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 + ξ₁² 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
  • ξ₁ is the adjusted ratio of bond strength as calculated below
• 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.

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

ξ₁ is the adjusted ratio of bond strength taking into account the different diameters of prestressing and reinforcing steel:

• ξ₁ = (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 = k₃ c + k₁ k₂ k₄ 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 = (n₁ ø₁² + n₂ ø₂²) / (n₁ ø₁ + n₂ ø₂)
• c is the cover to the longitudinal reinforcement,
• k₁ = 0,8 for high bond bars, 1,6 for plain bars (e.g. prestressing tendons),
• k₂ = 0,5 for bending, 1,0 for pure tension,
• k₃ = 3,4
• k₄ = 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