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 1

Nonlinear Newmark-Beta method for nonlinear truss element formulation.

	Input:
	Unstrained element length: ; Poisson’s ratio: ; Initial element cross section area: ;
	Initial element strain: ; Elastic modulus: ; Line initial coordinate ;
	Mass matrix: ; Initial displacement, velocity, and acceleration matrix , , and ;
	Damping matrix ; 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 From , calculate the element length
	3.8.2 Calculate element length incremental:
	3.8.3 Computing axial elongation factor using equation (5a)
	3.8.4 Computing logarithmic strain using equation (5b)
	3.8.5 Computing the current state stress using equations (5c) and (5d)
	3.8.6 Computing the cross section area using equation (6)
	3.8.7 Calculate the tangent stiffness at iteration using equation (2)
	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 truss element