4.7—ANALYSIS OF SPAN-WIRE STRUCTURES C4.7 The analysis of span-wire structures shall be performed

using methods based on commonly accepted principles of mechanics. Tensions and deflections of span wires shall satisfy equilibrium and compatibility. Loads applied to span-wire structures shall be computed according to Section 3,

“Loads.” Gravity and wind loads induced by attachments (i.e., signs, signals, and accessories) may be applied as equivalent point loads. Gravity and wind loads induced by the span wire may be applied as a series of concentrated loads.

Refined analytical methods based on large deflection theory or finite element formulations should be considered in the analysis of complex span-wire configurations, because the behavior of these structures may not be adequately explained using small deflection theory.

Because of the nonlinear relationship between geometry and forces in span wires, superposition principles should not be applied to combine the effects of different loads.

Therefore, the analysis should be performed considering a single load case with all loads acting simultaneously.

Appendix A provides two methods to compute tensions on span wires. The simplified method, outlined in Appendix A, is intended to consider the case of rigid vertical supports. The detailed method outlined in the same appendix is intended to consider the case of flexible vertical supports.

4.8—SECOND-ORDER EFFECTS C4.8

For vertical cantilever support structures (pole-type), the secondary bending moment caused by the axial load shall be accounted for by provisions of Article 4.8.1, unless a more detailed calculation is made in accordance with Article 4.8.2.

When a member is subjected to axial compressive stresses acting simultaneously with bending stresses, a second-order moment equal to the product of the resulting eccentricity times the applied axial load is generated. To account for this effect on vertical pole-type members, two methods of evaluating bending stresses are presented: the simplified method outlined in Article 4.8.1, and the detailed method outlined in Article 4.8.2.

The simplified method is intended primarily for hand computations, whereas the detailed method is intended for situations where a refined analysis is desired and a computer is available. The simplified method is conservative with respect to the detailed method.

4-4 STANDARD SPECIFICATIONS FOR STRUCTURAL SUPPORTS FOR HIGHWAY SIGNS,LUMINAIRES, AND TRAFFIC SIGNALS

4.8.1—Simplified Method C4.8.1

In the combined stress ratio equations for steel and aluminum (Eq. 5-16 in Article 5.12.1 and Eq. 6-30 in Article 6.7.1), the bending stress fb shall be divided by the coefficient for amplification CA to account for the secondary moment. The coefficient for amplification CA shall be taken as:

The coefficient for amplification CA was included in the Specifications to be used mainly for vertical cantilever supports over 15 m (49.2 ft) in height or where other conditions are such that secondary P-delta effects are significant. It is derived based on linear structural analysis.

Previous editions of the Specifications included the following equation to compute the coefficient CA:

3

This equation provided an overestimation of the P-delta effect for certain combinations of lateral and vertical loads.

Eq. 4-1 presents a revised equation to compute the coefficient of amplification CA where the term 1/0.52 was eliminated (Fouad et al., 1998). The revised equation improves the accuracy of the coefficient of amplification CA

to estimate the second-order effects on steel and aluminum vertical cantilever supports.

Eq. 4-1 is limited to values of kL/r greater than or equal to:

2π2 y

E F

to ensure that the maximum allowable axial stress Fa is limited to 0.26Fy, where Fy is the specified minimum yield strength. This requirement is intended to keep the axial stresses sufficiently low such that effects of residual stresses on the buckling behavior of the pole can be ignored. The radius of gyration r may be calculated at a distance of 0.50L for a tapered column.

4.8.2—Detailed Method C4.8.2 In lieu of the approximate procedure of Article 4.8.1, a

detailed second-order elastic analysis that is applicable to all materials covered by the Specifications may be performed considering the final deflected position of the vertical support. In using that approach, the following procedure shall be used: Multiply the applied loads of a specific group load combination by 1.45.

Perform a second-order structural analysis using these loads. This analysis will provide amplified bending moments, and hence bending stresses, that account for the combined effects of axial loads and deflected shape of the structure. For the case of prestressed concrete structures, the effects of cracking and reinforcement on the member stiffness shall be taken into consideration.

The magnified forces resulting from analysis (i.e., bending moment, axial force, torsional moment, and shear force) shall be divided by 1.45 for use with the combined stress ratio equations of the Specifications.

As an alternative procedure, a more exact method of analysis is presented, whereby the member is analyzed considering the actual deflected shape of the structure. With this method, the coefficient of amplification CAis taken as 1.0 because the secondary moment is already considered in the analysis. This method implies a nonlinear relationship between the applied loads and the resulting deflections.

Therefore, no superposition of results should be performed;

all dead, wind, and ice loads from a group loading should be considered as being applied simultaneously in a single load case.

Because the second-order analysis is nonlinear in nature, loads computed according to Section 3, “Loads,” are increased by 1.45 to provide the same safety factor as that of the linear elastic analysis. The nonlinear analysis is then performed with the magnified loads. Forces and moments resulting from the analysis are divided by 1.45 to bring the forces into the working load range for computing the actual stresses in the vertical support.

The 1.45 is the inherent safety factor for the case of Group II or III loads. It is calculated as the ratio of the ultimate flexural strength of the cross-section divided by the allowable moment, after being multiplied by the allowed stress increase of 1.33. The safety factor may, therefore, be derived as follows:

1.925

where 1.925 is the safety factor against failure that is used to determine the allowable stress for the case of dead loads only (i.e., Group I loads).

The member is analyzed in its final (stable) deflected position considering all axial and lateral loads. The analysis is nonlinear because of the nonlinear relationship that results from the combined effects of the loads and resulting deflections on the final deflected position. The final secondary moment is the product of the axial loads and their eccentricities, and the eccentricities are dependent on the horizontal deflections produced by the wind loads and the eccentric axial loads. Because these interactions result in a nonlinear analysis, the loads have to be factored, and the iterative analysis performed to achieve stable equilibrium under these loads, to ensure that the prescribed safety factor has been furnished for all axial loads and lateral loads. The analysis should be performed with all axial loads (including pole weight) and the wind loads increased by 1.45 for Group II and III load combinations. Results from the analysis may then be divided by 1.45 to obtain the working forces and stresses to be used in the combined stress ratio equations.

4-6 STANDARD SPECIFICATIONS FOR STRUCTURAL SUPPORTS FOR HIGHWAY SIGNS,LUMINAIRES, AND TRAFFIC SIGNALS

4.9—REFERENCES

Fouad, F. H., E. A. Calvert, and E. Nunez. 1998. Structural Supports for Highway Signs, Luminaires, and Traffic Signals, NCHRP Report 411. Transportation Research Board, National Research Council, Washington, DC.

United States Steel Corp. 1973. Axial Buckling Loads of Steel Lighting Standards. Applied Research Laboratory, United States Steel Corporation. Pittsburgh, PA, February 16, 1973. Bulletin.