-
Section 4 Structural Analysis and Evaluation (SI)
SPECIFICATIONS COMMENTARY
the cross-section resulting from eccentric loading or
unsymmetrical structure geometry. )
The provisions of Articles 4.6.2.1 and 4.6.3.2
influence surfaces, such as those by Homberg (1968)
and Pucher (1964), or other elastic analysis procedures
may be used to evaluate live load plus impact moment
effects in the top flange of the box section.
Transverse elastic and creep shortening due to
prestressing and shrinkage shall be considered in the
transverse analysis.
The effect of secondary moments due to
prestressing shall be included in stress calculations at
the service limit state and construction evaluation. At the
strength limit state, the secondary force effects induced
by prestressing, with a load factor of 1.0, shall be added
algebraically to the force effects due to factored dead
and live loads and other applicable loads.
The transverse design of beam-type segmental
bridge decks may be in accordance with the provisions
of Article 4.6.2.
4.6.2.9.5 Longitudinal Analysis
4.6.2.9.5a General
Longitudinal analysis of segmental concrete bridges
shall consider a specific construction method and
construction schedule as well as the time-related effects )
of concrete creep, shrinkage, and prestress losses.
The effect of secondary moments due to
prestressing shall be included in stress calculations at
the service limit state. At the strength limit state, the
secondary force effects induced by prestressing, with a
load factor of 1.0, shall be added algebraically to other
applicable factored loads.
4.6.2.9.5b Erection Analysis
Analysis of the structure during any construction
stage shall consider the construction load combinations,
stresses, and stability considerations specified in Article
5.14.2.3.
4.6.2.9.5c Analysis ofthe Final Structural System
The provisions ofArticle 5.14.2.2.3 shall apply.
4.6.3 Refined Methods of Analysis
4.6.3.1 GENERAL C4.6.3.1
Refined methods,listed in Article 4.4, may be used
for the analysis of bridges. In such analyses,
consideration shall be given to aspect ratios of elements, )
positioning and number of nades, and other features of
4 - 56
-
Section 4 Structural Analysis and Evaluation (SI)
SPECIFICATIONS COMMENTARY
topology that may affect the accuracy of the analytical
solution.
A structurally continuous railing, barrier, or median, This provision reflects the experimentally observed
acting compositely with the supporting components, may response of bridges. This source of stiffness has
be considered to be structurally active at service and traditionally been neglected but exists and may be
fatigue limit states. included, provided that full composite behavior is
assured.
When a refined method of analysis is used, a table These live load distribution coefficients should be
of live load distribution coefficients for extreme force provided for each combination of component and lane.
effects in each span shall be provided in the contract
documents to aid in permit issuance and rating of the
bridge.
4.6.3.2 DECKS
4.6.3.2.1 General C4.6.3.2.1
Unless otherwise specified, flexural and torsional In many salid decks, the wheel load-carrying
deformation of the deck shall be considered in the contribution of torsion is comparable to that of flexure.
analysis, but vertical shear deformation may be Large torsional moments exist in the end zones of
neglected. skewed gírder bridges due to differential deflection. In
Locations of flexural discontinuity through which most deck types, shear stresses are rather low, and their
shear may be transmitted should be modeled as hinges. contribution to vertical deflection is not significant. In-
In the analysis of decks that may crack and/or plane shear deformations, which gave rige to the concept
separate along element boundaries when loaded, of effective width for composite bridge decks, should not
Poisson's ratio may be neglected. The wheel loads shall be neglected.
be modeled as patch loads distributed ayer an area, as
) specified in Article 3.6.1.2.5, extended by half of the deck
depth on all tour sides.
4.6.3.2.2 Isotropic Plate Model C4.6.3.2.2
For the purpose of this section, bridge decks that are Analysis is rather insensitive to small deviations in
sol id, have uniform or clase to uniform depth, and whose constant depth, such as those due to superelevation,
stiffness is clase to equal in every in-plane direction shall crown, and haunches. In slightly cracked concrete slabs,
be considered isotropic. even a large difference in the reinforcement ratio will not
cause significant changes in load distribution.
The torsional stiffness of the deck may be estimated
using Equation C4.6.2.2.1-1 where b = unity.
4.6.3.2.3 Orthotropic Plate Model C4.6.3.2.3
In orthotropic plate modeling, the flexural rigidity of The accuracy of the orthotropic plate analysis is
the elements may be uniformly distributed along the sharply reduced for systems consisting of a small
'1. cross-sectionof the deck. Where the torsionalstiffness numberof elementssubjectedto concentratedloads.
of the deck is not contributed solely by a salid plate of
uniform thickness, the torsional rigidity should be
established by physical testing, three-dimensional
analysis, or generally accepted and verified
approximations.
4.6.3.3 BEAM-SLAB BRIDGES C4.6.3.3
The aspect ratio of finite elements and grid panels More restrictive limits for aspect ratio may be
should not exceed 5.0. Abrupt changes in size and/or specified for the software used.
shape of finite elements and grid panels should be In the absence of other information, the following
avoided. guidelines may be used at the discretion of the Engineer:
4 - 57
'"."é::t~-:c':'" ":~,;c,:i"&~'."':~;
:'
I
-
Section 4 Structural Analysis and Evaluation (SI)
SPECIFICATIONS COMMENTARY
Nodal loads shall be statically equivalent to the 8 A mínimum of five, and preferably nine, nades per '
actualloads being applied. beam span should be used. )
8 For finite element analyses involving plate and beam
elements, it is preferable to maintain the relative
vertical distances between various elements. If thís
is not possible, longitudinal and transverse elements
may be positioned at the midthickness of the plate-
bending elements, provided that the eccentricities
are included in the equivalent properties of those
sections that are composite.
8 For grid analysis or finite element and finite
difference analyses of live load, the slab shall be
assumed to be effective for stiffness in both positive
and negative flexure. In a filled or partially filled grid
system, composite section properties should be
used.
8 In finite element analysis, an element should have
membrane capability with discretization sufficient to
properly account for shear lag. The force effects so
computed should be applied to the appropriate
composite or noncomposite section for computing
resistance.
8 For longitudinal composite members in grid
analyses, stiffness should be computed by assuming )
a width of the slab to be effective, but it need not be
less than as specified in Article 4.6.2.6.
8 For K-frame and X-trame diaphragms, equivalent
beam flexure and shear stiffnesses should be
computed. For bridges with widely spaced
diaphragms, it may be desirable to use notional
transverse beam members to model the deck. The
number of such beams is to some extent
discretionary. The significance of shear lag in the
transverse beam-slab width as it relates to lateral
load distribution can be evaluated qualitatively by
varying the stiffness of the beam-slab elements
within reasonable limits and observing the results.
Such a sensitivity study often shows that this effect
is not significant.
\.
I 8 Live load force effects in diaphragms should be
calculated by the grid or finite element analysis. The
easiest way to establish extreme force effects is by
using influence surfaces analogous to those
developed for the main longitudinal members.
8 The Sto Venant torsional inertia may be determined
using the equation in Article C4.6.2.2.1.
Transformation of concrete and steel to a common
material should be on the basis of shear modulus, G, '
J
4 - 58
~5~~,?k~
Section 4 - Structural Analysis and Evaluation (SI)
SPECIFICATIONS COMMENTARY
,
I
which can be taken as G O.5E/(1+1.1). It is
recommended that the Sto Venant rigidity of
=
composite sections utilize only one-half of the
effective width of the flexural section, as described
above, before transformation.
4.6.3.4 CELLULAR AND BOX BRIDGES
A refined analysis of cellular bridges may be made
by any of the analytic methods specified in Article 4.4,
except the yield line method, which accounts for the two
dimensions seen in plan view and for the modeling of
boundary conditions. Models intended to quantify
torsional warping and/or transverse trame action should
be fully three-dimensional.
For single box cross-sections, the superstructure
may be analyzed as a spine beam for both flexural and
torsional effects. A steel box should not be considered
to be torsionally rigid unless internal bracing is provided
to maintain the box cross-section. The transverse
position of bearings shall be modeled.
4.6.3.5 TRUSS BRIDGES C4.6.3.5
A refined plane trame or space trame analysis shall Load applied to deck or floorbeams instead of to
include consideration for the following: truss joints will yield results that more completely
quantify out-of-plane actions.
) . Composite action with the deck or deck system; Experience has shown that dead load force effects
calculated using either plane trame or space trame
. Continuity among the components; analysis in a truss with properly cambered primary and
secondary members and detailed to minimize
. Force effects due to self-weight of components, eccentricity at joints, will be quite Glose to those
change in geometry due to deformation, and axial calculated by the conventional approximations. In many
I offset at panel points;and cases,a completethree-dimensionaltrame analysismay
be the only way to accurately calculate forces in
. In-plane and out-of-plane buckling of components secondary members, particularly live load force effects.
including original out-of-straightness, continuity
among the components and the effect axial forces
present in those components.
Out-of-plane buckling of the upper chords of pony
truss bridges shall be investigated. If the truss derives
its lateral stability from transverse trames, of which the
floorbeams are a part, the deformation of the floorbeams
I due to vehicularloadingshall be considered.
4.6.3.6 ARCH BRIDGES C4.6.3.6
The provisions of Article 4.6.3.5 shall apply where
applicable.
The effect of the extension of cable hangers shall be
considered in the analysis of an arch tie.
Where not controlled through proper detailing, rib Rib shortening and arch design and construction are
shortening should be investigated. discussed by Nettleton (1977).
The use of large deflection analysis of arches of Any single-step correction factor cannot be expected
longer spans should be considered in lieu of the moment to accurately model deflectioneffectsover a wide range
magnification correction as specified in Article4.5.3.2.2c. of stiffnesses.
4-59
;;¡;;J,~;:~" ",t";?;é'c'::c;;"'c",:~,~,;::,;~~;;;:;;~~~:r;¡".~~!:'~""i7f' ':)1
-
Section 4 Structural Analysis and Evaluation (SI)
SPECIFICATIONS COMMENTARY
If a hinge is provided at the crown of the rib in
addition to hinges at the abutment,the arch becomes )
statically determinate,and stresses due to change of
temperature and rib shortening are essentially
eliminated.
Arches may be analyzed, designed, and constructed
as hinged under dead load or portions of dead load and
as fixed at some hinged locations for the remaining
design loads.
When the distribution of stresses between the top In trussed arches, considerable latitude is available
and bottomchords of trussed arches is dependent on in design for distribution of stresses between the top and
the manner of erection, the manner of erection shall be bottom chords dependent on the manner of erection. In
indicated in the contract documents. such cases, the manner of erection should be indicated
in the contract documents.
4.6.3.7 CABLE-STAYED BRIDGES C4.6.3.7
The distribution of force effects to the components of Nonlinear effects on cable-stayed bridges are treated
a cable-stayed bridge may be determined by either in several texts,e.g., Podolny and Scalzi (1976), Troitsky
spatial or planar structural analysis if justified by (1977), and a report by the ASCE Committee on Cable
consideration of tower geometry, number of planes of Suspended Bridges (ASCE 1991), from which the
stays, and the torsional stiffness of the deck particular forms of Equations 1 and 2 were taken.
superstructure.
Cable-stayed bridges shall be investigated for
nonlinear effects that may result from:
. The change in cable gag at alllimit states,
,
. Deformation of deck superstructure and towers at all
limit states, and
. Material nonlinearity at the extreme event limit
states.
Cable gag may be investigated using an equivalent
member modeled as a chord with modified modulus of
elasticity given by Equation 1 for instantaneous stiffness
and Equation 2, applied iteratively, for changing cable
loads.
]-1 (4.6.3.7-1)
E = E [1 + EAW2(COSO)5
MOD 12H3
\
- [ (H1 + H2) EAW2(COSO)5 j -1
EMOD - E 1 + (4.6.3.7-2)
24 H21 H2
2
where:
E = modulus of elasticity of the cable (MPa)
W = total weight of cable (N) I
I 4-60
-
Section 4 Structural Analysis and Evaluation (SI)
¡
I
SPECIFICATIONS COMMENTARY
) A = cross-sectional area of cable (mm2)
a = angle between cable and horizontal
(DEG)
H, H1' H2 = horizontal componentof cable force (N)
The change in force effects due to deflection may be
investigated using any method that satisfies the
provisions of Article 4.5.3.2.1 and accounts for the
change in orientation of the ends of cable stays.
Cable-stayed bridges shall be investigated for the
1055of any one cable stay.
4.6.3.8 SUSPENSION BRIDGES C4.6.3.8
Force effects in suspension bridges shall be In the past, short suspension bridges have been
analyzed by the large deflection theory for vertical loads. analyzed by conventional small deflection theories.
The effects of wind loads shall be analyzed, with Correction factor methods have been used on short- to
consideration of the tension stiffening of the cables. The moderate-span bridges to account for the effect of
torsional rigidity of the deck may be neglected in deflection, which is especially significant for calculating
assigning forces to cables, suspenders, and components deck system moments. Any contemporary suspension
of stiffening trusses. bridge would have a span such that the large deflection
theory should be used. Suitable computer programs are
commercially available. Therefore, there is little rationale
to use anything other than the large deflection solution.
For the same economic reasons, the span would
) probably be long enough that the influence of the
torsional rigidity of the deck, combined with the relatively
small effect of live load compared to dead load, will make
the simple sum-of-moments technique suitable to assign
loads to the cables and suspenders and usually even to
the deck system, e.g., a stiffening truss.
4.6.4 Redistribution of Negative Moments in
Continuous Beam Bridges
4.6.4.1 GENERAL
The Owner may permit the redistribution of force
effects in multispan, multibeam, or girder
superstructures. Inelastic behavior shall be restricted to
the flexure of beams or girders, and inelastic behavior
due to shear and/or uncontrolled buckling shall not be
permitted. Redistribution of loads shall not be
considered in the transverse direction.
The reduction of negative moments ayer the internal
supports due to the redistribution shall be accompanied
by a commensurate increase in the positive moments in
the spans.
,
,
!
i
I
4 - 61
