′ are the shear forces; Mi′ and Mi+1′ are the bending moments; Nig is the axial force coming from the dead load of the structure above node i; and Ni+1g is the dead load above node i+1.Figure 3
Free-body diagrams of segmentl.Following the beam theory considering shear deformation and axial force, we have(5)Ni+1′=Ni′=EAlhlwi−wi+1,(6)Vi+1′=−Vi′=K1lui+1−ui+K2lθi+θi+1,(7)Mi′=K2lui+1−ui+K3lθi+K4lθi+1,(8)Mi+1′=K2lui+1−ui+K4lθi+K3lθi+1,where K1l=12EIl/hl31+2αl−6Nl/5hl, K2l=−6EIl/hl21+2αl+Nl/10, K3l=41+0.5αlEIl/hl1+2αl−2Nlhl/15, and K4l=21−αlEIl/hl1+2αl+Nlhl/30. Subscript l denotes the lth segment. αl=6EIl/hl2GAl is the coefficient of shear deformation in the transverse direction. EIl, GAl, and EAl are the bending stiffness, shear stiffness, and tensile/compressive stiffness of segment l, respectively. Nl=Nig+Ni′, where Nig is the axial force coming from the dead weight of the chimney above node i. Al is the cross-sectional area of segment l. wi and θi are the vertical displacement and rotational angle of node i, respectively. wi+1 and θi+1 are the vertical displacement and rotational angle of node i+1, respectively.Subtracting equation (8) from equation (7) yields(9)Mi′−Mi+1′=2EIlhl−Nlhl6θi−θi+1.Given the equilibrium of the upper half of segmentl, we have(10)Mi+1′=Ml−Vlhl2−Nlui+1−ui2,(11)Vl=K1lui+1−ui+K2lθi+θi+1.Given the equilibrium of the lower half of segmentl, we have(12)Mi′=−Ml−Vlhl2−Nlui+1−ui2,where Nl, Vl, and Ml are the central axial force, central shear force, and central bending moment acting on the central section of segment l, respectively.We substitute equations (10) and (12) into equation (9), and we obtain the central bending moment of segment l, shown as follows:(13)Ml=K5lθi+1−θi,where K5l=EIl/hl−Nlhl/12.


### 2.3. Implementation of Algorithm

Horizontal and vertical earthquake waves are inputted from the foundation of the tall RC chimney. The algorithm description diagram is shown in Figure4, and the calculating procedure is as follows.Step 1. Calculate the angular acceleration θ¨i of investigated lump i at time t using equation (3), the vertical acceleration w¨ci at time t using equation (1), w¨i at time t by considering w¨i being nearly equal to w¨ci, the horizontal acceleration u¨ci at time t using equation (2), and u¨i using equation (4).Step 2. Calculate angle θi, vertical displacement wi, and horizontal displacement ui for investigated lump i at time t+Δt by integration in the time domain.Step 3. Calculate axial force Ni′ at time t+Δt using equation (5) and then obtain central axial force Nl using the formula Nl=Nig+Ni′. Central shear force Vl and central bending moment Ml are calculated using equations (11) and (13), respectively, at time t+Δt.Step 4. Calculate θ¨i, w¨ci, and u¨ci at time t+Δt as performed in Step 1.Figure 4
Algorithm description diagram.The circulation from Step 1 to Step 4 is implemented in the time domain by a computer program. The numerical algorithm is provided to calculate the earthquake response of the tall RC chimney, considering eccentricitieszci (i=1,2,…,n).Wilsonθ method is used to solve the conventional dynamic equations of equilibrium in which eccentricities are not considered. Their results are called the conventional results in this study.Here, the rotation of the foundation is not considered.


## 2.1. Governing Equations of Investigated Lumps

A chimney with variable cross section can be discretized into many segments of different cross section areas along the vertical direction in practical calculation based on the stick MDOF model. These segments are used to construct the investigated lumps. Figure1 shows a typical investigated lump i (the nodal numbering i is also used to describe the investigated lump) consisting of half the upper segment and half the lower segment around the discrete node i. Sign C is the centroid of the investigated lump. The centroid of the investigated lump is commonly not at the same point of the corresponding discrete node i for the chimney with variable cross-sectional area. Here, the distance between the discrete node and the centroid of the investigated lump within the same investigated lump is called eccentricity.Figure 1
Sketch of deformation of investigated lumpi.The vertical equilibrium equation for the centroid of investigated lumpi is(1)miw¨ci=Nl−1−Nl−mig,where mi is the mass of investigated lump i, w¨ci is the vertical acceleration of centroid C, and Nl−1 and Nl are the axial forces acting on investigated lump i.The transverse equilibrium equation for the centroid of investigated lumpi is(2)miu¨ci=Vl−Vl−1,where u¨ci is the horizontal acceleration of the centroid of investigated lump i and Vl and Vl−1 are the shear forces acting on investigated lump i.On the basis of mechanics theory, the rotational equilibrium equation should be established about the horizontal axis (y-axis) through centroid C of investigated lump i, that is,(3)Jciθ¨i=Ml−Ml−1+Vlhl2+zci+Vl−1hl−12−zci+Nlui+1−ui2+zciui−ui−1hl−1+Nl−112−zcihl−1ui−ui−1,where Jci is the moment of inertia of investigated lump i about the y-axis through its centroid; θ¨i and zci are the angular acceleration and the eccentricity of investigated lump i, respectively; Ml and hl are the bending moments and the length of segment l, respectively; and ui is the transverse displacement of node i.Figure2 shows that the following relation holds:(4)u¨i=u¨ci+zciθ¨i,where u¨i is the horizontal acceleration of node i. In Figure 2, w¨i is the vertical acceleration of node i.Figure 2
Accelerations of nodei and centroid C of the investigated lump.


## 2.2. Forces and Bending Moment on the Central Cross Section of the Segment

Figure3 shows the free-body diagrams of the upper and lower halves of segment l by an artificial cut at the central section of segment l. In Figure 3, Ni′ and Ni+1′ are the axial compressive forces at the bottom and top ends of segment l, respectively, because of its vertical deformation; Vi′ and Vi+1′ are the shear forces; Mi′ and Mi+1′ are the bending moments; Nig is the axial force coming from the dead load of the structure above node i; and Ni+1g is the dead load above node i+1.Figure 3
Free-body diagrams of segmentl.Following the beam theory considering shear deformation and axial force, we have(5)Ni+1′=Ni′=EAlhlwi−wi+1,(6)Vi+1′=−Vi′=K1lui+1−ui+K2lθi+θi+1,(7)Mi′=K2lui+1−ui+K3lθi+K4lθi+1,(8)Mi+1′=K2lui+1−ui+K4lθi+K3lθi+1,where K1l=12EIl/hl31+2αl−6Nl/5hl, K2l=−6EIl/hl21+2αl+Nl/10, K3l=41+0.5αlEIl/hl1+2αl−2Nlhl/15, and K4l=21−αlEIl/hl1+2αl+Nlhl/30. Subscript l denotes the lth segment. αl=6EIl/hl2GAl is the coefficient of shear deformation in the transverse direction. EIl, GAl, and EAl are the bending stiffness, shear stiffness, and tensile/compressive stiffness of segment l, respectively. Nl=Nig+Ni′, where Nig is the axial force coming from the dead weight of the chimney above node i. Al is the cross-sectional area of segment l. wi and θi are the vertical displacement and rotational angle of node i, respectively. wi+1 and θi+1 are the vertical displacement and rotational angle of node i+1, respectively.Subtracting equation (8) from equation (7) yields(9)Mi′−Mi+1′=2EIlhl−Nlhl6θi−θi+1.Given the equilibrium of the upper half of segmentl, we have(10)Mi+1′=Ml−Vlhl2−Nlui+1−ui2,(11)Vl=K1lui+1−ui+K2lθi+θi+1.Given the equilibrium of the lower half of segmentl, we have(12)Mi′=−Ml−Vlhl2−Nlui+1−ui2,where Nl, Vl, and Ml are the central axial force, central shear force, and central bending moment acting on the central section of segment l, respectively.We substitute equations (10) and (12) into equation (9), and we obtain the central bending moment of segment l, shown as follows:(13)Ml=K5lθi+1−θi,where K5l=EIl/hl−Nlhl/12.


## 2.3. Implementation of Algorithm

Horizontal and vertical earthquake waves are inputted from the foundation of the tall RC chimney. The algorithm description diagram is shown in Figure4, and the calculating procedure is as follows.Step 1. Calculate the angular acceleration θ¨i of investigated lump i at time t using equation (3), the vertical acceleration w¨ci at time t using equation (1), w¨i at time t by considering w¨i being nearly equal to w¨ci, the horizontal acceleration u¨ci at time t using equation (2), and u¨i using equation (4).Step 2. Calculate angle θi, vertical displacement wi, and horizontal displacement ui for investigated lump i at time t+Δt by integration in the time domain.Step 3. Calculate axial force Ni′ at time t+Δt using equation (5) and then obtain central axial force Nl using the formula Nl=Nig+Ni′. Central shear force Vl and central bending moment Ml are calculated using equations (11) and (13), respectively, at time t+Δt.Step 4. Calculate θ¨i, w¨ci, and u¨ci at time t+Δt as performed in Step 1.Figure 4
Algorithm description diagram.The circulation from Step 1 to Step 4 is implemented in the time domain by a computer program. The numerical algorithm is provided to calculate the earthquake response of the tall RC chimney, considering eccentricitieszci (i=1,2,…,n).Wilsonθ method is used to solve the conventional dynamic equations of equilibrium in which eccentricities are not considered. Their results are called the conventional results in this study.Here, the rotation of the foundation is not considered.


## 3. Stability Condition

### 3.1. Corresponding Relations

For simplicity in deriving the stability condition of the algorithm, we assume thatEIl=EI, hl=h, and αl=α for all segments. Axial force Nl is omitted in Kjl j=1,2,…,5. Eccentric phenomena are neglected in equation (3); and mi=m and Jci=Ji=J for all investigated lumps. On the basis of these assumptions, vertical governing equations are decoupled from the governing equations of transverse and rotational deformations. Thus, we only study the coupling equations of transverse and rotational deformations to provide the stability condition for the algorithm presented in this study.On the basis of the above assumption, the corresponding relations between the coefficients used in Timoshenko beam theory and those used in the proposed algorithm are as follows (see AppendixA):(14)ρ0A0⟺mh,GAs⟺K⋅h,ρ0I⟺Jh,EI⟺EI.


### 3.2. Stability Condition Based on Timoshenko Beam Model

The flexural wave equation of Timoshenko beam is(15)∂2u∂2t+C02R2∂4u∂4z−R21+C02CQ2∂4u∂2z∂2t+R2CQ2∂4u∂4t=0,where C02=E/ρ0, R2=I/A0, and CQ2=GAs/ρ0A0.Substituting a flexural waveu=u0eiωt−kz into the finite difference formula of Equation (15), we obtain the following equation (see equation (A.12) in Appendix A):(16)4C02R2h4sin4kh2−R21+C02CQ2⋅4h2Δt2sin2ωΔt2sin2kh2+R2CQ24Δt4sin4ωΔt2−1Δt2sin2ωΔt2=0.Solving equation (16) with sin2ωΔt/2 as unknown yields the following:(17)sin2ωΔt2=Δt2C02+CQ24/h2sin2kh/2+CQ2/R2±C02+CQ24/h2sin2kh/2+CQ2/R22−8C0CQ/h2sin2kh/228.Givensin2ωΔt/2≤1, we approximately have the following equation (see equation (A.16) in Appendix A):(18)Δt≤hC02+CQ2sin2kh/2+CQ2h2/4R2≤hC02+CQ2+CQ2h2/4R2.


### 3.3. Stability Condition for the Proposed Algorithm

The stability condition used in the algorithm presented in this paper can be provided by using equation (18) and the corresponding relations (14) as follows:(19)Δt≤hEIh/J+12EI/mh1+2α+3EIh/J1+2α.


## 3.1. Corresponding Relations

For simplicity in deriving the stability condition of the algorithm, we assume thatEIl=EI, hl=h, and αl=α for all segments. Axial force Nl is omitted in Kjl j=1,2,…,5. Eccentric phenomena are neglected in equation (3); and mi=m and Jci=Ji=J for all investigated lumps. On the basis of these assumptions, vertical governing equations are decoupled from the governing equations of transverse and rotational deformations. Thus, we only study the coupling equations of transverse and rotational deformations to provide the stability condition for the algorithm presented in this study.On the basis of the above assumption, the corresponding relations between the coefficients used in Timoshenko beam theory and those used in the proposed algorithm are as follows (see AppendixA):(14)ρ0A0⟺mh,GAs⟺K⋅h,ρ0I⟺Jh,EI⟺EI.


## 3.2. Stability Condition Based on Timoshenko Beam Model

The flexural wave equation of Timoshenko beam is(15)∂2u∂2t+C02R2∂4u∂4z−R21+C02CQ2∂4u∂2z∂2t+R2CQ2∂4u∂4t=0,where C02=E/ρ0, R2=I/A0, and CQ2=GAs/ρ0A0.Substituting a flexural waveu=u0eiωt−kz into the finite difference formula of Equation (15), we obtain the following equation (see equation (A.12) in Appendix A):(16)4C02R2h4sin4kh2−R21+C02CQ2⋅4h2Δt2sin2ωΔt2sin2kh2+R2CQ24Δt4sin4ωΔt2−1Δt2sin2ωΔt2=0.Solving equation (16) with sin2ωΔt/2 as unknown yields the following:(17)sin2ωΔt2=Δt2C02+CQ24/h2sin2kh/2+CQ2/R2±C02+CQ24/h2sin2kh/2+CQ2/R22−8C0CQ/h2sin2kh/228.Givensin2ωΔt/2≤1, we approximately have the following equation (see equation (A.16) in Appendix A):(18)Δt≤hC02+CQ2sin2kh/2+CQ2h2/4R2≤hC02+CQ2+CQ2h2/4R2.


## 3.3. Stability Condition for the Proposed Algorithm

The stability condition used in the algorithm presented in this paper can be provided by using equation (18) and the corresponding relations (14) as follows:(19)Δt≤hEIh/J+12EI/mh1+2α+3EIh/J1+2α.


## 4. Method Validation

We study bending wave propagation of a beam to verify the effectiveness of the new algorithm. The beam with equal cross-sectional area is designed to avoid eccentricity. The result given by the finite difference method [29] is used to test the proposed numerical algorithm.The elastic modulus of the beam isE = 209 GPa, density is ρ = 8000 kg/m3, Poisson’s ratio is v = 0.3, and shear cross-sectional correction factor is k′ = 0.886. The length of the beam is X = 6.35 m, and the radius is 0.127 m. The number of discrete segments is 500 in the length direction for the uniform beam. On the basis of the calculation parameters of the beam, the time step of the beam satisfying the stability condition of the method is Δt≤1.51μs obtained by equation (19), and the calculation time step is Δt=1.25μs.The length from the left end of the beam isx, and the bending moment at the section is M. After normalization, the time t¯ is 1/XE/ρt, the length is x¯=x/X, and the bending moment is M¯=MX/EI.The inclination moment of the left end of the beam is(20)M¯1=t¯t¯0,0≤t¯≤t¯0,1,t¯>t¯0.Figure5 shows the comparison curves of the bending moments M¯ at different cross sections of the beam at t¯=7.0554.Figure 5
Moment distribution along the length of the uniform beam att¯ = 7.0554.The calculation results of the new numerical method are nearly the same as those of the finite difference method. The comparison results show that the presented algorithm is valid for calculating the earthquake responses of tall RC chimney.


## 5. Earthquake Responses of Tall RC Chimneys

### 5.1. Earthquake Records Used

Eleven earthquake acceleration records on four types of soil are used as earthquake load to calculate the structural earthquake response. Four types of soil are denoted by rock, soft rock, stiff soil, and soft soil. Soil type is characterized by shear wave velocity. The corresponding earthquake records are selected from the Pacific Earthquake Engineering Research Center website [30]. Table 1 shows the 11 earthquake records.Table 1
Earthquake records used.
Serial number
Soil types
Earthquake
Date
Direction
Station and component
PGA (g)
No. 1
Rock
Landers
1992/06/28
Horizontal
LANDERS/ABY000
0.115
Vertical
LANDERS/ABY-UP
0.090
No. 2
Soft rock
Loma Prieta
1989/10/18
Horizontal
LOMAP/CLS000
0.644
Vertical
LOMAP/CLS-UP
0.455
No. 3
Stiff soil
Borrego
1942/10/21
Horizontal
BORREGO/B-ELC000
0.033
Vertical
BORREGO/B-ELC-UP
0.068
No. 4
Soft soil
Chichi
1999/09/20
Horizontal
CHICHI/CHY004-N
0.100
Vertical
CHICHI/CHY004-V
0.041
No. 5
Rock
Morgan Hill
1984/04/24
Horizontal
MORGAN/G01230
0.069
Vertical
MORGAN/G01-UP
0.092
No. 6
Soft rock
Loma Prieta
1989/10/18
Horizontal
LOMAP/GGB360
0.123
Vertical
LOMAP/GGB-UP
0.056
No. 7
Stiff soil
Chichi
1999/09/20
Horizontal
CHICHI CHY028-N
0.821
Vertical
CHICHI CHY028-V
0.337
No. 8
Stiff soil
Chichi
1999/09/20
Horizontal
CHICHI TCU076-N
0.416
Vertical
CHICHI TCU076-V
0.281
No. 9
Stiff soil
Morgan Hill
1984/04/24
Horizontal
MORGAN/G03090
0.200
Vertical
MORGAN/G03-UP
0.395
No. 10
Stiff soil
Morgan Hill
1984/04/24
Horizontal
MORGAN HVR240
0.312
Vertical
MORGAN HVR-UP
0.110
No. 11
Soft soil
Imperial Valley
1079/10/15
Horizontal
IMPVALL H-E03140
0.266
Vertical
IMPVALL H-E03-UP
0.127According to the peak value of design earthquake acceleration of 8 degrees seismic intensity in the literature [31], the peaks of the horizontal earthquake accelerations are adjusted to 0.2 g in practical calculations. The peaks of the vertical earthquake accelerations are adjusted to 0.13 g. The damping is omitted.


### 5.2. Two Models of RC Chimneys

Two RC chimneys [32] of 150 m and 210 m height are studied. Figure 6 shows the two models of RC chimneys. (1) The concrete strength grade of the 150 m high RC chimney is C25, and the steel strength grade is HRB335. The outer radius of the bottom is 8.46 m, and the wall thickness is 0.4 m. The outer radius of the top is 3.0 m, and the wall thickness is 0.16 m. The gradient c is 0.1 below 25 m, 0.03 ranging from 25 m to 75 m, and 0.015 ranging from 75 m to 150 m. Table 2 shows the parameters of the 150 m high RC chimney. (2) The concrete strength grade of the 210 m high RC chimney is C35, and the steel strength grade is HRB335. The outer radius of the bottom is 10.6 m, and the wall thickness is 0.6 m. The outer radius of the top is 3.85 m, and the wall thickness is 0.18 m. The gradient c is 0.1 below 30 m, 0.03 ranging from 30 m to 100 m, and 0.015 ranging from 100 m to 210 m. Table 3 shows the parameters of the 210 m high RC chimney. According to the literature [33], the standard value of the compressive strength of C25 concrete is 16.7 N/mm2, the design value is 11.9 N/mm2, and the elastic modulus is EC = 2.8 × 104 N/mm2. The standard value of the compressive strength of C35 concrete is 23.4 N/mm2, the design value is 16.7 N/mm2, and the elastic modulus is EC = 3.15 × 104 N/mm2. The elastic modulus of HRB335 reinforcement is ES = 2.0 × 105 N/mm2, the standard value of strength (yield strength) is 335 N/mm2, and the design value of strength is 300 N/mm2.Figure 6
Model of (a) 150 m and (b) 210 m high chimneys.
(a)
(b)Table 2
Parameters of 150 m high RC chimney.
Elevation (m)
Outer radius (m)
Wall thickness (m)
Reinforcement ratio (%)
150
3.00
0.16
0.45
137.5
3.52
0.16
0.47
125
3.71
0.16
0.47
112.5
3.89
0.18
0.42
100
4.08
0.18
0.42
87.5
4.27
0.22
0.47
75
4.46
0.22
0.47
62.5
4.83
0.26
0.52
50
5.21
0.26
0.52
37.5
5.58
0.34
0.50
25
5.96
0.34
0.50
5
7.96
0.40
0.52
0
8.46
0.40
0.52Table 3
Parameters of 210 m high RC chimney.
Elevation (m)
Outer radius (m)
Wall thickness (m)
Reinforcement ratio (%)
210
3.50
0.18
0.57
190
4.15
0.18
0.57
170
4.45
0.20
0.51
150
4.75
0.20
0.67
130
5.05
0.20
0.67
120
5.20
0.23
0.74
110
5.35
0.26
0.65
100
5.50
0.29
0.58
90
5.80
0.32
0.65
80
6.10
0.35
0.60
70
6.40
0.38
0.67
60
6.70
0.41
0.62
50
7.00
0.44
0.74
40
7.30
0.47
0.70
30
7.60
0.50
0.65
16.5
8.95
0.53
0.77
0
10.60
0.60
0.68The discrete segments are all 10 m long in the vertical direction for each of the two chimneys. Each chimney will become part of such calculated model consisting of several cylinders of different cross-sectional areas because an actual chimney structure has variable cross-sectional areas. Thus, the phenomena of eccentricities between the nodes and the corresponding centroids of the investigated lumps appear in the calculation of the earthquake response of the actual chimney. The mass of each investigated lump consists of half the upper segment and half the lower segment around the discrete node. The moment of inertia of each investigated lump about the horizontal axis through its centroid is calculated. On the basis of the calculation parameters of the 150 m and 210 m high chimneys, the time step satisfying the stability condition of the method is obtained by equation (19), Δt≤3.4ms and Δt≤3.1ms in the 150 m and 210 m high chimneys, respectively. The time step is 1.0 ms.


#### 5.2.1. Effect of Eccentricity

Eleven earthquake waves are used. The effect of eccentricity on normal earthquake stress of the chimney are studied when horizontal and vertical components are inputted simultaneously.Figure7 shows the spatial distribution curves of the maximum earthquake normal stresses of different cross sections for the 150 m high chimney under the actions of Landers (No. 1 in Table 1), Loma Prieta (No. 2), Borrego (No. 3), and Chichi (No. 4) waves. Figure 8 shows the comparison curves of the 210 m high chimney. “New” denotes that the phenomena of eccentricities are considered. “Conventional” denotes the conventional results, in which the phenomena of eccentricities are neglected, for the earthquake responses of the real RC chimneys. Dashed lines (“tensile”) denote tensile stresses, whereas solid lines (“press”) denote compressive stresses.Figure 7
Spatial distributions of maximum earthquake normal stresses at different cross sections of the 150 m high chimney under four earthquake wave actions: (a) Landers wave (No. 1 in Table1); (b) Loma Prieta wave (No. 2); (c) Borrego wave (No. 3); (d) Chichi wave (No. 4).
(a)
(b)
(c)
(d)Figure 8
Spatial distributions of maximum earthquake normal stresses at different cross sections of the 210 m high chimney under four earthquake wave actions: (a) Landers wave (No. 1); (b) Loma Prieta wave (No. 2); (c) Borrego wave (No. 3); (d) Chichi wave (No. 4).
(a)
(b)
(c)
(d)A comparison of the red solid lines and the blue solid lines in Figures7 and 8 shows that the increased phenomena of peak values of the normal compressive stresses appear when eccentricities are considered.A comparison of the red dashed lines and the blue dashed lines in Figures7 and 8 shows that the increased phenomena of peak values of the normal tensile stresses appear when eccentricities are considered.Table4 shows the details of the increase in percentage of maximum earthquake normal tensile/compressive stresses when phenomena of eccentricities are considered.Table 4
Percentages of increase in maximum earthquake normal stresses.
Seismic waves
Maximum earthquake normal stresses increased
150 m
210 m
Landers (No. 1)
7% (compressive)
29% (tensile)
30% (compressive)
31% (tensile)
Loma Prieta (No. 2)
4% (compressive)
4% (tensile)
−6% (compressive)
3% (tensile)
Borrego (No. 3)
40% (compressive)
45% (tensile)
−5% (compressive)
−11% (tensile)
Chichi (No. 4)
15% (compressive)
11% (tensile)
31% (compressive)
17% (tensile)Table5 shows maximum earthquake compressive stress, mean value, and standard deviation of different sections of the 150 m high chimney under 11 earthquake wave actions when vertical eccentricities are considered. Table 6 shows maximum earthquake compressive stress, mean value, and standard deviation when vertical eccentricities are ignored.Table 5
Maximum earthquake compressive stresses of the 150 m high chimney under the action of 11 earthquake waves considering vertical eccentricities.
High
Serial number
1
2
3
4
5
6
7
8
9
10
11
Mean
Standard deviation
5
17.010
7.717
11.246
14.230
3.482
10.097
5.921
7.997
3.671
6.339
7.160
8.625
4.215
15
18.103
7.988
9.306
14.845
2.561
10.899
6.311
8.594
3.176
6.847
7.752
8.762
4.583
25
17.811
7.554
9.795
16.047
2.853
11.025
6.220
7.992
3.318
6.901
9.003
8.956
4.662
35
16.475
6.128
10.006
15.363
3.864
9.321
5.186
7.083
4.455
6.180
8.760
8.438
4.190
45
19.225
5.138
12.582
16.774
5.095
6.611
4.861
6.217
4.728
4.847
8.994
8.643
5.222
55
21.856
5.972
14.454
18.917
4.458
7.092
5.830
6.942
4.052
4.971
8.281
9.348
6.172
65
24.280
8.720
13.826
18.302
4.572
12.040
7.450
9.123
4.556
6.848
9.922
10.876
6.010
75
23.119
11.202
16.489
18.571
5.310
16.889
8.408
11.441
5.503
10.154
11.935
12.638
5.573
85
20.135
12.336
18.254
18.691
4.395
19.644
8.477
12.966
5.251
11.897
12.518
13.142
5.579
95
23.456
14.853
17.752
22.674
5.197
21.044
9.936
15.339
6.017
13.187
12.684
14.740
6.205
105
23.388
15.798
19.462
25.328
5.928
20.219
10.333
17.028
6.852
13.349
11.428
15.374
6.429
115
23.024
13.572
24.448
23.733
7.695
16.853
9.306
14.868
7.790
12.157
10.025
14.861
6.370
125
19.287
10.223
23.603
18.134
8.715
11.734
7.027
11.008
7.465
9.020
8.885
12.282
5.508
135
10.645
5.170
15.509
9.135
6.332
5.451
3.760
6.105
4.277
4.523
5.318
6.929
3.522
145
2.028
0.928
3.245
1.614
1.427
0.880
0.714
1.370
0.909
0.823
1.085
1.366
0.740Table 6
Maximum earthquake compressive stresses of the 150 m chimney under the action of 11 earthquake waves ignoring vertical eccentricities.
High
Serial number
1
2
3
4
5
6
7
8
9
10
11
Mean
Standard deviation
5
13.289
7.369
8.554
12.633
3.384
10.305
5.134
7.313
3.331
5.934
6.745
7.636
3.341
15
14.345
7.691
8.419
13.151
2.551
11.081
5.622
7.949
2.885
5.974
7.481
7.923
3.773
25
14.663
7.413
9.033
13.475
2.504
11.043
5.951
7.640
3.002
5.274
8.561
8.051
3.902
35
14.349
5.850
8.187
13.705
2.992
9.354
6.185
6.321
3.458
5.063
8.813
7.661
3.731
45
17.519
4.976
8.059
13.816
4.116
7.107
6.016
6.577
3.657
4.618
8.719
7.744
4.307
55
20.530
5.451
9.178
15.920
4.751
7.269
6.060
7.221
3.133
4.536
8.345
8.400
5.264
65
20.660
7.658
12.663
17.034
3.858
11.333
7.614
8.218
3.604
6.526
9.731
9.900
5.263
75
19.725
10.469
15.268
20.713
4.605
15.857
8.474
9.739
5.055
8.741
11.681
11.848
5.426
85
17.501
12.267
16.931
22.010
4.934
18.636
8.167
10.660
5.284
10.439
12.392
12.656
5.552
95
20.104
14.349
17.435
21.680
5.011
20.800
8.797
12.931
5.334
11.404
12.197
13.640
5.891
105
21.951
15.170
16.700
19.318
5.453
20.830
9.370
15.129
5.800
10.514
11.032
13.752
5.745
115
20.385
14.337
16.688
18.629
7.109
18.066
8.969
14.270
6.014
8.398
9.452
12.938
5.122
125
17.655
11.136
14.798
15.712
7.554
13.324
7.386
10.924
5.278
6.956
8.047
10.797
4.106
135
10.348
5.974
9.327
9.096
5.879
6.914
4.598
6.387
3.223
4.267
4.705
6.429
2.300
145
3.091
1.632
3.315
2.693
2.384
1.875
1.460
2.273
1.196
1.333
1.554
2.073
0.728In Table5, the comparison of the data shows that the maximum earthquake compressive stress obtained under different earthquake waves is significantly different, and the degree of the difference can be expressed by the standard deviation. To quantify the degree of difference, its standard deviation S is(21)S=∑l=1nσl−σ¯2n−1,where n is the number of samples, that is, the number of selected earthquake waves; σl is normal stress on the central section of segment l, that is, the sum of bending stress and axial stress, which is σl=Ml/Wl+Nl/Al; Wl is the bending-resistance modulus of the segment l; and σ¯ is the mean normal stress of the central section.The right side of Figure9(a) is the comparison of the mean values of earthquake compressive stresses between Tables 5 and 6 and the corresponding standard deviation. The left side of Figure 9(a) is the comparisons of the mean values of earthquake tensile stresses and the corresponding standard deviation. The data in Tables 7 and 8 (see Appendix B) are provided regarding maximum earthquake tensile stress, mean value, and standard deviation of the 150 m high chimney when considering vertical eccentricities and neglecting vertical eccentricities. In Figure 9(a), the red and pink columns are, respectively, the mean values of earthquake compressive and tensile stresses considering vertical eccentricities. The blue and green columns are, respectively, the mean values of earthquake compressive and tensile stress neglecting vertical eccentricities. The red and pink lines are, respectively, the standard deviations of different earthquake compressive and tensile stresses considering vertical eccentricities. The blue and green lines are the standard deviations of different earthquake compressive and tensile stresses neglecting vertical eccentricities. Figure 9(b) shows the case of the 210 m high chimney (data in Tables 9–12).Figure 9
Comparisons of mean and standard deviation of the normal stress of chimney: (a) 150 m high chimney; (b) 210 m high chimney.
(a)
(b)Table 7
Maximum earthquake tensile stresses of the 150 m high chimney considering vertical eccentricities.
High
Serial number
1
2
3
4
5
6
7
8
9
10
11
Mean
Standard deviation
5
−14.899
−8.142
−11.107
−14.432
−3.545
−10.385
−5.515
−7.845
−3.068
−7.032
−7.351
−8.484
3.918
15
−14.643
−8.224
−9.130
−15.498
−3.170
−11.141
−6.012
−8.139
−2.637
−7.093
−7.417
−8.464
4.079
25
−15.184
−7.429
−10.161
−15.337
−2.699
−11.614
−6.114
−8.303
−2.496
−6.597
−8.329
−8.569
4.290
35
−16.726
−6.233
−10.023
−14.233
−2.904
−9.964
−5.454
−7.328
−2.928
−5.911
−8.414
−8.193
4.345
45
−20.149
−5.441
−13.105
−15.379
−3.684
−7.288
−5.441
−5.944
−3.742
−5.080
−8.275
−8.502
5.371
55
−23.087
−5.802
−14.819
−17.339
−3.907
−7.209
−6.053
−6.328
−4.228
−4.772
−9.252
−9.345
6.296
65
−24.106
−8.842
−13.522
−18.325
−4.194
−11.620
−6.900
−8.679
−4.588
−6.925
−10.773
−10.770
6.021
75
−23.367
−11.458
−17.079
−20.100
−5.300
−16.431
−8.021
−11.427
−5.875
−10.037
−11.643
−12.794
5.798
85
−18.712
−12.305
−16.848
−20.900
−5.449
−19.142
−8.437
−12.990
−5.350
−12.505
−12.969
−13.237
5.330
95
−23.568
−15.257
−17.401
−22.463
−5.488
−20.761
−9.536
−14.479
−5.326
−13.449
−12.741
−14.588
6.229
105
−26.781
−16.007
−19.081
−23.512
−6.008
−19.810
−10.048
−15.935
−6.174
−12.991
−12.563
−15.356
6.669
115
−26.059
−13.898
−24.943
−22.161
−7.670
−16.399
−9.067
−14.409
−6.982
−11.603
−10.365
−14.869
6.797
125
−21.997
−10.105
−24.736
−16.982
−8.222
−11.353
−6.804
−10.998
−7.044
−8.738
−8.935
−12.356
6.136
135
−12.284
−5.071
−15.064
−8.473
−5.732
−5.321
−3.660
−5.796
−4.293
−4.246
−4.992
−6.812
3.667
145
−2.349
−0.883
−3.139
−1.509
−1.387
−0.863
−0.744
−1.380
−0.862
−0.817
−0.948
−1.353
0.757Table 8
Maximum earthquake tensile stresses of the 150 m high chimney ignoring vertical eccentricities.
High
Serial number
1
2
3
4
5
6
7
8
9
10
11
Mean
Standard deviation
5
−12.713
−7.405
−8.992
−13.313
−3.442
−10.154
−5.158
−7.275
−2.661
−5.482
−6.441
−7.549
3.476
15
−13.689
−7.593
−8.485
−14.183
−2.785
−10.792
−5.623
−7.339
−2.313
−5.516
−7.092
−7.765
3.883
25
−14.737
−7.445
−9.195
−14.689
−2.622
−11.050
−5.718
−7.167
−2.953
−5.404
−7.922
−8.082
4.101
35
−16.038
−5.984
−9.391
−15.677
−2.783
−9.446
−5.863
−6.125
−3.703
−4.969
−8.241
−8.020
4.412
45
−19.887
−5.361
−9.031
−15.521
−4.038
−7.266
−5.923
−6.521
−3.986
−4.520
−8.731
−8.253
5.066
55
−22.010
−5.296
−8.157
−14.138
−4.718
−7.033
−6.395
−7.338
−3.525
−4.671
−8.345
−8.330
5.348
65
−20.442
−7.589
−9.998
−16.058
−3.955
−11.176
−7.868
−8.750
−3.820
−6.332
−9.822
−9.619
4.957
75
−18.744
−10.206
−14.035
−20.320
−4.984
−15.659
−8.765
−10.409
−5.016
−8.449
−11.561
−11.650
5.078
85
−17.127
−12.362
−14.803
−21.183
−4.995
−18.267
−8.545
−10.835
−4.926
−9.392
−11.886
−12.211
5.259
95
−19.308
−14.363
−15.150
−21.096
−5.158
−20.466
−8.701
−12.565
−4.347
−9.829
−11.888
−12.988
5.778
105
−20.735
−15.441
−17.201
−19.442
−5.141
−21.169
−9.351
−14.565
−4.676
−9.461
−10.996
−13.471
5.938
115
−20.512
−14.054
−17.158
−18.152
−6.723
−18.408
−9.040
−13.521
−5.411
−8.833
−9.374
−12.835
5.232
125
−16.682
−10.576
−13.862
−14.924
−6.909
−13.614
−7.317
−10.586
−5.691
−7.308
−7.275
−10.431
3.814
135
−9.702
−5.606
−9.130
−8.663
−5.585
−7.096
−4.223
−6.238
−4.007
−4.296
−4.774
−6.302
2.067
145
−3.090
−1.589
−3.339
−2.557
−2.419
−1.928
−1.258
−2.233
−1.350
−1.345
−1.680
−2.072
0.717Table 9
Maximum earthquake compressive stresses of the 210 m high chimney considering vertical eccentricities.
High
Serial number
1
2
3
4
5
6
7
8
9
10
11
Mean
Standard deviation
5
9.073
4.481
8.898
10.293
2.941
9.218
3.664
7.445
6.273
6.974
6.646
6.901
2.416
15
9.938
4.449
8.687
10.863
3.229
9.964
3.797
8.021
6.456
7.002
6.773
7.198
2.591
25
10.466
4.260
7.814
11.223
3.642
10.552
3.865
8.428
6.349
6.810
6.874
7.298
2.707
35
10.942
4.536
6.342
10.703
3.143
10.389
3.755
8.574
5.477
6.347
7.981
7.108
2.806
45
10.664
4.682
7.363
9.732
4.033
9.170
3.919
8.081
4.383
5.379
8.613
6.911
2.501
55
10.790
5.288
9.082
10.740
4.553
7.951
4.411
7.829
4.681
5.806
9.083
7.292
2.452
65
10.391
5.523
9.869
11.456
4.298
7.003
4.417
7.768
5.244
5.932
8.403
7.300
2.480
75
11.058
5.397
9.500
11.388
4.244
5.592
4.090
7.069
5.027
6.239
8.873
7.134
2.653
85
12.283
5.330
8.956
13.619
4.035
7.998
4.624
8.426
5.976
8.010
9.112
8.034
3.010
95
13.669
5.604
11.904
16.621
4.800
11.195
5.281
10.340
7.300
9.797
9.709
9.656
3.704
105
14.484
7.044
14.138
16.663
5.771
13.921
5.765
11.962
9.226
10.333
11.088
10.945
3.708
115
16.080
8.297
16.374
15.511
6.376
17.064
6.771
14.798
11.033
10.732
12.649
12.335
3.950
125
18.201
9.514
17.908
19.468
7.501
20.617
7.767
17.874
12.290
11.079
13.899
14.192
4.836
135
20.731
10.387
17.076
25.177
7.543
23.166
8.884
19.191
15.434
13.947
14.108
15.968
5.754
145
22.279
8.930
16.182
26.017
6.539
23.188
8.091
17.090
15.283
14.884
12.716
15.564
6.365
155
22.605
9.904
20.913
24.714
6.672
21.495
8.269
17.399
13.326
15.063
12.582
15.722
6.154
165
21.942
11.283
22.329
23.957
6.477
19.121
8.648
17.038
12.612
13.700
15.753
15.715
5.766
175
18.729
10.621
19.787
19.615
5.954
15.222
8.110
14.354
10.513
11.494
15.196
13.600
4.668
185
13.872
8.191
14.943
14.117
5.435
10.425
6.618
10.449
8.133
8.744
12.918
10.349
3.233
195
7.892
4.429
8.716
7.440
4.429
5.082
3.846
5.557
4.556
4.826
7.781
5.868
1.735
205
1.547
0.811
1.607
1.321
1.283
0.827
0.769
1.046
0.833
0.868
1.611
1.138
0.343Table 10
Maximum earthquake compressive stresses of the 210 m high chimney ignoring vertical eccentricities.
High
Serial number
1
2
3
4
5
6
7
8
9
10
11
Mean
Standard deviation
5
7.913
4.726
10.278
9.045
3.297
7.885
3.325
8.324
5.691
5.556
6.690
6.612
2.301
15
8.491
4.613
8.632
9.591
3.192
8.644
3.575
8.693
5.534
5.858
6.674
6.681
2.270
25
8.928
4.295
7.073
9.821
3.250
9.291
3.594
8.592
6.273
6.019
6.922
6.733
2.306
35
9.236
4.388
7.835
9.172
3.702
9.105
3.628
7.755
6.311
5.957
8.086
6.834
2.166
45
9.185
4.857
7.109
7.923
4.515
8.191
3.885
6.749
5.297
5.549
9.094
6.578
1.871
55
9.400
5.456
7.823
8.987
4.993
7.663
4.080
6.935
4.754
5.110
9.630
6.803
2.022
65
9.131
6.071
9.355
9.790
4.614
6.873
4.019
7.968
5.305
5.143
9.256
7.048
2.137
75
9.085
5.875
11.357
10.901
5.045
5.885
3.444
9.261
5.258
5.004
9.866
7.362
2.765
85
9.804
5.437
11.261
11.823
5.110
8.655
4.133
10.770
6.123
6.690
9.679
8.135
2.726
95
10.652
5.793
12.132
14.150
5.420
11.524
4.778
12.735
7.346
8.406
10.092
9.366
3.211
105
13.107
7.555
13.799
14.715
6.191
13.593
5.764
13.224
8.968
8.992
11.902
10.710
3.289
115
15.388
8.611
14.686
14.126
6.467
15.260
7.159
12.609
9.625
9.462
13.769
11.560
3.359
125
16.887
10.171
15.816
16.068
7.341
17.337
7.958
13.671
9.509
10.761
15.444
12.815
3.741
135
17.439
10.288
16.448
18.511
7.162
19.932
7.987
17.367
12.785
13.289
16.818
14.366
4.357
145
16.328
8.887
20.181
19.843
6.132
20.116
6.912
19.538
14.792
14.446
13.972
14.650
5.294
155
14.935
10.856
23.421
19.606
6.668
18.987
7.166
19.462
15.096
14.097
13.785
14.916
5.274
165
14.959
11.991
23.222
18.400
7.429
16.611
7.161
18.445
14.258
12.321
14.319
14.465
4.759
175
12.915
11.174
20.729
15.953
6.914
13.073
6.393
14.984
11.878
9.970
14.322
12.573
4.082
185
11.388
9.186
19.862
12.126
7.682
9.282
5.618
11.792
9.435
7.435
12.891
10.609
3.793
195
7.752
5.577
13.811
6.922
6.482
5.034
3.681
7.045
5.408
4.287
8.232
6.748
2.736
205
2.573
1.600
4.560
1.897
2.434
1.349
1.189
2.327
1.491
1.240
2.438
2.100
0.967Table 11
Maximum earthquake tensile stresses of the 210 m high chimney considering vertical eccentricities.
High
Serial number
1
2
3
4
5
6
7
8
9
10
11
Mean
Standard deviation
5
−10.262
−4.948
−8.135
−10.034
−2.895
−8.930
−3.943
−7.722
−5.298
−6.667
−6.588
−6.857
2.424
15
−10.435
−4.938
−7.568
−11.181
−2.919
−9.723
−4.201
−8.217
−5.552
−6.907
−6.881
−7.138
2.630
25
−10.935
−4.554
−6.246
−12.106
−3.241
−10.269
−4.221
−8.691
−6.122
−6.946
−7.151
−7.317
2.884
35
−10.553
−4.145
−6.472
−11.522
−3.273
−10.373
−3.642
−8.527
−5.875
−6.486
−7.779
−7.150
2.868
45
−10.408
−4.240
−7.468
−9.756
−4.018
−9.579
−3.551
−7.277
−4.746
−5.330
−8.867
−6.840
2.562
55
−11.024
−4.832
−8.860
−10.900
−4.600
−8.365
−3.709
−6.914
−4.711
−5.920
−9.001
−7.167
2.610
65
−10.567
−5.284
−9.795
−11.400
−4.375
−6.960
−3.870
−7.157
−4.987
−6.342
−8.445
−7.198
2.563
75
−9.261
−5.017
−10.066
−11.198
−4.483
−5.807
−3.879
−7.801
−5.100
−6.042
−9.195
−7.077
2.520
85
−13.145
−4.857
−9.754
−14.902
−4.693
−8.210
−4.959
−7.609
−6.291
−8.202
−9.445
−8.370
3.332
95
−16.522
−6.118
−12.903
−17.681
−4.622
−11.390
−5.345
−9.800
−7.187
−10.105
−10.256
−10.175
4.304
105
−17.546
−7.618
−15.135
−17.703
−5.666
−14.323
−6.034
−12.327
−8.284
−10.474
−11.492
−11.509
4.327
115
−20.278
−8.779
−15.214
−17.611
−6.720
−16.912
−6.769
−15.784
−8.848
−10.066
−12.580
−12.687
4.725
125
−22.597
−9.647
−15.636
−19.295
−6.777
−19.980
−7.368
−18.825
−9.581
−9.736
−15.350
−14.072
5.641
135
−24.483
−10.272
−15.168
−25.018
−7.579
−22.450
−8.291
−19.846
−11.195
−12.573
−16.906
−15.798
6.393
145
−22.866
−9.103
−16.355
−25.713
−5.767
−22.516
−8.095
−17.358
−14.228
−14.451
−15.153
−15.600
6.365
155
−20.992
−9.940
−20.354
−25.657
−5.673
−22.139
−8.594
−16.513
−15.972
−14.672
−14.746
−15.932
6.137
165
−19.689
−11.736
−21.039
−25.636
−7.114
−20.421
−9.126
−15.506
−15.183
−13.670
−12.456
−15.598
5.594
175
−17.762
−11.194
−21.276
−21.932
−6.327
−16.354
−8.317
−13.811
−12.043
−11.622
−13.407
−14.004
4.944
185
−13.414
−8.601
−17.871
−15.897
−6.273
−11.122
−6.563
−10.575
−8.266
−8.356
−12.082
−10.820
3.745
195
−7.845
−4.509
−10.141
−8.223
−4.973
−5.357
−3.606
−5.460
−4.672
−4.489
−7.163
−6.040
2.015
205
−1.591
−0.770
−1.794
−1.524
−1.424
−0.845
−0.694
−0.972
−0.819
−0.753
−1.387
−1.143
0.403Table 12
Maximum earthquake tensile stresses of the 210 m high chimney ignoring vertical eccentricities.
High
Serial number
1
2
3
4
5
6
7
8
9
10
11
Mean
Standard deviation
5
−7.404
−4.719
−8.533
−8.527
−3.335
−8.004
−3.635
−7.965
−6.483
−5.570
−7.629
−6.528
1.923
15
−7.867
−4.672
−7.619
−9.499
−3.416
−8.788
−3.696
−8.663
−6.613
−6.051
−7.548
−6.767
2.080
25
−8.776
−4.683
−8.208
−10.373
−3.163
−9.459
−3.540
−9.161
−6.331
−6.333
−6.843
−6.988
2.443
35
−9.086
−4.199
−8.700
−10.468
−3.866
−9.371
−3.766
−8.596
−5.128
−6.099
−7.874
−7.014
2.463
45
−8.183
−4.540
−7.346
−9.431
−4.291
−8.534
−4.033
−7.092
−4.514
−5.405
−8.582
−6.541
2.022
55
−7.768
−5.168
−8.978
−8.164
−4.662
−7.902
−4.170
−6.826
−4.667
−5.156
−9.280
−6.613
1.895
65
−8.262
−5.869
−8.851
−9.653
−4.784
−7.242
−4.073
−7.791
−5.180
−5.147
−9.472
−6.939
2.008
75
−8.876
−5.860
−9.759
−10.681
−4.735
−6.168
−3.488
−8.404
−5.163
−4.789
−9.311
−7.021
2.444
85
−9.957
−5.203
−10.615
−12.112
−5.577
−8.535
−3.543
−10.392
−6.100
−6.789
−8.921
−7.977
2.703
95
−11.990
−6.905
−12.809
−13.458
−5.783
−11.466
−4.556
−12.103
−7.473
−8.771
−9.768
−9.553
3.052
105
−13.475
−8.542
−13.539
−14.303
−6.060
−13.615
−5.865
−12.736
−8.416
−9.708
−11.819
−10.734
3.137
115
−14.847
−9.219
−14.785
−14.387
−6.629
−15.528
−7.442
−12.490
−8.892
−9.946
−13.041
−11.564
3.232
125
−16.430
−10.839
−14.144
−16.638
−7.599
−17.232
−8.242
−15.041
−11.332
−11.355
−15.373
−13.111
3.404
135
−17.538
−10.834
−18.645
−19.642
−6.946
−20.560
−7.990
−17.535
−13.396
−13.107
−15.292
−14.680
4.638
145
−16.080
−8.417
−21.186
−20.686
−6.343
−20.739
−7.896
−19.356
−13.220
−13.720
−12.230
−14.534
5.500
155
−18.273
−9.907
−21.025
−21.991
−5.937
−19.264
−8.263
−19.261
−14.030
−12.955
−13.363
−14.933
5.417
165
−18.640
−11.394
−23.870
−20.157
−6.959
−17.581
−8.305
−17.337
−13.363
−11.784
−15.212
−14.964
5.159
175
−15.885
−11.241
−21.453
−15.522
−7.398
−14.219
−7.484
−14.848
−11.109
−9.557
−15.198
−13.083
4.197
185
−11.686
−8.918
−19.105
−12.072
−6.373
−10.024
−5.830
−10.987
−8.906
−6.633
−12.945
−10.316
3.779
195
−7.186
−5.468
−13.557
−7.196
−5.199
−5.307
−3.738
−6.277
−5.492
−3.635
−8.202
−6.478
2.734
205
−2.180
−1.625
−4.540
−2.291
−2.251
−1.383
−1.265
−2.016
−1.692
−1.025
−2.601
−2.079
0.950Figure9 shows that the earthquake normal stresses increase evidently when considering vertical eccentricities. Significant differences of the maximum earthquake normal stress are observed under different seismic waves actions.Figure10 is the boxplot of the maximum earthquake compressive stress distribution at different cross sections of 150 m and 210 m high chimneys under different seismic wave actions when considering eccentricities. They show the mean value, minimum, maximum, and 25%–75% of earthquake compressive stresses. The distribution characteristics of the earthquake compressive stress data are also shown in the figures. The stress distributions near the bottom of the chimney are relatively intensive, whereas those at the middle and upper parts of the chimney are relatively scattered.Figure 10
Boxplot of the compressive stress distribution of chimney considering eccentricities: (a) 150 m high chimney; (b) 210 m high chimney.
(a)
(b)The results studied above show that significant differences of maximum earthquake normal stress under different earthquake wave actions. The comparisons demonstrate that the conventional results (eccentricities neglected) might be less than their actual tensile/compressive stresses in the seismic control cross section of the chimneys under the actions of several specific earthquake waves. The calculation results show that eccentricities between the spatial discrete nodes and the corresponding centroids of investigated lumps should be considered in earthquake response of tall RC chimneys to obtain accurate calculation results.


#### 5.2.2. Effects of Vertical Earthquake

The phenomena of eccentricities are considered to study the effect of vertical earthquake on earthquake normal stresses. Two types of seismic load are applied for each of the four earthquake waves. In the first case, the combined horizontal and vertical earthquake waves (see red lines in Figures11 and 12) are inputted. In the second case, only the horizontal earthquake wave (see blue lines in Figures 11 and 12) is inputted.Figure 11
Spatial distributions of maximum earthquake normal stresses at different cross sections for the 150 m high chimney under two types of seismic load applied for each of the four earthquake wave actions: (a) Landers wave (No. 1); (b) Loma Prieta wave (No. 2); (c) Borrego wave (No. 3); (d) Chichi wave (No. 4).
(a)
(b)
(c)
(d)Figure 12
Spatial distributions of maximum earthquake normal stresses at different cross sections for the 210 m high chimney under two types of seismic load applied for each of the four earthquake wave actions: (a) Landers wave (No. 1); (b) Loma Prieta wave (No. 2); (c) Borrego wave (No. 3); (d) Chichi wave (No. 4).
(a)
(b)
(c)
(d)For the 150 m high chimney, Figure11 shows that the peak normal tensile stress increases by 4% for the Landers wave (No. 1) and 10% for the Loma Prieta wave (No. 2) if vertical earthquake waves are considered. The peak normal compressive stress increases by 7% for the Loma Prieta wave (No. 2) and 10% for the Chichi wave (No. 4) if vertical earthquake waves are considered.For the 210 m high chimney, Figure12 shows that the peak normal tensile stress increases by 12% for the Landers wave (No. 1) and 9% for the Loma Prieta wave (No. 2) if vertical earthquake waves are considered. The peak normal compressive stress increases by 9% for Landers wave (No. 1) and 13% for Loma Prieta wave (No. 2) if vertical earthquake waves are considered.Figure13 shows the spatial comparison curves of the mean values of the maximum normal earthquake stresses for the 150 m and 210 m high chimneys under two types of seismic load. Figure 13(a) is obtained from the average values of Tables 5, 7, 13, and 14. Figure 13(b) is drawn from the mean values of Tables 9, 11, 15, and 16. Figure 13 shows that the normal stress of the chimney can be increased considering the vertical seismic acceleration.Figure 13
Comparison curve of the mean value of maximum earthquake normal stresses for chimneys under two types of seismic load: (a) 150 m high chimney; (b) 210 m high chimney.
(a)
(b)Table 13
Maximum earthquake compressive stresses of the 150 m high chimney when only horizontal seismic acceleration is inputted.
High
Serial number
1
2
3
4
5
6
7
8
9
10
11
Mean
Standard deviation
5
15.762
6.884
11.475
13.226
2.984
9.720
5.314
7.104
3.694
6.267
7.235
8.151
3.986
15
16.533
6.912
9.673
14.154
2.374
10.375
5.697
7.298
3.395
6.244
7.812
8.224
4.261
25
15.970
6.204
9.464
14.907
2.194
10.292
5.644
6.853
3.049
6.014
9.059
8.150
4.379
35
17.119
4.713
9.862
13.772
2.836
8.412
4.903
6.279
4.110
5.139
9.057
7.837
4.407
45
20.107
3.593
12.133
15.772
4.133
5.642
4.755
5.276
4.437
3.707
8.936
8.045
5.603
55
21.071
4.229
14.646
18.041
3.645
6.161
5.083
6.183
3.713
3.242
8.740
8.614
6.337
65
21.786
7.321
13.780
17.612
3.645
10.790
6.299
8.381
4.823
5.491
9.217
9.922
5.686
75
20.897
9.792
16.022
18.742
4.307
15.863
7.209
10.533
5.855
8.858
11.386
11.769
5.406
85
18.671
10.349
17.184
18.626
3.973
18.579
7.799
11.399
5.396
11.049
12.025
12.277
5.361
95
21.849
13.139
17.782
21.304
4.313
19.716
8.952
13.522
5.751
12.327
12.051
13.701
5.966
105
22.592
14.578
19.457
22.671
4.985
19.364
9.133
14.541
6.688
12.095
11.287
14.308
6.136
115
23.369
12.785
24.161
21.098
7.016
16.249
8.339
12.902
7.520
11.069
9.754
14.024
6.313
125
19.263
9.922
22.706
16.703
7.430
11.274
6.289
9.926
7.222
8.303
7.947
11.544
5.502
135
10.624
5.025
14.653
8.506
5.419
5.190
3.297
5.773
4.065
4.181
4.662
6.490
3.436
145
1.946
0.838
3.020
1.463
1.236
0.789
0.615
1.260
0.848
0.766
0.892
1.243
0.705Table 14
Maximum earthquake tensile stresses of the 150 m high chimney when only horizontal seismic acceleration is inputted.
High
Serial number
1
2
3
4
5
6
7
8
9
10
11
Mean
Standard deviation
5
−14.019
−7.073
−10.794
−13.376
−3.065
−9.883
−5.107
−7.404
−2.917
−6.109
−7.146
−7.899
3.748
15
−13.568
−6.946
−8.764
−14.184
−2.128
−10.529
−5.319
−7.538
−2.438
−6.015
−7.459
−7.717
3.918
25
−13.396
−6.049
−9.673
−14.268
−2.217
−10.586
−5.134
−7.319
−2.751
−5.799
−7.994
−7.744
3.943
35
−14.760
−4.852
−9.829
−14.309
−2.644
−8.895
−4.731
−6.290
−3.085
−5.040
−7.879
−7.483
4.145
45
−18.971
−3.749
−12.950
−15.797
−3.562
−6.153
−4.989
−5.575
−3.317
−3.663
−7.614
−7.849
5.499
55
−20.940
−3.856
−14.872
−16.458
−3.938
−5.793
−5.151
−5.917
−3.675
−3.466
−8.690
−8.432
6.123
65
−22.465
−7.073
−14.037
−17.207
−3.647
−10.650
−6.376
−8.108
−4.479
−5.484
−10.224
−9.977
5.836
75
−22.721
−9.433
−16.977
−18.920
−4.644
−15.423
−7.684
−10.420
−5.601
−8.588
−11.544
−11.996
5.779
85
−20.043
−10.301
−16.678
−19.349
−4.152
−17.961
−8.200
−11.508
−4.973
−11.015
−12.732
−12.447
5.513
95
−22.737
−13.287
−16.842
−21.924
−5.111
−19.726
−9.137
−12.996
−4.815
−11.834
−12.481
−13.717
6.121
105
−24.485
−14.367
−18.594
−22.963
−5.152
−18.972
−9.762
−13.993
−5.724
−11.670
−11.579
−14.297
6.413
115
−23.410
−12.647
−24.355
−21.682
−6.705
−15.693
−8.900
−12.975
−6.598
−11.093
−9.727
−13.980
6.490
125
−20.313
−9.584
−23.734
−16.823
−7.219
−10.906
−6.540
−9.631
−6.598
−8.641
−8.727
−11.701
5.879
135
−11.408
−4.711
−15.196
−8.484
−5.553
−4.981
−3.266
−5.171
−4.017
−4.446
−5.039
−6.570
3.671
145
−2.166
−0.757
−3.138
−1.419
−1.330
−0.772
−0.624
−1.151
−0.790
−0.775
−0.950
−1.261
0.764Table 15
Maximum earthquake compressive stresses of the 210 m high chimney when only horizontal seismic acceleration is inputted.
High
Serial number
1
2
3
4
5
6
7
8
9
10
11
Mean
Standard deviation
5
8.481
4.199
8.705
10.156
2.007
9.332
3.272
7.522
5.873
6.495
6.213
6.569
2.599
15
8.901
4.019
8.216
10.622
2.225
10.012
3.285
8.055
5.978
6.514
5.906
6.703
2.746
25
9.582
3.469
7.243
10.684
2.446
10.362
3.266
8.297
5.919
6.143
5.912
6.666
2.869
35
10.568
3.052
6.345
9.797
2.526
9.397
2.767
7.913
5.484
5.128
6.954
6.357
2.870
45
9.972
3.302
6.902
8.454
3.223
7.488
2.873
7.388
4.133
4.051
7.458
5.931
2.467
55
10.485
3.886
8.502
9.475
3.529
7.237
3.392
6.915
4.759
4.188
8.276
6.422
2.573
65
10.190
4.154
9.585
9.633
3.223
6.894
3.521
6.782
5.247
4.515
8.447
6.563
2.599
75
9.191
3.988
8.965
9.777
3.536
5.854
3.329
6.980
4.994
4.741
8.090
6.313
2.395
85
11.177
3.643
8.928
12.420
3.799
7.406
3.792
8.005
5.956
6.387
8.570
7.280
2.943
95
13.516
4.035
11.145
14.322
3.448
10.454
4.424
9.272
7.536
7.673
9.883
8.701
3.683
105
14.580
5.302
13.069
14.196
4.576
12.836
5.224
10.651
8.796
8.504
10.711
9.859
3.671
115
16.062
6.498
16.367
13.919
5.366
15.006
6.185
13.698
10.445
8.645
11.724
11.265
4.087
125
18.730
8.256
17.638
18.152
5.275
19.078
7.159
16.475
11.612
10.651
13.545
13.325
5.028
135
20.661
8.608
16.365
23.615
5.444
21.778
8.219
17.456
14.630
14.181
13.650
14.964
5.838
145
19.716
6.626
15.624
24.576
4.835
21.610
7.150
16.670
14.439
14.995
11.521
14.342
6.352
155
20.779
8.409
20.741
25.679
4.079
20.345
6.788
16.586
13.115
14.038
9.833
14.581
6.862
165
19.550
9.803
22.386
24.808
5.578
18.092
7.307
16.231
12.737
12.603
13.035
14.739
6.095
175
17.347
9.451
19.978
20.117
5.357
14.149
7.215
13.845
10.595
11.112
13.379
12.959
4.840
185
13.642
7.253
15.301
14.461
5.160
9.390
5.906
9.742
7.723
8.592
12.240
9.946
3.487
195
7.997
3.857
8.696
7.489
4.402
4.434
3.330
4.899
4.067
4.728
7.289
5.563
1.906
205
1.474
0.668
1.583
1.222
1.160
0.650
0.567
0.936
0.698
0.824
1.336
1.011
0.360Table 16
Maximum earthquake tensile stresses of the 210 m high chimney when only horizontal seismic acceleration is inputted.
High
Serial number
1
2
3
4
5
6
7
8
9
10
11
Mean
Standard deviation
5
−9.499
−3.914
−8.402
−10.528
−2.271
−8.311
−3.147
−6.884
−5.485
−5.603
−6.195
−6.385
2.643
15
−10.088
−3.759
−7.888
−11.120
−2.170
−9.016
−3.276
−7.374
−5.517
−5.617
−5.575
−6.491
2.872
25
−10.735
−3.268
−6.696
−12.154
−2.343
−9.816
−3.359
−8.020
−5.630
−5.495
−6.296
−6.710
3.196
35
−9.917
−3.148
−6.027
−11.745
−2.324
−9.532
−2.986
−7.852
−5.311
−5.431
−7.037
−6.483
3.071
45
−9.763
−3.035
−6.814
−9.538
−3.218
−7.901
−3.112
−7.034
−4.432
−4.657
−8.188
−6.154
2.563
55
−9.887
−3.853
−8.522
−9.456
−3.515
−6.972
−3.159
−6.434
−4.759
−4.555
−8.436
−6.323
2.486
65
−9.276
−4.106
−8.968
−10.471
−3.061
−6.145
−3.460
−7.597
−5.049
−4.824
−8.064
−6.457
2.550
75
−8.813
−4.006
−9.201
−10.360
−3.433
−5.116
−3.316
−8.168
−5.097
−4.562
−8.403
−6.407
2.593
85
−11.696
−3.733
−9.048
−13.089
−3.214
−7.460
−4.127
−7.880
−6.340
−6.498
−9.220
−7.482
3.172
95
−15.283
−5.286
−12.218
−16.060
−3.427
−10.971
−4.632
−9.692
−7.143
−8.267
−10.285
−9.388
4.147
105
−17.060
−6.333
−14.403
−16.566
−4.330
−13.582
−5.322
−11.064
−8.223
−8.519
−10.820
−10.566
4.430
115
−19.059
−6.495
−14.769
−16.340
−5.136
−15.559
−5.968
−14.179
−8.929
−7.669
−12.054
−11.469
4.823
125
−21.359
−7.756
−15.261
−20.501
−5.311
−17.463
−6.259
−17.193
−9.471
−8.103
−13.228
−12.900
5.807
135
−21.802
−8.174
−15.876
−26.142
−5.335
−20.549
−7.745
−18.417
−11.458
−11.799
−13.529
−14.621
6.555
145
−19.722
−7.568
−15.281
−26.594
−4.215
−21.400
−7.706
−16.488
−13.112
−13.520
−12.850
−14.405
6.567
155
−18.995
−9.012
−19.516
−23.940
−4.076
−21.179
−8.472
−16.200
−15.026
−13.612
−12.953
−14.816
5.995
165
−18.022
−10.638
−21.577
−22.969
−5.468
−19.116
−8.724
−16.437
−14.438
−12.906
−11.733
−14.730
5.466
175
−16.832
−10.263
−21.403
−20.010
−5.569
−15.119
−7.902
−14.741
−11.533
−11.518
−11.998
−13.353
4.844
185
−13.024
−7.875
−17.879
−14.764
−5.138
−10.190
−5.783
−11.224
−8.565
−8.835
−10.257
−10.321
3.791
195
−6.944
−4.092
−10.140
−7.591
−4.379
−4.781
−3.151
−5.887
−4.848
−4.741
−5.925
−5.680
1.956
205
−1.261
−0.689
−1.797
−1.221
−1.265
−0.696
−0.591
−1.091
−0.865
−0.805
−1.125
−1.037
0.353The results studied above show that the vertical ground motion should not be neglected, especially for tall RC chimneys built on rock sites or soft rock sites.


## 5.1. Earthquake Records Used

Eleven earthquake acceleration records on four types of soil are used as earthquake load to calculate the structural earthquake response. Four types of soil are denoted by rock, soft rock, stiff soil, and soft soil. Soil type is characterized by shear wave velocity. The corresponding earthquake records are selected from the Pacific Earthquake Engineering Research Center website [30]. Table 1 shows the 11 earthquake records.Table 1
Earthquake records used.
Serial number
Soil types
Earthquake
Date
Direction
Station and component
PGA (g)
No. 1
Rock
Landers
1992/06/28
Horizontal
LANDERS/ABY000
0.115
Vertical
LANDERS/ABY-UP
0.090
No. 2
Soft rock
Loma Prieta
1989/10/18
Horizontal
LOMAP/CLS000
0.644
Vertical
LOMAP/CLS-UP
0.455
No. 3
Stiff soil
Borrego
1942/10/21
Horizontal
BORREGO/B-ELC000
0.033
Vertical
BORREGO/B-ELC-UP
0.068
No. 4
Soft soil
Chichi
1999/09/20
Horizontal
CHICHI/CHY004-N
0.100
Vertical
CHICHI/CHY004-V
0.041
No. 5
Rock
Morgan Hill
1984/04/24
Horizontal
MORGAN/G01230
0.069
Vertical
MORGAN/G01-UP
0.092
No. 6
Soft rock
Loma Prieta
1989/10/18
Horizontal
LOMAP/GGB360
0.123
Vertical
LOMAP/GGB-UP
0.056
No. 7
Stiff soil
Chichi
1999/09/20
Horizontal
CHICHI CHY028-N
0.821
Vertical
CHICHI CHY028-V
0.337
No. 8
Stiff soil
Chichi
1999/09/20
Horizontal
CHICHI TCU076-N
0.416
Vertical
CHICHI TCU076-V
0.281
No. 9
Stiff soil
Morgan Hill
1984/04/24
Horizontal
MORGAN/G03090
0.200
Vertical
MORGAN/G03-UP
0.395
No. 10
Stiff soil
Morgan Hill
1984/04/24
Horizontal
MORGAN HVR240
0.312
Vertical
MORGAN HVR-UP
0.110
No. 11
Soft soil
Imperial Valley
1079/10/15
Horizontal
IMPVALL H-E03140
0.266
Vertical
IMPVALL H-E03-UP
0.127According to the peak value of design earthquake acceleration of 8 degrees seismic intensity in the literature [31], the peaks of the horizontal earthquake accelerations are adjusted to 0.2 g in practical calculations. The peaks of the vertical earthquake accelerations are adjusted to 0.13 g. The damping is omitted.


## 5.2. Two Models of RC Chimneys

Two RC chimneys [32] of 150 m and 210 m height are studied. Figure 6 shows the two models of RC chimneys. (1) The concrete strength grade of the 150 m high RC chimney is C25, and the steel strength grade is HRB335. The outer radius of the bottom is 8.46 m, and the wall thickness is 0.4 m. The outer radius of the top is 3.0 m, and the wall thickness is 0.16 m. The gradient c is 0.1 below 25 m, 0.03 ranging from 25 m to 75 m, and 0.015 ranging from 75 m to 150 m. Table 2 shows the parameters of the 150 m high RC chimney. (2) The concrete strength grade of the 210 m high RC chimney is C35, and the steel strength grade is HRB335. The outer radius of the bottom is 10.6 m, and the wall thickness is 0.6 m. The outer radius of the top is 3.85 m, and the wall thickness is 0.18 m. The gradient c is 0.1 below 30 m, 0.03 ranging from 30 m to 100 m, and 0.015 ranging from 100 m to 210 m. Table 3 shows the parameters of the 210 m high RC chimney. According to the literature [33], the standard value of the compressive strength of C25 concrete is 16.7 N/mm2, the design value is 11.9 N/mm2, and the elastic modulus is EC = 2.8 × 104 N/mm2. The standard value of the compressive strength of C35 concrete is 23.4 N/mm2, the design value is 16.7 N/mm2, and the elastic modulus is EC = 3.15 × 104 N/mm2. The elastic modulus of HRB335 reinforcement is ES = 2.0 × 105 N/mm2, the standard value of strength (yield strength) is 335 N/mm2, and the design value of strength is 300 N/mm2.Figure 6
Model of (a) 150 m and (b) 210 m high chimneys.
(a)
(b)Table 2
Parameters of 150 m high RC chimney.
Elevation (m)
Outer radius (m)
Wall thickness (m)
Reinforcement ratio (%)
150
3.00
0.16
0.45
137.5
3.52
0.16
0.47
125
3.71
0.16
0.47
112.5
3.89
0.18
0.42
100
4.08
0.18
0.42
87.5
4.27
0.22
0.47
75
4.46
0.22
0.47
62.5
4.83
0.26
0.52
50
5.21
0.26
0.52
37.5
5.58
0.34
0.50
25
5.96
0.34
0.50
5
7.96
0.40
0.52
0
8.46
0.40
0.52Table 3
Parameters of 210 m high RC chimney.
Elevation (m)
Outer radius (m)
Wall thickness (m)
Reinforcement ratio (%)
210
3.50
0.18
0.57
190
4.15
0.18
0.57
170
4.45
0.20
0.51
150
4.75
0.20
0.67
130
5.05
0.20
0.67
120
5.20
0.23
0.74
110
5.35
0.26
0.65
100
5.50
0.29
0.58
90
5.80
0.32
0.65
80
6.10
0.35
0.60
70
6.40
0.38
0.67
60
6.70
0.41
0.62
50
7.00
0.44
0.74
40
7.30
0.47
0.70
30
7.60
0.50
0.65
16.5
8.95
0.53
0.77
0
10.60
0.60
0.68The discrete segments are all 10 m long in the vertical direction for each of the two chimneys. Each chimney will become part of such calculated model consisting of several cylinders of different cross-sectional areas because an actual chimney structure has variable cross-sectional areas. Thus, the phenomena of eccentricities between the nodes and the corresponding centroids of the investigated lumps appear in the calculation of the earthquake response of the actual chimney. The mass of each investigated lump consists of half the upper segment and half the lower segment around the discrete node. The moment of inertia of each investigated lump about the horizontal axis through its centroid is calculated. On the basis of the calculation parameters of the 150 m and 210 m high chimneys, the time step satisfying the stability condition of the method is obtained by equation (19), Δt≤3.4ms and Δt≤3.1ms in the 150 m and 210 m high chimneys, respectively. The time step is 1.0 ms.


### 5.2.1. Effect of Eccentricity

Eleven earthquake waves are used. The effect of eccentricity on normal earthquake stress of the chimney are studied when horizontal and vertical components are inputted simultaneously.Figure7 shows the spatial distribution curves of the maximum earthquake normal stresses of different cross sections for the 150 m high chimney under the actions of Landers (No. 1 in Table 1), Loma Prieta (No. 2), Borrego (No. 3), and Chichi (No. 4) waves. Figure 8 shows the comparison curves of the 210 m high chimney. “New” denotes that the phenomena of eccentricities are considered. “Conventional” denotes the conventional results, in which the phenomena of eccentricities are neglected, for the earthquake responses of the real RC chimneys. Dashed lines (“tensile”) denote tensile stresses, whereas solid lines (“press”) denote compressive stresses.Figure 7
Spatial distributions of maximum earthquake normal stresses at different cross sections of the 150 m high chimney under four earthquake wave actions: (a) Landers wave (No. 1 in Table1); (b) Loma Prieta wave (No. 2); (c) Borrego wave (No. 3); (d) Chichi wave (No. 4).
(a)
(b)
(c)
(d)Figure 8
Spatial distributions of maximum earthquake normal stresses at different cross sections of the 210 m high chimney under four earthquake wave actions: (a) Landers wave (No. 1); (b) Loma Prieta wave (No. 2); (c) Borrego wave (No. 3); (d) Chichi wave (No. 4).
(a)
(b)
(c)
(d)A comparison of the red solid lines and the blue solid lines in Figures7 and 8 shows that the increased phenomena of peak values of the normal compressive stresses appear when eccentricities are considered.A comparison of the red dashed lines and the blue dashed lines in Figures7 and 8 shows that the increased phenomena of peak values of the normal tensile stresses appear when eccentricities are considered.Table4 shows the details of the increase in percentage of maximum earthquake normal tensile/compressive stresses when phenomena of eccentricities are considered.Table 4
Percentages of increase in maximum earthquake normal stresses.
Seismic waves
Maximum earthquake normal stresses increased
150 m
210 m
Landers (No. 1)
7% (compressive)
29% (tensile)
30% (compressive)
31% (tensile)
Loma Prieta (No. 2)
4% (compressive)
4% (tensile)
−6% (compressive)
3% (tensile)
Borrego (No. 3)
40% (compressive)
45% (tensile)
−5% (compressive)
−11% (tensile)
Chichi (No. 4)
15% (compressive)
11% (tensile)
31% (compressive)
17% (tensile)Table5 shows maximum earthquake compressive stress, mean value, and standard deviation of different sections of the 150 m high chimney under 11 earthquake wave actions when vertical eccentricities are considered. Table 6 shows maximum earthquake compressive stress, mean value, and standard deviation when vertical eccentricities are ignored.Table 5
Maximum earthquake compressive stresses of the 150 m high chimney under the action of 11 earthquake waves considering vertical eccentricities.
High
Serial number
1
2
3
4
5
6
7
8
9
10
11
Mean
Standard deviation
5
17.010
7.717
11.246
14.230
3.482
10.097
5.921
7.997
3.671
6.339
7.160
8.625
4.215
15
18.103
7.988
9.306
14.845
2.561
10.899
6.311
8.594
3.176
6.847
7.752
8.762
4.583
25
17.811
7.554
9.795
16.047
2.853
11.025
6.220
7.992
3.318
6.901
9.003
8.956
4.662
35
16.475
6.128
10.006
15.363
3.864
9.321
5.186
7.083
4.455
6.180
8.760
8.438
4.190
45
19.225
5.138
12.582
16.774
5.095
6.611
4.861
6.217
4.728
4.847
8.994
8.643
5.222
55
21.856
5.972
14.454
18.917
4.458
7.092
5.830
6.942
4.052
4.971
8.281
9.348
6.172
65
24.280
8.720
13.826
18.302
4.572
12.040
7.450
9.123
4.556
6.848
9.922
10.876
6.010
75
23.119
11.202
16.489
18.571
5.310
16.889
8.408
11.441
5.503
10.154
11.935
12.638
5.573
85
20.135
12.336
18.254
18.691
4.395
19.644
8.477
12.966
5.251
11.897
12.518
13.142
5.579
95
23.456
14.853
17.752
22.674
5.197
21.044
9.936
15.339
6.017
13.187
12.684
14.740
6.205
105
23.388
15.798
19.462
25.328
5.928
20.219
10.333
17.028
6.852
13.349
11.428
15.374
6.429
115
23.024
13.572
24.448
23.733
7.695
16.853
9.306
14.868
7.790
12.157
10.025
14.861
6.370
125
19.287
10.223
23.603
18.134
8.715
11.734
7.027
11.008
7.465
9.020
8.885
12.282
5.508
135
10.645
5.170
15.509
9.135
6.332
5.451
3.760
6.105
4.277
4.523
5.318
6.929
3.522
145
2.028
0.928
3.245
1.614
1.427
0.880
0.714
1.370
0.909
0.823
1.085
1.366
0.740Table 6
Maximum earthquake compressive stresses of the 150 m chimney under the action of 11 earthquake waves ignoring vertical eccentricities.
High
Serial number
1
2
3
4
5
6
7
8
9
10
11
Mean
Standard deviation
5
13.289
7.369
8.554
12.633
3.384
10.305
5.134
7.313
3.331
5.934
6.745
7.636
3.341
15
14.345
7.691
8.419
13.151
2.551
11.081
5.622
7.949
2.885
5.974
7.481
7.923
3.773
25
14.663
7.413
9.033
13.475
2.504
11.043
5.951
7.640
3.002
5.274
8.561
8.051
3.902
35
14.349
5.850
8.187
13.705
2.992
9.354
6.185
6.321
3.458
5.063
8.813
7.661
3.731
45
17.519
4.976
8.059
13.816
4.116
7.107
6.016
6.577
3.657
4.618
8.719
7.744
4.307
55
20.530
5.451
9.178
15.920
4.751
7.269
6.060
7.221
3.133
4.536
8.345
8.400
5.264
65
20.660
7.658
12.663
17.034
3.858
11.333
7.614
8.218
3.604
6.526
9.731
9.900
5.263
75
19.725
10.469
15.268
20.713
4.605
15.857
8.474
9.739
5.055
8.741
11.681
11.848
5.426
85
17.501
12.267
16.931
22.010
4.934
18.636
8.167
10.660
5.284
10.439
12.392
12.656
5.552
95
20.104
14.349
17.435
21.680
5.011
20.800
8.797
12.931
5.334
11.404
12.197
13.640
5.891
105
21.951
15.170
16.700
19.318
5.453
20.830
9.370
15.129
5.800
10.514
11.032
13.752
5.745
115
20.385
14.337
16.688
18.629
7.109
18.066
8.969
14.270
6.014
8.398
9.452
12.938
5.122
125
17.655
11.136
14.798
15.712
7.554
13.324
7.386
10.924
5.278
6.956
8.047
10.797
4.106
135
10.348
5.974
9.327
9.096
5.879
6.914
4.598
6.387
3.223
4.267
4.705
6.429
2.300
145
3.091
1.632
3.315
2.693
2.384
1.875
1.460
2.273
1.196
1.333
1.554
2.073
0.728In Table5, the comparison of the data shows that the maximum earthquake compressive stress obtained under different earthquake waves is significantly different, and the degree of the difference can be expressed by the standard deviation. To quantify the degree of difference, its standard deviation S is(21)S=∑l=1nσl−σ¯2n−1,where n is the number of samples, that is, the number of selected earthquake waves; σl is normal stress on the central section of segment l, that is, the sum of bending stress and axial stress, which is σl=Ml/Wl+Nl/Al; Wl is the bending-resistance modulus of the segment l; and σ¯ is the mean normal stress of the central section.The right side of Figure9(a) is the comparison of the mean values of earthquake compressive stresses between Tables 5 and 6 and the corresponding standard deviation. The left side of Figure 9(a) is the comparisons of the mean values of earthquake tensile stresses and the corresponding standard deviation. The data in Tables 7 and 8 (see Appendix B) are provided regarding maximum earthquake tensile stress, mean value, and standard deviation of the 150 m high chimney when considering vertical eccentricities and neglecting vertical eccentricities. In Figure 9(a), the red and pink columns are, respectively, the mean values of earthquake compressive and tensile stresses considering vertical eccentricities. The blue and green columns are, respectively, the mean values of earthquake compressive and tensile stress neglecting vertical eccentricities. The red and pink lines are, respectively, the standard deviations of different earthquake compressive and tensile stresses considering vertical eccentricities. The blue and green lines are the standard deviations of different earthquake compressive and tensile stresses neglecting vertical eccentricities. Figure 9(b) shows the case of the 210 m high chimney (data in Tables 9–12).Figure 9
Comparisons of mean and standard deviation of the normal stress of chimney: (a) 150 m high chimney; (b) 210 m high chimney.
(a)
(b)Table 7
Maximum earthquake tensile stresses of the 150 m high chimney considering vertical eccentricities.
High
Serial number
1
2
3
4
5
6
7
8
9
10
11
Mean
Standard deviation
5
−14.899
−8.142
−11.107
−14.432
−3.545
−10.385
−5.515
−7.845
−3.068
−7.032
−7.351
−8.484
3.918
15
−14.643
−8.224
−9.130
−15.498
−3.170
−11.141
−6.012
−8.139
−2.637
−7.093
−7.417
−8.464
4.079
25
−15.184
−7.429
−10.161
−15.337
−2.699
−11.614
−6.114
−8.303
−2.496
−6.597
−8.329
−8.569
4.290
35
−16.726
−6.233
−10.023
−14.233
−2.904
−9.964
−5.454
−7.328
−2.928
−5.911
−8.414
−8.193
4.345
45
−20.149
−5.441
−13.105
−15.379
−3.684
−7.288
−5.441
−5.944
−3.742
−5.080
−8.275
−8.502
5.371
55
−23.087
−5.802
−14.819
−17.339
−3.907
−7.209
−6.053
−6.328
−4.228
−4.772
−9.252
−9.345
6.296
65
−24.106
−8.842
−13.522
−18.325
−4.194
−11.620
−6.900
−8.679
−4.588
−6.925
−10.773
−10.770
6.021
75
−23.367
−11.458
−17.079
−20.100
−5.300
−16.431
−8.021
−11.427
−5.875
−10.037
−11.643
−12.794
5.798
85
−18.712
−12.305
−16.848
−20.900
−5.449
−19.142
−8.437
−12.990
−5.350
−12.505
−12.969
−13.237
5.330
95
−23.568
−15.257
−17.401
−22.463
−5.488
−20.761
−9.536
−14.479
−5.326
−13.449
−12.741
−14.588
6.229
105
−26.781
−16.007
−19.081
−23.512
−6.008
−19.810
−10.048
−15.935
−6.174
−12.991
−12.563
−15.356
6.669
115
−26.059
−13.898
−24.943
−22.161
−7.670
−16.399
−9.067
−14.409
−6.982
−11.603
−10.365
−14.869
6.797
125
−21.997
−10.105
−24.736
−16.982
−8.222
−11.353
−6.804
−10.998
−7.044
−8.738
−8.935
−12.356
6.136
135
−12.284
−5.071
−15.064
−8.473
−5.732
−5.321
−3.660
−5.796
−4.293
−4.246
−4.992
−6.812
3.667
145
−2.349
−0.883
−3.139
−1.509
−1.387
−0.863
−0.744
−1.380
−0.862
−0.817
−0.948
−1.353
0.757Table 8
Maximum earthquake tensile stresses of the 150 m high chimney ignoring vertical eccentricities.
High
Serial number
1
2
3
4
5
6
7
8
9
10
11
Mean
Standard deviation
5
−12.713
−7.405
−8.992
−13.313
−3.442
−10.154
−5.158
−7.275
−2.661
−5.482
−6.441
−7.549
3.476
15
−13.689
−7.593
−8.485
−14.183
−2.785
−10.792
−5.623
−7.339
−2.313
−5.516
−7.092
−7.765
3.883
25
−14.737
−7.445
−9.195
−14.689
−2.622
−11.050
−5.718
−7.167
−2.953
−5.404
−7.922
−8.082
4.101
35
−16.038
−5.984
−9.391
−15.677
−2.783
−9.446
−5.863
−6.125
−3.703
−4.969
−8.241
−8.020
4.412
45
−19.887
−5.361
−9.031
−15.521
−4.038
−7.266
−5.923
−6.521
−3.986
−4.520
−8.731
−8.253
5.066
55
−22.010
−5.296
−8.157
−14.138
−4.718
−7.033
−6.395
−7.338
−3.525
−4.671
−8.345
−8.330
5.348
65
−20.442
−7.589
−9.998
−16.058
−3.955
−11.176
−7.868
−8.750
−3.820
−6.332
−9.822
−9.619
4.957
75
−18.744
−10.206
−14.035
−20.320
−4.984
−15.659
−8.765
−10.409
−5.016
−8.449
−11.561
−11.650
5.078
85
−17.127
−12.362
−14.803
−21.183
−4.995
−18.267
−8.545
−10.835
−4.926
−9.392
−11.886
−12.211
5.259
95
−19.308
−14.363
−15.150
−21.096
−5.158
−20.466
−8.701
−12.565
−4.347
−9.829
−11.888
−12.988
5.778
105
−20.735
−15.441
−17.201
−19.442
−5.141
−21.169
−9.351
−14.565
−4.676
−9.461
−10.996
−13.471
5.938
115
−20.512
−14.054
−17.158
−18.152
−6.723
−18.408
−9.040
−13.521
−5.411
−8.833
−9.374
−12.835
5.232
125
−16.682
−10.576
−13.862
−14.924
−6.909
−13.614
−7.317
−10.586
−5.691
−7.308
−7.275
−10.431
3.814
135
−9.702
−5.606
−9.130
−8.663
−5.585
−7.096
−4.223
−6.238
−4.007
−4.296
−4.774
−6.302
2.067
145
−3.090
−1.589
−3.339
−2.557
−2.419
−1.928
−1.258
−2.233
−1.350
−1.345
−1.680
−2.072
0.717Table 9
Maximum earthquake compressive stresses of the 210 m high chimney considering vertical eccentricities.
High
Serial number
1
2
3
4
5
6
7
8
9
10
11
Mean
Standard deviation
5
9.073
4.481
8.898
10.293
2.941
9.218
3.664
7.445
6.273
6.974
6.646
6.901
2.416
15
9.938
4.449
8.687
10.863
3.229
9.964
3.797
8.021
6.456
7.002
6.773
7.198
2.591
25
10.466
4.260
7.814
11.223
3.642
10.552
3.865
8.428
6.349
6.810
6.874
7.298
2.707
35
10.942
4.536
6.342
10.703
3.143
10.389
3.755
8.574
5.477
6.347
7.981
7.108
2.806
45
10.664
4.682
7.363
9.732
4.033
9.170
3.919
8.081
4.383
5.379
8.613
6.911
2.501
55
10.790
5.288
9.082
10.740
4.553
7.951
4.411
7.829
4.681
5.806
9.083
7.292
2.452
65
10.391
5.523
9.869
11.456
4.298
7.003
4.417
7.768
5.244
5.932
8.403
7.300
2.480
75
11.058
5.397
9.500
11.388
4.244
5.592
4.090
7.069
5.027
6.239
8.873
7.134
2.653
85
12.283
5.330
8.956
13.619
4.035
7.998
4.624
8.426
5.976
8.010
9.112
8.034
3.010
95
13.669
5.604
11.904
16.621
4.800
11.195
5.281
10.340
7.300
9.797
9.709
9.656
3.704
105
14.484
7.044
14.138
16.663
5.771
13.921
5.765
11.962
9.226
10.333
11.088
10.945
3.708
115
16.080
8.297
16.374
15.511
6.376
17.064
6.771
14.798
11.033
10.732
12.649
12.335
3.950
125
18.201
9.514
17.908
19.468
7.501
20.617
7.767
17.874
12.290
11.079
13.899
14.192
4.836
135
20.731
10.387
17.076
25.177
7.543
23.166
8.884
19.191
15.434
13.947
14.108
15.968
5.754
145
22.279
8.930
16.182
26.017
6.539
23.188
8.091
17.090
15.283
14.884
12.716
15.564
6.365
155
22.605
9.904
20.913
24.714
6.672
21.495
8.269
17.399
13.326
15.063
12.582
15.722
6.154
165
21.942
11.283
22.329
23.957
6.477
19.121
8.648
17.038
12.612
13.700
15.753
15.715
5.766
175
18.729
10.621
19.787
19.615
5.954
15.222
8.110
14.354
10.513
11.494
15.196
13.600
4.668
185
13.872
8.191
14.943
14.117
5.435
10.425
6.618
10.449
8.133
8.744
12.918
10.349
3.233
195
7.892
4.429
8.716
7.440
4.429
5.082
3.846
5.557
4.556
4.826
7.781
5.868
1.735
205
1.547
0.811
1.607
1.321
1.283
0.827
0.769
1.046
0.833
0.868
1.611
1.138
0.343Table 10
Maximum earthquake compressive stresses of the 210 m high chimney ignoring vertical eccentricities.
High
Serial number
1
2
3
4
5
6
7
8
9
10
11
Mean
Standard deviation
5
7.913
4.726
10.278
9.045
3.297
7.885
3.325
8.324
5.691
5.556
6.690
6.612
2.301
15
8.491
4.613
8.632
9.591
3.192
8.644
3.575
8.693
5.534
5.858
6.674
6.681
2.270
25
8.928
4.295
7.073
9.821
3.250
9.291
3.594
8.592
6.273
6.019
6.922
6.733
2.306
35
9.236
4.388
7.835
9.172
3.702
9.105
3.628
7.755
6.311
5.957
8.086
6.834
2.166
45
9.185
4.857
7.109
7.923
4.515
8.191
3.885
6.749
5.297
5.549
9.094
6.578
1.871
55
9.400
5.456
7.823
8.987
4.993
7.663
4.080
6.935
4.754
5.110
9.630
6.803
2.022
65
9.131
6.071
9.355
9.790
4.614
6.873
4.019
7.968
5.305
5.143
9.256
7.048
2.137
75
9.085
5.875
11.357
10.901
5.045
5.885
3.444
9.261
5.258
5.004
9.866
7.362
2.765
85
9.804
5.437
11.261
11.823
5.110
8.655
4.133
10.770
6.123
6.690
9.679
8.135
2.726
95
10.652
5.793
12.132
14.150
5.420
11.524
4.778
12.735
7.346
8.406
10.092
9.366
3.211
105
13.107
7.555
13.799
14.715
6.191
13.593
5.764
13.224
8.968
8.992
11.902
10.710
3.289
115
15.388
8.611
14.686
14.126
6.467
15.260
7.159
12.609
9.625
9.462
13.769
11.560
3.359
125
16.887
10.171
15.816
16.068
7.341
17.337
7.958
13.671
9.509
10.761
15.444
12.815
3.741
135
17.439
10.288
16.448
18.511
7.162
19.932
7.987
17.367
12.785
13.289
16.818
14.366
4.357
145
16.328
8.887
20.181
19.843
6.132
20.116
6.912
19.538
14.792
14.446
13.972
14.650
5.294
155
14.935
10.856
23.421
19.606
6.668
18.987
7.166
19.462
15.096
14.097
13.785
14.916
5.274
165
14.959
11.991
23.222
18.400
7.429
16.611
7.161
18.445
14.258
12.321
14.319
14.465
4.759
175
12.915
11.174
20.729
15.953
6.914
13.073
6.393
14.984
11.878
9.970
14.322
12.573
4.082
185
11.388
9.186
19.862
12.126
7.682
9.282
5.618
11.792
9.435
7.435
12.891
10.609
3.793
195
7.752
5.577
13.811
6.922
6.482
5.034
3.681
7.045
5.408
4.287
8.232
6.748
2.736
205
2.573
1.600
4.560
1.897
2.434
1.349
1.189
2.327
1.491
1.240
2.438
2.100
0.967Table 11
Maximum earthquake tensile stresses of the 210 m high chimney considering vertical eccentricities.
High
Serial number
1
2
3
4
5
6
7
8
9
10
11
Mean
Standard deviation
5
−10.262
−4.948
−8.135
−10.034
−2.895
−8.930
−3.943
−7.722
−5.298
−6.667
−6.588
−6.857
2.424
15
−10.435
−4.938
−7.568
−11.181
−2.919
−9.723
−4.201
−8.217
−5.552
−6.907
−6.881
−7.138
2.630
25
−10.935
−4.554
−6.246
−12.106
−3.241
−10.269
−4.221
−8.691
−6.122
−6.946
−7.151
−7.317
2.884
35
−10.553
−4.145
−6.472
−11.522
−3.273
−10.373
−3.642
−8.527
−5.875
−6.486
−7.779
−7.150
2.868
45
−10.408
−4.240
−7.468
−9.756
−4.018
−9.579
−3.551
−7.277
−4.746
−5.330
−8.867
−6.840
2.562
55
−11.024
−4.832
−8.860
−10.900
−4.600
−8.365
−3.709
−6.914
−4.711
−5.920
−9.001
−7.167
2.610
65
−10.567
−5.284
−9.795
−11.400
−4.375
−6.960
−3.870
−7.157
−4.987
−6.342
−8.445
−7.198
2.563
75
−9.261
−5.017
−10.066
−11.198
−4.483
−5.807
−3.879
−7.801
−5.100
−6.042
−9.195
−7.077
2.520
85
−13.145
−4.857
−9.754
−14.902
−4.693
−8.210
−4.959
−7.609
−6.291
−8.202
−9.445
−8.370
3.332
95
−16.522
−6.118
−12.903
−17.681
−4.622
−11.390
−5.345
−9.800
−7.187
−10.105
−10.256
−10.175
4.304
105
−17.546
−7.618
−15.135
−17.703
−5.666
−14.323
−6.034
−12.327
−8.284
−10.474
−11.492
−11.509
4.327
115
−20.278
−8.779
−15.214
−17.611
−6.720
−16.912
−6.769
−15.784
−8.848
−10.066
−12.580
−12.687
4.725
125
−22.597
−9.647
−15.636
−19.295
−6.777
−19.980
−7.368
−18.825
−9.581
−9.736
−15.350
−14.072
5.641
135
−24.483
−10.272
−15.168
−25.018
−7.579
−22.450
−8.291
−19.846
−11.195
−12.573
−16.906
−15.798
6.393
145
−22.866
−9.103
−16.355
−25.713
−5.767
−22.516
−8.095
−17.358
−14.228
−14.451
−15.153
−15.600
6.365
155
−20.992
−9.940
−20.354
−25.657
−5.673
−22.139
−8.594
−16.513
−15.972
−14.672
−14.746
−15.932
6.137
165
−19.689
−11.736
−21.039
−25.636
−7.114
−20.421
−9.126
−15.506
−15.183
−13.670
−12.456
−15.598
5.594
175
−17.762
−11.194
−21.276
−21.932
−6.327
−16.354
−8.317
−13.811
−12.043
−11.622
−13.407
−14.004
4.944
185
−13.414
−8.601
−17.871
−15.897
−6.273
−11.122
−6.563
−10.575
−8.266
−8.356
−12.082
−10.820
3.745
195
−7.845
−4.509
−10.141
−8.223
−4.973
−5.357
−3.606
−5.460
−4.672
−4.489
−7.163
−6.040
2.015
205
−1.591
−0.770
−1.794
−1.524
−1.424
−0.845
−0.694
−0.972
−0.819
−0.753
−1.387
−1.143
0.403Table 12
Maximum earthquake tensile stresses of the 210 m high chimney ignoring vertical eccentricities.
High
Serial number
1
2
3
4
5
6
7
8
9
10
11
Mean
Standard deviation
5
−7.404
−4.719
−8.533
−8.527
−3.335
−8.004
−3.635
−7.965
−6.483
−5.570
−7.629
−6.528
1.923
15
−7.867
−4.672
−7.619
−9.499
−3.416
−8.788
−3.696
−8.663
−6.613
−6.051
−7.548
−6.767
2.080
25
−8.776
−4.683
−8.208
−10.373
−3.163
−9.459
−3.540
−9.161
−6.331
−6.333
−6.843
−6.988
2.443
35
−9.086
−4.199
−8.700
−10.468
−3.866
−9.371
−3.766
−8.596
−5.128
−6.099
−7.874
−7.014
2.463
45
−8.183
−4.540
−7.346
−9.431
−4.291
−8.534
−4.033
−7.092
−4.514
−5.405
−8.582
−6.541
2.022
55
−7.768
−5.168
−8.978
−8.164
−4.662
−7.902
−4.170
−6.826
−4.667
−5.156
−9.280
−6.613
1.895
65
−8.262
−5.869
−8.851
−9.653
−4.784
−7.242
−4.073
−7.791
−5.180
−5.147
−9.472
−6.939
2.008
75
−8.876
−5.860
−9.759
−10.681
−4.735
−6.168
−3.488
−8.404
−5.163
−4.789
−9.311
−7.021
2.444
85
−9.957
−5.203
−10.615
−12.112
−5.577
−8.535
−3.543
−10.392
−6.100
−6.789
−8.921
−7.977
2.703
95
−11.990
−6.905
−12.809
−13.458
−5.783
−11.466
−4.556
−12.103
−7.473
−8.771
−9.768
−9.553
3.052
105
−13.475
−8.542
−13.539
−14.303
−6.060
−13.615
−5.865
−12.736
−8.416
−9.708
−11.819
−10.734
3.137
115
−14.847
−9.219
−14.785
−14.387
−6.629
−15.528
−7.442
−12.490
−8.892
−9.946
−13.041
−11.564
3.232
125
−16.430
−10.839
−14.144
−16.638
−7.599
−17.232
−8.242
−15.041
−11.332
−11.355
−15.373
−13.111
3.404
135
−17.538
−10.834
−18.645
−19.642
−6.946
−20.560
−7.990
−17.535
−13.396
−13.107
−15.292
−14.680
4.638
145
−16.080
−8.417
−21.186
−20.686
−6.343
−20.739
−7.896
−19.356
−13.220
−13.720
−12.230
−14.534
5.500
155
−18.273
−9.907
−21.025
−21.991
−5.937
−19.264
−8.263
−19.261
−14.030
−12.955
−13.363
−14.933
5.417
165
−18.640
−11.394
−23.870
−20.157
−6.959
−17.581
−8.305
−17.337
−13.363
−11.784
−15.212
−14.964
5.159
175
−15.885
−11.241
−21.453
−15.522
−7.398
−14.219
−7.484
−14.848
−11.109
−9.557
−15.198
−13.083
4.197
185
−11.686
−8.918
−19.105
−12.072
−6.373
−10.024
−5.830
−10.987
−8.906
−6.633
−12.945
−10.316
3.779
195
−7.186
−5.468
−13.557
−7.196
−5.199
−5.307
−3.738
−6.277
−5.492
−3.635
−8.202
−6.478
2.734
205
−2.180
−1.625
−4.540
−2.291
−2.251
−1.383
−1.265
−2.016
−1.692
−1.025
−2.601
−2.079
0.950Figure9 shows that the earthquake normal stresses increase evidently when considering vertical eccentricities. Significant differences of the maximum earthquake normal stress are observed under different seismic waves actions.Figure10 is the boxplot of the maximum earthquake compressive stress distribution at different cross sections of 150 m and 210 m high chimneys under different seismic wave actions when considering eccentricities. They show the mean value, minimum, maximum, and 25%–75% of earthquake compressive stresses. The distribution characteristics of the earthquake compressive stress data are also shown in the figures. The stress distributions near the bottom of the chimney are relatively intensive, whereas those at the middle and upper parts of the chimney are relatively scattered.Figure 10
Boxplot of the compressive stress distribution of chimney considering eccentricities: (a) 150 m high chimney; (b) 210 m high chimney.
(a)
(b)The results studied above show that significant differences of maximum earthquake normal stress under different earthquake wave actions. The comparisons demonstrate that the conventional results (eccentricities neglected) might be less than their actual tensile/compressive stresses in the seismic control cross section of the chimneys under the actions of several specific earthquake waves. The calculation results show that eccentricities between the spatial discrete nodes and the corresponding centroids of investigated lumps should be considered in earthquake response of tall RC chimneys to obtain accurate calculation results.


### 5.2.2. Effects of Vertical Earthquake

The phenomena of eccentricities are considered to study the effect of vertical earthquake on earthquake normal stresses. Two types of seismic load are applied for each of the four earthquake waves. In the first case, the combined horizontal and vertical earthquake waves (see red lines in Figures11 and 12) are inputted. In the second case, only the horizontal earthquake wave (see blue lines in Figures 11 and 12) is inputted.Figure 11
Spatial distributions of maximum earthquake normal stresses at different cross sections for the 150 m high chimney under two types of seismic load applied for each of the four earthquake wave actions: (a) Landers wave (No. 1); (b) Loma Prieta wave (No. 2); (c) Borrego wave (No. 3); (d) Chichi wave (No. 4).
(a)
(b)
(c)
(d)Figure 12
Spatial distributions of maximum earthquake normal stresses at different cross sections for the 210 m high chimney under two types of seismic load applied for each of the four earthquake wave actions: (a) Landers wave (No. 1); (b) Loma Prieta wave (No. 2); (c) Borrego wave (No. 3); (d) Chichi wave (No. 4).
(a)
(b)
(c)
(d)For the 150 m high chimney, Figure11 shows that the peak normal tensile stress increases by 4% for the Landers wave (No. 1) and 10% for the Loma Prieta wave (No. 2) if vertical earthquake waves are considered. The peak normal compressive stress increases by 7% for the Loma Prieta wave (No. 2) and 10% for the Chichi wave (No. 4) if vertical earthquake waves are considered.For the 210 m high chimney, Figure12 shows that the peak normal tensile stress increases by 12% for the Landers wave (No. 1) and 9% for the Loma Prieta wave (No. 2) if vertical earthquake waves are considered. The peak normal compressive stress increases by 9% for Landers wave (No. 1) and 13% for Loma Prieta wave (No. 2) if vertical earthquake waves are considered.Figure13 shows the spatial comparison curves of the mean values of the maximum normal earthquake stresses for the 150 m and 210 m high chimneys under two types of seismic load. Figure 13(a) is obtained from the average values of Tables 5, 7, 13, and 14. Figure 13(b) is drawn from the mean values of Tables 9, 11, 15, and 16. Figure 13 shows that the normal stress of the chimney can be increased considering the vertical seismic acceleration.Figure 13
Comparison curve of the mean value of maximum earthquake normal stresses for chimneys under two types of seismic load: (a) 150 m high chimney; (b) 210 m high chimney.
(a)
(b)Table 13
Maximum earthquake compressive stresses of the 150 m high chimney when only horizontal seismic acceleration is inputted.
High
Serial number
1
2
3
4
5
6
7
8
9
10
11
Mean
Standard deviation
5
15.762
6.884
11.475
13.226
2.984
9.720
5.314
7.104
3.694
6.267
7.235
8.151
3.986
15
16.533
6.912
9.673
14.154
2.374
10.375
5.697
7.298
3.395
6.244
7.812
8.224
4.261
25
15.970
6.204
9.464
14.907
2.194
10.292
5.644
6.853
3.049
6.014
9.059
8.150
4.379
35
17.119
4.713
9.862
13.772
2.836
8.412
4.903
6.279
4.110
5.139
9.057
7.837
4.407
45
20.107
3.593
12.133
15.772
4.133
5.642
4.755
5.276
4.437
3.707
8.936
8.045
5.603
55
21.071
4.229
14.646
18.041
3.645
6.161
5.083
6.183
3.713
3.242
8.740
8.614
6.337
65
21.786
7.321
13.780
17.612
3.645
10.790
6.299
8.381
4.823
5.491
9.217
9.922
5.686
75
20.897
9.792
16.022
18.742
4.307
15.863
7.209
10.533
5.855
8.858
11.386
11.769
5.406
85
18.671
10.349
17.184
18.626
3.973
18.579
7.799
11.399
5.396
11.049
12.025
12.277
5.361
95
21.849
13.139
17.782
21.304
4.313
19.716
8.952
13.522
5.751
12.327
12.051
13.701
5.966
105
22.592
14.578
19.457
22.671
4.985
19.364
9.133
14.541
6.688
12.095
11.287
14.308
6.136
115
23.369
12.785
24.161
21.098
7.016
16.249
8.339
12.902
7.520
11.069
9.754
14.024
6.313
125
19.263
9.922
22.706
16.703
7.430
11.274
6.289
9.926
7.222
8.303
7.947
11.544
5.502
135
10.624
5.025
14.653
8.506
5.419
5.190
3.297
5.773
4.065
4.181
4.662
6.490
3.436
145
1.946
0.838
3.020
1.463
1.236
0.789
0.615
1.260
0.848
0.766
0.892
1.243
0.705Table 14
Maximum earthquake tensile stresses of the 150 m high chimney when only horizontal seismic acceleration is inputted.
High
Serial number
1
2
3
4
5
6
7
8
9
10
11
Mean
Standard deviation
5
−14.019
−7.073
−10.794
−13.376
−3.065
−9.883
−5.107
−7.404
−2.917
−6.109
−7.146
−7.899
3.748
15
−13.568
−6.946
−8.764
−14.184
−2.128
−10.529
−5.319
−7.538
−2.438
−6.015
−7.459
−7.717
3.918
25
−13.396
−6.049
−9.673
−14.268
−2.217
−10.586
−5.134
−7.319
−2.751
−5.799
−7.994
−7.744
3.943
35
−14.760
−4.852
−9.829
−14.309
−2.644
−8.895
−4.731
−6.290
−3.085
−5.040
−7.879
−7.483
4.145
45
−18.971
−3.749
−12.950
−15.797
−3.562
−6.153
−4.989
−5.575
−3.317
−3.663
−7.614
−7.849
5.499
55
−20.940
−3.856
−14.872
−16.458
−3.938
−5.793
−5.151
−5.917
−3.675
−3.466
−8.690
−8.432
6.123
65
−22.465
−7.073
−14.037
−17.207
−3.647
−10.650
−6.376
−8.108
−4.479
−5.484
−10.224
−9.977
5.836
75
−22.721
−9.433
−16.977
−18.920
−4.644
−15.423
−7.684
−10.420
−5.601
−8.588
−11.544
−11.996
5.779
85
−20.043
−10.301
−16.678
−19.349
−4.152
−17.961
−8.200
−11.508
−4.973
−11.015
−12.732
−12.447
5.513
95
−22.737
−13.287
−16.842
−21.924
−5.111
−19.726
−9.137
−12.996
−4.815
−11.834
−12.481
−13.717
6.121
105
−24.485
−14.367
−18.594
−22.963
−5.152
−18.972
−9.762
−13.993
−5.724
−11.670
−11.579
−14.297
6.413
115
−23.410
−12.647
−24.355
−21.682
−6.705
−15.693
−8.900
−12.975
−6.598
−11.093
−9.727
−13.980
6.490
125
−20.313
−9.584
−23.734
−16.823
−7.219
−10.906
−6.540
−9.631
−6.598
−8.641
−8.727
−11.701
5.879
135
−11.408
−4.711
−15.196
−8.484
−5.553
−4.981
−3.266
−5.171
−4.017
−4.446
−5.039
−6.570
3.671
145
−2.166
−0.757
−3.138
−1.419
−1.330
−0.772
−0.624
−1.151
−0.790
−0.775
−0.950
−1.261
0.764Table 15
Maximum earthquake compressive stresses of the 210 m high chimney when only horizontal seismic acceleration is inputted.
High
Serial number
1
2
3
4
5
6
7
8
9
10
11
Mean
Standard deviation
5
8.481
4.199
8.705
10.156
2.007
9.332
3.272
7.522
5.873
6.495
6.213
6.569
2.599
15
8.901
4.019
8.216
10.622
2.225
10.012
3.285
8.055
5.978
6.514
5.906
6.703
2.746
25
9.582
3.469
7.243
10.684
2.446
10.362
3.266
8.297
5.919
6.143
5.912
6.666
2.869
35
10.568
3.052
6.345
9.797
2.526
9.397
2.767
7.913
5.484
5.128
6.954
6.357
2.870
45
9.972
3.302
6.902
8.454
3.223
7.488
2.873
7.388
4.133
4.051
7.458
5.931
2.467
55
10.485
3.886
8.502
9.475
3.529
7.237
3.392
6.915
4.759
4.188
8.276
6.422
2.573
65
10.190
4.154
9.585
9.633
3.223
6.894
3.521
6.782
5.247
4.515
8.447
6.563
2.599
75
9.191
3.988
8.965
9.777
3.536
5.854
3.329
6.980
4.994
4.741
8.090
6.313
2.395
85
11.177
3.643
8.928
12.420
3.799
7.406
3.792
8.005
5.956
6.387
8.570
7.280
2.943
95
13.516
4.035
11.145
14.322
3.448
10.454
4.424
9.272
7.536
7.673
9.883
8.701
3.683
105
14.580
5.302
13.069
14.196
4.576
12.836
5.224
10.651
8.796
8.504
10.711
9.859
3.671
115
16.062
6.498
16.367
13.919
5.366
15.006
6.185
13.698
10.445
8.645
11.724
11.265
4.087
125
18.730
8.256
17.638
18.152
5.275
19.078
7.159
16.475
11.612
10.651
13.545
13.325
5.028
135
20.661
8.608
16.365
23.615
5.444
21.778
8.219
17.456
14.630
14.181
13.650
14.964
5.838
145
19.716
6.626
15.624
24.576
4.835
21.610
7.150
16.670
14.439
14.995
11.521
14.342
6.352
155
20.779
8.409
20.741
25.679
4.079
20.345
6.788
16.586
13.115
14.038
9.833
14.581
6.862
165
19.550
9.803
22.386
24.808
5.578
18.092
7.307
16.231
12.737
12.603
13.035
14.739
6.095
175
17.347
9.451
19.978
20.117
5.357
14.149
7.215
13.845
10.595
11.112
13.379
12.959
4.840
185
13.642
7.253
15.301
14.461
5.160
9.390
5.906
9.742
7.723
8.592
12.240
9.946
3.487
195
7.997
3.857
8.696
7.489
4.402
4.434
3.330
4.899
4.067
4.728
7.289
5.563
1.906
205
1.474
0.668
1.583
1.222
1.160
0.650
0.567
0.936
0.698
0.824
1.336
1.011
0.360Table 16
Maximum earthquake tensile stresses of the 210 m high chimney when only horizontal seismic acceleration is inputted.
High
Serial number
1
2
3
4
5
6
7
8
9
10
11
Mean
Standard deviation
5
−9.499
−3.914
−8.402
−10.528
−2.271
−8.311
−3.147
−6.884
−5.485
−5.603
−6.195
−6.385
2.643
15
−10.088
−3.759
−7.888
−11.120
−2.170
−9.016
−3.276
−7.374
−5.517
−5.617
−5.575
−6.491
2.872
25
−10.735
−3.268
−6.696
−12.154
−2.343
−9.816
−3.359
−8.020
−5.630
−5.495
−6.296
−6.710
3.196
35
−9.917
−3.148
−6.027
−11.745
−2.324
−9.532
−2.986
−7.852
−5.311
−5.431
−7.037
−6.483
3.071
45
−9.763
−3.035
−6.814
−9.538
−3.218
−7.901
−3.112
−7.034
−4.432
−4.657
−8.188
−6.154
2.563
55
−9.887
−3.853
−8.522
−9.456
−3.515
−6.972
−3.159
−6.434
−4.759
−4.555
−8.436
−6.323
2.486
65
−9.276
−4.106
−8.968
−10.471
−3.061
−6.145
−3.460
−7.597
−5.049
−4.824
−8.064
−6.457
2.550
75
−8.813
−4.006
−9.201
−10.360
−3.433
−5.116
−3.316
−8.168
−5.097
−4.562
−8.403
−6.407
2.593
85
−11.696
−3.733
−9.048
−13.089
−3.214
−7.460
−4.127
−7.880
−6.340
−6.498
−9.220
−7.482
3.172
95
−15.283
−5.286
−12.218
−16.060
−3.427
−10.971
−4.632
−9.692
−7.143
−8.267
−10.285
−9.388
4.147
105
−17.060
−6.333
−14.403
−16.566
−4.330
−13.582
−5.322
−11.064
−8.223
−8.519
−10.820
−10.566
4.430
115
−19.059
−6.495
−14.769
−16.340
−5.136
−15.559
−5.968
−14.179
−8.929
−7.669
−12.054
−11.469
4.823
125
−21.359
−7.756
−15.261
−20.501
−5.311
−17.463
−6.259
−17.193
−9.471
−8.103
−13.228
−12.900
5.807
135
−21.802
−8.174
−15.876
−26.142
−5.335
−20.549
−7.745
−18.417
−11.458
−11.799
−13.529
−14.621
6.555
145
−19.722
−7.568
−15.281
−26.594
−4.215
−21.400
−7.706
−16.488
−13.112
−13.520
−12.850
−14.405
6.567
155
−18.995
−9.012
−19.516
−23.940
−4.076
−21.179
−8.472
−16.200
−15.026
−13.612
−12.953
−14.816
5.995
165
−18.022
−10.638
−21.577
−22.969
−5.468
−19.116
−8.724
−16.437
−14.438
−12.906
−11.733
−14.730
5.466
175
−16.832
−10.263
−21.403
−20.010
−5.569
−15.119
−7.902
−14.741
−11.533
−11.518
−11.998
−13.353
4.844
185
−13.024
−7.875
−17.879
−14.764
−5.138
−10.190
−5.783
−11.224
−8.565
−8.835
−10.257
−10.321
3.791
195
−6.944
−4.092
−10.140
−7.591
−4.379
−4.781
−3.151
−5.887
−4.848
−4.741
−5.925
−5.680
1.956
205
−1.261
−0.689
−1.797
−1.221
−1.265
−0.696
−0.591
−1.091
−0.865
−0.805
−1.125
−1.037
0.353The results studied above show that the vertical ground motion should not be neglected, especially for tall RC chimneys built on rock sites or soft rock sites.


## 5.2.1. Effect of Eccentricity

Eleven earthquake waves are used. The effect of eccentricity on normal earthquake stress of the chimney are studied when horizontal and vertical components are inputted simultaneously.Figure7 shows the spatial distribution curves of the maximum earthquake normal stresses of different cross sections for the 150 m high chimney under the actions of Landers (No. 1 in Table 1), Loma Prieta (No. 2), Borrego (No. 3), and Chichi (No. 4) waves. Figure 8 shows the comparison curves of the 210 m high chimney. “New” denotes that the phenomena of eccentricities are considered. “Conventional” denotes the conventional results, in which the phenomena of eccentricities are neglected, for the earthquake responses of the real RC chimneys. Dashed lines (“tensile”) denote tensile stresses, whereas solid lines (“press”) denote compressive stresses.Figure 7
Spatial distributions of maximum earthquake normal stresses at different cross sections of the 150 m high chimney under four earthquake wave actions: (a) Landers wave (No. 1 in Table1); (b) Loma Prieta wave (No. 2); (c) Borrego wave (No. 3); (d) Chichi wave (No. 4).
(a)
(b)
(c)
(d)Figure 8
Spatial distributions of maximum earthquake normal stresses at different cross sections of the 210 m high chimney under four earthquake wave actions: (a) Landers wave (No. 1); (b) Loma Prieta wave (No. 2); (c) Borrego wave (No. 3); (d) Chichi wave (No. 4).
(a)
(b)
(c)
(d)A comparison of the red solid lines and the blue solid lines in Figures7 and 8 shows that the increased phenomena of peak values of the normal compressive stresses appear when eccentricities are considered.A comparison of the red dashed lines and the blue dashed lines in Figures7 and 8 shows that the increased phenomena of peak values of the normal tensile stresses appear when eccentricities are considered.Table4 shows the details of the increase in percentage of maximum earthquake normal tensile/compressive stresses when phenomena of eccentricities are considered.Table 4
Percentages of increase in maximum earthquake normal stresses.
Seismic waves
Maximum earthquake normal stresses increased
150 m
210 m
Landers (No. 1)
7% (compressive)
29% (tensile)
30% (compressive)
31% (tensile)
Loma Prieta (No. 2)
4% (compressive)
4% (tensile)
−6% (compressive)
3% (tensile)
Borrego (No. 3)
40% (compressive)
45% (tensile)
−5% (compressive)
−11% (tensile)
Chichi (No. 4)
15% (compressive)
11% (tensile)
31% (compressive)
17% (tensile)Table5 shows maximum earthquake compressive stress, mean value, and standard deviation of different sections of the 150 m high chimney under 11 earthquake wave actions when vertical eccentricities are considered. Table 6 shows maximum earthquake compressive stress, mean value, and standard deviation when vertical eccentricities are ignored.Table 5
Maximum earthquake compressive stresses of the 150 m high chimney under the action of 11 earthquake waves considering vertical eccentricities.
High
Serial number
1
2
3
4
5
6
7
8
9
10
11
Mean
Standard deviation
5
17.010
7.717
11.246
14.230
3.482
10.097
5.921
7.997
3.671
6.339
7.160
8.625
4.215
15
18.103
7.988
9.306
14.845
2.561
10.899
6.311
8.594
3.176
6.847
7.752
8.762
4.583
25
17.811
7.554
9.795
16.047
2.853
11.025
6.220
7.992
3.318
6.901
9.003
8.956
4.662
35
16.475
6.128
10.006
15.363
3.864
9.321
5.186
7.083
4.455
6.180
8.760
8.438
4.190
45
19.225
5.138
12.582
16.774
5.095
6.611
4.861
6.217
4.728
4.847
8.994
8.643
5.222
55
21.856
5.972
14.454
18.917
4.458
7.092
5.830
6.942
4.052
4.971
8.281
9.348
6.172
65
24.280
8.720
13.826
18.302
4.572
12.040
7.450
9.123
4.556
6.848
9.922
10.876
6.010
75
23.119
11.202
16.489
18.571
5.310
16.889
8.408
11.441
5.503
10.154
11.935
12.638
5.573
85
20.135
12.336
18.254
18.691
4.395
19.644
8.477
12.966
5.251
11.897
12.518
13.142
5.579
95
23.456
14.853
17.752
22.674
5.197
21.044
9.936
15.339
6.017
13.187
12.684
14.740
6.205
105
23.388
15.798
19.462
25.328
5.928
20.219
10.333
17.028
6.852
13.349
11.428
15.374
6.429
115
23.024
13.572
24.448
23.733
7.695
16.853
9.306
14.868
7.790
12.157
10.025
14.861
6.370
125
19.287
10.223
23.603
18.134
8.715
11.734
7.027
11.008
7.465
9.020
8.885
12.282
5.508
135
10.645
5.170
15.509
9.135
6.332
5.451
3.760
6.105
4.277
4.523
5.318
6.929
3.522
145
2.028
0.928
3.245
1.614
1.427
0.880
0.714
1.370
0.909
0.823
1.085
1.366
0.740Table 6
Maximum earthquake compressive stresses of the 150 m chimney under the action of 11 earthquake waves ignoring vertical eccentricities.
High
Serial number
1
2
3
4
5
6
7
8
9
10
11
Mean
Standard deviation
5
13.289
7.369
8.554
12.633
3.384
10.305
5.134
7.313
3.331
5.934
6.745
7.636
3.341
15
14.345
7.691
8.419
13.151
2.551
11.081
5.622
7.949
2.885
5.974
7.481
7.923
3.773
25
14.663
7.413
9.033
13.475
2.504
11.043
5.951
7.640
3.002
5.274
8.561
8.051
3.902
35
14.349
5.850
8.187
13.705
2.992
9.354
6.185
6.321
3.458
5.063
8.813
7.661
3.731
45
17.519
4.976
8.059
13.816
4.116
7.107
6.016
6.577
3.657
4.618
8.719
7.744
4.307
55
20.530
5.451
9.178
15.920
4.751
7.269
6.060
7.221
3.133
4.536
8.345
8.400
5.264
65
20.660
7.658
12.663
17.034
3.858
11.333
7.614
8.218
3.604
6.526
9.731
9.900
5.263
75
19.725
10.469
15.268
20.713
4.605
15.857
8.474
9.739
5.055
8.741
11.681
11.848
5.426
85
17.501
12.267
16.931
22.010
4.934
18.636
8.167
10.660
5.284
10.439
12.392
12.656
5.552
95
20.104
14.349
17.435
21.680
5.011
20.800
8.797
12.931
5.334
11.404
12.197
13.640
5.891
105
21.951
15.170
16.700
19.318
5.453
20.830
9.370
15.129
5.800
10.514
11.032
13.752
5.745
115
20.385
14.337
16.688
18.629
7.109
18.066
8.969
14.270
6.014
8.398
9.452
12.938
5.122
125
17.655
11.136
14.798
15.712
7.554
13.324
7.386
10.924
5.278
6.956
8.047
10.797
4.106
135
10.348
5.974
9.327
9.096
5.879
6.914
4.598
6.387
3.223
4.267
4.705
6.429
2.300
145
3.091
1.632
3.315
2.693
2.384
1.875
1.460
2.273
1.196
1.333
1.554
2.073
0.728In Table5, the comparison of the data shows that the maximum earthquake compressive stress obtained under different earthquake waves is significantly different, and the degree of the difference can be expressed by the standard deviation. To quantify the degree of difference, its standard deviation S is(21)S=∑l=1nσl−σ¯2n−1,where n is the number of samples, that is, the number of selected earthquake waves; σl is normal stress on the central section of segment l, that is, the sum of bending stress and axial stress, which is σl=Ml/Wl+Nl/Al; Wl is the bending-resistance modulus of the segment l; and σ¯ is the mean normal stress of the central section.The right side of Figure9(a) is the comparison of the mean values of earthquake compressive stresses between Tables 5 and 6 and the corresponding standard deviation. The left side of Figure 9(a) is the comparisons of the mean values of earthquake tensile stresses and the corresponding standard deviation. The data in Tables 7 and 8 (see Appendix B) are provided regarding maximum earthquake tensile stress, mean value, and standard deviation of the 150 m high chimney when considering vertical eccentricities and neglecting vertical eccentricities. In Figure 9(a), the red and pink columns are, respectively, the mean values of earthquake compressive and tensile stresses considering vertical eccentricities. The blue and green columns are, respectively, the mean values of earthquake compressive and tensile stress neglecting vertical eccentricities. The red and pink lines are, respectively, the standard deviations of different earthquake compressive and tensile stresses considering vertical eccentricities. The blue and green lines are the standard deviations of different earthquake compressive and tensile stresses neglecting vertical eccentricities. Figure 9(b) shows the case of the 210 m high chimney (data in Tables 9–12).Figure 9
Comparisons of mean and standard deviation of the normal stress of chimney: (a) 150 m high chimney; (b) 210 m high chimney.
(a)
(b)Table 7
Maximum earthquake tensile stresses of the 150 m high chimney considering vertical eccentricities.
High
Serial number
1
2
3
4
5
6
7
8
9
10
11
Mean
Standard deviation
5
−14.899
−8.142
−11.107
−14.432
−3.545
−10.385
−5.515
−7.845
−3.068
−7.032
−7.351
−8.484
3.918
15
−14.643
−8.224
−9.130
−15.498
−3.170
−11.141
−6.012
−8.139
−2.637
−7.093
−7.417
−8.464
4.079
25
−15.184
−7.429
−10.161
−15.337
−2.699
−11.614
−6.114
−8.303
−2.496
−6.597
−8.329
−8.569
4.290
35
−16.726
−6.233
−10.023
−14.233
−2.904
−9.964
−5.454
−7.328
−2.928
−5.911
−8.414
−8.193
4.345
45
−20.149
−5.441
−13.105
−15.379
−3.684
−7.288
−5.441
−5.944
−3.742
−5.080
−8.275
−8.502
5.371
55
−23.087
−5.802
−14.819
−17.339
−3.907
−7.209
−6.053
−6.328
−4.228
−4.772
−9.252
−9.345
6.296
65
−24.106
−8.842
−13.522
−18.325
−4.194
−11.620
−6.900
−8.679
−4.588
−6.925
−10.773
−10.770
6.021
75
−23.367
−11.458
−17.079
−20.100
−5.300
−16.431
−8.021
−11.427
−5.875
−10.037
−11.643
−12.794
5.798
85
−18.712
−12.305
−16.848
−20.900
−5.449
−19.142
−8.437
−12.990
−5.350
−12.505
−12.969
−13.237
5.330
95
−23.568
−15.257
−17.401
−22.463
−5.488
−20.761
−9.536
−14.479
−5.326
−13.449
−12.741
−14.588
6.229
105
−26.781
−16.007
−19.081
−23.512
−6.008
−19.810
−10.048
−15.935
−6.174
−12.991
−12.563
−15.356
6.669
115
−26.059
−13.898
−24.943
−22.161
−7.670
−16.399
−9.067
−14.409
−6.982
−11.603
−10.365
−14.869
6.797
125
−21.997
−10.105
−24.736
−16.982
−8.222
−11.353
−6.804
−10.998
−7.044
−8.738
−8.935
−12.356
6.136
135
−12.284
−5.071
−15.064
−8.473
−5.732
−5.321
−3.660
−5.796
−4.293
−4.246
−4.992
−6.812
3.667
145
−2.349
−0.883
−3.139
−1.509
−1.387
−0.863
−0.744
−1.380
−0.862
−0.817
−0.948
−1.353
0.757Table 8
Maximum earthquake tensile stresses of the 150 m high chimney ignoring vertical eccentricities.
High
Serial number
1
2
3
4
5
6
7
8
9
10
11
Mean
Standard deviation
5
−12.713
−7.405
−8.992
−13.313
−3.442
−10.154
−5.158
−7.275
−2.661
−5.482
−6.441
−7.549
3.476
15
−13.689
−7.593
−8.485
−14.183
−2.785
−10.792
−5.623
−7.339
−2.313
−5.516
−7.092
−7.765
3.883
25
−14.737
−7.445
−9.195
−14.689
−2.622
−11.050
−5.718
−7.167
−2.953
−5.404
−7.922
−8.082
4.101
35
−16.038
−5.984
−9.391
−15.677
−2.783
−9.446
−5.863
−6.125
−3.703
−4.969
−8.241
−8.020
4.412
45
−19.887
−5.361
−9.031
−15.521
−4.038
−7.266
−5.923
−6.521
−3.986
−4.520
−8.731
−8.253
5.066
55
−22.010
−5.296
−8.157
−14.138
−4.718
−7.033
−6.395
−7.338
−3.525
−4.671
−8.345
−8.330
5.348
65
−20.442
−7.589
−9.998
−16.058
−3.955
−11.176
−7.868
−8.750
−3.820
−6.332
−9.822
−9.619
4.957
75
−18.744
−10.206
−14.035
−20.320
−4.984
−15.659
−8.765
−10.409
−5.016
−8.449
−11.561
−11.650
5.078
85
−17.127
−12.362
−14.803
−21.183
−4.995
−18.267
−8.545
−10.835
−4.926
−9.392
−11.886
−12.211
5.259
95
−19.308
−14.363
−15.150
−21.096
−5.158
−20.466
−8.701
−12.565
−4.347
−9.829
−11.888
−12.988
5.778
105
−20.735
−15.441
−17.201
−19.442
−5.141
−21.169
−9.351
−14.565
−4.676
−9.461
−10.996
−13.471
5.938
115
−20.512
−14.054
−17.158
−18.152
−6.723
−18.408
−9.040
−13.521
−5.411
−8.833
−9.374
−12.835
5.232
125
−16.682
−10.576
−13.862
−14.924
−6.909
−13.614
−7.317
−10.586
−5.691
−7.308
−7.275
−10.431
3.814
135
−9.702
−5.606
−9.130
−8.663
−5.585
−7.096
−4.223
−6.238
−4.007
−4.296
−4.774
−6.302
2.067
145
−3.090
−1.589
−3.339
−2.557
−2.419
−1.928
−1.258
−2.233
−1.350
−1.345
−1.680
−2.072
0.717Table 9
Maximum earthquake compressive stresses of the 210 m high chimney considering vertical eccentricities.
High
Serial number
1
2
3
4
5
6
7
8
9
10
11
Mean
Standard deviation
5
9.073
4.481
8.898
10.293
2.941
9.218
3.664
7.445
6.273
6.974
6.646
6.901
2.416
15
9.938
4.449
8.687
10.863
3.229
9.964
3.797
8.021
6.456
7.002
6.773
7.198
2.591
25
10.466
4.260
7.814
11.223
3.642
10.552
3.865
8.428
6.349
6.810
6.874
7.298
2.707
35
10.942
4.536
6.342
10.703
3.143
10.389
3.755
8.574
5.477
6.347
7.981
7.108
2.806
45
10.664
4.682
7.363
9.732
4.033
9.170
3.919
8.081
4.383
5.379
8.613
6.911
2.501
55
10.790
5.288
9.082
10.740
4.553
7.951
4.411
7.829
4.681
5.806
9.083
7.292
2.452
65
10.391
5.523
9.869
11.456
4.298
7.003
4.417
7.768
5.244
5.932
8.403
7.300
2.480
75
11.058
5.397
9.500
11.388
4.244
5.592
4.090
7.069
5.027
6.239
8.873
7.134
2.653
85
12.283
5.330
8.956
13.619
4.035
7.998
4.624
8.426
5.976
8.010
9.112
8.034
3.010
95
13.669
5.604
11.904
16.621
4.800
11.195
5.281
10.340
7.300
9.797
9.709
9.656
3.704
105
14.484
7.044
14.138
16.663
5.771
13.921
5.765
11.962
9.226
10.333
11.088
10.945
3.708
115
16.080
8.297
16.374
15.511
6.376
17.064
6.771
14.798
11.033
10.732
12.649
12.335
3.950
125
18.201
9.514
17.908
19.468
7.501
20.617
7.767
17.874
12.290
11.079
13.899
14.192
4.836
135
20.731
10.387
17.076
25.177
7.543
23.166
8.884
19.191
15.434
13.947
14.108
15.968
5.754
145
22.279
8.930
16.182
26.017
6.539
23.188
8.091
17.090
15.283
14.884
12.716
15.564
6.365
155
22.605
9.904
20.913
24.714
6.672
21.495
8.269
17.399
13.326
15.063
12.582
15.722
6.154
165
21.942
11.283
22.329
23.957
6.477
19.121
8.648
17.038
12.612
13.700
15.753
15.715
5.766
175
18.729
10.621
19.787
19.615
5.954
15.222
8.110
14.354
10.513
11.494
15.196
13.600
4.668
185
13.872
8.191
14.943
14.117
5.435
10.425
6.618
10.449
8.133
8.744
12.918
10.349
3.233
195
7.892
4.429
8.716
7.440
4.429
5.082
3.846
5.557
4.556
4.826
7.781
5.868
1.735
205
1.547
0.811
1.607
1.321
1.283
0.827
0.769
1.046
0.833
0.868
1.611
1.138
0.343Table 10
Maximum earthquake compressive stresses of the 210 m high chimney ignoring vertical eccentricities.
High
Serial number
1
2
3
4
5
6
7
8
9
10
11
Mean
Standard deviation
5
7.913
4.726
10.278
9.045
3.297
7.885
3.325
8.324
5.691
5.556
6.690
6.612
2.301
15
8.491
4.613
8.632
9.591
3.192
8.644
3.575
8.693
5.534
5.858
6.674
6.681
2.270
25
8.928
4.295
7.073
9.821
3.250
9.291
3.594
8.592
6.273
6.019
6.922
6.733
2.306
35
9.236
4.388
7.835
9.172
3.702
9.105
3.628
7.755
6.311
5.957
8.086
6.834
2.166
45
9.185
4.857
7.109
7.923
4.515
8.191
3.885
6.749
5.297
5.549
9.094
6.578
1.871
55
9.400
5.456
7.823
8.987
4.993
7.663
4.080
6.935
4.754
5.110
9.630
6.803
2.022
65
9.131
6.071
9.355
9.790
4.614
6.873
4.019
7.968
5.305
5.143
9.256
7.048
2.137
75
9.085
5.875
11.357
10.901
5.045
5.885
3.444
9.261
5.258
5.004
9.866
7.362
2.765
85
9.804
5.437
11.261
11.823
5.110
8.655
4.133
10.770
6.123
6.690
9.679
8.135
2.726
95
10.652
5.793
12.132
14.150
5.420
11.524
4.778
12.735
7.346
8.406
10.092
9.366
3.211
105
13.107
7.555
13.799
14.715
6.191
13.593
5.764
13.224
8.968
8.992
11.902
10.710
3.289
115
15.388
8.611
14.686
14.126
6.467
15.260
7.159
12.609
9.625
9.462
13.769
11.560
3.359
125
16.887
10.171
15.816
16.068
7.341
17.337
7.958
13.671
9.509
10.761
15.444
12.815
3.741
135
17.439
10.288
16.448
18.511
7.162
19.932
7.987
17.367
12.785
13.289
16.818
14.366
4.357
145
16.328
8.887
20.181
19.843
6.132
20.116
6.912
19.538
14.792
14.446
13.972
14.650
5.294
155
14.935
10.856
23.421
19.606
6.668
18.987
7.166
19.462
15.096
14.097
13.785
14.916
5.274
165
14.959
11.991
23.222
18.400
7.429
16.611
7.161
18.445
14.258
12.321
14.319
14.465
4.759
175
12.915
11.174
20.729
15.953
6.914
13.073
6.393
14.984
11.878
9.970
14.322
12.573
4.082
185
11.388
9.186
19.862
12.126
7.682
9.282
5.618
11.792
9.435
7.435
12.891
10.609
3.793
195
7.752
5.577
13.811
6.922
6.482
5.034
3.681
7.045
5.408
4.287
8.232
6.748
2.736
205
2.573
1.600
4.560
1.897
2.434
1.349
1.189
2.327
1.491
1.240
2.438
2.100
0.967Table 11
Maximum earthquake tensile stresses of the 210 m high chimney considering vertical eccentricities.
High
Serial number
1
2
3
4
5
6
7
8
9
10
11
Mean
Standard deviation
5
−10.262
−4.948
−8.135
−10.034
−2.895
−8.930
−3.943
−7.722
−5.298
−6.667
−6.588
−6.857
2.424
15
−10.435
−4.938
−7.568
−11.181
−2.919
−9.723
−4.201
−8.217
−5.552
−6.907
−6.881
−7.138
2.630
25
−10.935
−4.554
−6.246
−12.106
−3.241
−10.269
−4.221
−8.691
−6.122
−6.946
−7.151
−7.317
2.884
35
−10.553
−4.145
−6.472
−11.522
−3.273
−10.373
−3.642
−8.527
−5.875
−6.486
−7.779
−7.150
2.868
45
−10.408
−4.240
−7.468
−9.756
−4.018
−9.579
−3.551
−7.277
−4.746
−5.330
−8.867
−6.840
2.562
55
−11.024
−4.832
−8.860
−10.900
−4.600
−8.365
−3.709
−6.914
−4.711
−5.920
−9.001
−7.167
2.610
65
−10.567
−5.284
−9.795
−11.400
−4.375
−6.960
−3.870
−7.157
−4.987
−6.342
−8.445
−7.198
2.563
75
−9.261
−5.017
−10.066
−11.198
−4.483
−5.807
−3.879
−7.801
−5.100
−6.042
−9.195
−7.077
2.520
85
−13.145
−4.857
−9.754
−14.902
−4.693
−8.210
−4.959
−7.609
−6.291
−8.202
−9.445
−8.370
3.332
95
−16.522
−6.118
−12.903
−17.681
−4.622
−11.390
−5.345
−9.800
−7.187
−10.105
−10.256
−10.175
4.304
105
−17.546
−7.618
−15.135
−17.703
−5.666
−14.323
−6.034
−12.327
−8.284
−10.474
−11.492
−11.509
4.327
115
−20.278
−8.779
−15.214
−17.611
−6.720
−16.912
−6.769
−15.784
−8.848
−10.066
−12.580
−12.687
4.725
125
−22.597
−9.647
−15.636
−19.295
−6.777
−19.980
−7.368
−18.825
−9.581
−9.736
−15.350
−14.072
5.641
135
−24.483
−10.272
−15.168
−25.018
−7.579
−22.450
−8.291
−19.846
−11.195
−12.573
−16.906
−15.798
6.393
145
−22.866
−9.103
−16.355
−25.713
−5.767
−22.516
−8.095
−17.358
−14.228
−14.451
−15.153
−15.600
6.365
155
−20.992
−9.940
−20.354
−25.657
−5.673
−22.139
−8.594
−16.513
−15.972
−14.672
−14.746
−15.932
6.137
165
−19.689
−11.736
−21.039
−25.636
−7.114
−20.421
−9.126
−15.506
−15.183
−13.670
−12.456
−15.598
5.594
175
−17.762
−11.194
−21.276
−21.932
−6.327
−16.354
−8.317
−13.811
−12.043
−11.622
−13.407
−14.004
4.944
185
−13.414
−8.601
−17.871
−15.897
−6.273
−11.122
−6.563
−10.575
−8.266
−8.356
−12.082
−10.820
3.745
195
−7.845
−4.509
−10.141
−8.223
−4.973
−5.357
−3.606
−5.460
−4.672
−4.489
−7.163
−6.040
2.015
205
−1.591
−0.770
−1.794
−1.524
−1.424
−0.845
−0.694
−0.972
−0.819
−0.753
−1.387
−1.143
0.403Table 12
Maximum earthquake tensile stresses of the 210 m high chimney ignoring vertical eccentricities.
High
Serial number
1
2
3
4
5
6
7
8
9
10
11
Mean
Standard deviation
5
−7.404
−4.719
−8.533
−8.527
−3.335
−8.004
−3.635
−7.965
−6.483
−5.570
−7.629
−6.528
1.923
15
−7.867
−4.672
−7.619
−9.499
−3.416
−8.788
−3.696
−8.663
−6.613
−6.051
−7.548
−6.767
2.080
25
−8.776
−4.683
−8.208
−10.373
−3.163
−9.459
−3.540
−9.161
−6.331
−6.333
−6.843
−6.988
2.443
35
−9.086
−4.199
−8.700
−10.468
−3.866
−9.371
−3.766
−8.596
−5.128
−6.099
−7.874
−7.014
2.463
45
−8.183
−4.540
−7.346
−9.431
−4.291
−8.534
−4.033
−7.092
−4.514
−5.405
−8.582
−6.541
2.022
55
−7.768
−5.168
−8.978
−8.164
−4.662
−7.902
−4.170
−6.826
−4.667
−5.156
−9.280
−6.613
1.895
65
−8.262
−5.869
−8.851
−9.653
−4.784
−7.242
−4.073
−7.791
−5.180
−5.147
−9.472
−6.939
2.008
75
−8.876
−5.860
−9.759
−10.681
−4.735
−6.168
−3.488
−8.404
−5.163
−4.789
−9.311
−7.021
2.444
85
−9.957
−5.203
−10.615
−12.112
−5.577
−8.535
−3.543
−10.392
−6.100
−6.789
−8.921
−7.977
2.703
95
−11.990
−6.905
−12.809
−13.458
−5.783
−11.466
−4.556
−12.103
−7.473
−8.771
−9.768
−9.553
3.052
105
−13.475
−8.542
−13.539
−14.303
−6.060
−13.615
−5.865
−12.736
−8.416
−9.708
−11.819
−10.734
3.137
115
−14.847
−9.219
−14.785
−14.387
−6.629
−15.528
−7.442
−12.490
−8.892
−9.946
−13.041
−11.564
3.232
125
−16.430
−10.839
−14.144
−16.638
−7.599
−17.232
−8.242
−15.041
−11.332
−11.355
−15.373
−13.111
3.404
135
−17.538
−10.834
−18.645
−19.642
−6.946
−20.560
−7.990
−17.535
−13.396
−13.107
−15.292
−14.680
4.638
145
−16.080
−8.417
−21.186
−20.686
−6.343
−20.739
−7.896
−19.356
−13.220
−13.720
−12.230
−14.534
5.500
155
−18.273
−9.907
−21.025
−21.991
−5.937
−19.264
−8.263
−19.261
−14.030
−12.955
−13.363
−14.933
5.417
165
−18.640
−11.394
−23.870
−20.157
−6.959
−17.581
−8.305
−17.337
−13.363
−11.784
−15.212
−14.964
5.159
175
−15.885
−11.241
−21.453
−15.522
−7.398
−14.219
−7.484
−14.848
−11.109
−9.557
−15.198
−13.083
4.197
185
−11.686
−8.918
−19.105
−12.072
−6.373
−10.024
−5.830
−10.987
−8.906
−6.633
−12.945
−10.316
3.779
195
−7.186
−5.468
−13.557
−7.196
−5.199
−5.307
−3.738
−6.277
−5.492
−3.635
−8.202
−6.478
2.734
205
−2.180
−1.625
−4.540
−2.291
−2.251
−1.383
−1.265
−2.016
−1.692
−1.025
−2.601
−2.079
0.950Figure9 shows that the earthquake normal stresses increase evidently when considering vertical eccentricities. Significant differences of the maximum earthquake normal stress are observed under different seismic waves actions.Figure10 is the boxplot of the maximum earthquake compressive stress distribution at different cross sections of 150 m and 210 m high chimneys under different seismic wave actions when considering eccentricities. They show the mean value, minimum, maximum, and 25%–75% of earthquake compressive stresses. The distribution characteristics of the earthquake compressive stress data are also shown in the figures. The stress distributions near the bottom of the chimney are relatively intensive, whereas those at the middle and upper parts of the chimney are relatively scattered.Figure 10
Boxplot of the compressive stress distribution of chimney considering eccentricities: (a) 150 m high chimney; (b) 210 m high chimney.
(a)
(b)The results studied above show that significant differences of maximum earthquake normal stress under different earthquake wave actions. The comparisons demonstrate that the conventional results (eccentricities neglected) might be less than their actual tensile/compressive stresses in the seismic control cross section of the chimneys under the actions of several specific earthquake waves. The calculation results show that eccentricities between the spatial discrete nodes and the corresponding centroids of investigated lumps should be considered in earthquake response of tall RC chimneys to obtain accurate calculation results.


## 5.2.2. Effects of Vertical Earthquake

The phenomena of eccentricities are considered to study the effect of vertical earthquake on earthquake normal stresses. Two types of seismic load are applied for each of the four earthquake waves. In the first case, the combined horizontal and vertical earthquake waves (see red lines in Figures11 and 12) are inputted. In the second case, only the horizontal earthquake wave (see blue lines in Figures 11 and 12) is inputted.Figure 11
Spatial distributions of maximum earthquake normal stresses at different cross sections for the 150 m high chimney under two types of seismic load applied for each of the four earthquake wave actions: (a) Landers wave (No. 1); (b) Loma Prieta wave (No. 2); (c) Borrego wave (No. 3); (d) Chichi wave (No. 4).
(a)
(b)
(c)
(d)Figure 12
Spatial distributions of maximum earthquake normal stresses at different cross sections for the 210 m high chimney under two types of seismic load applied for each of the four earthquake wave actions: (a) Landers wave (No. 1); (b) Loma Prieta wave (No. 2); (c) Borrego wave (No. 3); (d) Chichi wave (No. 4).
(a)
(b)
(c)
(d)For the 150 m high chimney, Figure11 shows that the peak normal tensile stress increases by 4% for the Landers wave (No. 1) and 10% for the Loma Prieta wave (No. 2) if vertical earthquake waves are considered. The peak normal compressive stress increases by 7% for the Loma Prieta wave (No. 2) and 10% for the Chichi wave (No. 4) if vertical earthquake waves are considered.For the 210 m high chimney, Figure12 shows that the peak normal tensile stress increases by 12% for the Landers wave (No. 1) and 9% for the Loma Prieta wave (No. 2) if vertical earthquake waves are considered. The peak normal compressive stress increases by 9% for Landers wave (No. 1) and 13% for Loma Prieta wave (No. 2) if vertical earthquake waves are considered.Figure13 shows the spatial comparison curves of the mean values of the maximum normal earthquake stresses for the 150 m and 210 m high chimneys under two types of seismic load. Figure 13(a) is obtained from the average values of Tables 5, 7, 13, and 14. Figure 13(b) is drawn from the mean values of Tables 9, 11, 15, and 16. Figure 13 shows that the normal stress of the chimney can be increased considering the vertical seismic acceleration.Figure 13
Comparison curve of the mean value of maximum earthquake normal stresses for chimneys under two types of seismic load: (a) 150 m high chimney; (b) 210 m high chimney.
(a)
(b)Table 13
Maximum earthquake compressive stresses of the 150 m high chimney when only horizontal seismic acceleration is inputted.
High
Serial number
1
2
3
4
5
6
7
8
9
10
11
Mean
Standard deviation
5
15.762
6.884
11.475
13.226
2.984
9.720
5.314
7.104
3.694
6.267
7.235
8.151
3.986
15
16.533
6.912
9.673
14.154
2.374
10.375
5.697
7.298
3.395
6.244
7.812
8.224
4.261
25
15.970
6.204
9.464
14.907
2.194
10.292
5.644
6.853
3.049
6.014
9.059
8.150
4.379
35
17.119
4.713
9.862
13.772
2.836
8.412
4.903
6.279
4.110
5.139
9.057
7.837
4.407
45
20.107
3.593
12.133
15.772
4.133
5.642
4.755
5.276
4.437
3.707
8.936
8.045
5.603
55
21.071
4.229
14.646
18.041
3.645
6.161
5.083
6.183
3.713
3.242
8.740
8.614
6.337
65
21.786
7.321
13.780
17.612
3.645
10.790
6.299
8.381
4.823
5.491
9.217
9.922
5.686
75
20.897
9.792
16.022
18.742
4.307
15.863
7.209
10.533
5.855
8.858
11.386
11.769
5.406
85
18.671
10.349
17.184
18.626
3.973
18.579
7.799
11.399
5.396
11.049
12.025
12.277
5.361
95
21.849
13.139
17.782
21.304
4.313
19.716
8.952
13.522
5.751
12.327
12.051
13.701
5.966
105
22.592
14.578
19.457
22.671
4.985
19.364
9.133
14.541
6.688
12.095
11.287
14.308
6.136
115
23.369
12.785
24.161
21.098
7.016
16.249
8.339
12.902
7.520
11.069
9.754
14.024
6.313
125
19.263
9.922
22.706
16.703
7.430
11.274
6.289
9.926
7.222
8.303
7.947
11.544
5.502
135
10.624
5.025
14.653
8.506
5.419
5.190
3.297
5.773
4.065
4.181
4.662
6.490
3.436
145
1.946
0.838
3.020
1.463
1.236
0.789
0.615
1.260
0.848
0.766
0.892
1.243
0.705Table 14
Maximum earthquake tensile stresses of the 150 m high chimney when only horizontal seismic acceleration is inputted.
High
Serial number
1
2
3
4
5
6
7
8
9
10
11
Mean
Standard deviation
5
−14.019
−7.073
−10.794
−13.376
−3.065
−9.883
−5.107
−7.404
−2.917
−6.109
−7.146
−7.899
3.748
15
−13.568
−6.946
−8.764
−14.184
−2.128
−10.529
−5.319
−7.538
−2.438
−6.015
−7.459
−7.717
3.918
25
−13.396
−6.049
−9.673
−14.268
−2.217
−10.586
−5.134
−7.319
−2.751
−5.799
−7.994
−7.744
3.943
35
−14.760
−4.852
−9.829
−14.309
−2.644
−8.895
−4.731
−6.290
−3.085
−5.040
−7.879
−7.483
4.145
45
−18.971
−3.749
−12.950
−15.797
−3.562
−6.153
−4.989
−5.575
−3.317
−3.663
−7.614
−7.849
5.499
55
−20.940
−3.856
−14.872
−16.458
−3.938
−5.793
−5.151
−5.917
−3.675
−3.466
−8.690
−8.432
6.123
65
−22.465
−7.073
−14.037
−17.207
−3.647
−10.650
−6.376
−8.108
−4.479
−5.484
−10.224
−9.977
5.836
75
−22.721
−9.433
−16.977
−18.920
−4.644
−15.423
−7.684
−10.420
−5.601
−8.588
−11.544
−11.996
5.779
85
−20.043
−10.301
−16.678
−19.349
−4.152
−17.961
−8.200
−11.508
−4.973
−11.015
−12.732
−12.447
5.513
95
−22.737
−13.287
−16.842
−21.924
−5.111
−19.726
−9.137
−12.996
−4.815
−11.834
−12.481
−13.717
6.121
105
−24.485
−14.367
−18.594
−22.963
−5.152
−18.972
−9.762
−13.993
−5.724
−11.670
−11.579
−14.297
6.413
115
−23.410
−12.647
−24.355
−21.682
−6.705
−15.693
−8.900
−12.975
−6.598
−11.093
−9.727
−13.980
6.490
125
−20.313
−9.584
−23.734
−16.823
−7.219
−10.906
−6.540
−9.631
−6.598
−8.641
−8.727
−11.701
5.879
135
−11.408
−4.711
−15.196
−8.484
−5.553
−4.981
−3.266
−5.171
−4.017
−4.446
−5.039
−6.570
3.671
145
−2.166
−0.757
−3.138
−1.419
−1.330
−0.772
−0.624
−1.151
−0.790
−0.775
−0.950
−1.261
0.764Table 15
Maximum earthquake compressive stresses of the 210 m high chimney when only horizontal seismic acceleration is inputted.
High
Serial number
1
2
3
4
5
6
7
8
9
10
11
Mean
Standard deviation
5
8.481
4.199
8.705
10.156
2.007
9.332
3.272
7.522
5.873
6.495
6.213
6.569
2.599
15
8.901
4.019
8.216
10.622
2.225
10.012
3.285
8.055
5.978
6.514
5.906
6.703
2.746
25
9.582
3.469
7.243
10.684
2.446
10.362
3.266
8.297
5.919
6.143
5.912
6.666
2.869
35
10.568
3.052
6.345
9.797
2.526
9.397
2.767
7.913
5.484
5.128
6.954
6.357
2.870
45
9.972
3.302
6.902
8.454
3.223
7.488
2.873
7.388
4.133
4.051
7.458
5.931
2.467
55
10.485
3.886
8.502
9.475
3.529
7.237
3.392
6.915
4.759
4.188
8.276
6.422
2.573
65
10.190
4.154
9.585
9.633
3.223
6.894
3.521
6.782
5.247
4.515
8.447
6.563
2.599
75
9.191
3.988
8.965
9.777
3.536
5.854
3.329
6.980
4.994
4.741
8.090
6.313
2.395
85
11.177
3.643
8.928
12.420
3.799
7.406
3.792
8.005
5.956
6.387
8.570
7.280
2.943
95
13.516
4.035
11.145
14.322
3.448
10.454
4.424
9.272
7.536
7.673
9.883
8.701
3.683
105
14.580
5.302
13.069
14.196
4.576
12.836
5.224
10.651
8.796
8.504
10.711
9.859
3.671
115
16.062
6.498
16.367
13.919
5.366
15.006
6.185
13.698
10.445
8.645
11.724
11.265
4.087
125
18.730
8.256
17.638
18.152
5.275
19.078
7.159
16.475
11.612
10.651
13.545
13.325
5.028
135
20.661
8.608
16.365
23.615
5.444
21.778
8.219
17.456
14.630
14.181
13.650
14.964
5.838
145
19.716
6.626
15.624
24.576
4.835
21.610
7.150
16.670
14.439
14.995
11.521
14.342
6.352
155
20.779
8.409
20.741
25.679
4.079
20.345
6.788
16.586
13.115
14.038
9.833
14.581
6.862
165
19.550
9.803
22.386
24.808
5.578
18.092
7.307
16.231
12.737
12.603
13.035
14.739
6.095
175
17.347
9.451
19.978
20.117
5.357
14.149
7.215
13.845
10.595
11.112
13.379
12.959
4.840
185
13.642
7.253
15.301
14.461
5.160
9.390
5.906
9.742
7.723
8.592
12.240
9.946
3.487
195
7.997
3.857
8.696
7.489
4.402
4.434
3.330
4.899
4.067
4.728
7.289
5.563
1.906
205
1.474
0.668
1.583
1.222
1.160
0.650
0.567
0.936
0.698
0.824
1.336
1.011
0.360Table 16
Maximum earthquake tensile stresses of the 210 m high chimney when only horizontal seismic acceleration is inputted.
High
Serial number
1
2
3
4
5
6
7
8
9
10
11
Mean
Standard deviation
5
−9.499
−3.914
−8.402
−10.528
−2.271
−8.311
−3.147
−6.884
−5.485
−5.603
−6.195
−6.385
2.643
15
−10.088
−3.759
−7.888
−11.120
−2.170
−9.016
−3.276
−7.374
−5.517
−5.617
−5.575
−6.491
2.872
25
−10.735
−3.268
−6.696
−12.154
−2.343
−9.816
−3.359
−8.020
−5.630
−5.495
−6.296
−6.710
3.196
35
−9.917
−3.148
−6.027
−11.745
−2.324
−9.532
−2.986
−7.852
−5.311
−5.431
−7.037
−6.483
3.071
45
−9.763
−3.035
−6.814
−9.538
−3.218
−7.901
−3.112
−7.034
−4.432
−4.657
−8.188
−6.154
2.563
55
−9.887
−3.853
−8.522
−9.456
−3.515
−6.972
−3.159
−6.434
−4.759
−4.555
−8.436
−6.323
2.486
65
−9.276
−4.106
−8.968
−10.471
−3.061
−6.145
−3.460
−7.597
−5.049
−4.824
−8.064
−6.457
2.550
75
−8.813
−4.006
−9.201
−10.360
−3.433
−5.116
−3.316
−8.168
−5.097
−4.562
−8.403
−6.407
2.593
85
−11.696
−3.733
−9.048
−13.089
−3.214
−7.460
−4.127
−7.880
−6.340
−6.498
−9.220
−7.482
3.172
95
−15.283
−5.286
−12.218
−16.060
−3.427
−10.971
−4.632
−9.692
−7.143
−8.267
−10.285
−9.388
4.147
105
−17.060
−6.333
−14.403
−16.566
−4.330
−13.582
−5.322
−11.064
−8.223
−8.519
−10.820
−10.566
4.430
115
−19.059
−6.495
−14.769
−16.340
−5.136
−15.559
−5.968
−14.179
−8.929
−7.669
−12.054
−11.469
4.823
125
−21.359
−7.756
−15.261
−20.501
−5.311
−17.463
−6.259
−17.193
−9.471
−8.103
−13.228
−12.900
5.807
135
−21.802
−8.174
−15.876
−26.142
−5.335
−20.549
−7.745
−18.417
−11.458
−11.799
−13.529
−14.621
6.555
145
−19.722
−7.568
−15.281
−26.594
−4.215
−21.400
−7.706
−16.488
−13.112
−13.520
−12.850
−14.405
6.567
155
−18.995
−9.012
−19.516
−23.940
−4.076
−21.179
−8.472
−16.200
−15.026
−13.612
−12.953
−14.816
5.995
165
−18.022
−10.638
−21.577
−22.969
−5.468
−19.116
−8.724
−16.437
−14.438
−12.906
−11.733
−14.730
5.466
175
−16.832
−10.263
−21.403
−20.010
−5.569
−15.119
−7.902
−14.741
−11.533
−11.518
−11.998
−13.353
4.844
185
−13.024
−7.875
−17.879
−14.764
−5.138
−10.190
−5.783
−11.224
−8.565
−8.835
−10.257
−10.321
3.791
195
−6.944
−4.092
−10.140
−7.591
−4.379
−4.781
−3.151
−5.887
−4.848
−4.741
−5.925
−5.680
1.956
205
−1.261
−0.689
−1.797
−1.221
−1.265
−0.696
−0.591
−1.091
−0.865
−0.805
−1.125
−1.037
0.353The results studied above show that the vertical ground motion should not be neglected, especially for tall RC chimneys built on rock sites or soft rock sites.


## 6. Conclusions

A numerical algorithm that can consider the existing phenomena of eccentricities is presented to calculate the earthquake responses of tall RC chimneys. The numerical results demonstrate that the tensile and compressive stresses may increase in the seismic control cross sections of the chimneys under the actions of several specific earthquake waves when existing eccentricities are considered. The phenomena of eccentricities should be considered to obtain accurate calculation results of earthquake responses for the seismic designs of tall RC chimneys. The maximum normal stresses of earthquake response appear within the upper half of the chimneys, and their values are considerably larger than those within the lower half of the chimneys. Vertical ground motion should not be neglected, especially for chimneys on rock sites and soft rock sites.



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*Source: 1417969-2020-03-11.xml*