Hanging Cable

A cable is a structural member that has stiffness only in its tangential direction, but virtually no bending stiffness. When supported only at its two ends, it deflects under gravitational load. An idealized hanging cable can be analyzed analytically and forms a deflection curve known as a catenary (derived from the Latin word catena for chain). In COMSOL Multiphysics, hanging cables, chains, or strings can be modeled with the Wire interface.

Model Definition

The example consists of a cable carrying two street lamps. The model setup, the global coordinate system and relevant properties are indicated in Figure 1. The cable has an initial length L, and it is supported at two points spaced by a length d < L apart from each other. Three load cases are analyzed:

Cable only with gravitational load

Cable and lamps with gravitational load

Cable and lamps with gravitational and wind loads

Figure 1: Cable carrying two lamps. The gravitational acceleration is parallel to the z direction, and the wind is parallel to the y direction.

The lamps are assumed to be spherical in shape and mounted at distances 0.3L and 0.8L along the cable’s initial length. The wind vector is assumed to point in the positive y direction, and the forces acting on the lamps and the cable are estimated as

where cD is the drag coefficient, ρ the air density, and A the reference area for the drag.

The cable is made of steel (see Table 1), and behaves linear elastically.

Table 1: Material Properties.
Property	Cable
Young’s modulus	200 GPa
Poisson's ratio	0.3
Density	7850 kg/m3

An analytical solution can be derived for the first load case (a cable with gravitational load only). The cable setup and relevant properties are indicated in Figure 2. To the right, a free-body diagram is shown for the cable section between points A and B.

Figure 2: Deformed cable under its self-weight. A free-body diagram for the cable segment between points A and B is shown to the right.

As before, the cable has an initial length L, and its support points are spaced by a distance d apart from each other. The analytical treatment assumes that the stiffness in tangential direction is high, and any cable elongation is negligible; that is, the length of the deformed cable is still equal to the original length L.

The analytical treatment of a hanging cable is a classical problem of calculus of variations. The derivation of the catenary can be found in many text books. Typically, the curve is derived as a constrained variation problem, by constructing a functional that incorporates the potential energy of the system plus a constraint equation (the arc length of the deflected cable is L) with a Lagrange multiplier. The application of the Euler–Lagrange equations then lead to a differential equation which when solved yields the catenary curve.

From a mechanical perspective, the catenary can also be derived by simply formulating the equilibrium equations using a free-body diagram in a deformed configuration, as shown on the right in Figure 2. Choosing point A to be the lowest deflection point simplifies the analysis. Force balance in the x and y directions yield

and

where N is the tangential force at point B, FA the tangential force at point A (only acting in the horizontal direction). The variable W is the weight given as μgs, where μ is the mass per length, g the gravitational acceleration, and s is the arc length of the cable segment. Rearranging and dividing the equations yields

Here, we have also used that tan(

) is the slope of the curve dy/dx. Differentiation with respect to x yields a 2nd-order nonlinear differential equation for the deflection y(x):

In the last step, the constant FA/(μg) was replaced by a different constant, a. The general solution for this differential equation is

where the constants c1 and c2 can be found from the boundary and symmetry conditions. This leads to the final expression for the catenary:

The expression contains the unknown parameter a, which can be found by using the constraint that the length of the catenary must equal L:

This equation can be only be solved numerically for a. Note that the solution does not depend on any material properties. Cables of equal lengths and uniform density take on the same deformed shape, independently of their respective weights.

Results and Discussion

The computed and analytical solutions agree very well when considering the first load case (cable deforming under its own weight); see Figure 3.

Figure 3: Cable sag due to self-weight (load case 1). Comparison of the analytical and the solution computed using the Wire interface.

Figure 4 shows the vertical sag (w) for all three load cases, as well as the deflection magnitude (in the y and z directions) for the last load case which includes the wind load in the y direction. The lamps are hung asymmetrically, leading to the nonsymmetric deflection curves. The cable is stiff in its axial direction and does not exhibit any significant elongation. With gravity and wind loads acting on the lamps, the maximum cable deflection, or sag, decreases.

Figure 4: Vertical and total cable sag for the three load cases.

The distribution of the axial forces are compared in Figure 5.

Figure 5: Axial force along the cable for the three load cases.

Notes About the COMSOL Implementation

The analytic expression for the catenary includes a scaling factor a. As shown above, its value is expressed in terms of a transcendental function; that is, no closed-form solution exists. However, the value for a can be easily obtained numerically. To do so in COMSOL Multiphysics, a interface is added to the model.

Most cable problems are geometrically nonlinear. A wire which is not in tension is numerically unstable. Physically, it wrinkles in an unpredictable manner. In order to start the analysis in this case, an initial stress is added by giving an approximate expression for the expected deflection using the feature.

Analyzing this type of problem poses some numerical challenges. In a cable which is not pretensioned, the strains will typically be very small. Still the force balance is computed from the stress in each element, a stress that is proportional to the strain. The strains are computed as derivatives of the displacements, which typically are large. The numerics will thus contain computing a small difference between two large numbers. This is the reason why it is recommended to use linear element shape functions for this type of analysis.

Cables and wires can be modeled using Wire interface in COMSOL. Additional information and tips about the modeling of cables can be found in the section Modeling with Truss Elements in the Structural Mechanics Module User’s Guide.

Structural_Mechanics_Module/Verification_Examples/hanging_cable

Modeling Instructions

From the menu, choose .

New

In the window, click

Model Wizard

In the window, click

In the tree, select .

Click .

Click

In the tree, select .

Click

Global Definitions

Parameters 1

In the window, under click .

In the window for , locate the section.

In the table, enter the following settings:


Name	Expression	Value	Description
L_cable	5.4[m]	5.4 m	Cable length
D_cable	5[mm]	0.005 m	Cable diameter
A_cable	pi/4*D_cable^2	1.9635E-5 m²	Cable cross section area
u_support	0.2[m]	0.2 m	End point displacement
d_support	L_cable - 2*u_support	5 m	Support distance
m_lamp	0.5[kg]	0.5 kg	Lamp mass
D_lamp	400[mm]	0.4 m	Lamp diameter
E	2.0e11[Pa]	2E11 Pa	Young's modulus
rho	7850[kg/m^3]	7850 kg/m³	Density
q_dyn	(1.225[kg/m^3]/2)*(15[m/s])^2	137.81 Pa	Dynamic pressure

To easily activate or deactivate gravity or wind loads, add two load groups.

Lamp Weight

In the window, right-click and choose .

In the window for , type Lamp Weight in the text field.

In the text field, type lgLamp.

Wind Load

In the window, right-click and choose .

In the window for , type Wind Load in the text field.

In the text field, type lgWind.

Geometry 1

Polygon 1 (pol1)

In the window, expand the node.

Right-click and choose .

In the window for , locate the section.

In the table, enter the following settings:


x (m)	y (m)	z (m)
-L_cable/2	0	0
L_cable/2	0	0

Partition Edges 1 (pare1)

In the toolbar, click

and choose .

On the object , select Edge 1 only.

In the window for , locate the section.

In the table, enter the following settings:


Relative arc length parameters
0.3
0.8

Click

Materials

Material 1 (mat1)

In the window, under right-click and choose .

In the window for , locate the section.

In the table, enter the following settings:


Property	Variable	Value	Unit	Property group
Axial stiffness	k_A	A_cable*E	N	Elastic wire
Mass per unit length	rho_L	rho*A_cable	kg/m	Elastic wire

Wire (wire)

Elastic Wire 1

In the window, under click .

In the window for , locate the section.

In the A text field, type A_cable.

Click the

button in the toolbar.

In the dialog box, in the tree, select the check box for the node .

Click .

In the window for , click to expand the section.

In the ecl text field, type -5e-3.

Prescribed Displacement 1

In the toolbar, click

and choose .

Select Points 1 and 4 only.

In the window for , locate the section.

From the list, choose .

In the u 0 x text field, type -sign(X)*u_support.

From the list, choose .

Gravity 1

In the toolbar, click

and choose .

Edge Load: Wind on Cable

In the toolbar, click

and choose .

In the window for , type Edge Load: Wind on Cable in the text field.

Locate the section. From the list, choose .

Specify the Ftot vector as


0	x
1.1q_dyn(L_cable*D_cable)	y
0	z

In the toolbar, click

and choose .

Point Load: Wind on Lamps

In the toolbar, click

and choose .

In the window for , type Point Load: Wind on Lamps in the text field.

Select Points 2 and 3 only.

Locate the section. Specify the FP vector as


0	x
0.45q_dynpi*(D_lamp/2)^2	y
0	z

In the toolbar, click

and choose .

Point Mass: Lamps

In the toolbar, click

and choose .

In the window for , type Point Mass: Lamps in the text field.

Select Points 2 and 3 only.

The lamp mass can be included in load group 1 by conditionally activating the mass using the load group’s weight factor.

Locate the section. In the m text field, type if(group.lgLamp, group.lgLamp*m_lamp, 0).

Initial Values 1

In the window, click .

In the window for , locate the section.

Specify the u vector as


0	X
0	Y
(-L_cable/2 + X)(L_cable/2 + X)(2/L_cable^2)*sqrt(L_cable^2 - d_support^2)	Z

A cable which is not in tension is not numerically stable. To improve solver convergence an initial guess is provided in terms of a parabolic function.

Mesh 1

Edge 1

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

Size

In the window, click .

In the window for , locate the section.

Click the button.

Locate the section. In the text field, type L_cable/60.

In the text field, type L_cable/60.

Click

Global Definitions

Analytic 1 (an1)

In the toolbar, click

and choose .

In the window for , type y_catenary in the text field.

Locate the section. In the text field, type a*(cosh(X/a) - cosh(0.5*d_support/a)).

In the text field, type X, a, d_support.

Locate the section. In the text field, type m.

In the table, enter the following settings:


Argument	Unit
X	m
a	m
d_support	m

The analytic solution, y_catenary, takes the parameter a as an input. Its value can be obtained from a transcendental function which can be easily solved for with a interface.

Add Component

In the window, right-click the root node and choose .

Add Physics

In the toolbar, click