Eurocode 2 EN 1992-1-1:2004 and EN 1992-2:

This section describes the algorithms applied in accordance with the Eurocode 2 EN 1992-1-1:2004 and EN 1992-2:2005 for design of superstructure decks that includes cast-in-place multi-cell concrete box.

For MulticellConcBox design in CSiBridge, each web and its tributary slabs are designed separately. Moments and shears due to live load are distributed to individual webs in accordance with the live load distribution method specified in the Design Request (Chapter 4). Torsion effects are ignored.

6.3.1 Stress Design

The following design parameters are defined by the user in the design request:

– FactorCompLim – fck multiplier; Default Value = 0.6. The fck is multiplied by the FactorCompLim to obtain concrete compression limit.

– FactorTensLim - fctk multiplier; Default Value = 0.4. The fctk is multiplied by the FactorTensLim to obtain concrete tension limit.

The stresses are evaluated at three points at the top fiber of the top slab and three points at the bottom fiber of the bottom slab: the left corner, the center- line web, and the right corner of the relevant slab tributary area. The locations are labeled in the output plots and tables.

Concrete compressive and tensile strengths are read at every point, and com- pression and tension limits are evaluated using the FactorCompLim - fck mul- tiplier and FactorTensLim - fctk multiplier.

6 - 28 Eurocode 2 EN 1992-1-1:2004 and EN 1992-2:2005

The stresses assume linear distribution and take into account axial (P) and ei- ther both bending moments (M2 and M3) or only P and M3, depending on which method for determining LLDF has been specified in the design request (see Chapters 3 and 4).

The stresses are evaluated for each demand set (Chapter 4). If the demand set contains live load, the program positions the load to capture extreme stress at each of the evaluation points.

Extremes are found for each point and the controlling demand set name is rec- orded.

6.3.2 Flexure Design

The following design parameters are defined by the user in the Design Request: – γc– Partial safety factor for concrete; Default Value = 1.5.

– γsreb– Partial safety factor for reinforcing steel; Default Value = 1.15. – γsPT– Partial safety factor for prestressing steel; Default Value = 1.15.

– εprePT– Factor to estimate pre-strain in PT. Multiplies fpk to obtain stress in tendons after losses. Typical values between 0.4 and 0.9.

6.3.2.1 Design Process

The derivation of the moment resistance of the section is based on assumptions specified in Section 6.1:

– Plane sections remain plane.

– The strain in bonded reinforcement or bonded prestressing tendons, whether in tension or in compression, is the same as that in the surrounding concrete. – The tensile strength of the concrete is ignored.

– The stresses in the concrete in compression are derived from the rectangular design stress/strain relationship given in EN 1992-1-1 clause 3.1. (Figure 6.1). The factor λ, defining the effective height of the compression zone and the factor η, defining the effective strength, follow from:

Eurocode 2 EN 1992-1-1:2004 and EN 1992-2:2005 6 - 29

λ = 0.8 for fck ≤ 50 MPa (EN 1992-1-1 3.19)

λ = 0.8 − (fck − 50)/400 for 50 < fck ≤ 90 MPa (EN 1992-1-1 3.20) and

η = 1.0 for fck ≤ 50 MPa (EN 1992-1-1 3.21)

η = 1.0 − (fck -50)/200 for 50< fck ≤90 MPa (EN 1992-1-1 3.22)

Figure 6-1 Rectangular Stress Distribution, Eurocode 2 EN 1992-1-1

– The stresses in the reinforcing or prestressing steel are derived from the de- sign curves in EN 1992-1-1, Figures 3.2 and 3.3 (Figures 6.2 and 6.3).

Figure 6-2 Idealized and Design Stress-Strain Diagrams for Reinforcing Steel (for Tension and Compression)

Eurocode 2 EN 1992-1-1:2004

A

c

A

s

F

s

d

x

λx

3 cu

ε

s

ε

cd

t

η

B A B A Idealized Design

σ

(

t _{y k}

)

k= f f yk s kf γ yk kf yd s f E εud εuk ε yd yk s f = f γ yk f yk kf

6 - 30 Eurocode 2 EN 1992-1-1:2004 and EN 1992-2:2005

Figure 6-3 Idealized and Design Stress-Strain Diagrams for

Prestressing Steel (Absolute Values are Shown for Tensile Stress and Strain) Eurocode 2 EN 1992-1-1:2004

– The initial strain in prestressing tendons is taken into account when assessing the stresses in the tendons. CSiBridge determines the initial strain by multiplying the prestressing steel tensile strength fpk by thr user specified factor εprePT and dividing it by Young’s modulus

– The limit on mean compressive strain in accordance with EN 1992-1-1, clause 6.1 (5) for sections in concentric loadings is not considered in the CSiBridge algorithm.

6.3.2.2 Algorithms

At each section and each web:

– The equivalent slab thickness is evaluated based on the slab tributary area and the slab width assuming a rectangular shape.

slab slabeq slab A t b =

– The tendon and rebar location, area, and material are read. Only bonded ten- dons are processed; unbonded tendons are ignored.

B

A

B A Idealized Design

σ

pk s k γ pd p f E εud εuk ε 0.1 pd p k s f = f γ 0.1 p k f pk k

Eurocode 2 EN 1992-1-1:2004 and EN 1992-2:2005 6 - 31 – The section properties are calculated for the section before skew, grade, and

superelevation are applied. This is consistent with the demands being re- ported in the section local axis. The entire top and bottom slab tributary areas are considered as effective in compression.

The ultimate moment resistance of a section is determined using the strain compatibility method and an iterative approach. The following steps are used: 1) The position of neutral axis is assumed, and strains in individual rebars and

tendons are calculated. Bars and tendons within the concrete compression zone are ignored.

2) The distance x from the extreme compression fiber to the neutral axis is compared to the equivalent slab thickness tslabeq to determine if the section is a T-section or a rectangular section. If λ x > tslabeq, the section is a T-section.

3) The steel stresses appropriate to the calculated steel strains are calculated from the stress-strain idealization.

4) The concrete stresses appropriate to the strains associated with the assumed neutral axis depth are calculated from the stress-strain idealization.

5) The net tensile and compressive forces at the section are calculated. If these are not equal (the acceptance criterion is abs F

{

conc−

[

Frebar +FPT

]}

<=0.001*Fconc), the neural axis depth is adjusted accordingly, and the procedure returns to Step 1.

6) When the net tensile force is equal to the net compressive force, the mo- ments are taken about the center of gravity of the concrete compressive block to determine the ultimate moment resistance.

The resistance is evaluated only for bending about horizontal axis 3. Separate capacity is calculated for positive and negative moment. The capacity is based on bonded tendons and mild steel located in the tension zone as defined in the Bridge Object. Tendons and mild steel reinforcement located in the compres- sion zone are not considered. It is assumed that all defined tendons in a section, stressed or not, have effective stress after loses equal to εprePT * fpk. If a certain tendon should not be considered for the flexural capacity calculation, its area must be set to zero.

6 - 32 Eurocode 2 EN 1992-1-1:2004 and EN 1992-2:2005

6.3.3 Shear Design

The following design parameters are defined by the user in the design request: − γc– Partial safety factor for concrete; Default Value = 1.5.

− γsreb– Partial safety factor for reinforcing steel; Default Value = 1.15. − γsPT– Partial safety factor for prestressing steel; Default Value = 1.15. − angle θ - The angle between concrete compression strut and the beam axis

perpendicular to the shear force.

− Factor Duct Dia - Factor that multiplies PT duct diameter when evaluating effective web thickness bw in accordance with EN 1992-1-1, clause 6.2.3 (6).

− αl - Factor for the transmission length of PT, used in shear resistance equa- tion (EN 1992-1-1 6.4).

− Inner Arm Method - Method that will be used to calculate the inner lever arm z of the section. Options are based on defined PT; based on defined rebar; based on defined PT and rebar; multiplier of section depth.

− Inner Arm Limit - Factor that multiplies the depth of the section to get the lower limit of the inner lever arm z of the section. (z ≥ Inner Arm Limit *

Section Depth).

− Effective depth limit - Factor that multiplies the depth of the section to get the lower limit of the effective depth to the tensile reinforcement d of the section (d = Effective depth limit * Section Depth).

− Type of section – Type of section for shear design; options are program de- termined; prestressed; non-prestressed. If the program determined option is used and at least one bonded tendon (regardless if it is stressed or not) is defined in the section cut, the section is classified as prestressed.

− Determining Factor ν1 - Method that will be used to calculate the factor ν1; options are program determined or user defined. If the program determined option is used, the algorithm assumes the factor ν1 = ν; where ν is deter- mined as follows:

Eurocode 2 EN 1992-1-1:2004 and EN 1992-2:2005 6 - 33

(

)

0.6 1 in MPa 250 ck ck f v= _ − _ f  

If the design stress of the shear reinforcement is below 80% of the charac- teristic yield stress fyk, ν1 is taken as:

ν1 = 0.6 for fck ≤ 60 MPa (EN 1992-1-16.10.aN)

ν1 = 0.9 – fck / 200 > 0.5 for fck ≥ 60 MPa (EN 1992-1-16.10.bN) − Factor ν1 – user defined value of factor ν1

− Determining Factor αcw - Method that will be used to calculate the factor

αcw . Options are program determined or user defined. If the program de- termined option is used, the algorithm assumes the factor αcw as follows:

(

)

(

)

1.0 for non-prestressed structures

1 for 0 0.25 1.25 for 0.25 0.5 2.5 1 for 0.5 1.0 cp cd cp cd cd cp cd cp cd cd cp cd f f f f f f f + σ < σ ≤ < σ < − σ < σ ≤

− Factor αcw- User defined value for factor αcw used to take account of com- pression in the shear area.

− Factor fywk - Multiplier of vertical shear rebar characteristic yield strength to obtain a stress limit in shear rebar used in equation (EN 1992-1-1 6.10aN). Typical values 0.8 to 1.0.

− Shear Rebar Material – A previously defined rebar material label that will be used to determine the required area of the transverse rebar in the girder. − Longitudinal Rebar Material - A previously defined rebar material label

that will be used to determine the required area of longitudinal rebar in the girder.

6.3.3.1 Variables

Ak Area enclosed by the centerlines of the connecting exterior webs and top and bottom slabs, including inner hollow area

6 - 34 Eurocode 2 EN 1992-1-1:2004 and EN 1992-2:2005

Arebarbot, Arebartop Area of reinforcing steel on the flexural tension side of the member

APTbot, APTtop Area of prestressing steel on the flexural tension side of the member

Ast Area of required closed transverse torsion reinforcement per unit length in accordance with EN 1992-1-1, clause 6.3 (3)

Asw Area of transverse shear reinforcement per unit length

Aswmin Minimum area of transverse shear reinforcement per unit length in

accordance with EN 1992-1-1, clause 9.2.2 (5)

b Minimum web width

bw Effective web width adjusted for presence of prestressing ducts in accordance with EN 1992-1-1, clause 6.2.3 (6)

d Effective section depth girder

d Depth of the girder

dPTbot Distance from the top fiber to the center of the prestressing steel near

the bottom fiber.

dPTtop Distance from the bottom fiber to the center of the prestressing steel

near the top fiber

fcd Design compression strength of the concrete

fyd Design yield strength of the steel reinforcement

fyk Characteristic yield strength of the steel reinforcement

MEd Ultimate design moment demand

NEd Applied factored axial force, taken as positive in compression

VEd Ultimate design shear force demand per web excluding force in ten- dons

Eurocode 2 EN 1992-1-1:2004 and EN 1992-2:2005 6 - 35

Vp Component in the direction of the applied shear of the effective pre- stressing force; if Vp has the same sign as VEd, the component is re- sisting the applied shear.

c

In document Bridge Superstructure Design (Page 168-176)