EN 1992-1-1:2004 and EN 1992-2:2005 5 49 The stresses in the concrete in compression are derived from the rectangular

design stress/strain relationship given inEN 1992-1-1 clause 3.1.7 (Figure 5- 7).

Figure 5-7 Rectangular Stress Distribution, Eurocode 2 EN 1992-1-1

The factor λ, defining the effective height of the compression zone, and the factor η, defining the effective strength, follow from:

λ = 0.8 for fck ≤ 50 MPa (EN 1992-1-13.19)

λ = 0.8 − (fck − 50)/400 for 50 < fck ≤ 90 MPa (EN 1992-1-13.20) and

η = 1.0 for fck ≤ 50 MPa (EN 1992-1-13.21)

η = 1.0 − (fck − 50)/200 for 50< fck≤ 90 MPa (EN 1992-1-13.22) 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 5-8 and 5-9). – The initial strain in prestressing tendons is taken into account when assessing

the stresses in the tendons. CSiBridge determines the initial strain by multip- lying the prestressing steel tensile strength fpk by the user-specified factor

εprePT and dividing it by Young’s modulus.

3 cu

ε

s

ε

cd

t

η

A

s

d

x

λx

F

s

A

c

5 - 50 Eurocode 2 EN 1992-1-1:2004 and EN 1992-2:2005

Figure 5-8 Idealized and Design Stress-Strain Diagrams for Reinforcing Steel for Tension and Compression

Eurocode 2 EN 1992-1-1

Figure 5-9 Idealized and Design Stress-Strain Diagrams for

Prestressing Steel, Absolute Values are Shown for Tensile Stress and Strain Eurocode 2 EN 1992-1-1 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

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 5 - 51 – The limit on the mean compressive strain in accordance with EN 1992-1-1, clause 6.1 (5) for a section in concentric loadings is not considered in the CSiBridge algorithm.

5.3.2.2 Algorithms

At each section:

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

slab slabeq slab A t b =

– The tendon and rebar locations, areas, and materials are read. Only bonded tendons are processed; unbonded tendons are ignored.

– The section properties are calculated for the section before skew, grade, and superelevation have been applied. This is consistent with the demands being reported in the section local axis. The entire top and bottom slabs are consi- dered 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 the neutral axis is assumed, and the strains in individual re-

bar and tendons are calculated. Bars and tendons within the concrete com- pression 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 rectangular section. If λx > 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.

5 - 52 Eurocode 2 EN 1992-1-1:2004 and EN 1992-2:2005

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 for bending about horizontal axis 3 only. 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.

5.3.3 Shear Design

The following design parameters are defined by the user in the Design Request (see Chapter 4):

– 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 the 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 – The method 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 – The 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 *

Eurocode 2 EN 1992-1-1:2004 and EN 1992-2:2005 5 - 53 – Effective depth limit – The 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 – The type of section for shear design; options are program determined; prestressed; non-prestressed. If the program determined option is used and at least one bonded tendon (regardless if stressed or not) is defined in the section cut, the section is classified as prestressed.

– Determining Factor ν1 – The method 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 determined as fol- lows:

(

)

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

If the design stress of the shear reinforcement is below 80% of the characte- ristic 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 – The method to calculate the factor αcw . Options are program determined or user defined. If the program determined 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– The user defined value for factor αcw used to take account of compression in the shear area.

5 - 54 Eurocode 2 EN 1992-1-1:2004 and EN 1992-2:2005

– Factor fywk – The multiplier of the vertical shear rebar characteristic yield strength to obtain a stress limit in the shear rebar used in equation (EN 1992- 1-1 6.10aN). The typical value is in the range of 0.8 to 1.0.

– Shear Rebar Material – A previously defined rebar material definition that can be used to determine the required area of transverse rebar in the girder. – Longitudinal Rebar Material – A previously defined rebar material definition

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

5.3.3.1 Variables

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

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 the presence of prestressing ducts in accordance with EN 1992-1-1, clause 6.2.3 (6)

d Effective section depth girder

d Depth of girder

dPTBot Distance from the top fiber to the center of prestressing steel near the bottom fiber

Eurocode 2 EN 1992-1-1:2004 and EN 1992-2:2005 5 - 55

dPTTop Distance from the bottom fiber to the center of prestressing steel near the top fiber

fcd Design compression strength of concrete

fyd Design yield strength of steel reinforcement

fyk Characteristic yield strength of steel reinforcement

MEd Ultimate design moment demand per section cut

NEd Applied factored axial force per section cut, taken as positive if compression

TEd Ultimate design torsion per section cut

VEd Ultimate design shear force demand per section cut excluding the force in the tendons

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

V₂ Shear in section cut excluding force in tendons

2Tot

V Shear in section cut including force in tendons

z Inner arm length

5.3.3.2 Design Process

The shear resistance is determined in accordance with EN 1992-1-1, clause 6.2. The procedure assumes that the concrete shear stresses are distributed uniform- ly over an area b wide and d deep, that the direction of principal compressive stresses (defined by angle θ) remains constant over d, and that the shear strength of the section can be determined by considering the biaxial stress con- ditions at just one location in the web. For design, the user should select only those sections that comply with these assumptions by defining appropriate sta- tion ranges in the Design Request (see Chapter 4).

The effective web width is taken as the minimum web width, measured parallel to the neutral axis. In determining the effective web width at a particular level,

5 - 56 Eurocode 2 EN 1992-1-1:2004 and EN 1992-2:2005

a fraction of the diameter of grouted ducts at that level is subtracted from the web width. The fraction is defined in the design parameter Factor Duct Dia. All defined tendons in a section, stressed or not, are assumed to be grouted. Each tendon at a section is checked for presence in the web, and the minimum controlling effective web thicknesses are evaluated.

The tendon duct is considered to have an effect on the web effective thickness even if only part of the duct is within the web boundaries. In such cases, the en- tire fraction of the tendon duct diameter is subtracted from the element thick- ness.

If several tendon ducts overlap in one web (when projected on the vertical axis), the diameters of the ducts are added for the sake of evaluation of the ef- fective thickness. The effective web thickness is calculated at the top and bot- tom of each duct.

The Shear and Torsion Design is completed on a per web basis. The D/C ratio is calculated and the required area of rebar is reported for each web. The sec- tion design shear force is distributed into individual webs assuming that the vertical shear that is carried by a web decreases with increased inclination of the web from vertical. Section torsion moments are assigned to external webs and slabs.

The rebar area and ratio are calculated using measurements normal to the web. Thus, vertical shear forces are divided by cos αweb. The rebar area calculated is the actual, normal cross-section of the bars.

5.3.3.3 Algorithm

 All section properties and demands are converted from CSiBridge model units to N, mm.

 For every COMBO specified in the Design Request that contains envelopes, a new force demand set is generated. The new force demand set is built up from the maximum tension values of P and the maximum absolute values of V2 and M3 of the two StepTypes (Max and Min) present in the envelope COMBO case. The StepType of this new force demand set is named ABS and the signs of the P, V2, and M3 are preserved. The ABS case follows the industry practice where sections are designed for extreme shear and moments

Eurocode 2 EN 1992-1-1:2004 and EN 1992-2:2005 5 - 57

In document Bridge Superstructure Design (Page 129-137)