Name: finding cable shape under distributed load
Uploaded: 2025-05-22T09:06:04.000Z
Duration: 1 min 24 s
Channel: JoVE

In suspension bridges, the main cables experience distributed loads, which can be transferred to the supporting towers and anchorages. Determine the cable&#39;s shape and tension.
The cable is assumed to be flexible, inextensible, and with negligible weight for the analysis.
Consider a small segment of the cable and draw the free-body diagram of the segment showing the distributed load and changes in tensile force.
Next, apply the equilibrium equations and by substituting the horizontal and vertical forces and moments acting on the segment, a set of equations is obtained.
Dividing each equation by &Delta;x and by taking each limit tending to zero, a new set of three equations is obtained.
Integration of the first equation yields the horizontal component of tensile force at any point along the cable.
Next, integrate the second equation. Dividing it by the component of tensile force and substituting the value of 	tan&#952;, the expression for the slope of the cable is obtained.
Finally, performing a second integration on the derived equation determines the curve of the cable.

cable subjected to a distributed load

Cable Subjected to a Distributed Load

The analysis of suspension bridges is a complex and critical process that involves multiple factors, including the shape and tension of the main cables. The main cables of suspension bridges are subjected to distributed loads, which result in changes in tensile forces and deformation of the cable. These loads must be carefully considered to ensure that the bridge is safe and capable of supporting the weight of different loads.
<img alt="Freebody diagram, tension balance, ΔF=w(x)Δx, Δk=k(Δx), bridge structure, mechanical analysis." src="/files/ftp_upload/14308/14308_Figure_1v2.svg" style="text-align: center; height: 350px;" data-altupdatedat="1754904545000" data-alt="Static equilibrium principles; freebody diagram with ∆F=w(x)∆x, ∆k=k(∆x) near bridge scene.">
To analyze a cable subjected to distributed loads, consider a small segment of the cable and draw a free-body diagram showing the distributed load and changes in tensile force. A set of equations is obtained by applying equilibrium equations and substituting the horizontal and vertical forces and moments acting on the segment. Dividing each equation by delta Δx and taking each limit tending to zero yields a new set of three equations. The first equation obtained from this analysis helps determine the tensile force's horizontal component at any point along the cable.
<img alt="Static equilibrium formula \(T \cos \theta = \text{constant} = F_H\), mechanical analysis." src="/files/ftp_upload/14308/14308_Equation_1v2.svg" style="text-align: center; height: 13px;" data-altupdatedat="1754904545000" data-alt="Static equilibrium equation, Tcosθ=constant=FH, physics formula, mathematical expression.">
It also allows calculating the cable's tension at any point along its length. The expression for the slope of the cable is obtained by dividing the second equation, which is the vertical component of tensile force, with the component of horizontal tensile force and substituting the value of tan θ.
<img alt="Tangent slope equation, tan θ = dy/dx = 1/FH ∫w(x)dx, calculus formula, mathematical analysis." src="/files/ftp_upload/14308/14308_Equation_2v2.svg" style="text-align: center; height: 35px;" data-altupdatedat="1754904545000" data-alt="Tangent slope equation; derivative integral formula for mathematical analysis.">
Finally, performing a second integration on the derived equation determines the curve of the cable.
<img alt="Mathematical integration formula: y=(1/F_H)∫(∫w(x)dx)dx for analysis of weighted functions." src="/files/ftp_upload/14308/14308_Equation_3v2.svg" style="text-align: center; height: 35px;" data-altupdatedat="1754904545000" data-alt="Mathematical formula for double integration, equation: y = (1/F_H) ∫(∫w(x)dx)dx.">
Thus, the process helps us understand the cable's behavior when subjected to distributed loads. It is also useful in designing the bridge's support system and ensuring that the cables can handle the weight of traffic and other stresses.

The analysis of suspension bridges is a complex and critical process that involves multiple factors, including the shape and tension of the main cables. ...

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finding cable shape under distributed load

Cable shape under a distributed load is found by analyzing the forces along a small piece of the cable. This method is important in suspension bridge design because the main cables must carry changing loads safely. The load causes the tensile force in the cable to change, and it also creates cable deformation.

The analysis starts with a free-body diagram of a short cable segment. The diagram shows the distributed load and the change in tensile force across that segment. Equilibrium equations are then applied to the horizontal force, vertical force, and moments acting on the cable piece.

Next, each equation is divided by delta x, and the limit is taken as delta x approaches zero. This gives a new set of three equations. The first equation helps determine the horizontal component of tensile force at any point along the cable, and it can also be used to find the cable tension along its length.

The second equation is used to find the slope of the cable. It relates the vertical component of tensile force to the horizontal component. By substituting tan theta, the cable angle can be linked to the shape of the cable. A second integration of the final equation gives the curve of the cable. This analysis helps engineers understand cable behavior under distributed loads and design bridge support systems that can handle traffic weight and other stresses.

Finding Cable Shape Under Distributed Load

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key terms and concepts

Analyze how a suspension bridge cable responds to distributed loads using free-body diagrams, equilibrium equations, slope, tension, and cable curve.

Finding Cable Shape Under Distributed Load | JoVE K12

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Finding Cable Shape Under Distributed Load

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