Direct deformation analysis of a reinforced concrete beam, considering the long-term effects of creep and shrinkage

This technical article presents a direct deformation analysis of a reinforced concrete beam, additionally considering the long-term effects of creep and shrinkage. An example is used to show how these effects influence the deformation of a structural component and are taken into account in the calculation. It explains which data are necessary in RFEM 6 to correctly consider all relevant factors and how the distribution factor affects the stiffness of the structural component.

Input Data

Geometry, reinforcement, and loading are described by the following parameters:

system

• Beam type: Single-span beam
• Span length: l = 4.210 m

Cross-section

• Slab thickness: h = 7.8 in
• Plate width b = 100 cm
• Material: Concrete C20/25 with E_cm = 30.000 MN/m² and B 500A
• Reinforcement: A_{s,-z,(bottom)} = 4.45 cm² with 7 ∅ 9 and d₁ = 30 mm
• Effective depth of bottom reinforcement: d_{def,+z (bottom)} = 17 cm

Permanent loads

• Self-weight: g_s = 0.20 m ⋅ 1 m ⋅ 25 kN/m³ = 5.00 kN/m
• Plaster and flooring: g_bp = 1.50 kN/m
• Total: g_k,total = 6.5 kN/m

Variable loads

• Live load (Office): q_b = 2.00 kN/m with ψ₂ = 0.3
• Partition compensation: q_t = 1.25 kN/m with ψ₂ = 1.0

Quasi-permanent load

• 6.5 kN/m + 0.3 ⋅ 2.00 kN/m + 1.0 ⋅ 1.25 kN/m = 8.35 kN/m

Design bending moment for deflection calculation

• My_,Ed,def = 8.35 kN/m ⋅ (4.21 m)²/8 = 18.50 kNm

Initial Values of Deformation Analysis

• Mean modulus of elasticity of concrete: E_cm = 30,000 MN/m²

• Longitudinal Reinforcement Ratio: ρ = A_s/A_c = 4.45 cm²/(20 cm ⋅ 100 cm) = 0.223%

• Shrinkage strain: ε_sh = -0.5‰
• Creep coefficient: φ = 2

It is necessary to activate the "Advanced time-dependent properties of concrete" option in the cross-section settings in order to specify the creep coefficient by user-defined settings.

In the tab now available, select the “Creep” and “Shrinkage” options first in order to view and edit the “Basic Values of Time-Dependent Properties”. The creep coefficient φ has been specified by entering φ₀, ϵ_cd,0, and ϵ_ca(∞).

Creep

The creep effects are determined by a reduction of the concrete's modulus of elasticity E_c.

The effective modulus of elasticity E_c,eff takes into account the long-term effects of the concrete, especially the creep. The creep describes the long-term strain of the concrete under a constant load. The creep coefficient φ reduces the modulus of elasticity E_cm (mean modulus of elasticity of concrete) so that the actual stiffness of concrete is represented over a long period of time. This value is used in further calculations, such as for the moment of inertia or the ratio of stiffnesses.

E_c,eff = 30,000 MN/m²/( 1 + 2 ) = 10,001.2 MN/m²

Effective shear modulus of concrete G_c,eff
The effective shear modulus describes the concrete's resistance to shear deformations and is determined with the ratio of the transversal strain to the longitudinal strain (the Poisson's ratio v of concrete). This value is especially important for the calculation of cross-section deformations and for shear design checks.

G_c,eff = 10,001.2 MN/m²/( 2 ⋅ ( 1 + 0.2 ) ) = 4,167.180 MN/m²

Effective modular ratio for uncracked state (long-term loading) α_e,l
The ratio α_e,l indicates how much stiffer the steel is in comparison with the concrete under long-term loading. E_s is the modulus of elasticity of steel, E_c,l is the effective modulus of elasticity of concrete in the uncracked state (identical to E_c,eff ). Since concrete has a lower stiffness due to long-term effects such as creep, the value of α_e,l is higher in this state. This ratio is used in the calculation of the center of gravity and the effective section properties.

α_e,l = 2 ⋅ 10⁵ MN/m²/10,001.2 MN/m² = 20

Effective modular ratio for uncracked state (short-term loading) α_e,I,st
The ratio α_e,I,st describes the ratio of the stiffness of steel to concrete under short-term loading. In contrast to α_e,l, the mean modulus of elasticity E_cm is used here without considering creep effects. This reflects the actual load situation when the concrete is only loaded for a short time. This value is particularly relevant for the design of short-term loadings.

α_e,I,st = 2 ⋅ 10⁵ MN/m²/30,000 MN/m² = 6.67

Effective modular ratio for cracked state α_e,II
In the cracked state, the concrete in the tension zone is not considered as load-bearing. The ratio α_e,II takes this into account by only including the concrete's effective modulus of elasticity E_c,eff. This value shows that the stiffness of the steel is higher in the cracked state compared to the concrete, underlining the importance of the reinforcement in such cases.

α_e,II = 2 ⋅ 10⁵ MN/m²/10,001.2 MN/m² = 20.00

Geometric Parameters uncracked

The centroidal distance of the ideal cross-section in the uncracked state under long-term loading, z_I , describes the position of the centroid, taking into account the concrete area and the reinforcement. The effect of the reinforcement is scaled by a conversion factor α_e,l, which represents the ratio between the modulus of elasticity of steel and the effective modulus of elasticity of concrete. This is particularly important because long-term loading, such as creep, weakens the concrete. The center of gravity affects the calculation of moments and deformations in the cross-section and is thus a central parameter for the structural analysis.

The effective area of section in uncracked state at long-term loading, A_I , represents the effective area bearing loads. In addition to the concrete area, the reinforcement area is also taken into account, which is supplemented by the factor α_e,l. This way, the cross-section stiffness is represented more realistically. This value is decisive for the evaluation of the load-bearing capacity and for the calculation of the deformation of the structural component.

A_I = 1,000 mm ⋅ 200 mm + 20 ⋅ (4.45 cm² + 0 cm²) = 2,089.05 cm²

The effective moment of inertia of the ideal center of gravity in the uncracked state under long-term loading,_II , describes the cross-section's resistance to bending. It takes into account both the concrete surface and the reinforcement, although the latter generates additional moments due to its position relative to the centroid. This moment of inertia is a key factor in the deformation analysis and it shows how much bending moments the cross-section can withstand.

The eccentricity of the ideal center of gravity of the cross-section in the uncracked state, e_I , indicates the deviation of the center of gravity from the geometric center of the cross-section. This eccentricity is important since it affects the moments developed in the cross-section, which directly affect the deformations.

e_I = 103 mm - 200 mm/2 = 3 mm

The centroidal distance of the ideal cross-section in the uncracked state under short-term loading, z_I,st describes the centroidal position under loadings that do not consider the creep or shrinkage effects. Therefore, the conversion factor α_e,I,st used in the short-term calculation is smaller than for the long-term loadings. This centroidal distance is decisive for the distribution of loads and the determination of moments in short-term loadings.

The effective section area in the uncracked state at short-term loading, A_I,st , is similar to the area A_I but adjusted by the conversion factor α_e,I,st, which does not account for long-term effects. This will result in a smaller area and have an impact on the calculation of resistance for short-term loads.

A_I,st = 1,000 mm ⋅ 200 mm + 6.67 ⋅ (4.45 cm² + 0 cm²) = 2,029.69 cm²

The effective moment of inertia of the ideal center of gravity in the uncracked state under short-term loading, I_I,st, represents the cross-section’s resistance to bending without the influence of long-term effects. It takes into account both the concrete and the reinforcement area and their distances from the centroid, which are crucial for calculating the deformation under short-term loads.

Geometrical parameters cracked

The centroidal distance of the ideal cross-section in the cracked state, z_II , takes into account the modified resistance of the cross-section since the concrete tension zone no longer supports any loads after crack formation. The centroid location will be recalculated taking into account only the concrete's compression zone and reinforcement. This parameter is central to the analysis of the cross-section after the crack and affects the resistance and deformation.

The effective cross-section area in the cracked state, A_II , represents the remaining area after crack formation. Here, only the concrete compression zone and the reinforcement area are considered, which greatly reduces the stiffness of the cross-section. This value is crucial for the ultimate limit state design in case of cracked cross-sections.

A_II = 1,000 mm ⋅ 46.8 mm + 20 ⋅ (4.45 cm² + 0) = 557.41 cm²

The effective moment of inertia of the ideal centroid in the cracked state, I_II , describes the resistance to bending after cracking. Since the tension zone is no longer load-bearing, the moment of inertia is reduced significantly. This value is an essential factor for calculating the deformation and for evaluating the resistance of cracked sections.

The eccentricity of the ideal cross-section's center of gravity in the cracked state, e_II , describes the shift of the center of gravity due to the crack formation. This displacement affects the resulting moments and the deformation of the cross-section, and is therefore an important parameter for the structural analysis.

e_II = 46.8 mm - 200 mm/2 = -53.2 mm

Shrinkage

The axial force due to shrinkage, N_sh , arises because the reinforcement does not carry the strain of the concrete caused by the shrinkage and thus absorbs the forces. These forces result from the interaction between the concrete tension force and the reinforcement's response. The calculated value shows how heavily the reinforcement is loaded by shrinkage. At this point, the shrinkage strain ε_sh = -0.5‰, which is user-defined indirectly, is used.

N_sh = -2 ⋅ 10⁵ MN/m² ⋅ ( -0.000.5 ) ⋅ ( 4.45 cm² + 0.00 ) = 44.532 kN

The eccentricity of the shrinkage force to the center of gravity of the ideal cross-section in the uncracked state, e_sh,I , describes the position of the resulting shrinkage force relative to the center of gravity of the cross-section. A larger eccentricity leads to higher moments and larger deformations.

e_sh,I = (4.45 cm² ⋅ 170 mm + 0)/(4.45 cm² + 0) - 103 mm = 67 mm

The shrinkage moment for uncracked state M_sh,I results from the shrinkage force N_sh and the eccentricity e_sh,I. It shows how the shrinkage force creates a moment through its effect on the cross-section. This moment has a significant influence on the deformations and stresses in the cross-section, and must be considered in the design.

M_sh,I = 44.532 kN ⋅ 67 mm = 2.98 kNm

The curvature coefficient for uncracked state k_sh,I indicates how the shrinkage moment acts in relation to the axial force and eccentricity. It shows how the distribution of the shrinkage force and the centroid position influence the deformations of the structural component. This value is crucial to completely describe the deformations of the cross-section due to shrinkage.

k_sh,I = ( 2.98 kNm + 18.5 kNm - 0 )/( 18.50 kNm – 0 ) = 1.161

The eccentricity of the shrinkage force to the center of gravity of the ideal cross-section in the cracked state, e_sh,II , describes the position of the resulting shrinkage force in relation to the center of gravity of the cross-section in the cracked state. The area moments of the reinforcement, A_{s,def, +z,(bottom)} and A_{s,def, -z,(top)}, are determined with respect to their position, d_{ef, +z,(bottom)} and d_{ef, -z,(top)} and divides it by the total reinforcement area. The centroidal distance of the cracked cross-section, z_II, is subtracted from the result. This eccentricity affects the shrinkage moment, because a larger eccentricity leads to a larger moment.

e_sh,II = (4.45 cm² ⋅ 170 mm + 0)/(4.45 cm² + 0) – 46.8 mm = 123.2 mm

The bending moment by the axial force N_sh for the cracked state M_sh,II is obtained by multiplying the shrinkage force N_sh by the previously calculated eccentricity e_sh,II. This moment describes the additional bending stress acting on the cross-section due to the shrinkage force. This value is particularly relevant in the cracked state, where the concrete tension zone no longer supports any loads.

M_sh,II = 44.532 kN ⋅ 123.2 mm = 5.48 kNm

The curvature coefficient for the cracked state, k_sh,II , indicates how strongly the deformation of the cross-section is influenced by the shrinkage moment and the other acting forces. The shrinkage moment M_sh,II, the existing bending moment My_,Ed,def, as well as the axial force N_Ed and its eccentricity e_II are taken into account. The calculation places the resulting moment in proportion to the moment without shrinkage and thus provides a measure for the influence of the shrinkage force.

k_sh,II = ( 5.48 kNm + 18.50 kNm - 0 )/( 18.50 kNm - 0 ) = 1.296

Deformation of the cross-section

A cross-section deformation is the curvature of a structural component caused by external actions, taking into account its material and state parameters.

The calculation of the deformation of the cross-section in the uncracked state, κ_I , describes the curvature of the cross-section caused by the shrinkage moment and the elastic properties of the material. The shrinkage moment My_,Ed,def is taken into account as well as the axial force_NEd and its eccentricity e_I. These values are multiplied by the factor k_sh,I, which describes the influence of the shrinkage moment in the uncracked state. The denominator includes the concrete's effective modulus of elasticity, E_c,eff, and the moment of inertia of the uncracked cross-section, I_I, which determine the cross-section's stiffness.

κ_I = (1.161 ⋅ (18.50 kNm - 0))/(10,001.2 MN/m² ⋅ 70,844.30 cm⁴)
= 3 mrad/m

The calculation of the deformation of the section in the cracked state, κ_II , shows the curvature of the section after the crack formation, taking into account the shrinkage moment and the reduced resistance of the cracked section. Here, the shrinkage moment My_,Ed,def, the axial force_NEd, and its eccentricity e_II are multiplied by the factor k_sh,II, which describes the influence of the shrinkage moment in the cracked state. In the denominator, the concrete's effective modulus of elasticity, E_c,eff, and the reduced moment of inertia of the cracked cross‑section, I_II, are lost, which reflect the cross‑section's lesser stiffness. The deformation of the cross-section in the cracked state is significantly greater than in the uncracked state because the stiffness of the cracked cross-section is reduced.

κ_II = (1.296 ⋅ (18.50 kNm - 0))/(10,001.2 MN/m² ⋅ 16,933.50 cm⁴) = 14.2 mrad/m

Final State

The final state describes the maximum stresses that can occur in the uncracked cross-section under both long-term and short-term loadings in order to ensure the load-bearing capacity and serviceability of the structural component.

The maximum stress in the uncracked state under long-term loading, σ_max,It describes the maximum stress that can occur in the uncracked cross-section as a result of long-term loading. It is composed of two parts:

• the contribution of the axial forces_NEd and_Nsh
• the contribution of the bending moments My_,Ed,def, M_sh,I, and the moment arising from the eccentricity (z_I - h/2) of the axial force_NEd.

The second part is amplified by the moment of inertia I_I and the distance (h - z_I ).

The maximum stress in uncracked state under short-term loading, σ_max,st indicates the greatest stress in the cross-section under short-term loadings. In contrast to the long-term loading, only the axial force_NEd and My_,Ed,def are considered here since there are no internal forces from shrinkage.

The maximum stress in the uncracked state, σ_max , is the greater of the two stress values due to the long-term and short-term loading. It ensures that the highest possible loading of the cross-section is taken into account.

σ_max = max ( 3.155 MN/m²; 2.689 MN/m² )
= 3.155 MN/m²

The distribution coefficient (damage parameter) ζ_d describes the transition between the behavior of the cross-section in the uncracked and cracked states. It is calculated by the ratio of the characteristic concrete tensile strength, f_ctm, to the maximum stress, σ_max. In this case, the nonlinearity is taken into account by the exponential relation.

ζ_d = 1 – 0.5 ⋅ (2.200 MN/m²/3.155 MN/m²)²
= 0.757 ≤ 1
with:
β = 1.0 (short-term loading)
β = 0.5 (long-term loading or many cycles of repetitive loading)
If the distribution coefficient ζ_d = 1, the structural component is completely in the cracked state. On the other hand, if ζ_d is equal to 0, the concrete is completely uncracked.

For the calculation of the distribution coefficient ζ_d, it is important which option is selected for the crack state detection. When you select the "Crack state calculated from associated load" option, the crack state (distribution coefficient ζ_d ) is calculated exclusively from the current load (load combination), as in this example. The other options are described in the manual.

The curvature of the section, κ_f , is calculated by interpolation between the cracked (κ_II ) and uncracked (κ_I ) state, weighted by the distribution coefficient, ζ_d. This allows for a realistic description of the curvature behavior in the transition state.

κ_f = 0.757 ⋅ 14.2 mrad/m + (1 – 0.757) ⋅ 3 mrad/m
= 11.5 mrad/m
The ideal cross-section area, A_f , describes the transition between the uncracked cross-section area, A_I, and the cracked cross-section area, A_II. Again, the weighting is carried out by the distribution coefficient, ζ_d.


The ideal moment of inertia I_y,f describes the moment of the cross-section taking into account the distribution coefficient ζ_d, as well as the moments of inertia in the uncracked state I_I and in the cracked state I_II. Additional factors such as k_sh,II and k_sh,I consider the effects of shrinkage in the respective state.


The eccentricity of the center of gravity,_ef , describes the position of the resulting centroid of the cross-section, based on the transition between the uncracked and cracked states. It takes into account the distribution coefficient, ζ_d, as well as the respective moduli of elasticity, E_c,eff, and moments of inertia, I_I and I_II.


The ideal moment of inertia to the geometric center of the cross-section I_y,0,f , considers in addition to the ideal moment of inertia I_y,f, and the ideal cross-section area A_f, also the shift of the center of gravity due to the eccentricity e_f. This displacement is taken into account by the Steiner component of A_f.

I_y,0,f = 16,145.50 cm⁴ + 678.30 cm² ⋅ (-49.2 mm)²
= 32,538.80 cm⁴

Final stiffnesses

The final stiffnesses of a structural component describe its resistance to deformations and rotations under different types of loads. The program considers axial and flexural stiffnesses as well as torsional and shear stiffnesses. These values are used as a basis for the analysis of the structural behavior and serviceability of a structural component.
The tangent membrane stiffness EA_f describes the axial stiffness of the cross-section taking into account the concrete's effective modulus of elasticity E_c,eff and the ideal cross-section area A_f.

EA_f = 10,001.2 MN/m² ⋅ 678.30 cm²
= 678,387 kN
The tangent bending stiffness, EI_y,0,f , describes the cross-section's resistance to bending about the ideal center of gravity. It is determined by the effective modulus of elasticity of concrete E_c,eff and the ideal moment of area I_y,0,f.

EI_y,0,f = 10,001.2 MN/m² ⋅ 32,538.80 cm⁴
= 3,254.28 kNm²
The tangent flexural resistance, EI_z,0,f , describes the cross-section's resistance to bending about the local z-axis. It is defined by the concrete's effective modulus of elasticity E_c,eff and the second moment of area about the z-axis I_z.

EI_z,0,f = 10,001.2 MN/m² ⋅ 1,666,670 cm⁴
= 166,687 kNm²

The factor r describes the reduction of the shear stiffness based on the ratio of the ideal moments of inertia I_f and I_I.

r = 16,145.50 cm⁴/70,844.30 cm⁴
= 0.228
The shear stiffness to the y-axis GA_y,f takes into account the concrete's effective shear modulus G_c,eff, the cross-sectional area A_c,y, and the reduction factor r.

GA_y,f = 4,167.18 MN/m² ⋅ 1,666.67 cm² ⋅ 0.228
= 158,284 kN
The shear stiffness to the z-axis, GA_z,f , is calculated in the same way as for the y-axis.

GA_z,f = 4,167.18 MN/m² ⋅ 1,666.67 cm² ⋅ 0.228
= 158,284 kN
The torsional stiffness GI_T,f corresponds in the case considered to the torsional stiffness in the uncracked state GI_T,I.

GI_T,f = 7,770 kNm²
The eccentric stiffness element ES_y describes the additional loading of the cross-section caused by the eccentricity e_f. It is calculated by using the axial stiffness EA_f and the eccentricity e_f.

ES_y = 678,387 kN ⋅ ( -49.2 mm )
= -33,350.20 kNm

Deflection

To ensure the serviceability, the actual deformation is compared to the allowable limit values. The total deflection around the precamber is corrected and checked against the specified limit values.
When calculating the deflection, the main load combination is considered without time-dependent effects such as creep and shrinkage (short-term), whereas the corresponding load combinations are always calculated with time-dependent properties (long-term). If more than one corresponding load is available, the deflection of the load with the highest value is taken as a basis.
The limit deflection in z-direction uz_,lim is calculated with the reference length in z-direction L_z,ref, and the limit deflection criterion L_z,ref/u_z,lim.

u_z,lim = 4.210 m/250 = 16.8 mm
The deflection in the z-direction,_uz , results from the difference of the total deflection uz_,tot, and the precamber at the location x,_uz,c.

u_z = 19.4 mm - 0 = 19.4 mm

Design check

η = max (19.4 mm/16.8 mm; 0.0 mm/16.8 mm) = 1.155 Since η = 1.155 > 1, the allowable deflection is exceeded!

Conclusion

The deformation calculation according to the approximation methods defined in the standards, such as the deformation analysis according to Section 7.4.3 of EN 1992-1-1, is carried out using effective stiffnesses that are calculated in the finite elements according to the limit state (cracked or uncracked). These effective stiffnesses form the basis for the subsequent calculation of the deformation of the structural component using a further FEM analysis.

The reinforced concrete cross-section is considered for the determination of the effective stiffnesses, whereby the reinforced concrete cross-section is classified into "cracked" or "uncracked" for the serviceability limit state on the basis of the determined internal forces. The effect of concrete between the cracks is considered by a distribution coefficient, for example according to Equation 7.19 (EN 1992-1-1). The material behavior of the concrete is assumed to be linear-elastic up to the tensile strength of concrete, which is sufficiently precise for serviceability.

The long-term effects of creep and shrinkage are taken into account when determining the effective stiffnesses at the cross-section level of the structural component in order to ensure a realistic representation of the deformations under long-term loads.