Approximate and refined analysis methods - Computational Analysis and Design of Bridge Structures - page 55

where:
M is bending moment in stiffening girder
M ′ is bending moment in the unsuspended girder for the live loads
( h = 0)
h is horizontal tension in cable
y is cable sag

This theory is only used in a preliminary design for estimating cable quantities.
As a consequence of large displacements of long suspension spans, the elastic
theory results in underestimated moments, shears, and deflections. A deflec-
tion theory is then developed and referred to as second-order theory with the
expression of

M M hy H h

= ′ − − ( + ) ν

(2.16)

where:
ν is cable deflection under loads

As displacements affect structural geometry, Equation 2.16 is not linear,
and linear superposition technically is not applicable. There would be diffi-
culty in using the influence line concept. For this type of analysis, programs
that can handle large deflection and material nonlinearity should be used.
Large deflection analysis is necessary for structures, such as suspension
bridge, that undergo a large translation and rotation, and where their load-
carrying path is altered as the load is increased. The nonlinear procedure
for the suspension bridge is tedious and time consuming. With simplifica-
tion to a quasi-linear theory, an average value of H ( H max and H min ) may
be used as a basis of linearized influence line as in the case of first-order
theory. There may be two sets of influence line generated, one by H max and
another by H min , to establish the most critical live load effect. More detailed
procedures to handle the nonlinearity in computation with modern tech-
nology, especially on live load, will be deferred to Chapter 12, which is
designated for suspension bridges.