Nonlinear buckling analysis of cat-head transmission tower under strong wind load

Based on ANSYS APDL finite element analysis software and B-R criterion combined with displacement equality criterion, the dynamic stability of the cat-head transmission tower is analyzed. In this paper, the static characteristics of the cat-head transmission tower are analyzed first, and the wind load is gradually increased according to the wind speed gradient of 5 m/s. The results show that the cat-head transmission tower cannot be normally served under the wind speed of 45 m/s. In order to analyze the specific instability wind speed of the transmission tower in detail, the nonlinear buckling analysis is used to analyze the cat-head transmission tower. The nonlinear buckling analysis results of the cat-head transmission tower are as follows: the wind speed of instability is V10=43.65 m/s. Therefore, the influence of wind load fluctuation on the dynamic stability of the transmission tower should not be ignored, and the influence of fluctuating wind should be considered in the design of the transmission tower.


Introduction
Buckling analysis is a failure mode identification technique in engineering calculation.Buckling is a common cause of instability of transmission tower structures, and the instability of tower structures is often sudden, so it is more harmful.The buckling analysis includes eigenvalue buckling analysis and nonlinear buckling analysis [1].In this paper, nonlinear buckling analysis is used to analyze the critical buckling load of transmission towers under the average wind.
Many scholars at home and abroad have proved that the key reason for the collapse of the transmission tower caused by wind load is the deformation of the tower structure, and the deformation of the structure is mainly due to the stress concentration of local rods, which leads to the plastic deformation of the rods.

Principle of nonlinear buckling analysis
Through nonlinear static analysis of the gradually increasing load of the structure, the critical load when the structure is unstable is obtained.As shown in Figure 1, the first limit point is the instability load of the transmission tower.Because the incremental tangential stiffness matrix quickly becomes a singular matrix when the structure is unstable, the equilibrium iterative equation is difficult to converge [2].Therefore, in order to obtain a complete structural load displacement curve, nonlinear buckling analysis methods are mainly divided into the displacement increment method, arc length method and artificial spring method.After a large number of scholars' comparative experimental research, the arc length method is universal and can accurately calculate the results.Therefore, the arc length method is used to analyze the nonlinear buckling of the cat-head transmission tower.The calculation process of the arc length method is as follows: it is assumed that the bit shift solution has been obtained in a certain period of time, and the structure changes from stable equilibrium to unstable equilibrium in the time increment step [3].In order to obtain the exact solution, it is necessary to make the time increment step size as small as possible.
As shown in Figure 1, when the load displacement curve reaches the first extreme point, the slope of the curve is 0, that is, the determinant of the tangent stiffness matrix of the nonlinear balance equation is 0, then the curve reaches the critical load point [4].

Static analysis of cat-head transmission tower
Statics analysis of the cat-head transmission tower is the basis of nonlinear buckling analysis of the cathead transmission tower under wind load [5].By statics analysis of the cat-head transmission tower, the weak position of the cat-head transmission tower under wind load and the instability of the cat-head transmission tower under wind speed can be obtained.It provides the basis for nonlinear buckling analysis of the cat-head transmission tower.

Modeling of cat-head transmission tower
This paper uses a 2B10-ZMC1 220 kV cat-head transmission tower as the research object.The total height of the tower is 30m, the call height is 27 m, the root opening is 6.379 m, the main material is Q345, the auxiliary material is Q235, and all the iron parts are anticorrosive by hot-dip galvanizing.Table 1 shows the structural parameters of the cat-head transmission tower.According to the analysis of the actual structure of the cat-head transmission tower, this paper uses beam 188 unit to construct the beam model of the transmission tower [6], and accurately establishes the cat-head transmission tower model by defining the spatial coordinates and the size of each member unit in detail.The model of cat-head transmission tower are shown as Figure 2.

Modeling of cat-head transmission tower
According to GB 50135-2019 Code for Structural Design of Towering Structures [7], the standard value of wind load acting on the transmission tower is calculated according to Formula (1): where k  is the standard value of wind load acting on the unit projected area at the height of z of the transmission tower (KN/m 2 ), z  is the wind vibration coefficient at z height, and 0  is basic wind pressure (KN/m 2 ).Calculated from the wind speed 10 V at the reference height [8], the expression is Formula(2): where s  is shape coefficient of wind load and z  is the coefficient of wind pressure height change at z height.s  and z  are known from the parameter table of structural design specification.

Statics analysis of cat-head transmission tower under strong wind load
Taking 5 m/s as the wind speed gradient and increasing wind speed step by step, the wind speed of different strong wind levels is applied to the transmission tower.It can be found that when the wind speed at the reference height is 45 m/s, the cat-head transmission tower will become unstable.The displacement diagram and stress diagram under this wind load are shown as Figure 3.By analyzing the cat-head transmission tower at 45 m/s wind speed, it is found that the maximum displacement of the cat-head transmission tower at this wind speed is 0.347 m, and the maximum stress is 356 Mpa.Compared with the yield strength of the main material, it is found that the maximum stress exceeds the yield strength of the main material of the cat-head transmission tower at 345 Mpa, and plastic deformation occurs.At this time, it is considered that the cat-head transmission tower has been unable to serve normally.

Nonlinear buckling analysis results of cat-head transmission tower
There are many methods in nonlinear buckling analysis to determine the failure of the tower structure.The Budiansky-Roth criterion is generally applicable to the nonlinear buckling analysis of the transmission tower through the verification of scholars at home and abroad.Budiansky-Roth criterion describes that when the structure is subjected to a small load, but the structure changes dramatically, then the structure becomes unstable.According to the B-R criterion, during the static analysis of the transmission tower, when the wind speed is 45 m/s, the transmission tower cannot serve normally.ANSYS APDL finite element software is used to analyze the nonlinear buckling of the cat-head transmission tower, and the displacement of the top node of the transmission tower is selected as the analysis object to observe its response.The corresponding relationship between load, wind speed and displacement is shown in Table 2 2, the top displacement curve, the top displacement-load curve and the top displacement-wind velocity curve as shown in Figure 4 can be obtained.According to the figure, when the load is 4.1962, V10= 43.46 m/s, and when V10=0~43.46 m/s, the load and displacement on the structure change linearly.At this time, the structure is in a non-buckling state.When V10 > 43.46 m/s, the load and displacement of the structure do not change linearly, and the wind speed-displacement curve appears horizontal segment.According to the Budiansky-Roth criterion, the structure is in a state of buckling instability at this time, and the maximum displacement at the top of the tower is 0.341 m.At this time, the maximum stress of the main material of the tower is 347 MPa, which is greater than the tensile strength of the main material of the transmission tower.The plastic deformation occurs and the rod is in an abnormal working state.

Conclusion
According to the B-R criterion, the unstable wind speed of the transmission tower is 45 m/s when the static analysis is carried out.ANSYS APDL finite element software is used to analyze the nonlinear buckling of the cat-head transmission tower, and the displacement of the top node of the transmission tower is selected as the analysis object.When the load is 4.1962, V10= 43.46 m/s, and when V10 = 0~43.46m/s, the load and displacement of the structure change linearly.At this time, the structure is in a non-buckling state.When V10 > 43.46 m/s, the load and displacement of the structure do not change linearly, and the wind speed-displacement curve appears horizontal segment.According to the Budiansky-Roth criterion, the structure is in a state of buckling instability at this time, and the maximum displacement at the top of the tower is 0.341 m.At this time, the maximum stress of the main material of the tower is 347 MPa, which is greater than the tensile strength of the main material of the transmission tower.The plastic deformation occurs and the rod is in an abnormal working state.

Figure 2 .
Figure 2. Model of cat-head transmission tower.
(a) Displacement diagram (b) Stress diagram Figure 3. Mechanical analysis diagram of cat-head transmission tower.

Table 1 .
Steel parameters of cat-head transmission tower.

Table 2 .
. Corresponding table of wind speed, load and displacement.