CHAPTER II
LITERATURE REVIEW
2.1
Introduction
Rapid development in construction resulting a construction of various types
of bridge. The advancement in material science lead to new materials used for
constructing bridges. Hence, the effects of the wind on the bridge structures should
be determined. Unlike dead loads, which are permanent and changing slowly, live
loads and wind loads change rapidly and creating effects much larger than if the
same loads applied gradually for a suspension bridge structure. Therefore, there is a
need to understand the nature of wind and its effects on the bridge structural system.
In designing bridge to withstand wind forces, several important factors must
be considered such as the strength and stability requirements of structural system,
buffeting wind effects and resonance of the bridge oscillations. The conventional
methods to obtain the wind effects data are acquired through wind tunnel test. The
development of computer applications in finite elements analysis enable simulation
of wind flows effects that usually done in wind tunnel tests.
11
2.2
Winds
Wind is term use for air in motion and is usually applied to the natural
horizontal motion of the atmosphere. Winds are produced by differences in
atmospheric pressure, which are primarily attributable to differences in temperature.
In fact, warmer air has less density and thus rises, whereas cold air falls with respect
to the adjacent air. In addition to such vertical motion, there is a horizontal one that is
due to the movement of cold, dense air toward zones of lower pressure and warmer
air.
Winds that are of interest in the design can be classified into three major
types: the prevailing winds, seasonal winds and local winds. The characteristics of
prevailing winds and seasonal winds are analytically studied together, while local
winds are studied separately. The grouping is on account of widely differing scale of
fluctuations period.
2.2.1
Types of Wind Flows
Flow of wind, unlike other fluids, is not steady and fluctuates in random
fashion. Winds are classified into steady, unsteady, laminar and turbulent (eddying)
forms.
Steady winds have the steady mean incident flow and constant in velocity. An
unsteady wind flow is one in which the incident mean stream velocity is fluctuating
in some way, either with time or in direction.
12
The flow that is smooth and regular is associated with name laminar. In
contrast, individuals particles and groups of particles spinning and rotating and
moving first in one direction then in another with no order except that the whole
aggregate is proceeding in the streaming direction is described as turbulent. The
change from laminar to turbulent flow is of relevance to the aerodynamics forces and
it is governed by three factors:
i)
Reynolds Number
ii)
Turbulent of the Wind
iii)
Roughness of the surface
At low value of Reynolds number, the viscous forces are large compared to
the acceleration of the air. Viscous forces always oppose motion and hence random
motions of the air are damped out and laminar flow persists. At some critical value of
Reynolds number the damping force becomes insufficient to damp out oscillations of
the flows and turbulent flows occurred.
2.3
Characteristics of Wind
Wind is a phenomenon of great complexity because of the many flow
situations arising from the interaction of wind structures. Wind engineering
simplifications are made to arrive at meaningful predictions of wind behaviour by
distinguishing the characteristics of wind.
13
2.3.1
Variation of Wind Velocity with Height
The variation of velocity with height can be considered as a gradual
retardation of the wind speed nearer the ground as a result of surface frictions. The
velocity increase is curvilinear varying from zero at the ground surface to a
maximum at some distance above the ground. The height at which the velocity
ceases to increase is called the gradient height and the corresponding velocity, the
gradient velocity.
The wind speed profile within the atmospheric boundary level can be
expressed by mathematical relationships based on fundamental equations of
continuum mechanics. The mean wind profile in horizontally homogeneous terrain
was proposed by Hellman in 1916 and is called the power law, which is expressed as
 Z
VZ  Vg 
Z
 g




α
(Eq. 2.1)
where
VZ = mean wind velocity at height z above the ground
Vg = gradient wind velocity at any reference height Zg above ground
Z = height above ground
Zg = depth of boundary layer
 = power law exponent coefficient
14
2.3.2
Turbulent Nature of Wind
The motion of wind is turbulent. Any movement of air at speeds greater than
0.9 to 1.3 m/s is turbulent, causing particles of air to move randomly in all directions.
The scale and intensity of turbulences can be likened to the size and rotating speed of
eddies or vortices that make up the turbulence. The properties of wind are studied
statistically. The velocity of wind can be considered as having two components: a
mean velocity whose value increases with height and turbulent velocity fluctuations.
Spectral analysis techniques provide a convenient method for dealing with random
turbulences of wind.
2.3.3
Probabilistic Approach to Wind Load
The intensity and speed of wind is considered as a function of duration
recurrence (return period). A 50-year return period of 30 m/s means that the area will
experience a wind faster than 30 m/s within a period of 50 years. The probability of
occurrence of the speed exceeding 30 m/s within a given year is 1/50 (0.02 or 2
percent). However, there is still high probability of wind exceeding the design speed
within the service lifetime of the structure.
2.3.4
Dynamics Nature of Wind
The gust response factor is a measure of energy transfer from wind to
structures it hits. Wind loads associated with gustiness change rapidly and creating
effects much larger than if the same loads were applied gradually. A structure could
15
have significant dynamics amplification of response to wind depend on how the gust
frequency correlates with the nature frequency of the structure and also on the size of
the gust in relation to building size.
2.3.5
Vortex Theory
When a steady wind blows perpendicularly across the width of bridge deck
with slender bluff, a zone of turbulent fluid flow, called wake, whose nature depends
on the Reynolds number, is created on the leeward side or past the trailing edge of
the airfoil. Vortices shed from the top and bottom bridge deck with a define
periodicity. This oscillating streamline pattern caused by alternate vortex shedding
cause a fluctuating pressure on the bridge deck that is dynamic in nature. The
frequency of the fluctuating force is equal to the frequency of the vortex shedding
and is proportional to the wind velocity.
According to Strouhal, the vortex-shedding phenomenon can be described in
terms of a non-dimensional parameter, called the Strouhal number, defined as
S
NsD
U
(Eq. 2.2)
where
S = Strouhal number
NS = the frequency of full cycles of vortex shedding
D = the characteristic dimension of the body projected on a plane
normal to the mean flow velocity
U = Velocity of oncoming flow, assumed laminar
16
The mechanism of vortex shedding of bridge deck is similar to that for an
airfoil. The fluctuating pattern of wakes exerts a vertical dynamic force on bridge
deck. The bridge will vibrate in a resonant manner with large amplitudes when the
wind velocity is such that the frequency of vortex shedding approaches that of the
natural frequency of the bridge.
2.3.6
Flutter Theory
Flutter refers to an oscillating motion in which two or more modes of
oscillation, usually bending and torsion are combined. As wind velocity increases, a
critical value is reached, which triggers the flutter motion. It is characterised by a
rapid build-up of amplitude with little or no further increases in wind speed. The
amplitude may reach catastrophic proportions in a few cycles of motion.
The bridge design should be such that the critical velocity at which this
motion occurs is high, and it should be ensured that wind speeds of this magnitude
will not occurs at the bridge site. In addition to aerodynamic factors, the flutter
velocity depends on the elastic and dynamic properties of the bridge.
2.3.7
Velocity Pressure or Dynamics Pressure
The velocity pressure or dynamic pressure, q is the pressure generated by the
wind on a building by the wind assuming that the moving air comes to a complete
stop as it hits the structure. Therefore, velocity pressure is the total kinetic energy
possessed by the air that is transferred from the air to the structure.
17
However, the real pressure at any given point will be lesser than the
theoretical value of q on account of the reality of air deviates from its original path as
it approaches and passed over structure, transferring only part of its energy to the
structure. Hence, the dynamic pressure represents the total kinetic energy of the air
that strikes the structure.
Considering the P and V is the pressure and velocity respectively, at a point
from a distance from structure where air is undisturbed and at the point on the
structure stricken by the wind, P1 and V1. ρ is the density of air. The expression for
dynamic pressure is derived from the Bernoulli equation.
P
1
1
2
ρV 2  P1  ρV1
2
2
(Eq. 2.3)
Assuming that air stops completely when hits structure, V1=0 and (P-P1)=q,
the Equation 2.3 becomes
q
2.3.8
1
ρV 2
2
(Eq. 2.4)
Wind Pressure or Static Velocity Pressure
The real wind pressure, p at any point given point of a structure is the product
of dynamic pressure, q with a coefficient of shape factor, Cp. Thus,
p  Cpq
(Eq. 2.5)
18
2.3.9
Wind Force or Drag Force
Consider as a whole, the total wind force or drag force, F on the structure can
be determined by two methods. In theory, such force can be calculated by summing
vector all the individual pressure forces. In practice, it is computed as the product of
the projected surface area to wind direction, A, the dynamic pressure q and the shape
factor CF. Thus,
F  qAC F
(Eq. 2.6)
The difference between Cp and CF is that Cp refers to a single point while CF
refers to the whole structure. An overall shape factor, CF can be obtained in a wind
tunnel tests and it can divide into two basic components, which are lift coefficient,
CL and drag coefficient, CD.
2.3.10 Pressure Coefficient, CP
Pressure coefficients, CP determined experimentally by undergoing the wind
tunnel tests. Gauges are placed at the point where the pressure is to be measured on
the structure model and shape coefficients can be derived from the readings.
The coefficients are dimensionless. Positive values are for positive pressure
while negative values indicate suction. The coefficient CP depends basically on the
shape or geometrical characteristics of the structure and at times on the Reynolds
number, Re.
19
Apart from wind tunnel testing, large number of pressure coefficients for
variety of building shapes can be obtained by gathering the research works
conducted by Ackeret (1936) from Institute of Aerodynamics, Switzerland. His
extensive works were incorporated into Swiss Building Code and American Society
of Civil Engineers (ASCE).
2.3.11 Reynolds Number (Re)
The Reynolds number, Re is the dimensionless quantity as a fundamental
parameter in aerodynamics for determining the shape factors of structures and
consequently the wind pressure.
The Reynolds number is defined as the ratio of the internal force to the
viscous drag force. The former is the obstructing force generated by the frontal
impact as the air collides against an obstacle. The latter is instead the force developed
by the viscosity existing between air strata just like friction between solid lamina.
Hence,
V 2 h 2 Vh
Re 

Vh

where
ρ = air density
V = Wind Velocity
h = diameter or width of structure
μ = air viscosity
(Eq. 2.7)
20
Considering  

, then Equation 2.7 becomes

Re 
2.4
Vh

(Eq. 2.8)
Wind Loads for Design
Wind effects on structures include the direct application of wind-induced
forces, movement of the structure, and the flow of the wind around the structure,
which may affect pedestrians or the function of the building. Normally, wind effects
are grouped according to limit states and safety and serviceability considerations.
The selection of structural systems based on their ability to resist wind-induced
stresses with appropriate margins of safety is an example of a design for safety or an
ultimate limit state. Limiting deflections caused by the wind loads to prevent
excessive vibrations and limiting the motion of the structure to prevent occupant
discomfort are examples of serviceability limit state design.
The design of bridge structures is based on estimates of overall wind effects,
which must be taken into account in the structural systems, local wind effects and
type of usage, which govern the design of components.
Aeroelastic and aerodynamics information needed to estimate overall as well
as local wind effects cannot be determined from first principles and must be obtained
from wind tunnel tests. However, in the case of bridge structure that have geometric
shapes that are not unusual aerodynamically or structurally, the information can be
incorporated with existing information. The existing information provides in the
21
design Codes of Practice is based upon the extensive wind tunnel tests on similar
structures.
Wind flowing past a body must be diverted from its original path and such
deflections lead to changes in the speed of the air. The pressure exerted by the air
body on the body is thereby altered from that of the undisturbed stream. The
viscosity of the air leads to frictional forces tending locally to reduce the wind speed.
As the results of these processes, the body experience an aerodynamic force and
moment. The wind force on a body may be resolved into three components, which
are normal drag forces, transverse forces and uplift.
The normal force or drag force is the total force tending to drag the body in
the direction of the wind. The total drag may be separated into a number of
components, which have different mechanisms such as surface friction drag and
normal pressure drag. Surface friction drag arises from the resolved components of
the tangential stresses on the surface of the body. This component of drag is directly
attributable to viscosity.
The large transverse force may be obtained from a small deviation in wind
direction. The transverse force arises directly from and is proportional to the rotation
induced in the flow-streaming past the body by the shape of the body offers to the
flow.
Some conclusions of a critical study on standard provisions developed for
wind loads on structures indicated that standard provision for wind loads tend to
overestimate peak wind loads on the structural systems. The poor respect to risk
consistency arises from the conventional method.
22
The shortcoming of the standard provisions are attributed to the fact that the
specified wind load induces stresses governing the design that may differ from those
induced by the actual load, which depend on winds speed, gravity load and type of
structural system.
Wind loads depend on the wind flow and on the aerodynamics and structural
properties of the structure being designed. The wind effects which should be
considered in the design of a particular structure vary depending on the following
factors:
2.5
i)
Basic wind speed
ii)
Mean recurrence of wind speed
iii)
The characteristics of the terrain surrounding the structure
iv)
Directional properties of the wind climate
v)
Dynamic effects and aerodynamics instabilities of the structure
vi)
Structural properties of the structural system
vii)
Position of the areas acted on by the wind flow
viii)
Pressure coefficients and load factors
Wind Loads on Bridges
The inherent static strength of heavy masonry structures to resists wind forces
were not considered dangerous until major failures began to occur in slender bridges.
During 1870s, bridge collapses, caused mainly by poor lateral resistance against
wind loads, reached the severe rate per year in United States and Europe.
The collapse of Dryburgh Abbey Suspension Bridge in Scotland in 1818 was
caused by aerodynamic instability. The Brighton Chain Pier Bridge, in Brighton,
23
England, crashed in 1836 due to dynamic wind forces similar to those that destroyed
the Tacoma Narrows Bridge in Washington in 1940. The other major disaster due to
wind loads is the failure of Tay Bridge in Scotland in 1879. Investigation has
indicated that the weakness of bridge is its low resistance to wind forces. Another
catastrophic bridge disaster was the failure of the Quebec Bridge in Canada- a huge
3300 feet long bridge, on August 29, 1907. The bridge was suddenly collapsed
during construction and taking with it more than 18000 tons of steel and 86 men
working on the bridge.
The famous disaster in suspension bridges history is the collapse of Tacoma
Narrows Bridge at Washington, as shown in Figure 2.1. On November 7, 1940, it
collapsed under action of a moderate 35-42 mph wind (equivalent to 5 lb/ft2 of static
wind forces) that caused excessive vibration and twisting of the deck, as shown in
Figure 1.2. Studies attributed the bridge failure to excessive oscillation caused by
wind action, which were made possible by the extraordinary degree of flexibility of
the structure and its relatively small capacity to absorb dynamic forces.
Figure 2.1: Collapse of Tacoma Narrows Bridge due to Wind Loads
24
Engineers learned valuable lessons from past bridge failures. However, it was
not until the collapse of the Tacoma Narrows Bridge in 1940 that full scientific
attention was given to wind effect problems on the bridge, especially the flexible
type of suspension bridges.
Many researches has been done on the wind loads on the bridges because
they are frequently built on exposed sites and subjected to very severe wind
conditions. The long span and lightweight construction of suspension bridges has
made these bridges particularly susceptible to wind induced oscillations.
Wind induced oscillations are caused either by aerodynamic instability or by
buffeting. The oscillations caused by aerodynamic instability are set up in a steady
wind by the interaction of the structure with the air stream.
Because the oscillations induced depend on the bridge characteristics, they
can be reduced in magnitude by changing the bridge shape, frequency or damping.
National Physical Laboratory [Frazer and Structon, 1971] have drawn some
conclusions about the effects of dynamics of bridge to the wind induced forces.
Among the conclusions is truss-stiffened bridges do not suffer from vertical bending
oscillations. Torsional oscillations can be reduced by incorporating features, which
break up the continuity of the flow pattern.
2.6
Bridges
A bridge is a structure providing passage over an obstacle without closing the
way beneath. The required passage may be for a road, a railway, pedestrian, a canal
or a pipeline. The obstacle to be crossed may be a river, a road, railway or a valley.
25
Bridges had been constructed over thousand of years. The history of development of
bridge can be linked with the history of the evolution of civil engineering.
The maximum span that can be bridged depends on the structural form
adopted for a particular bridge. Beginning with the Brooklyn Bridge (completed in
1883), bridges with increasing span lengths have been built over the world. With the
development of new materials and construction techniques, the emergence of great
arch, suspension and cable stayed bridges have been built. These stand as symbols of
civil engineering triumph.
2.6.1
Types of Bridges
Rapid development in construction resulted various types of bridge. Bridges
can be characterised into several ways depending on the objective of classification.
Bridges are always classified according to the characteristics such as materials of
construction, span length, span types, usage, load path characteristics, structural
forms, deck type and position.
Bridges can be identified by the materials used for construction their
superstructures, namely timber, concrete and steel. Often, a combination of materials
is used in bridge building. Recently, a new breed of materials called advanced
composites materials has been developed for structural applications in bridges.
Medium and long span bridges, such as cable stayed and suspension bridges, is
usually consists of fully steel structures.
Bridges can be classified by their structural form from an engineering
perspective in the reason of the methods of analysis in depend on the structural form.
26
Structural form refers to the load resisting mechanism of a bridge by which it
transfers gravitational and lateral loads from the deck to the foundation. It can be
classified into slab-stringer bridges, orthotropic bridges, truss bridges, rigid frame
bridges, arch bridges, cantilever bridges, cable stayed bridges and suspension
bridges.
2.6.2
Suspension Bridge
Suspension bridges are recognised for spanning the longest distances. It
consists of four essentials parts: the towers, the anchorages, the cables, and the deck.
The deck of the bridges are supported by stiffen trusses or girders and hang from
suspension cables or hangers. The anchorages at ends secured the suspension cables,
which are usually built of masonry or concrete. Suspension bridges can classify by
the type of cable anchorage, which is external and internal anchorage.
Conventional suspension bridges consist of two parallel cables as the main
load-carrying element. Loads from the bridge superstructure are transferred to the
main cable through a continuous pattern of diagonal suspenders or hangers.
Railroads suspension bridges always suffer the excessive deflection
problems. The vertical component of tension in the side span helps reduce the dead
load reaction at the end span support; the horizontal component of tension creates
compression in the stiffening trusses or girders.
27
2.7
Loads on Bridges
Bridge structures must be designed to resist various types of loads: gravity as
well as lateral. Generally, the major components of loads acting on bridges are dead
and live loads, environmental loads (temperature, wind, earthquake) and other loads
arising from braking of vehicles and collisions. Gravity loads are caused by the dead
weight of the bridge itself, the superimposed dead load, and the live load, whereas
the lateral loads are caused by the environmental phenomenon such as wind and
earthquake.
Bridges structures are subjected to dynamic loads of moving vehicles besides
the static loads. In the case the bridges built over waterways, the bridges are
subjected to earth pressures, water pressure, and stream flow pressure and ice
pressure at substructures. Two major components of bridge are substructure and
superstructure. In design purposes, the forces acting on the bridge can be divided into
loads acting on superstructure and loads acting on substructure.
2.7.1
Gravity Loads
Gravity loads are caused by the deadweight of the bridge itself, the
superimposed dead load and the live load. The dead load on a bridge structure
consists of the self-weight of the superstructure, substructure and the weight of other
items such as utilities services. An important consideration in dead load
computations is to include anticipated future wearing surface, widening of the
roadway for additional traffic lanes and additional utilities that the bridge may have
to carry.
28
For bridges, live loads refer to the loads due to moving vehicles and
pedestrian those are dynamic. The live loads of bridges are complex due to the
dynamics movements of loadings and the effect of the live load on a bridge is a
function of several parameters. The effects of dynamic live loads are much greater
compared to static live loads.
2.7.2
Impact
Impact loading can be defined as a suddenly applied load. For bridges, the
phenomenon of impact is related to the bridge-vehicle interaction. The interaction of
moving loads and the bridge superstructure results in dynamic amplification of the
moving loads, resulting in vibrations and increased stresses. This dynamic response
is considered in design, according to American Association of State Highway and
Transportation Officials (AASHTO) specifications, by ascribing impact factor (I), in
terms of static equivalent of dynamic and vibratory effects. Consideration of the
dynamic responses is important for several reasons:
i)
Stresses are increased above those due to static-load applications, a cause
of primary concern in bridge design.
ii)
Excessive vibration may cause, in the minds of vehicle occupants, a
psychological fear of driving over an unsound bridge.
iii)
Excessive deck vibrations may cause discomfort to the pedestrian traffic.
The maximum dynamic deflection of a bridge deck due to vehicle impact has
been found to be a function of the initial amplitude of bounce of the vehicle and its
natural frequency. The dynamic response of long span and cable suspended bridge
are more complex.
29
2.7.3
Pedestrian Loading
On most bridges, sidewalks and curbs are provided. Hence, the live load on
them should be provided due consideration in design. Pedestrian crossing produces
complex dynamic responses, involving different modes of vibration, as well as
motion due time variation of static deflection.
2.7.4
Longitudinal Forces
The longitudinal forces refer to forces that act in the direction of the
longitudinal axis of the bridge, specifically in the direction of the traffic. These
forces develop result of braking effort and traction effort. The vehicle’s inertia force
is transfer to the deck through friction between the deck and the wheels.
2.7.5
Wind Loads
Wind loads form a major component of lateral loads that act on bridge.
Bridges are frequently built on exposed sites and are subject to severe wind
conditions. Wind loads on the bridges depending on parameters such as wind
velocity, types of bridge, angle of attack, the size and shape of the bridge, the terrain
and the gust characteristics.
Wind effects on the bridge structures may be threefold as static wind
pressures, dynamic wind movement and buffeting between adjacent structures. Static
30
wind pressures are those that cause a bridge to deflect or deform. Dynamic wind
movements affect long span flexible bridge and prone to oscillate in number of
modes. A distinction should be made between the static effects of wind and the
dynamic effects of wind on flexible structures such as suspension and cable stayed
bridge.
Aerodynamic instability means that the effect of steady wind, acting on a
flexible structure of conventional cross section, to produce a fluctuating force
automatically synchronizing in timing and direction with the harmonic motions of
those of the structure so as to cause a progressive amplification of these motions to
dangerous destructive amplitudes. A significant force that is caused by an
aerodynamic phenomenon is the wind uplift or the vertical component of wind. The
Tacoma Narrows Bridge is the failure caused by this effect.
Static wind force, the main wind force acting on a bridge structure, develops
as a result of a steady wind that exerts a fairly constant pressure in the general
direction of the wind. Pressure due to wind is calculated by applying the principles of
fluid mechanics. Bernoulli’s theorem stated when fluid strikes an object; the increase
in the static pressure equals the decrease in the dynamic pressure.
2.7.6
Temperature Induced Forces
Temperature induced forces is a kind of environmental load. Temperature
variations cause bridges to expand and contract. The temperature-induced forces are
generated in bridges as a result of repeated cycles of heating and cooling from solar
radiation and the surrounding air.
31
2.7.7
Forces From Stream
Columns and piers in streams and rivers are subjected to forces due to the
stream flow. The force of stream current against the pier depends on the velocity of
the flowing water and the cross-sectional shape of pier. The forces acting on the pier
increase with existence of debris and ice.
2.7.8
Seismic Loads
Seismic loads are the lateral forces induced by the earthquake. The seismic
forces are proportion of the bridge’s dead load. There are three methods for analysis
and design of bridges for seismic loads, namely response spectrum modal analysis,
the time history method and the equivalent static force method. The first two
methods are called dynamics analysis.
2.7.9
Miscellaneous Loads
Other loads may be present as a result of the use of special construction
material and method. Construction, handling and erection loads are vital for pre-cast
or pre-stressed bridges. Temporary loads imposed by the sequence of construction
stages, construction equipments and false works should be considered.
32
Forces caused by displacement of supports and by shrinkage and creep of
concrete should be carefully considered in analysis. The forces are of special
significance in the construction of reinforced and prestressed concretes bridges.
2.8
Loads on Bridge According to BS 5400: Part 2 (1978)
A bridge and its entire component should be designed to safely resist all loads
to which it may be subjected during its services life. Many of the described forces
and loads act on the bridge structure simultaneously.
The design loads to be considered for bridge construction are as stipulated in
BS 5400 Part 2: Specification of Loads. In bridge design, the nominal loads should
be multiply by the appropriate value of partial load factor, fL to derive the design
load to be used in the calculation of moments, shears, total loads and other effect for
each of the limit states under consideration.
The loads applied to a bridge structure are regarded as either permanent or
transient. Dead loads, superimposed dead loads, loads due to filling materials,
shrinkage and creep and differential settlement are considered as permanent loads in
the Standard. All loads other than permanent ones are considered as transient loads.
The maximum of effects of certain transient loads including primary live loads and
secondary live loads do not coexist with the maximum effects of certain others.
Therefore, reduced factor can be applied based on provision by the standard.
The BS 5400 Part 2 also provides values for the partial load factor for each
force, to be used for ultimate and serviceability limit states under five force
33
combinations. The combinations of loads provided in BS 5400 Part 2 for bridge can
be summarised as below:
i)
Load Combination 1
For highway and foot/cycle track bridges, the loads to be considered
are the permanent loads, together with the appropriate primary live
loads, and for railway bridges, the permanent loads, together with the
appropriate primary and secondary live loads.
ii)
Load Combination 2
For all bridges, the loads to be considered are the loads in combination
1 together with those due to wind, and where erection is being
considered temporary erection loads.
iii)
Load Combination 3
For all bridges, the loads to be considered are the loads in combination
1, together with those arising from restraint due to the effects of
temperature range and difference, and, where erection is being
considered, temporary erection loads.
iv)
Load Combination 4
Combination 4 does not apply to railway bridge except for vehicle
collision loading on the bridge supports. For highway bridges, the
loads to be considered are the permanent loads and the secondary live
loads, together with the appropriate primary live loads associated with
them. Secondary live loads are considered separately.
v)
Load Combination 5
For all bridges, the only loads to be considered are the permanent
loads, together with the loads due to friction at bearings.
34
However, for a railway bridge, only first 3 load combinations are considered
in design. All the load combinations are applied with factor of 1.0 under
consideration of serviceability limit states.
2.8.1
Wind Loads
The wind pressure on a bridge depends on the geographical location, the local
topography, and the height of the bridge above ground, and the horizontal
dimensions and cross section of the bridge or element under consideration. The
maximum pressures are due to gusts that cause local and transient fluctuations about
the mean wind pressure. Design gust pressures are derived from the isotachs of mean
hourly wind speed. These wind speeds are appropriate to a height above ground level
of 10 m in open level country and a 120-year return period.
Maximum gust speed, VC on the bridge on which the application of wind
loading increases the effects being considered shall be taken as
VC = v K1S1S2
where
V = mean hourly wind speed
K1= wind coefficient related to return period
S1= funnelling factor
S2= gust factor
(Eq. 2.9 )
35
The nominal transverse wind load, Pt shall be taken as acting at the centroids
of the appropriate areas and horizontally unless local conditions change the direction
of the wind. The nominal transverse wind load should derived as
Pt = qA1CD
(Eq. 2.10)
where
q = dynamic pressure head (  0.613Vc )
2
A1= solid area
CD= drag coefficient
The nominal vertical wind load, PV is whether an upward or downward load
acting at the centroids of the appropriate areas for the bridge super structure. The
value should be derived from
PV = qA3CL
(Eq. 2.11)
where
A3 = area in plan
CL = lift coefficient
2.8.2
Railway Bridge Live Load
For a monorail bridge, the live load considered is Type RL Loading, which is
a reduced loading for use only on passenger rapid transit railway systems on lines
36
where main line locomotives and rolling stock do not operate. Nominal Type RL
Loading consists of a single 200 kN concentrated load coupled with a uniformly
distributed load of 50 kN/m for loaded lengths up to 100 metres. For loaded lengths
in excess of 100 m the distributed nominal load shall be 50 kN/m for the first 100 m
and shall be reduced to 25 kN/m for lengths in excess of 100 m.
The standard railway loadings are equivalent static loadings and should be
multiplied by appropriate dynamic factors to allow for impact, oscillation and other
dynamic effects including those caused by track and wheel irregularities. The
dynamic factor for single track cross girders should be taken as 1.40.
All elements of bridges subject to railway loading should be checked against
the effects of fatigue caused by repeated cycles of live loading. The number of load
cycles should be based on a life expectancy of 120 years for bridges intended as
permanent structures. The load factor to be used in all cases when considering
fatigue is 1.0.
2.9
Serviceability Criteria
Functional considerations are the very reason for creating a structure.
Structural design can be considered optimal if it can simultaneously satisfy the
requirements of function, safety, economy and aesthetics [Kuzmanovic and Wilems,
1983]. Deflection, slenderness and flexibility of structure, and fatigue are three
elements of the design matrix that are related to the performance of a bridge structure
during its service life. Together, they form a part of the design criteria, generally
known as the serviceability criteria. There are other considerations that affect
serviceability and, in time, lead to functional obsolescence of a bridge. These include
37
changed traffic patterns, increased size and weight limitations of live load, and
material deterioration (corrosion of steel) due to poor maintenance.
2.9.1
Deflection
Deflection limitations for bridges appear to have evolved in the early 1930s
when reports of objectionable vibrations of steel girder bridges began to appear.
According to AASHTO, the allowable deflections due to live load plus impact were
limited to 1/800 of the span for bridges carry light traffic and to 1/1000 for bridges
carrying heavy traffic.
The ASCE Committee [ASCE, 1958] had pointed out two reasons for
limiting deflections for railway bridge:
i)
to avoid excessive vibration of the structure in resonance with the
recurring hammer blows of the locomotive driving wheels.
ii)
to avoid objectionable oscillation of the rolling stock induced when
the deflections of the successive spans tended to set up a harmonic
excitation of the sprung weight
2.9.2
Fatigue in Steel Bridges
The most disastrous failure of a bridge in recent times was that of the Pont
Pleasant Bridge, West Virginia, on December 15, 1967. the cause of the failure was
determined to be stress corrosion and fatigue. Fatigue can be broadly defined as a
38
phenomenon of reduced material resistance under fluctuating stresses. The fatigue
limit is the maximum stress that can repeated indefinitely without causing fatigue
failure when applying fluctuating loads.
The phenomenon of fatigue is related to the fact that a cyclic repetitive
stressing of steel can produce fracture by much smaller forces than are required for
failure under a static load. Cyclic stresses are known to initiate and then to propagate
cracks in structural members, eventually weakening them.
The potential of crack initiation and growth increases in the presence of stress
raisers. Stress raisers can characterised as imperfections that create local stress
concentrations. Some examples of stress raisers are sudden changes in the geometry
of a members cross section, welds or mechanical fasteners, and features that interrupt
the smooth flow of stresses in a member.
In bridge structures, load fluctuations, and hence stress reversal, are generally
caused by the random movement of vehicles. The fatigue criterion is further
aggravated by a phenomenon known as wind induced fatigue, which is caused by
turbulent winds. The 1940 collapse of Tacoma Narrows Bridge is a classic example
of wind induced fatigue failure. Prior to its collapse, the bridge, as shown in Figure
1.1, was seen having large amplitude vibrations, both in linear and torsional modes
caused by wind [Liu, 1992].
Fatigue problems arise frequently when the tensile stress are involved, so the
members need to be fatigue checked that are expected to undergo a large number of
stress reversals pr a large number of variations of tensile stress.
39
Tests [AISC Mkt, 1986] performed on a large number of different steels
indicate that:
1.
Up to tensile strength of 1.4106 kPa, the fatigue limit of steel is about
50 percent of its tensile strength.
2.
Fatigue strength of steel increases with tensile strength of steel; the
higher the tensile strength, the higher the fatigue strength
3.
The fatigue strength decreases with increasing number of loading
cycles.
4.
The presence of a weld in a specimen reduces its fatigue strength. The
decrease in fatigue strength is greater for steels with greater tensile
strength.
5.
For the same maximum stress, the number of stress cycles required to
produce failure increases with increasing stress ratio.
There several approaches to evaluate fatigue such as fracture mechanics
approach and stress-life-detail category approach. The detailed evaluation procedure
of fatigue specific to bridges is described in the Guide Specification for Fatigue
Evaluation of Existing Steel Bridges published by the American Association of State
Highway and Transportation Officials [AASHTO, 1990].
2.10
Finite Element Method (FEM)
2.10.1 Introduction
The finite element method (FEM) is a numerical procedure for solving
continuum mechanics problems. FEM is a method of structural analysis based on
dividing the structures into simple parts called elements. These elements are then
40
assembled into complex structure for which a solution can be obtained by simple
numerical methods.
The method provides engineers with a versatile tool, which has very wide
applicability. Finite elements methods could handle the complicated configurations
through the discretization process in this method. The method also can handle
complex boundary conditions in homogeneity in materials composition and mixed
structures with relative ease. One of the main attractions of the method is its
versatility with respect to the type of elements that can be used to define the
structure.
2.10.2 Concept of the Finite Element Method
Finite elements represented the piecewise approximation of a function , by
means of polynomials, each defined over a small region (element) and expressed in
terms of nodal values of the function. The nodal forces that produce the various
element deformation modes must be found to formulate an element.
The finite element method is not restricted to problems of structural
mechanics. A feature of stress, displacement or energy field is represented by a
function,  which varies smoothly in the actual structure. If model by triangles, the
 surface is approximated by flat facets. The  is a linear function of x and y, which
is
 = a1 + a2x + a3y
(Eq. 2.12)
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The four and eight node elements display warped and curved surfaces and
better approximate the actual function. The function is
 = a1 + a2x + a3y +a4xy
(Eq. 2.13)
The displacement method could be used for solution in the finite element
method. The displacement of any point in an element is defined with respect to the
displacement of the element nodes by:
U  Nr
(Eq. 2.14)
where [N] is the shape function matrix and {r} are discrete values of the nodal
displacement.
The strain energy expression can be written as
ε  Br
(Eq. 2.15)
where [B] is strain displacement matrix and defined as
0 
N x

B   0 N y
N y N x 
(Eq. 2.16)
42
Then the constitutive relation is formulated using the uni-axial stress strain
curve of the material. For the elastic case it is defined as
σ  Dε
(Eq. 2.17)
where [D] is the elasticity matrix, which defines the material property. By satisfying
the equilibrium of the system and utilising the principle of virtual work, it can be
shown that
Kr  R
(Eq. 2.18)
where [K] is the element stiffness matrix and defined as
K    BT DBdv
(Eq. 2.19)
R   NT Fs dA   NT f dv
(Eq. 2.20)
and where