Mechanical Properties and Stability of Single-Layer Reticulated Shell with Holes*

WANG Jianxing,LIU Ge,ZHANG Pengyu,ZHANG Zhaowei

(College of Civil Engineering,North China University of Technology,Beijing 100144,China)

Abstract：Taking the reticulated shell structure node as the main research object,the numerical model of the main reticulated shell structure of the dry coal shed of the power plant is constructed,the buckling mode of the perforated reticulated shell structure is studied,and the nonlinear buckling analysis is carried out to explore the static characteristics under vertical load and wind load.It is found that the stress concentration occurs at the opening of the reticulated shell.The maximum stress of the joint under the vertical load is 0.037 kPa,and the maximum stress of the joint under the wind load is 0.024 kPa.The maximum displacement of the node under the vertical load is 0.881 mm,and the maximum displacement of the node under the wind load is 1.692 mm,which is 69.34% and 19.57% higher than the average displacement of the node of the whole reticulated shell structure,respectively,which meets the requirements of the code.In the eigenvalue buckling analysis,some members of the rigid joint reticulated shell structure are deformed,which has a certain influence on the structural safety.After the re-optimization design of the reticulated shell,some joints adopt hinged joints,and the eigenvalue buckling load is 4.12% higher than that of pure rigid joints,which effectively improves the stability of the reticulated shell structure.

Key words：open-hole reticulated shell structure;mechanical activation behavior;buckling mode shape;numerical analysis;stability

In a variety of large-span spatial structures,the single-layer spherical reticulated shell structure has been widely concerned because of its changeable shape,curved surface composed of rods,transparent internal space,and large span.The reticulated shell structure has the characteristics of meeting the needs of storage and working space.Commonly used in coal yards,cement plants,and other projects.Domestic and foreign scholars mainly study single-layer reticulated shell structure through stability calculation and influencing factors.Kato et al[1]carried out the relevant theoretical research on the continuous analysis method of reticulated shell and buckling eigenvalue.Oilawa et al[2]studied various forms of reticulated shell structures after considering the material nonlinearity and initial defects of reticulated shell structures.Altuna et al[3]carried out the finite element analysis on the combined single-layer reticulated shell.Fan et al[4]carried out several studies on the stability of spherical reticulated shells and proposed the random defect mode analysis method.Tang et al[5]further improved the random defect mode method on this basis.

In the main body of reticulated shell structure in general engineering,the joints adopt pure rigid joints,while ignoring the practical application of hinge joints in reticulated shell structure.In this paper,the dry coal shed of a single-layer reticulated shell structure as the main body of the power plant is taken as an engineering example.The reticulated shell structure needs to be drilled to deal with practical engineering.The single-layer reticulated shell structure model was established.The internal forces of the two structures under the full-span uniform load were calculated before and after the drilling treatment.The axial force distribution characteristics were explored and summarized.The force difference and eigenvalue buckling analysis of the reticulated shell structures with pure rigid joints and hinge joints were compared.Finally,the influence of holes on the structure is obtained.

1 Engineering Overview and Model Establishment

1.1 Project Overview

The dry coal shed of a power plant takes the single-layer hemispherical reticulated shell structure as the main body,as shown in Fig. 1 and Fig. 2.The span of the reticulated shell is 197.6 m,the rise is 50 m,and the building area is 30 651 m2.According to the production process requirements,the lower part of the reticulated shell is perforated,and the model is shown in Fig. 1 and Fig. 2.The structure adopts the arrangement of fixed hinge support partition,and the reticulated shell support is fixed.The anchor bolt is rigidly connected to the concrete foundation.

Fig. 2 Model Side View

Reticulated shell supporting structure adopts reinforced concrete independent column,concrete strength grade C30.The design strength of Q235 steel thin-walled seamless circular pipe is 215 N/mm2and is controlled within the standard value of 0.95.The specific mechanical parameters are show in Table 1.

Table 1 Mechanical Properties of Materials

1.2 Load Numerical Analysis

The main loads on the lattice shell structure of a dry coal shed are gravity constant load,wind load,temperature gradient and ground vibration.The design live load of the roof is 0.50 kN/m2,the grey load of the roof area is 0.80 kN/m2,the seismic fortification intensity is VI,the design basic acceleration value is 0.05 g,and the site soil type is II;the basic wind pressure isw0= 0.4 kN/m2,and the surface roughness is class B.

Constant load:Gravity constant load mainly includes the weight of member and node in reticulated shell structure,the weight of the roof and related production equipment.The following equation[6]can be used for calculation:

wheregckis reticulated shell structure weight (kN/m2),qwis the load standard value outside the weight of the grid,L2is short span (m),ξis the section coefficient.

Wind load:Due to the large span and small self-weight of the reticulated shell structure,the members are prone to deformation under wind load resulting in damage.The standard value[7]is

wk=β2μ2μsw0,

wherewkis basic wind pressure (kN/m2),μsis wind load shape coefficient,μ2is height change factor,β2is gust response coefficient.

2 Analysis and Calculation of Mechanical Characteristics

2.1 Constitutive Model and Numerical Analysis

The stress-strain relationship of steel rod assembly of reticulated shell structure under monotonic loading is studied by using the constitutive model proposed by ref. [8].The expression is

whereris the shape constant of the compressive stress-strain curve,fcis compressive strength,εpis the strain corresponding to this strength,E0is the initial elastic modulus of the bar,Edis the slope of the stress-strain curve,εuis the compressive strain in the limit state.

To calibrate the accuracy of the results calculated by the above formula,the Mises yieldcriterion[9]is used to test the stress change of steel bars under external load as follow:

whereσcis compressive stress,εcis compressive strain,χis a constant representing the brittleness of the material.

The finite element software Midas Civil is used to analyze the structural internal force analysis of the above various states,the results are shown in Fig. 3 and Fig. 4.As shown in Table 2,under the vertical load,the maximum positive stress of the reticulated shell structure is 0.037 kPa,which is distributed at the opening of the reticulated shell.Under wind load,the maximum normal stress of the reticulated shell structure at the opening reaches 0.024 kPa.Both of them meet the design strength requirements.According to the requirements,they are lower than the allowable stress of steel tube materials,which is consistent with the stability of the reticulated shell structure in practical use.Considering the economic requirements,under various special working conditions,the maximum stress is concentrated around the opening,but there is a large safety technical distance from the specified limit.Steel materials with low strength and high elastic modulus can be used in structural design to reduce the deflection change of members and give full play to the practical value and economy of structural materials.

Fig. 3 Distribution of Vertical Load Stress

Fig. 4 Stress Distribution of Wind Load

Table 2 Stress Values of Reticulated Shell Structure Under Two Types of Loads kPa

2.2 Displacement of Reticulated Shell Structure

The reticulated shell is a symmetrical structure with 401 nodes.The treatment of displacement refers to the special structuralmechanical properties of swallowtail tenon[10].In the analysis of structural displacement,the structural connection node is taken as the research subject,and the nodes are numbered.The local elastic displacement change of the structure is shown by the displacement in each direction through finite element analysis.The displacement of each node under vertical load and wind load is shown in Fig. 5 and Fig. 6.

Fig. 5 Displacement Diagram of Vertical Load Node

Fig. 6 Displacement Diagram of Wind Load Node

Under the vertical load,the displacement distribution of each node of the reticulated shell structure is relatively uniform,and the maximum displacement is 0.881 mm.Among them,the displacement at node 196 is 69.34% higher than the average displacement of the node,and a large mutation occurs.Under wind load,the maximum displacement of the joint is 1.692 mm,the displacement distribution in thexandydirections is the same,and the displacement inzdirection is symmetrical about the center.In all kinds of working conditions,the maximum displacement is distributed at the opening of the reticulated shell,which meets the requirements of the specification and meets the practical requirements.

3 Eigenvalue Buckling Analysis

The theoretical buckling strength of the ideal elastic structure can be judged,and the Euler critical load of the structure can be preliminarily determined by eigenvalue buckling analysis.Firstly,the reticulated shell structure is discretized.Taking each member of the reticulated shell as the basic element,the stability analysis is carried out by using the nonlinear finite element method based on the discretization assumption.The following eigenvalue equations can be used to predict the theoretical stability coefficient[11-13]and buckling mode of an ideal elastic structure:

(K+λS)Φ=O,

whereKis the stiffness matrix,Sis the stress stiffness matrix,λis the buckling eigenvalue,Φis the displacement feature vector.

Considering the influence of nonlinearity and initial imperfection,the nonlinear buckling analysis of reticulated shell structure is needed for further accuratecalculation.The nonlinear finite element equation is

The first three buckling modes of reticulated shell structures are shown in Fig. 7～9.It can be seen from the figure that on the main rib,the first-order modal of the structure has a large vertical displacement in the edge area of the hole,and the region is first unstable.The second-order mode has incomplete symmetric instability in the structures on both sides of the center,and the vertical displacement near the hole is larger than that on the opposite side.In the third-order mode,the vertical displacement near the hole is still large,and the whole structure shows two depressions and two protrusions.

The bucking mode diagram after the application of hinged joints is shown in Fig. 10.After replacing the pure rigid joint with the hinged joint,the eigenvalue buckling load of the main structure of the reticulated shell is 445.43 kN/m2,which is 4.12% higher than that of the pure rigid joint.Both sides of the buckling mode center of the structure remain stable and small displacement,and only structural instability occurs at the edge nodes of the hole.The influence of vertical load and wind load on reticulated shell is effectively reduced,and the stability and structural safety of the project are improved.Referring to ref. [14],the conclusion is further demonstrated.

Fig. 7 First-Order Buckling Mode Diagram

Fig. 8 Second-Order Buckling Mode Diagram

Fig. 9 Third-Order Buckling Mode Diagram

Fig. 10 Buckling Mode Diagram of Hinged Joints

4 Conclusion

The main structure of the project adopts a single-layer hemispherical reticulated shell structure.In the eigenvalue buckling analysis,the deflection of some members changes,which has a certain impact on the structural safety of the reticulated shell structure,and further repair and transformation are needed.For example,the support of the reticulated shell structure is reformed to reduce the influence of structural stress unevenness on the super structure;the reticulated shell is treated to avoid the damage to the main structure of the building caused by the uneven settlement of the lower foundation.By comparing the displacement changes of the reticulated shell structure under the two types of loads,the maximum displacement value of the grid node is 0.881 mm under the vertical load,which is 69.34% higher than the average displacement of the node.Under the action of wind load,the maximum displacement value of the grid node is 1.69 mm,which is 19.57% higher than the average displacement of the node.In order to avoid the large-scale deflection change of the bar,the elastic restraint support can effectively improve the bearing capacity of the structure and reduce the sensitivity to the displacement change of the main building.In this paper,the eigenvalue buckling analysis and static characteristics analysis of the main reticulated shell structure of the dry coal shed of the power plant are carried out by using Midas Civil software modeling.Under various loads,the stress concentration phenomenon occurs at the opening of the reticulated shell.Under the vertical load,the maximum stress value of the structure is 0.037 kPa;under the action of wind load,the maximum stress value of the structure is 0.024 kPa,which meets the allowable stress of the material and leaves a large safety range.There are many kinds of spatial reticulated shell structures in practical use.In this paper,the open-hole single-layer reticulated shell is studied.Whether other structures meet the conclusions of this paper still needs to be calculated and discussed.