Investigation and Development of a Three-Dimensional Transmission Tower-Line System Model Using Nonlinear Truss and Elastic Catenary Elements for Wind Loading Dynamic Simulation

Algorithm 2

Nonlinear Newmark-Beta method for elastic catenary element formulation.

	Input:
	Unstrained element length: ; Elastic modulus: ; Linear thermal expansion coefficient:
	Temperature change: ; Distributed loads in each direction :
	Line initial coordinate ; Mass matrix ; Damping matrix ; Initial stiffness matrix
	Initial displacement, velocity and acceleration matrix , , and ;
	Newmark-Beta convergence criteria: ; Elastic catenary convergence criteria:
	Output:
	displacement ; velocity ; acceleration
	Begin:
	1.0 Initial parameters calculation
	1.1 State determinations: the force and the initial tangent stiffness matrix
	1.2 Solve  ⟶
	1.3 Select Newmark-Beta parameters and , and time interval
	1.4 Calculate ;
	2.0 Calculations for each time instant,
	2.1 Initialize , , and
	2.1
	3.0 For each iteration,
	3.1 The residual force
	3.2 Check convergence; if , skip the following steps and go to step 4.0; otherwise, implement the following steps:
	3.3
	3.4 Solve  ⟶
	3.5
	3.6 Update the line coordinate
	3.7 Update
	3.8 Update the line tangent stiffness
	3.8.1 For element
	3.8.2  Extract the two element ends coordinate from as and
	3.8.3  From (16), calculate
	3.8.4  From (25c), calculate the internal nodal force vector at
	3.8.5  Compute target length in each direction
	3.8.6  Initialize the convergent parameter
	3.8.7  While
	3.8.8   Calculate parameter using (16)
	3.8.9   Calculate the force in at using (24c)
	3.8.10   Calculate the end force at node and using (10)
	3.8.11    Calculate the projection length of the element at Cartesian coordinates using (15)
	3.8.12   Calculate
	3.8.13   Calculate the flexibility matrix using (19)
	3.8.14   Update the internal force incremental
	3.8.15   Update the internal force vector at node as
	3.8.16   Calculate the stiffness matrix using (21)
	3.8.17   Assemble the element stiffness using (22)
	3.8.18  Assemble the global stiffness matrix
	3.9 Replace by and repeat steps 3.1 to 3.8; after converge, denote final displacement value as , and the coordinate of the line as
	4.0 Calculate the velocity and acceleration for time instant
	,
	5.0 Replace by and implement steps 2.0 to 4.0 for time instant
	: the tangent stiffness of the transmission line using nonlinear truss element; refers to transmission line; refers to catenary element