Serviceability limit state

Serviceability limit states are verified according to the chapter 7 of EN 1992-1-1. Analysis of serviceability limit states checks the specified service requirements for a structure or structural member during its working life.

Cross-sections are assumed to be cracked for any tensile stress during the analysis of serviceability limit states. The user defined setting is able to consider cross-sections with tensile stress up to the mean value of axial tensile strength f_ctm as uncracked.

Stress limitation

The verification is based on the chapter 7.2. The analysis is performed for the load (combination) type "Characteristic".

The stress level under the characteristic load is be limited due to occurrence of longitudinal cracks. Maximum compressive stress is given by expression

Where is:	k₁	The coefficient, the value is 0.6
	*f_ck*	The characteristic compressive cylinder strength of concrete at 28 days

The verification is done according to the chapter 7.2(2) only for the environmental conditions XD, XF or XS.

Tensile stresses in the reinforcement is limited in order to avoid inelastic strain, unacceptable cracking or deformation. The maximum tensile stress is limited in accordance with 7.2(5) by the formula

Where is:	k₃	The coefficient, the value is 0.8
	*f_yk*	The characteristic yield strength of reinforcement

The ratio of stiffness of reinforcement and concrete may be specified for the design standard EN 1992-2. This ratio may take account of the degradation of modulus of elasticity of concrete due to creep and similar effects. Such procedure may be required by consequent standards (e.g CSN 73 6214, chapter 6).

Crack control

The verification is based on the chapter 7.3. Cracking limitation ensures the proper functioning and durability of the structure and keep the appearance in acceptable state. . The analysis is performed for the load (combination) type "Quasi-permanent".

Crack width is calculated according to the chapter 7.3.4. The crack width w_k is given by formula (7.8):

Where is:	*s_r,max*	The maximum crack spacing
	*ε_sm*	The mean strain in the reinforcement under the relevant combination of loads. Only the additional tensile strain beyond the state of zero strain of the concrete at the same level is considered
	*ε_cm*	The mean strain in the concrete between cracks

Expression ε_sm-ε_cm is given by (7.9):

Where is:	*σ_s*	The stress in the tension reinforcement assuming a cracked section
	*α_e*	The ratio E_s/E_cm
	*k_t*	The factor dependent on the duration of the load. The value is 0.6 for short-term loads and 0.4 for long-term loads

and

Where is:	*A_s*	The area of reinforcing steel
	*A_c,eff*	The effective area of concrete in tension surrounding the reinforcement

The maximum crack spacing s_r,max is given by (7.11):

Where is:	k₁	The coefficient which takes account of the bond properties of the bonded reinforcement. The value for high bond bars is 0.8.
	k₂	The coefficient which takes account of the distribution of strain. The value is 1.0 for pure tension and 0.5 for bending.
	k₃	The coefficient, the value is 3.4
	k₄	The coefficient, the value is 0.425
	c	The cover to the longitudinal reinforcement
	d	The bar diameter. Where a mixture of bar diameters appears, an equivalent diameter is used.

The equivalent diameter d is given by expression (7.12):

Where is:	n₁	The number of bars of the diameter d₁
	n₂	The number of bars of the diameter d₂

The value of maximum crack width w_max is based on the table 7.1N.

Deflection control (only program Concrete Beam)

The deflection is calculated using the exact analysis in accordance the recommendation in 7.4.3(7). First, the curvatures at frequent sections along the member are calculated. This is followed by the calculation of deflection using the numerical integration. The deformation parameters at points where the section isn't fully cracked are obtained using the expression (7.18) that is described in the chapter 7.4.3(3):

Where is:	α	The deformation parameter (e.g. a strain, a curvature, a rotation)
	ζ	The distribution coefficient
	*α_\|*	The parameter calculated for the uncracked conditions
	*α_\|\|*	The parameter calculated for the fully cracked conditions

The distribution coefficient ζ is given by (7.19):

Where is:	ζ	The distribution coefficient
	β	The coefficient taking account of the influence of the duration of the loading. The value is 1.0 for pure tension and 0.5 for bending.
	*σ_sr*	The stress in the tension reinforcement calculated on the basis of a cracked section under the loading conditions causing first cracking
	*σ_s*	The stress in the tension reinforcement calculated on the basis of a cracked section

The curvature due to shrinkage is given by the expression (7.21):

Where is:	*1/r_cs*	The curvature due to shrinkage
	*ε_cs*	The free shrinkage strain
	*α_e*	The effective modular ratio
	S	The first moment of area of the reinforcement about the centroid of the section
	I	The second moment of area of the section

The effective modular ratio α_e is given by:

Where is:	*α_e*	The effective modular ratio
	*E_s*	Tesign value of modulus of elasticity of reinforcing steel
	*E_c,eff*	The effective modulus of elasticity for concrete

The effective modulus of elasticity for concrete E_c,eff is calculated using the formula (7.20):

Where is:	*E_c,eff*	The effective modulus of elasticity for concrete
	*E_cm*	The secant modulus of elasticity of concrete
	*φ(∞,t₀)*	The creep coefficient relevant for the load and time interval

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