# Serviceability Limit State (Concrete Beam)

# Method of solution

Calculation at serviceability limit state starts with a calculation of moments and deflections using uncracked stiffness with load factors for total load. The analyses are performed using load combinations as described in Analysis.

Using the moment values the beam will be checked in the serviceability limit state for cracking and new stiffness values are decided at 20 section per span. A new analysis is performed using the new stiffness’s and new moments and deflection values are being calculated. The cracked calculation is repeated if necessary, see below.

The same calculation steps will be used in calculation of long term loading although here factors for long term load will be used. In long term analysis a section, which was cracked due to moments of total load, will also be considered cracked for long term moment regardless of this moment size.

# Cracking

## Cracking width calculation

The crack width wk may be calculated from:

- w
_{k}= S_{r,max}(ε_{sm}- ε_{cm})

where:

- S
_{r,max}is the maximum crack spacing - ε
_{sm}is the mean strain in the reinforcement - ε
_{cm}is the mean strain in the concrete between cracks

ε_{sm} - ε_{cm} may be calculated from:

- ε
_{sm}- ε_{cm}= [σ_{s}- k_{t}f_{ct,eff}/ ρ_{p,eff}(1 + α_{e}ρ_{p,eff})] / E_{s}0,6 σ_{s}/ E_{s}

where:

- σ
_{s}is the stress in the tension reinforcement assuming cracked section - α
_{e}is the ratio E_{s}/ E_{cm} - f
_{ct,eff}is the mean value of the tensile strength of the concrete when the first crack occur - f
_{ct,eff}= f_{ctm} - ρ
_{p,eff}= (A_{s}+ ξ_{1}^{2}A_{p}') / A_{c,eff}- A
_{p}' is the area of pre or post-tensioned tendons within A_{c,eff} - A
_{c,eff}is the effective area of concrete as calculated below - ξ
_{1}is the adjusted ratio of bond strength as calculated below

- A
- k
_{t}is a factor dependent on the duration of the load- k
_{t}= 0,6 for short term loading - k
_{t}= 0,4 for long term loading.

- k

**Effective area of concrete A _{c,eff}**

A_{c,eff} is the effective area of concrete of depth h_{c,ef},

where

- h
_{c,ef}is the lesser of:- 2,5 (h - d)
- (h - x) / 3
- h / 2

## Adjusted ratio of bond strength

ξ_{1} is the adjusted ratio of bond strength taking into account the different diameters of prestressing and reinforcing steel:

- ξ
_{1}= (ξ ø_{s}/ ø_{p})^{0,5}

where:

- ξ is the ratio of bond strength according to the below table
- ø
_{s}is the largest bar diameter - ø
_{p}is equivalent diameter of prestressing steel

## Crack spacing S_{r,max}

For bonded reinforcement with spacing 5(c + ø/2) the crack spacing is calculated as:

- S
_{r,max}= k_{3 }c + k_{1}k_{2}k_{4 }ø/ ρ_{p,eff}

where:

- ø is the bar diameter in mm. If more than one bar size is present an average bar size ø
_{eq}should be used, - ø
_{eq}= (n_{1}ø_{1}^{2}+ n_{2}ø_{2}^{2}) / (n_{1}ø_{1}+ n_{2}ø_{2}) - c is the cover to the longitudinal reinforcement,
- k
_{1}= 0,8 for high bond bars, 1,6 for plain bars (e.g. prestressing tendons), - k
_{2}= 0,5 for bending, 1,0 for pure tension, - k
_{3}= 3,4 - k
_{4}= 0,425 - ρ
_{p,eff}as above.

For not bonded reinforcement or reinforcement with spacing > 5 (c + ø / 2) the crack spacing is calculated as:

- S
_{r,max}= 1,3 (h - x)

where:

- x is the neutral axis depth

The crack spacing should be calculated in the direction of the principle tensile stress as:

- S
_{r,max}= 1 / (cos θ / S_{r,max,y}+ sin θ / S_{r,max,z})- θ is the angle between the reinforcement in the y-direction and the direction of the principal tensile stress,
- S
_{r,max,y}and S_{r,max,z}are the crack spacing’s calculated in the y and z directions respectively.

# Deflections

The calculation of the deflection of the beam is done iteratively and begins with that the average curvature is calculated using un-cracked cross-section in 20 sections per span. The calculation is performed under short term conditions using total load respectively under long-term conditions using long term load.

Using the moment values the beam will be checked for cracking and new effective stiffness values are decided at 20 sections per span using the curvatures. A new analysis is performed using the new stiffness’s and new moments and deflection values are being calculated. The cracked calculation is repeated until stiffness values are stabilized in such a way that the difference in deflection between two consecutive calculations is less then the user defined percentage of the first calculation. The calculation is also terminated when the user defined number of iteration steps is reached.

In long term calculation a section, which was cracked due to moments of total load, will also be considered cracked for long term moment regardless of this moment size.

# Effective stiffness

Calculation of curvature is normally done using the relation:

ψ = - M / EI

where:

- ψ is the curvature
- M is the moment
- E is the modulus of elasticity
- I is the moment of inertia.

In un-cracked state under short term condition:

E = E_{c,ef}

and I is moment of inertia of the composed section, i.e. concrete section adjusted for reinforcement according to:

- I = I
_{c}+ α I_{s}

where:

- I
_{c}is moment of inertia for concrete section with reference to centre-of-gravity of composed section, - I
_{s}is moment of inertia of reinforcement with reference to centre-of-gravity of composed section, - α is the quotation E
_{s}/ E_{c,ef}

At cracked analysis consideration must be taken to un-cracked parts of the beam between the cracks and under long term conditions also curvature changes due to creep and shrinking must be considered.

It is however recognized from geometry that:

- ψ = (ε
_{t}- ε_{b}) / H

where:

- ε
_{t}is strain in top of section - ε
_{b}is strain in bottom of section - H is section height.

Thus by calculating the strains in section it is easy to calculate the curvature generally. We get:

- E
_{Ieff}= M / ψ

and observe that this equation can be used if the curvature ψ is calculated using strains which include effects of cracking, creeping and shrinkage. Furthermore, if we mix un cracked and cracked calculation results using crack coefficient ζ we get:

- ψ = ψ
_{cracked}+ (1-ζ) ψ_{uncracked}

where:

- ζ is a distribution coefficient.
- ζ = 1 - β (σ
_{sr}/ σ_{s})^{2}

or for flexure: - ζ = 1 - β (M
_{cr}/ M)^{2}

All parameters are according to the crack width calculation described above:

- ζ is zero for the un-cracked condition,
- β is a coefficient taking account of the influence of the duration of the loading or of repeated loading on the average strain,
- β = 1,0 for a single short-term loading,
- β = 0,5 for sustained loads or many cycles of repeated loading,

- σ
_{s}is the stress in the tension reinforcement calculated on the basis of a cracked section - σ
_{sr}is the stress in the tension reinforcement calculated on the basis of a cracked section under the loading causing first cracking

**Creep**

Creep effects are considered by reducing the modulus of elasticity:

- E
_{c,eff}= E_{cm}/ (1 + φ)

where:

- φ is the creep coefficient.

**Shrinkage**

Effects of shrinkage are considered as:

- 1 / r
_{cs}= ε_{cs}α_{e}S / I

where:

- 1 / r
_{cs}is the curvature due to shrinkage - ε
_{cs}is the free shrinkage strain - α
_{e}is the effective modular ratio E_{s}/ E_{c,eff} - S is the first moment of area of both the top and bottom reinforcement about the center of gravity of the section
- I is the moment of inertia