Introduction and The General Cable Theorem - Civil Engineering (CE) PDF Download

Introduction 

Cables are flexible wire-like systems having no flexural (bending) stiffness, and they can carry only axial tension and no other type of force. Being fully flexible against bending the shape of a cable is determined by the external forces that are acting on the cable. Figure 3.1 illustrates how the shape of the cable between two supports A and B depends on the location and magnitude of the external forces P₁ and P₂ .

Figure 3.1 Shape of a cable is determined by external loads

A cable is unable to carry bending moment, shear force, torsion or axial compression. Nevertheless, cables can be very effectively used in achieving long-span light-weight systems, such as bridges or roofs for large arenas. Two kinds of bridge structural systems where cables are used are the suspension-cable systems and cablestayed systems . Figures 3.2 and 3.3 show examples of suspension-cable bridge and cable-stayed bridge, respectively.

Figure 3.2 A suspension-cable bridge (Golden gate bridge, San Francisco , USA)

Figure 3.3 A cable-stayed bridge (ANZAC bridge, Sydney , Australia )

Cables are usually made of multiple strands of cold-drawn high-strength steel wires twisted together. Generally, they have strength four to five times that of structural steel and practically inextensible under operating loading conditions. Since cables carry only axial tension, full potential of the cable cross-section can be utilized in transferring forces. Therefore, cables are able to carry the same amount of force with a much smaller cross-section compared to other structural systems. This high strength-to-weight ratio makes cables very useful where light-weight systems are needed. On the other hand, a beam over a very long span would require a very large (and deep) cross-section, and most of its potential will be used in carrying internal forces due to its own weight. If we use cables replacing this beam or in combination with a beam in stead, a lighter structure will be required, whose self-weight will not add significantly to load effects.

The primary disadvantage with cables is due to their flexible geometry. As the loading on a cable system changes (as in the case of moving loads on a bridge) there can also be large change in the cable geometry, and subsequently on forces acting in the cable. Unexpected forces may destabilize a cable system, causing excessive deformations. A designer should be very careful on this regard while designing a cable system, along with other issues such as, large forces at the anchors, large oscillations, etc.

The General Cable Theorem 

The general cable theorem helps us determine the shape of a cable supported at two ends when it is acted upon by vertical forces. It can be stated as: “At any point on a cable acted upon by vertical loads,the product of the horizontal component of cable tension and the vertical distance from that point to the cable chord equals the moment which would occur at that section if the loads carried by the cable were acting on an simply-supported beam of the same span as that of the cable.” 

Figure 3.4 Explanation of the general cable theorem: (a) Cable under vertical loads, and (b) Simply supported beam with equal span under the same set of loads

To explain, let us consider the cable AB in Figure 3.4a, which is acted upon by the vertical loads P₁, P₂, P₃ and P₄ at known locations.The line AB joining the two supports is known as the chord of the cable and the horizontal distance between the supports is known as its span .The vertical distance between the chord and the cable at any cross section is known as the dip .This is vertical distance that is mentioned in the general cable theorem .The cable in Figure 3.4a has a span L and the dip at a distance x from A is y.The horizontal reactions at supports A and B have to be equal to satisfy static equilibrium,and let it be H.The vertical reactions at supports A and B are A_y and B_y respectively. Figure 3.4b shows a simply-supported beam AB of same span ( L ) and acted upon by the same set of forces as the cable AB in Figure 3.4a.

For moment equilibrium about support B for the cable:

A_y L+ HL tan α= ∑M_BP                     (3.1)

where,∑M _BP is the summation of moments due to external forces ( P₁, P₂, P₃ and P₄) about point B Since the cable is totally flexible against bending, bending moment at any cross-section is zero. By equating bending moment at a distance x from A to zero, we get:

A_y x + Hz =∑M_xp                    (3.2a)

or,

A_y x + H(xtan α -y) = ∑M_xp                   (3.2b)

where, ∑M_XP is the summation of moments due to external forces ( P₁, P₂ and P₃  to the left of x) about section x .Substituting A_yx from Equations 3.1 and 3.2b:

                    (3.3)

Now, let us consider the simply-supported beam in Figure 3.4b. From moment equilibrium about support B ,we get the vertical reaction at support A :

                    (3.4)

So, the bending moment at a distance x from A is:

                    (3.5)

which, is same as the right side of Equation 3.3. Therefore:

H_y= Moment at x for the simply-supported beam                    (3.6)

which is the claim as per the general cable theorem .

Note that the horizontal component of the axial force at any section of a cable (under vertical external forces only) is same as the horizontal reaction (H) at the end supports. This can be proved considering the equilibrium of horizontal forces on any segment of the cable.

We can solve internal forces in a cable using the general cable theorem, and also we can obtain for the shape of the cable. If the cable length (not the span) is known to us, we can express this length in terms of the dip y .Using this information along with the general cable theorem we can solve for both the unknowns H and y .Alternatively, the dip at a certain point, instead of the total length of the cable, may be known to us. This information, along with the general cable theorem helps us solve for both H and y.