Initial calculations of main girder Bridges CE – educational materials for design exercise

Dr Mieszko KUŻAWA Wrocław, March 24, 2015

Faculty of Civil Engineering

Content of project report

1. Design assumptions

a) Objective of the project

b) Basic assumption concerning the design structure

c) Scope of the project

d) Codes, regulations and literature

2. Technical description of entire structure

3. Initial calculations of main girder

4. Detailed calculations of main girder of superstructure

• Cross section

Basic dimensions of analysed structure

• Lateral view / Longitudinal view and static system

3. Initial calculations of main girder of superstructure

A B C D

Analysed sections and influence functions of internal forces

For initial calculations only extreme values of bending moments in critical sections are evaluated using simple continuous beam model.

The most unfavorable location of loads for analyzed girder (A – exterior girder) will be applied according to the influence functions:

• Influence Lines of global bedning Moment in section α-α– IL Mα-α [m],

• Influence Lines of global bending Moment in section β-β – IL Mβ-β [m],

Xα-α = 0,425 Lt P = 1

P = 1

KA,A

KA,B

KA,C KA,D

• Influence Line of Lateral Load Distribution – ILLLD [-] P = 1

ILLLD is calculated using Courbon method which can be applied for relatively transversally stiff multi-girder bridge superstructures.

Characteristic values of ILLLD „i” ordinates are calculated using formula:

2,

1

i

ji

jiy

yy

nK where:

• n – number of girders,

• yi – denotes location of girder,

• yj – denotes location of P force,

ILLLD „i” [-] – function specifying action of

unit force, located in subsequent points of cross

section of span, on investigated girder „i”.

max min

Cross-sectional deformation after loading

Symmetrical part of the cross-sectional deformation after loading

Asymmetric part of the cross-sectional deformation after loading

Źródło [5]

• Principle of the Courbon method

a) The cross section of the span has a vertical axis of symmetry.

b) Beam bending stiffness and their spacing are equal.

c) Problem is static, linear-elastic, the principle of rigid cross-section is valid.

d) In the analyzed cross-section of the span infinitely rigid cross member is located.

e) Mechanical model allowing to analyze the behavior of the cross-section of the span subjected to P force is assumed in the form of an infinitely stiff cross beam with elastic Winkler-type supports.

• Symmetrical part of the cross-sectional deformation

The cross-section has moved evenly (translation) as a rigid body with a vector 𝑢(𝑠) which caused equal reactions in all elastic supports.

Conditions of equilibrium:

Symmetrical part of the cross-sectional deformation after loading

0

0

0

0

)(

M

H

n

PV s Symmetrical part of the cross-sectional deformation after

loading in real multi-girder bridge superstructures

ILLLD „A” [-]

Deformation

• Asymmetric part of the cross-sectional deformation

As a result of infinitely rigid cross beam the cross-section of the span rotated as a rigid body by an angle φ.

The rotation center is located on the vertical axis of symmetry of the system.

• Conditions of equilibrium:

Assuming hinged connection of girders with the cross beam!

• Deformation compatibility condition:

Asymmetric part of the cross-sectional deformation after loading

xPbbM

H

V

aa

1

)(

12

)(

20 20

0

0

1

)(

1

2

)(

2

1

)(

1

2

)(

2

bbb

u

b

u

tgaa

aa

Finally, the exemplary formula for reaction in edge springs if as

follows:

2

2

2

1

2)(

22 bb

bxPa

• In presented example rigid connection of girders with the cross beam can be assumed as well.

• Rotation of the cross beam will cause, in addition to the deflection, also the torsion of all main girders of φ angle.

• Conditions of equilibrium:

SSi

s

i MMyRM

H

V

60

0

0

1

0

Asymmetric part of the cross-sectional deformation after loading

y

z

u

• Uniform (pure) Torsion – ends are free to wrap

• Non-Uniform Torsion – warping deformation is constrained

• Comparison of ILLLD obtained using Courbon method and determined by means of FEM grillage model

• The influence of transverse stiffness of superstructure on load distribution in edge beams

x

yt

EI

EI

a

lZ

3

2

• Stiffness index of grillage

ILLLD „1” [-]

„1” „2” „3” „4”

„1” „2” „3” „4” Z = ∞ 0,70 0,40 0,10 -0,20

Z = 10 0,751 0,346 0,054 -0,151

Z = 1 0,89 0,19 -0,04 -0,03

Z = 0 1,00 0,00 0,00 0,00

• Values of ILLLD „1” in consecutive points „1” – „4”

EIx – bending stiffness of main girder,

EIy – bending stiffness of cross beam.

„1” „2” „3” „4”

„1” „2” „3” „4” Z = ∞ 0,40 0,30 0,20 0,10

Z = 10 0,346 0,363 0,237 0,054

Z = 1 0,19 0,58 0,28 -0,04

Z = 0 0,00 1,00 0,00 0,00

LWRPO „2” [-]

• Values of ILLLD „2” in consecutive points „1” – „4”

x

yt

EI

EI

a

lZ

3

2

• Stiffness index of grillage

EIx – bending stiffness of cross beam,

EIy – bending stiffness of main girder.

• The influence of transverse stiffness of superstructure on load distribution in inner beams

• Nominal weight of barriers – 0,5-1 kN/m of length of single element.

• Nominal density of construction materials (see Annex A of EN 1991-1-1)

Reinforced concrete (superstructure) – 25 kN/m3,

Normal concrete (sidewalks) – 24 kN/m3,

Hot rolled asphalt – 23 kN/m3,

Asphaltic concrete – 24 kN/m3,

Natural stones (curbs) – 27 kN/m3,

Insulation – 14 kN/m3.

For typical RC multi-girder spans nominal values (usually related to mean values or sometimes arbitrary chosen values) of densities can be assumed as characteristic ones:

γμ= γk,inf/sup

Permanent (dead) loads

• Load factors γF for determination of design permanent actions in ULS

Self weight of superstructure and non structural elements (unfavorable actions) – γG, sup = 1,35

Self weight of superstructure and non structural elements (favorable actions) – γG, inf = 1,00

Unfavorable actions

for exemplary

investigated

equilibrium

γf > 1

Favorable actions

for exemplary

investigated

equilibrium

γf ≤ 1

An example of application of the load factors γM to check the equilibrium

Calculations of dead loads acting on girder A

Lp Element Calculations

1.

Girders

(deck

included)

42,81 1,35 57,794 1,00 42,81 iAi Kg ,

KA,A

KA,B

KA,C KA,D

gk – characteristic action on girder A,

gmax – maximum action on girder A,

gmin – minimum action on girder A,

• Action of superstructure (without cross beams) on girder A

kg mkN / 1f maxg 1f ming mkN / mkN /

Lp Element Obliczenia

1. Road

pavement 2,84 1,35 3,834 1,00 2,84 g

• Action of road pavement on girder A

kg mkN / 1f maxg 1f igmin, mkN / mkN /

+ -

Lp Element Obliczenia

1. Barriers 0,50 1,35 0,675 1,00 0,50

ikg , mkN / 1f igmax, 1f igmin, mkN / mkN /

iAi Kg ,

• Action of barriers on girder A

• Calculation of total uniformly distributed dead loads actions on girder A

gk = 56,6 kN/m – Characteristic value of uniformly distributed total dead load acting on girder A,

gmax = 76,4 kN/m – Maximum value of uniformly distributed total dead load acting on girder A,

gmin = 56,6 kN/m – Minimum value of uniformly distributed total dead load acting on girder A,

Nr. Element Calculations gk,i

[kN/m] γ>1

gmax,i

[kN/m] γ=1

gmin,i

[kN/m]

1. Main girders 1,712m2*(0,7+0,4+0,1-0,2)*25kN/m3 42,81 1,35 57,794 1,00 42,81

2. Curbs 0,038m2*(0,550-0,050)*27kN/m3 0,51 1,35 0,6885 1,00 0,51

3. Sidewalks 0,25m*(2,120m-0,989m)*24kN/m3 6,78 1,35 9,153 1,00 6,78

4. Road

pavement 0,09m*(1,419m-0,047m)*23kN/m3 2,84 1,35 3,834 1,00 2,84

5. Barriers 0,5kN/m*(0,850+0,650-0,150-

0,350) 0,50 1,35 0,675 1,00 0,50

6. Edge beams 0,227m2*(0,850-0,350)*24kN/m3 2,73 1,35 3,6855 1,00 2,73

7. Insulation 0,01m*(3,825m-0,74m)*14kN/m3 0,43 1,35 0,5805 1,00 0,43

Total actions gk=56,6 gmax=76,4 gmin=56,6

kNmkNmGk 25,41/25)05,025,055,0(2,2 32

NkNGG fk 69,5535,125,41max

kNkNGG fk 25,420,125,41min

2

_ 2,2 mA BC

• Action of singular cross beam on girder A

Characteristic action of cross beam on girder A

Maximum action of cross beam on girder A

Minimum action of cross beam on girder A

• Location and numbering of the lanes for design

Traffic loads on roadway bridges according to EN 1991-2 C

arr

iag

ew

ay w

idth

– w

3

wIntn

Number of lanes:

Notional Lane Nr.

Notional Lane Nr.

Notional Lane Nr.

Remaining area

The location and numbering of the lanes should be determined in accordance with the following rules :

• The locations of notional lanes should not be necessarily related to the real roadway arrangement of lanes.

• For each individual verification (e.g. for a verification of the ultimate limit state of resistance of a selected girder cross-section to bending), the number of lanes, their location on the carriageway and their numbering should be so chosen independently, that the effects from the load models are the most adverse.

• The lane giving the most unfavourable effect is numbered Lane Number 1, the lane giving the second most unfavourable effect is numbered Lane Number 2, etc.

• Application of the load model LM1 on the individual lanes

Load Model 1 consists of two partial systems:

• Double-axle concentrated loads (tandem system : TS), each axle having the weight αQ∙Qk.

• Uniformly distributed loads (UDL system), having the weight per square metre of notional lane αq∙qk.

Load Model 1 should be applied on each notional lane as well as on the remaining areas.

On notional lane number „i”, the load magnitudes are referred to as:

Qik – magnitude of characteristic axle load on notional lane no i (i = 1, 2...),

qik – magnitude of characteristic vertical distributed load on notional lane no i (i = 1, 2...),

qrk – magnitude of characteristic vertical distributed load on the remaining area of the carriageway,

αQi, αqi – adjustment factors of some load models on lanes i.

Ca

rria

ge

wa

y w

idth

– w

Notional

Lane

Nr.

Notional

Lane

Nr.

Notional

Lane

Nr.

Remaining

area

Remaining

area

Load Model LM1: concentrated and uniformly distributed loads, which cover most of the effects of the traffic of lorries and cars. This model should be used for general and local verifications.

ikiq Qikiq Q

ikqi q

kq q11

kq q22

rkqr q

• Supplementary requirements for double-axle TS load:

No more than one tandem system should be taken into account per notional lane.

Only complete tandem systems should be taken into account.

For the assessment of general effects, each tandem system should be assumed to travel centrally along the axes of notional lanes (see figure below for local verifications).

Each axle of the tandem system should be taken into account with two identical wheels, the load per wheel being therefore equal to 0,5αQ∙Qk.

The contact surface of each wheel should be taken as square and of side 0,40 m.

Application of

tandem systems

for local

verifications

• Uniformly distributed loads (UDL system):

These loads should be applied only in the unfavourable parts of the influence surface, longitudinally and transversally.

• Adjustment factors αQi, αqi, αqr :

The values of adjustment factors should be selected depending on the expected traffic and possibly on different classes of routes.

Values of a factors may correspond, in the National Annex, to classes of traffic.

When they are taken equal to 1, they correspond to a traffic for which a heavy industrial international traffic is expected, representing a large part of the total traffic of heavy vehicles.

Suggested values of α (αQi=αqi= αqr) for design exercise purposes:

Load class A – α = 1,

Load class B – α = 0,8,

Load class C – α = 0,6.

• Load model 1 : characteristic values

• Crowd of pedestrians – characteristic combination value of load with traffic actions

pk = 3,0kN/m2

Vertical displacement of span at L/2 [mm]

Vertical acceleration at L/2 [m/s2]

Bending moment at L/2 [kNm]

• Dynamic behavior of bridge under traffic loads

The static stresses and deformations (and associated bridge deck acceleration) induced in a bridge are increased and decreased under the effects of moving traffic

• The dynamic amplification factor Ф

The dynamic factor Φ takes account of the dynamic magnification of stresses and vibration effects in the structure but does not take account of resonance effects.

stat

dyn

u

u

Identification of displacements used in

definition of Φ based on recorded dynamic

displacements and filtered quasi-static displacements

Definition of dynamic

amplification factor

The dynamic amplification is

included in the load models

of EN for roadway bridges

(fatigue excepted).

In EN code was established

for a medium pavement

quality (see annex B of EN

1991-2) and pneumatic

vehicle suspension.

Ф depends on various

parameters and on the action

effect under consideration.

Therefore, it cannot be

rationally represented by a

unique factor.

• Assessment of groups of traffic loads (characteristic values of the multi-component action)

• Actions of moving loads on girder A

• TS action on girder A:

Characteristic value

Design value

2,221,11max QkQQkQk QQP

kF PP max

• UDL (q1k + q2k + pk) action on girder A:

Characteristic values

Design values

12,221max,11max pkqkqqkqk pqqq

maxmax kF qq

232min pkqkk pqq

minmin kF qq

35,1F

max min

The combination rules, depending on the calculation to be undertaken, shall be in accordance with EN 1990.

Specific rules for the simultaneity with other actions for road bridges, are given in EN 1990:2002, A.2.

• Recommended values of ψ factors for road bridges

• The combination rules

• Maximum bending moment in section α-α acting on girder A

Extreme values of bending moments in critical sections α-α i β-β acting on girder A

mP

mGmGqgqgM

)804,3355,4(

655,0817,0157,2375,3

max

minmax2minmin1maxmaxmax

kNmM 5,4243max

• Minimum bending moment in section β-β acting on girder A

mPmGqgM BB )996,1993,1(925,1546,12 maxmax1maxmaxmin

kNmM BB 0,4607min

• Effective width of concrete slab

In initial calculations possible reduction of concrete slab due to shear lag effect will not be taken into consideration.

Initial calculations of cross section for bending moment

mmmmbm 1,275,06,01,1

201 bbbbm

Effective width effect in compression

plate

a) Flow of compressive forces stream in

the plate,

b) Distribution of compressive stresses

along the upper edge of concrete slab.

Assumed effective width:

Flow of compressive forces stream

• Parameters of the main girder

Concrete C40/50,

Steel A-II class (grade 18G2),

Diameter of longitudinal reinforcing bars,

Diameter of stirrups

Geometry of the main girder:

height h = 1,50m, web width bo = 0,6m, plate width b = 2,1m, plate thicknesss t = 0,25m

Concrete cover (stirrups),

Minimum distance in the light of longitudinal reinforcing bars

Useful height of cross section,

MPafcd 8,28

MPaf yd 0,295

mmmd 032,0322

mmmc 03,030

}5;max{ 2 mmdgdcc vu }30;32max{ mmmmcc vu mmcc vu 32

vcddchh 5,0211

mmh 41,1)032,05,0032,0012,003,050,1(1

mmmd 012,0121

• Dimensioning of section α-α –

Initial layout of Aa reinforcement

1 row can contain 8 bars of 32mm diameter,

2 rows were assumed,

Checking of applied h1 parameter.

Dimensioning for bending moment

kNmM 5,4243max

21max

xhxbfMM cdRd

if x < t – rectangular section

??x

Equilibrium equations of α-α cross section

Assessment of height of compressive area

Assessment of required steel area Aa

xbfAf cdayd ??aA

b = bm !!!

• Dimensioning of section β-β

kNmM BB 0,4607min

b = bo !!!

Thank you for your

attention!