E L

T
BRIDGE ENGINEERING
P
N
Prof. Piyali Sengupta
Department of Civil Engineering,
Indian Institute of Technology (ISM) Dhanbad

Module 05: Plate Girder Bridges

Lecture 13: Design of Plate Girder Bridges
Ø General Features

Ø Structural Configurations
E L
P T
N
Ø Design Principle
 Topic of Discussion

Ø General Features

Ø Structural Configurations

Ø Design Principle

E L
P T
N

Bridge Engineering
 General Features

• Plate girder bridges are the most common type of steel bridges
generally used for railway and high way crossings since late
18th century.

• The earliest forms of steel bridges constructed were plate girder

E L
bridges due to their simplicity and elegant aesthetics.

P T
N

Bridge Engineering
 General Features

• In general, plate girder bridges are economical for railway and

highway bridges of spans in the range of 15 to 40 m.

• Plate girder bridges may be very competitive for much longer

spans, when they are continuous in the range 50 to 200 m.

•
E L
For short span less than 10 m, plate girder bridges are

P T
uneconomical due to higher connection cost. Rolled I-sections
N
are preferred for short span steel beam bridges.

• Plate girder bridges are built up using two flange plates and a
web plate to form an I-shaped cross section.

Bridge Engineering
 General Features

• Before widespread use of welding, connecting the components

of the cross section was a major concern in the design of plate
girders. All of the connections were made by riveting or bolting.
• Plate girder bridges
constructed after 1960 are
E L
generally shop welded,
P T
N
replacing riveted fasteners.

• Riveted/bolted plate girders

are economically used for
15-30 m span. Welded plate
girders may be used up to
100 m span.

Bridge Engineering
 General Features

• The primary reasons of selection of the plate girder bridges are

the rapid assembling and lesser construction time in
comparison with the concrete bridges.

• However, maintenance
costs of steel plate girder
E L
bridges are quite high
P T
due to their susceptibility
to corrosion damages,
N
especially in coastal
areas, due to aggressive
exposure conditions.

Bridge Engineering
 General Features

• In case of railway bridges, the plate girders support the

sleepers over which the steel rails are fastened. Each rail is
supported on a plate girder so that the wheel loads are
transmitted directly to the plate girder without any torsion.

• Twin plate girders are

E L
braced laterally at the
P T
level of the bottom flange
to provide the lateral
N
stability. Cross bracings
and the end cross frames
resist the lateral loads on
the plate girder.

Bridge Engineering
 General Features

• The design of a plate girder is essentially an upgradation of an

ordinary beam design:
Ø the proportioning of a member with a section modulus
adequate to resist bending
Ø a web capable of resisting shear
E L
Ø sufficient stiffness
P T
• N
Designer has greater control over the dimensions of the section
and may make the web thinner in proportion to its depth than in
any of the rolled shapes.

• Local buckling of compression flange and shear buckling of

web may be of great concern.

Bridge Engineering
 General Features

• Practical alternatives to plate girders in the spans for which

they are economical, are trusses.

• Advantages over Trusses:

Ø Lower fabrication cost as compared to trusses

E L
Ø Erection is faster and cheaper than trusses
P T
N
Ø Require small vertical clearances than trusses

Ø Vibration and impact are not major problem due to

compactness

Bridge Engineering
 General Features

• Disadvantages over Trusses:

Ø Heavier than trusses for the same span and loads

Ø Larger exposed wind area as compared to truss

L
Ø Low torsional stiffness (box girder provides better torsional
E
stiffness)
P T
N
Ø Susceptibility to stability problems of the compression flange
during erection

Bridge Engineering
 Topic of Discussion

Ø General Features

Ø Structural Configurations

Ø Design Principle

E L
P T
N

Bridge Engineering
 Structural Configurations

Three types of plate girders are generally used in railway and high
way crossings.

The 1st figure shows the

simplest type of plate
girder built using cover
E L
plates to function as
P T
flanges and a web plate
connecting the flange
N
plates.

Bridge Engineering
 Structural Configurations

Three types of plate girders are generally used in railway and high
way crossings.

The 2nd figure shows a

plate girder used for
longer spans built by
E L
using more cover plates,
P T
to strengthen the flanges
and a web plate.
N

Bridge Engineering
 Structural Configurations

Three types of plate girders are generally used in railway and high
way crossings.

This figure shows a built

up plate girder generally
used for railway bridges
E L
of long spans using two
P T
plate girders connected
by lateral bracing.
N

Bridge Engineering
 Structural Configurations

• Plate girders used for longer span lengths require deeper web
plates.

• To prevent buckling, intermediate stiffeners comprising rolled

steel angles are used at regular intervals along the span and at
the supports.
E L
•
P T
If the span length is large, the web and flange plates are
N
connected by splices for structural integrity in resisting
external loads.

Bridge Engineering
 Structural Configurations

• Web plate: Vertical plate of the plate girder

• Flange or cover plates: Horizontal plates connected with the

web plate by welding or bolted through flange angles

L
• Flange angles: Rolled steel angles sections with or without

E
flange plates connected through rivets/bolts at the top and
T
• N P
bottom of the web plate, used in earlier time

Stiffeners: When web of a plate girder acting alone (without

stiffeners) proves inadequate, stiffeners may be provided for
various purposes.

Bridge Engineering
 Structural Configurations

• Types of Stiffeners: Intermediate transverse and longitudinal

stiffeners, load carrying and bearing stiffeners, diagonal,
torsional and tension stiffeners

• Intermediate transverse and longitudinal web stiffeners: The

E L
function of this stiffener is to improve the buckling strength of
a slender web due to shear.
P T
• N
Load carrying stiffener: The function of this stiffener is to
prevent local buckling of the web due to concentrated loading.

• Bearing stiffener: The function of this stiffener is to prevent

local crushing of the web due to concentrated loading.

Bridge Engineering
 Structural Configurations

• Torsional stiffener: The function of this stiffener is to provide

torsional restraint to beams and girders at supports.

• Diagonal stiffener: The function of this stiffener is to provide

local reinforcement to a web under shear and bearing

•
E L
Tension stiffener: The function of this stiffener is to transmit

P T
tensile forces applied to a web through a flange
N

Bridge Engineering
 Structural Configurations

• Flange Splices: Flange splices should preferably, not be located

at the points of maximum stress. Where splice plates are used,
their area shall be not less than 5% in excess of the area of the
flange element spliced. The centre of gravity of splice plate and

L
the flange element spliced shall coincide, as nearly as possible.

E
•
T
Web Splices: Splices and cutouts for service ducts in the webs
P
N
should preferably not be located at points of maximum shear
force and heavy concentrated loads. Splices in the webs of the
plate girders shall be designed to resist the shear force and
moments at the spliced section.

Bridge Engineering
Structural Configurations

E L
P T
N

Bolted Plate Girder

Bridge Engineering
Structural Configurations

E L
P T
N

Welded Plate Girder

Bridge Engineering
 Structural Configurations

E L
P T
N

Components of Welded Plate Girder

Bridge Engineering
 Structural Configurations

E L
P T
N

Plate Girder with Holes for Services

Bridge Engineering
 Topic of Discussion

Ø General Features

Ø Structural Configurations

Ø Design Principle

E L
P T
N

Bridge Engineering
 Design Principle of Plate Girder Bridges

The main structural elements in a plate girder to be designed are

the web and flanges plate together with the intermediate and the
end bearing stiffeners conforming to the specifications of IRC : 24-
2010 and IS: 800-2007.

The modes of failure of a plate girder are:

E L
• By yielding of the tension flange
P T
• N
By buckling of the compression flange

The compression flange buckling can take place in various ways,

such as vertical buckling in web, flange local buckling or lateral
torsional buckling.

Bridge Engineering
 Design Principle of Plate Girder Bridges

E L
P T
Bending Stress Distribution (left) and Shear Stress
N
Distribution (right) in an I-section

Based on the stress distribution diagrams, it is evident that the

flanges carry a major portion of the flexural load while the web
carries most of the shear load in the plate girder.

Bridge Engineering
 Design Principle of Plate Girder Bridges

• Proportioning of plate girder is to be done so as to maintain

minimum self-weight and high strength and stiffness.

• Curtailment of flange width or thickness can be done in low

bending moment zone.

•
E
Thicker web can be provided in high shear zone. L
P T
•
N
Hybrid girder an be constructed by utilizing plates of different
strengths for flanges and web, matching the strength
requirements of the structure.

• Flange to web connectors: The flanges of plate girders shall be

connected to the web by sufficient rivets, bolts or welds to
transmit the maximum horizontal shear force.

Bridge Engineering
 Design Principle of Plate Girder Bridges

• Maximum shear flow qw at the junction of web and flange is given

by qw = (V × Af × ӯ)/ 2Iz
Where Iz = (bf − tw) × d3/ 12,
Af = Area of flange = bf × tf,
bf = flange width,
E L
tf = flange thickness,
P T
d = depth of web, N
tw = thickness of web,
V = Factored shear force,
Af × ӯ = moment about neutral axis of the area between the
horizontal shear plane and the outside face of the section

Bridge Engineering
 Design Principle of Plate Girder Bridges

• It is economical to choose a relatively deep web to keep flange

areas at the greatest distance from the neutral axis.

• Thin slender web would be the consequence of this requirement.

L
• Slender web is prone to buckling at low values of applied shear,

E
necessitating provision of intermediate stiffeners.
T
•
N P
Web plate thickness should not be less than 6 mm if painted and
8 mm if unpainted.

• Thin stiffened web is economical and hence, used in the past.

• Recent practice is to provide thick web without stiffeners to

reduce fabrication time and cost.

Bridge Engineering
 Design Principle of Plate Girder Bridges

Minimum Web Thickness:

The thickness of the web of a plate girder section should meet the
following serviceability and compression flange buckling criteria.

L
• Serviceability Criteria:

T E
w
N P
w

Here, d = depth of web, tw = thickness of web

εw = yield stress ratio of web = √(250/fyw), fyw = yield stress of web

Bridge Engineering
 Design Principle of Plate Girder Bridges

Minimum Web Thickness:

• Serviceability Criteria:

E L
P T
N
Here, c = spacing of transverse stiffener
d = depth of web, tw = thickness of web
εw = yield stress ratio of web = √(250/fyw), fyw = yield stress of web

Bridge Engineering
 Design Principle of Plate Girder Bridges

Minimum Web Thickness:

• Serviceability Criteria:

E L
P T
N
Here, c = spacing of transverse stiffener
d = depth of web, tw = thickness of web
εw = yield stress ratio of web = √(250/fyw), fyw = yield stress of web

Bridge Engineering
 Design Principle of Plate Girder Bridges

Minimum Web Thickness:

• Serviceability Criteria:

E L
Here,
P T
d = depth of web, N tw = thickness of web
εw = yield stress ratio of web = √(250/fyw), fyw = yield stress of web

Bridge Engineering
 Design Principle of Plate Girder Bridges

Minimum Web Thickness:

• Compression Flange Buckling Criteria:

In order to avoid buckling of the compression flange into the web,

L
the web thickness shall satisfy the following.

T E
N P
Here, d = depth of web,
tw = thickness of web,
εf = yield stress ratio of the compression flange = √(250/fyf),
fyf = yield stress of the compression flange

Bridge Engineering
 Design Principle of Plate Girder Bridges

Minimum Web Thickness:

• Compression Flange Buckling Criteria:

E L
P T
Here, d = depth of web, N
tw = thickness of web,
c = spacing of the transverse stiffener,
εf = yield stress ratio of the compression flange = √(250/fyf),
fyf = yield stress of the compression flange

Bridge Engineering
 Design Principle of Plate Girder Bridges

Optimum Web Thickness:

If the moment M is assumed to be resisted entirely by the flanges,

then for an I-section beam, approximately

E L
P T
N

Bridge Engineering
 Design Principle of Plate Girder Bridges

Optimum Web Thickness:

E L
P T
N

Bridge Engineering
 Design Principle of Plate Girder Bridges

Optimum Web Thickness:

E L
P T
N
For trial girder section d/tw ratio of the web may be considered
somewhere between 135 and 240.

Bridge Engineering
 Design Principle of Plate Girder Bridges

Optimum Web Depth:

E L
P T
N

Bridge Engineering
 Design Principle of Plate Girder Bridges

Optimum Web Depth:

E L
P T
N
The optimum depth of the plate girder is determined based on the
area of steel used is minimum. It is desirable to know the optimum
depth for economy so that even if it cannot be adopted, it will serve
as a guide. Normally, a depth less than the optimum depth is
provided in case of design of plate girder.

Bridge Engineering
 Design Principle of Plate Girder Bridges

Shear Strength:

• The shear strength of a plate girder depends on the depth to

thickness ratio of the web and the spacing of the intermediate
stiffeners provided.

•
E L
The shear capacity of the web has two components, namely,

P T
shear strength before onset of buckling and shear strength at
post-buckling stage. N
• Prior to buckling, shear stress can be deduced from the simple
beam theory.

• Thin unstiffened web plate does not carry much load after
buckling.

Bridge Engineering
 Design Principle of Plate Girder Bridges

Shear Strength:

• As the shear force increases on

a stiffened web panel, the web
panel buckles. This load does
not indicate the maximum
E L
shear capacity of the web.
P T
• The shear force can be further
increased and the web panel
N
will continue to carry further
load relying on the tension field
Tension Field Action
action.

Bridge Engineering
 Design Principle of Plate Girder Bridges

Shear Strength:

• Part of the buckled web

takes the load in
tension. This tension
member action takes
E L
place across the web
P T
panel in an inclined
direction to the web
N
panel diagonal.

Tension Field Action

Bridge Engineering
 Design Principle of Plate Girder Bridges

Shear Strength:

• At this stage the girder acts like a N-type Pratt truss with the
compression forces being carried by the flanges and the
intermediate stiffeners. The buckled web resists the tension

E L
forces. This additional reserve strength is termed as tension field
action.
P T
• N
If no intermediate stiffeners are present or their spacing is large,
it is not possible for tension field action to take place. Then, the
shear capacity is restricted to the shear strength before
buckling.

Bridge Engineering
 Design Principle of Plate Girder Bridges

Shear Strength:

E L
P T
N

Bridge Engineering
 Design Principle of Plate Girder Bridges

Shear Strength of Web:

• Pre-buckling Behavior:

When a web plate is subjected to vertical shear, complementary

L
shear stresses are developed to satisfy equilibrium of the plate.

E
As a consequence, the plate develops diagonal tension and
T
diagonal compression.
N P
Critical elastic shear stress:
It is the shear stress τcr,e beyond which the
plate can not take any further compressive
stress along diagonal AC. V1 = t cr ,e dt w

Bridge Engineering
 Design Principle of Plate Girder Bridges

Critical elastic shear stress: For simply supported condition,

critical elastic shear stress τcr,e = Kvπ2E/ [12 × (1 − μ2) × (d/ tw)2]
Where μ = Poisson’s Ratio
d = Depth of web,
tw = Thickness of web
E L
Kv = Shear buckling Coefficient,
P T
= 5.35 when transverse stiffeners are N
provided only at supports
= 4.0 + 5.35/ (c/d)2 for c/d < 1.0
= 5.35 + 4.0/ (c/d)2 for c/d ≥ 1.0
c = Spacing of transverse stiffener

Bridge Engineering
 Design Principle of Plate Girder Bridges

Resistance to shear buckling shall be verified as specified in IS

800:2007, when

E L
P T
Where d = Depth of web,
tw = Thickness of web
N
Kv = Shear buckling Coefficient
ε = Yield stress ratio = √(250/fy)

Bridge Engineering
 Design Principle of Plate Girder Bridges

The nominal shear strength Vn of webs with or without intermediate

stiffeners as governed by buckling may be evaluated using one of
the following methods:

• Simple Post-critical Method

• Tension field method

E L
P T
N
The simple post-critical method, based on the shear buckling
strength can be used for web of I-section girders, with or without
intermediate transverse stiffeners, provided that the web has
transverse stiffeners at the supports.

The nominal shear strength is given by: Vn = Vcr

Bridge Engineering
 Design Principle of Plate Girder Bridges

The nominal shear strength Vn is given by: Vn = Vcr

= d tw tb

E L
P T
N

Bridge Engineering
Design Principle of Plate Girder Bridges

E L
P T
N

Bridge Engineering
 Design Principle of Plate Girder Bridges
• Tension field method

E L
P T
N

Bridge Engineering
 Design Principle of Plate Girder Bridges
• Tension field method

E L
P T
N

Bridge Engineering
 Design Principle of Plate Girder Bridges
• Tension field method

E L
P T
N

N f = axial force in the flanges = Mz/d1

M z = design bending moment, d1 = c/c distance between the flanges

Bridge Engineering
 Structural Configurations

E L
P T
N

Components of Welded Plate Girder

Bridge Engineering
 Design Principle of Plate Girder Bridges
Outstand of Web Stiffeners:

Unless the outer edge is continuously stiffened, the outstand from

the face of the web should not exceed 20 tqε. When the outstands
of web is between 14 tqε and 20 tqε, then the stiffener design should

L
be on the basis of a core section with an outstand of 14 tqε where tq
E
T
is thickness of the stiffener and ε is yield stress ratio = √(250/fy)
P
Eccentricity:
N
Where a load or reaction is applied eccentric to the centreline of the
web or where the centroid of the stiffener does not lie on the
centreline of the web, the resulting eccentricity of loading should
be accounted for in the design of the stiffener.

Bridge Engineering
 Design Principle of Plate Girder Bridges
Stiff Bearing Length:

The stiff bearing length of any element b1, is that length which
cannot deform appreciably in bending. To determine b1, the
dispersion of load through a steel bearing element should be taken

L
as 45° through solid material, such as bearing plates, flange plates,
E
etc.
P T
N

Bridge Engineering
 Design Principle of Plate Girder Bridges
Buckling Resistance of Stiffeners:

• The buckling resistance Fqd should be based on the design

compressive stress fcd of a strut, the radius of gyration being
taken about the axis parallel to the web.

E L
• The effective section is the full area or core area of the stiffener

P T
together with an effective length of web on each side of the

N
centre line of the stiffeners, limited to 20 times the web
thickness.

• The design strength used should be the minimum value

obtained for buckling about the web or the stiffener.

Bridge Engineering
 Design Principle of Plate Girder Bridges
• The effective length for intermediate transverse stiffeners used
in calculating the buckling resistance, Fqd should be taken as 0.7
times the length L of the stiffener.

• The effective length for load carrying web stiffeners used in

L
calculating the buckling resistance Fxd assumes that the flange
E
T
through which the load or reaction is applied is effectively
P
N
restrained against lateral movement relative to the other flange,
and should be taken as:
a) KL = 0.7 L, when flange is restrained against rotation in the
plane of the stiffener (by other structural elements)

b) KL = L, when flange is not so restrained where L = length of

the stiffener

Bridge Engineering
 Design Principle of Plate Girder Bridges
Intermediate Transverse Stiffener:

• Intermediate Transverse Stiffener can be termed as vertical

stiffeners also.

• It increases the buckling resistance of the web caused by shear.

E L
T
• It must be sufficiently stiff so as to not deform appreciably as
the web tends to buckle.
N P
• It must be sufficiently strong to withstand the shear transmitted
by the web.

• Angle sections, (in pairs or single), are provided for

riveted/bolted construction of plate girders and plate flat
sections for welded plate girders.

Bridge Engineering
 Design Principle of Plate Girder Bridges
Intermediate Transverse Stiffener:

• Minimum Stiffeners:

Transverse web stiffeners not subjected to external loads or

moments should have a second moment of area, Is about the centre

E L
line of the web, if stiffeners are on both sides of the web; and about

P T
the face of the web, if single stiffener on only one side of the web is
used such that:
N
Where d = Depth of web, tw = Minimum required web thickness,
c = actual stiffener spacing

Bridge Engineering
 Design Principle of Plate Girder Bridges
Intermediate Transverse Stiffener:

• Buckling Check:
This check is required only for intermediate stiffeners in webs
when tension field action is utilized. Stiffeners not subjected to

E L
external loads or moments should be checked for a stiffener force:

P T
N

Bridge Engineering
 Design Principle of Plate Girder Bridges
Intermediate Transverse Stiffener:

E L
P T
N

Bridge Engineering
 Design Principle of Plate Girder Bridges
Intermediate Transverse Stiffener:

E L
P T
N

Bridge Engineering
 Design Principle of Plate Girder Bridges
Intermediate Longitudinal Stiffener:

• Intermediate longitudinal stiffener can be termed as horizontal

stiffeners also.

• It increases the buckling resistance considerably as compared

E L
to transverse stiffeners when the web is subjected to bending

P T
• It consists of angle section for a riveted/ bolted plate girder and
N
plate section for a welded plate girder.

• Intermediate longitudinal stiffeners are provided in the

compression zone of the web.

Bridge Engineering
 Design Principle of Plate Girder Bridges
Intermediate Longitudinal Stiffener:

• First horizontal stiffener is provided at 1/5th of the distance from

the compression flange to the tension flange.

• Another stiffener is provided at the neutral axis if needed.

E L
P T
• It can be extended between the vertical stiffeners, however, it is
not required to be continuous over the vertical stiffeners.

•
N
Intermediate longitudinal stiffeners can be provided in pairs on
each side of the web, or single located on one side of the web.

Bridge Engineering
 Design Principle of Plate Girder Bridges
Load Carrying Stiffener:

• Load-carrying web stiffeners are provided where compressive

forces applied through a flange exceed the buckling strength
Fcdw of the unstiffened web.

•
E L
The design compressive strength Fcdw, of a member = Acdw × fcdw

•
P T
The area of cross section Acdw is taken as (b1 + n1) tw where

tw = thickness of web
N
b1 = stiff bearing length

n1 = length obtained by dispersion through the flange to the web

junction at a slope of 1:2.5 to the plane of the flange

Bridge Engineering
 Design Principle of Plate Girder Bridges
Load Carrying Stiffener:

Design compressive stress, fcdw of axially loaded compression

members can be calculated as follows.

fcdw = (fy/ γm0)/[φ + (φ2 – λ2)0.5]

E L
Where φ = 0.5 × [1 + α × (λ – 0.2) + λ2]

P T
N
λ = Non-dimensional effective slenderness ratio = √[fy × (KL/r)2/π2E]

α = Imperfection factor dependent on buckling class and flange

thickness

γm0 = Partial Safety Factor for material strength

Bridge Engineering
 Design Principle of Plate Girder Bridges
Load Carrying Stiffener:

KL/r = Effective slenderness ratio or ratio of effective length KL to

appropriate radius of gyration r

The effective length of the web for evaluating the slenderness ratio
is calculated as follows.
E L
P T
a) KL = 0.7 L, when flange is restrained against rotation in the
N
plane of the stiffener (by other structural elements)

b) KL = L, when flange is not so restrained where L = length of

the stiffener

Bridge Engineering
 Design Principle of Plate Girder Bridges
Load Carrying Stiffener:

• Load carrying web stiffeners should be of sufficient size to have

bearing strength of the stiffener Fpsd not less than the load
transferred, Fx

E L
P T
N

Bridge Engineering
 Design Principle of Plate Girder Bridges
Bearing Stiffener:

• This is used to transfer concentrated loads on the girder and

heavy reactions at supports to the full depth of the web.

• This is required when the web has insufficient strength for any

E L
of the limit states of web yielding, web crippling, or side sway
web buckling.
P T
• N
Where the web and the stiffener materials are of different
strengths the lesser value should be assumed to calculate the
capacity of the web and the stiffener.

Bridge Engineering
 Design Principle of Plate Girder Bridges
Bearing Stiffener:

• Bearing stiffeners should project nearly as much as the

overhang of the flange through which load is transferred.

E L
P T
N

Bridge Engineering
 Design Principle of Plate Girder Bridges
Bearing Stiffener:

Bearing stiffeners should be designed for the applied load or

reaction less the local capacity of the web, Fw given by

E L
P T
N

Bridge Engineering
 Design Principle of Plate Girder Bridges
Torsion Stiffeners:

When bearing stiffeners are required to provide torsional restraint

at the supports of the beam, they should meet the following criteria.

• Bearing stiffeners should be designed for the applied load or

E L
reaction less the local capacity of the web, Fw given by

P T
N

Bridge Engineering
 Design Principle of Plate Girder Bridges
Torsion Stiffeners:

• Second moment of area of the stiffener section about the centre

line of the web Is should be such that:

E L
P T
N

Bridge Engineering
 Design Principle of Plate Girder Bridges
Diagonal Stiffeners:

• Diagonal stiffeners should be designed to carry the portion of

the applied shear and bearing that exceeds the capacity of the
web.

•
E L
Where the web and the stiffener materials are of different

P T
strengths the lesser value should be assumed to calculate the

N
capacity of the web and the stiffener.

Bridge Engineering
 Design Principle of Plate Girder Bridges
Tension Stiffeners:

• Tension stiffeners should be designed to carry the portion of the

applied load or reaction less the capacity of the web Fw given by.

E L
P T
N

Bridge Engineering
 Design Principle of Plate Girder Bridges
Tension Stiffeners:

• Where the web and the stiffener materials are of different

strengths the lesser value should be assumed to calculate the
capacity of the web and the stiffener.

E L
P T
N

Bridge Engineering
 E L
P T
N

Bridge Engineering
v N. Subramanian, Design of Steel Structures: Limit States Method, Oxford

L
University Press.

T E
v N. Krishna Raju, Design of Bridges, Oxford & IBH Publishing Co. Pvt. Ltd.

N P
v D.J. Victor, Essentials of Bridge Engineering, Oxford & IBH Publishing Co. Pvt. Ltd.
v S. Ponnuswamy, Bridge Engineering, McGraw Hill Education.
v T.R. Jagadeesh and M.A. Jayaram, Design of Bridge Structures, PHI Learning Pvt.
Ltd.
v W.F. Chen, and L. Duan, Bridge Engineering Handbook, CRC Press, Taylor &
Francis Group.
v G. Parke and N. Hewson, ICE manual of Bridge Engineering, Thomas Telford
Publishing.
 E L
T
BRIDGE ENGINEERING
P
N
Prof. Piyali Sengupta
Department of Civil Engineering,
Indian Institute of Technology (ISM) Dhanbad

Module 05: Plate Girder Bridges

Lecture 14: Design Example of Plate Girder Bridges
 Design Example

E L
P T
N
 Design Example: Problem Statement
The cross-section of a stream is as shown in the Figure. Design a
plate girder bridge as railway crossing using the following data.
Effective Span L = 30 m; Material Grade Fe250
Broad Gauge rail track of gauge length = 1676 mm, Main Line,
Single Track. Dead load of track = 7.5 kN/m
Each rail is supported by a plate girder withE
L
P T cross bracings at
intervals of 6 m.
N

Cross-Section of Stream

Bridge Engineering
 Design Example: Solution

1. Given Data:

Effective Span L = 30 m.
Material Grade Fe 250 of yield strength 250 MPa
Broad Gauge rail track of gauge length = 1676 mm, Main Line,
Single Track.
E L
Dead load of track = 7.5 kN/m
P T
N
Each rail is supported by a plate girder with cross bracings at
intervals of 6 m.
The total loading is resisted by twin plate girders.

Bridge Engineering
 Design Example: Solution

2. Dead Loads:

Dead load of track = 7.5 kN/m

Self-weight of plate girder = (0.2 L + 1) = (0.2 × 30) +1 = 7 kN/m

Total dead load = (7.5 + 7) = 14.5 kN/m

E L
3. Live Loads:
P T
N
Equivalent Total live load for Bending Moment per Broad Gauge
track for 30 m span = 2727 kN

Total live load per girder = (2727/2) = 1363.5 kN

Bridge Engineering
Design Example: Solution

E L
P T
N

Bridge Engineering
 Design Example: Solution

Equivalent Uniformly Distributed Live Load for Bending Moment

per girder = 1363.5/30 = 45.45 kN/m

Equivalent Total live load for Shear Force per Broad Gauge track
for 30 m span = 2997 kN

Total live load per girder = (2997/2) = 1498.5 kN

E L
P T
Equivalent Uniformly Distributed Live Load for Shear Force per
girder = 1498.5/30 = 49.95 kN/m N
Coefficient of Dynamic Augment (CDA) = 0.372

Bridge Engineering
 Design Example: Solution

4. Design Bending Moments:

Bending Moment due to Dead Load = (14.5 × 302)/8 = 1631.25 kN.m

Bending Moment due to Live Load = (45.45 × 302)/8 = 5113.125 kN.m

L
Bending Moment due to Impact of Live Loads = (1.372 × 5113.25) =
E
7015.379 kN.m
P T
Design Bending Moment due to Dead Load
N
1631.25 + 1.5 × 7015.379) = 12969.943 kN.m
and Live Load = (1.5 ×

Bridge Engineering
 Design Example: Solution

5. Design Shear Forces:

Shear Force due to Dead Load = (14.5 × 30)/2 = 217.5 kN

Shear Force due to Live Load = (49.95 × 30)/2 = 749.25 kN
Shear Force due to Impact of Live Loads = (1.372 × 749.25) =
1027.971 kN
E L
P T
Design Shear Force due to Dead Load and Live Load = (1.5 × 217.5
+ 1.5 × 1027.971) = 1868.206 kN N

Bridge Engineering
 Design Example: Solution

6. Trial section of Web Plate:

Optimum depth of plate girder d = (Mk/fy)0.33 when intermediate

transverse stiffeners are not to be provided.

Where k = (d/tw) = 180 for thin webs and fy = 250 N/mm2

E L
T
Substituting, d = [(12969.943 × 106 × 180)/250]0.33 = 1950.695 mm

Optimum web thickness of plate girder t P

N = (M/f k )
w y
2 0.33

For k = (d/tw) = 180 for thin webs and fy = 250 N/mm2,

Substituting, tw = [(12969.943 × 106)/(250 × 1802)]0.33 = 11.415 mm

We can try to consider web plate of dimensions 1800 mm depth ×

14 mm thickness.

Bridge Engineering
 Design Example: Solution

7. Trial section of Flange Plate of Plate Girder:

Let us assume that the bending moment will be resisted by the
flanges and shear by the web.

Required area of flange of plate girder Af = (M × γm0)/ fyd

Where γm0 = Partial Safety Factor

E L
Substituting, d = 1800 mm, γ = 1.1 and fP
T
N = 250 N/mm 2
m0 y

A = [(12969.943 × 10 × 1.1)/(250 × 1800)] = 31704.305 mm

f
6 2

Assuming flange width bf = Span Length/40, flange width bf =

30000/40 = 750 mm

Bridge Engineering
 Design Example: Solution

Considering flange width bf = 750 mm, flange thickness tf = Af/ bf =

(31704.305/ 750) = 42.27 mm

We can try to consider flange plates of dimensions 750 mm width ×

50 mm thickness.

E L
Overall depth of plate girder D = d + 2 tf = 1800 + 2 × 50 = 1900 mm

Classification of Flanges: P T
N
Outstand of flange b = (bf − tw)/2 = (750 − 14)/2 = 368 mm

b/tf = (368/50) = 7.36 < 8.4

Hence, the flanges are plastic.

Bridge Engineering
Design Example: Solution
750

50

950
900

1800
E L
P T
14
N
50

Cross-section of a Plate Girder

Bridge Engineering
 Design Example: Solution

8. Check for Bending Strength:

Plastic Section Modulus Zp = 2 bf tf × (D − tf)/2 = 2 × 750 × 50 × (1900
− 50)/2 = 69375000 mm3

Moment capacity Md = (Zp× fy)/ γm0 = (69375000 × 250)/ 1.1 =

E L
15767045450 N-mm = 15767.045 kN-m > Ultimate Design Bending
Moment M = 12969.943 kN-m
P T
Hence, the section is safe. N
9. Check for Shear Strength:
d/ tw = 1800/14 = 128.57 < 200

Elastic critical shear stress τcr,e = Kvπ2E/ [12 × (1 − μ2) × (d/ tw)2]

Bridge Engineering
 Design Example: Solution

Where Kv = Shear buckling Coefficient,

= 5.35 when transverse stiffeners are provided only at supports
= 4.0 + 5.35/ (c/d)2 for c/d < 1.0
= 5.35 + 4.0/ (c/d)2 for c/d ≥ 1.0
c = Spacing of transverse stiffener
E L
P T
Assuming spacing of stiffeners c as equal to the depth of web plate
d, c/d = 1 N
Therefore, Kv = 5.35 + 4.0/ (c/d)2 = 9.35
μ = Poisson’s Ratio = 0.3

d/ tw = 1800/14 = 128.57

Bridge Engineering
 Design Example: Solution

Elastic critical shear stress τcr,e = Kvπ2E/ [12 × (1 − μ2) × (d/ tw)2] =
(9.35 × π2 × 2 × 105)/ [12 × (1 − 0.32) × 128.572] = 102.244 N/mm2

The nominal shear strength Vn is given by: Vn = Vcr

= d tw τ b

E L
P T
N

Bridge Engineering
 Design Example: Solution

Non-dimensional web slenderness ratio for shear buckling stress

λw = √[fyw/(√3 × τcr,e)] = √[250/(√3 × 102.244)] = 1.188 < 1.20

Shear stress corresponding to web buckling, for λw < 1.20,

τb = [1 – 0.8 × (λw – 0.8)] × (fyw/√3) = [1 – 0.8 × (1.188 – 0.8)] × (250/√3)
= 99.593 N/mm2
E L
P T
Shear force corresponding to web buckling Vcr = dtw τb = (1800 × 14
N
× 99.593) = 2509743.6 N = 2509.744 kN > Ultimate Design Shear
Force V = 1868.206 kN

Hence, the section is safe.

Bridge Engineering
 Design Example: Solution

10. Check for lateral torsional buckling:

Since the compression flange of the girder is laterally restrained
throughout, the check for lateral torsional buckling is not required.

11. Flange to Web Connection:

E L
Maximum shear force at the junction of web and flange is given by
qw = (V × Af × ӯ)/ 2Iz
P T
N
Iz = (bf − tw) × d3/ 12 = (750 − 14) × 18003/ 12 = 35.77 × 1010 mm4

qw = (V × Af × ӯ)/ 2Iz = [1868.206 × (750 × 50) × (900 + 50/2)]/ (2 ×

35.77 × 1010) = 0.091 kN/mm

Let us provide weld of size S = 6 mm. KS = 0.7 × 6 = 4.2 mm

Bridge Engineering
 Design Example: Solution

Strength of weld per unit length fwd = (4.2 × 250 × 10-3)/(√3 × 1.25) =
0.485 kN/mm > 0.091 kN/mm

6 mm continuous fillet welds can be provided on either side.

12. Intermediate Transverse Stiffeners:

E L
Since the ratio of (d/tw) = (1800/14) = 128.57 > 85, vertical stiffeners
are required.
P T
N
Adopting the spacing of stiffeners c = 1500 mm

Greater unsupported panel dimension of the web = 1500 mm < 270

tw < (270 × 14) = 3780 mm

Bridge Engineering
 Design Example: Solution

c/d = 1500/1800 = 0.833

the Intermediate transverse stiffeners are designed to have a

minimum moment of inertia Is = 1.5d3tw3/c2 = 1.5 × 18003 × 143/15002
= 10668672 mm4

E L
Using 12 mm thick plate, outstand of stiffener should not be
greater than 12 t = (12 × 12) = 144 mm
P T
N
We can adopt a plate of 12 mm × 140 mm with I = (12 × 1403)/3 =
10976000 mm4 which is greater than Is = 10668672 mm4

Bridge Engineering
 Design Example: Solution

Shear on welds connecting stiffener to web = (126t2/h)

Where t = web thickness = 14 mm

h = outstand of stiffener = 140 mm

L
Shear on welds connecting stiffener to web = (126t2/h) = (126 × 142)/
E
140 = 176.4 N/mm
P T
N
Size of weld s = [176.4/ (0.7 × 158)] = 1.59 mm

Effective Length of weld should be not less than 10t = (10 × 14) =
140 mm

Provide 160 mm long, 5 mm fillet welds alternately on either side.

Bridge Engineering
 Design Example: Solution

13. End Bearing Stiffeners:

The end bearing stiffener is designed as a column having the
ratio (h/t) not greater than 12.

Where h = outstand of stiffener and t = thickness of stiffener

If h = 300 mm, t = (300/12) = 25 mm

E L
Let us use 300 mm × 25 mm plate for end P
T
N bearing stiffener.

Permissible bearing stress = 0.8f = (0.8 × 250) = 200 N/mm 2

y

Bearing area required = (1868.206 × 103)/200 = 9341.03 mm2

If two plates of 300 mm × 25 mm are used, total area provided =

(300 × 25 × 2) = 15000 mm2 > 9341.03 mm2

Bridge Engineering
 Design Example: Solution

The length of web plate which acts

along with stiffener plates in bearing
reaction is specified as 20 t = (20 ×
14) = 280 mm 280 280

I = (25 × 6143)/12 + (2 × 280 × 143 )/12

E L 14

= 482 × 106 mm4

P T
Area of section Ac = [(614 × 25) +
(560 × 14)] = 23190 mm2
N
Radius of gyration r = √(I/A)= √[(482
× 106)/ 23190] = 144 mm
End Bearing Stiffener

Bridge Engineering
 Design Example: Solution

Effective length of stiffener KL = (0.7 × 1800) = 1260 mm

Effective slenderness ratio KL/r = (1260/144) = 8.75

For built-up sections, buckling class is c.

L
Design compressive stress fcd of axially loaded compression
E
P
10, buckling class c and fy = 250 MPa as 227 MPaT
members can be calculated from Table 9.3 of IS 800: 2007 for KL/r =

N
The design compressive strength P , of a member = A
d × fcd = c
(23190 × 227) = 5264130 N ≈ 5264 kN > Factored design
compression force P = 1868.206 kN

Hence, the design is safe.

Bridge Engineering
Design Example: Solution

E L
P T
N

Bridge Engineering
 Design Example: Solution

Length available for welding using alternate intermittent welds = 2

× (1800 – 50) = 3500 mm

Required strength of weld = [(1868.206 × 103)/3500] = 533.77 N/mm

Size of weld = [533.77/(0.7 × 158) ] = 4.83 mm

E L
both sides. P T
We can use 6 mm fillet welds of 160 mm length intermittently on

Bridge Engineering
 Design Example: Solution

14. Lateral Bracing:

For resisting, wind, racking and centrifugal forces, lateral bracing
is provided at intervals of 6 m along the span.

Wind load for Broad Gauge Bridges = 1.5 kN/m2

Depth of girder = (1.8 + 2 × 0.05) = 1.9 m

E L
P T
Span Length = 30 m
N
Coefficient for wind load on leeward girder = 0.25

Wind load on windward girder = (1.5 × 1.9 × 30) = 85.5 kN

Wind load on leeward girder = (0.25 × 85.5) = 21.375 kN

Bridge Engineering
 Design Example: Solution

Total wind load on windward girder and leeward girder = (85.5 +

21.375) = 106.875 kN

Lateral load due to racking forces = 5.9 kN/m

Total racking force for 30 m span length = (5.9 × 30) = 177 kN

E L
(177 + 106.875) = 283.875 kN ≈ 284 kN P T
Total lateral load on cross bracing due to wind and racking force =

N
This load is shared equally by 2 diagonal cross bracings.

Bridge Engineering
 Design Example: Solution

The plan layout of cross-bracing is shown in figure.

Maximum compression
force in the member U1L1
is P = (284/2) = 142 kN

Maximum tension force

E L
in the diagonal T =
P T
(284/2)/ sin θ = 142/ sin θ
N
Here, sin θ = 1.676/
√(1.6762 + 22) = 0.642

So, T = 142/0.642 = 284/2 = 142 kN

221.18 kN

Bridge Engineering
 Design Example: Solution

Design of Member U1L1:

Factored Design force P = 1.5 × 142 kN = 213 kN (Compression)

Let’s try ISA 80 × 80 × 8 with a cross sectional area Ac = 1220

mm2.

E L
T
The design compressive strength Pd, of a member = Ac × fcd
P
N
Design compressive stress, f of axially
members can be calculated as follows.
cd loaded compression

fcd = (fy/ γm0)/[φ + (φ2 – λ2)0.5]

Where φ = 0.5 × [1 + α × (λ – 0.2) + λ2]

Bridge Engineering
 Design Example: Solution

λ = Non-dimensional effective slenderness ratio = √[fy ×

(KL/r)2/π2E]

KL/r = Effective slenderness ratio or ratio of effective length KL

to appropriate radius of gyration r

α = Imperfection factor
E L
P T
γm0 = Partial Safety Factor for material strength = 1.1
N
For ISA 80 × 80 × 8 section, Radius of gyration r = 24.4 mm

Effective length KL = (0.7 × 1676) = 1173.2 mm

Effective slenderness ratio = (KL/r) = (1173.2/24.4) = 48.08

Bridge Engineering
 Design Example: Solution

For angle sections, buckling class is c as per Table 10 of IS

800:2007.
For buckling class c, Imperfection factor α = 0.49 as per Table 7
of IS 800:2007.

E L
λ = √[fy × (KL/r)2/π2E] = √[(250 × 48.082)/(π2 × 2 × 105)] = 0.54

P T
φ = 0.5 × [1 + α × (λ – 0.2) + λ2] = 0.5 × [1 + 0.54 × (0.54 – 0.2) +
0.542] = 0.74 N
fcd = (fy/ γm0)/[φ + (φ2 – λ2)0.5] = (250/ 1.1)/[0.74 + (0.742 – 0.542)0.5] =
182.41 MPa

Bridge Engineering
 Design Example: Solution

Pd = Ac × fcd = (1220 × 182.41) = 222540.2 N = 222.54 kN > Factored

design compression force P = 213 kN

Hence, the design is safe.

E L
Design of Diagonal Member:
P T
(Tension)
N
Factored Design force P = 1.5 × 221.18 kN = 331.77 kN ≈ 332 kN

Let’s try ISA 100 × 100 × 8 with a cross sectional area Ag = 1540
mm2.

Bridge Engineering
 Design Example: Solution

The design strength of a member under axial tension Td is the

lowest of the design strength due to yielding of gross section,
Tdg, rupture strength of critical section Tdn, and block shear Tdb.

Design strength of a member under axial tension due to yielding

of gross section, Tdg = (Ag × fy)/ γm0
E L
P T
γm0 = Partial safety factor for failure in tension by yielding = 1.1
N
Tdg = (Ag × fy)/ γm0 = (1540 × 250)/ 1.1 = 350000 N = 350 kN

Design rupture strength of critical section Tdn = (0.9An × fu)/ γm1

Bridge Engineering
 Design Example: Solution

An = Net root area at the threaded section = 1540 mm2

γm1 = Partial safety factor for failure governed by ultimate stress

= 1.25

fu = Ultimate stress of the material = 400 MPa (assumed)

E L
443.520 kN P T
Tdn = (0.9An × fu)/ γm1 = (0.9 × 1540 × 400)/ 1.25 = 443520 N =

N
The design strength of a member under axial tension T = 350 kN
d
> Factored design tension force T = 332 kN

Hence, the design is safe.

Bridge Engineering
 Design Example: Solution

15. Cross Frame:

End cross frames and

intermediate cross frames
are provided for spans
greater than 20 m.
E L 1.8 m

Lateral load to be resisted

P T
by one cross frame = 142 kN
142 kN N
Tension in diagonal = 142
secθ = 142 × (2.46/1.676) =
208.42 kN Cross Frame

Bridge Engineering
 Design Example: Solution

Design of Diagonal Member of Cross Frame:

Factored Design force P = 1.5 × 208.42 kN = 312.63 kN ≈ 313 kN

(Tension)

Let’s try ISA 100 × 100 × 8 with a cross sectional area Ag = 1540
mm2.
E L
P T
The design strength of a member under axial tension Td is the
N
lowest of the design strength due to yielding of gross section,
Tdg, rupture strength of critical section Tdn, and block shear Tdb.

Design strength of a member under axial tension due to yielding

of gross section, Tdg = (Ag × fy)/ γm0

Bridge Engineering
 Design Example: Solution

γm0 = Partial safety factor for failure in tension by yielding = 1.1

Tdg = (Ag × fy)/ γm0 = (1540 × 250)/ 1.1 = 350000 N = 350 kN

Design rupture strength of critical section Tdn = (0.9An × fu)/ γm1

L
An = Net root area at the threaded section = 1540 mm2
E
m1
P T
γ = Partial safety factor for failure governed by ultimate stress
= 1.25
N
fu = Ultimate stress of the material = 400 MPa (assumed)

Tdn = (0.9An × fu)/ γm1 = (0.9 × 1540 × 400)/ 1.25 = 443520 N =

443.520 kN

Bridge Engineering
 Design Example: Solution

The design strength of a member under axial tension Td = 350

kN > Factored design tension force T = 313 kN

Hence, the design is safe.

The cross frames are provided at 6 m intervals.

E L
shown in the subsequent figures. P T
Details of the plate girder, lateral bracings and stiffeners are

Bridge Engineering
 Design Example: Solution

ISA 100 ×
100 × 8

End
stiffeners

1800 with

E L
T
66 stiffeners

280
280 N P ISA 80 ×
80 × 8

280 750
560 × 25 800

Details of the Plate Girder

Bridge Engineering
 Design Example: Solution
End Bearing Stiffener 750 750

12
Intermediate
Stiffener 140
750
14 mm 140 140

ISA 100 × 100 × 8

14 14

ISA 80 ×

E L
T
ISA 80 ×
80 × 8 80 × 8

NP
140 750 750
750

Intermediate End Bearing

12
Stiffener Stiffener

Details of Lateral Bracings with Stiffeners

Bridge Engineering
 E L
P T
N

Bridge Engineering
 N. Subramanian, Design of Steel Structures: Limit States Method, Oxford

L
University Press.

E
 N. Krishna Raju, Design of Bridges, Oxford & IBH Publishing Co. Pvt. Ltd.
T
N P
 D.J. Victor, Essentials of Bridge Engineering, Oxford & IBH Publishing Co. Pvt. Ltd.
 S. Ponnuswamy, Bridge Engineering, McGraw Hill Education.
 T.R. Jagadeesh and M.A. Jayaram, Design of Bridge Structures, PHI Learning Pvt.
Ltd.
 W.F. Chen, and L. Duan, Bridge Engineering Handbook, CRC Press, Taylor &
Francis Group.
 G. Parke and N. Hewson, ICE manual of Bridge Engineering, Thomas Telford
Publishing.