Direct Deformation Analysis of Reinforced Concrete Beams Considering Long-Term Effects of Creep and Shrinkage

In this technical article, the direct deformation analysis of a reinforced concrete beam is performed, 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 how they are incorporated into the calculation. It is explained which inputs are necessary in RFEM 6 in order to correctly consider all relevant factors and how the distribution factor affects the stiffness of the structural component.

Input Data

The geometry, reinforcement, and loading are described by the following parameters:

Structural System

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

Cross-Section

• Plate thickness: h = 20 cm
• 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 the 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 as a user-defined value.

In the tab that is now available, first you have to select the check boxes for “Creep” and “Shrinkage” in order to be able 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 reducing the modulus of elasticity E_c of the concrete.

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 subjected to a constant load. The creep coefficient φ is used to reduce the modulus of elasticity E_cm (mean modulus of elasticity of the concrete), so that the actual concrete stiffness is represented over a long period of time. This value is used in further calculations, such as for the moment of inertia or the stiffness ratios.

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 resistance of the concrete to shear deformations and is determined using the ratio of the transversal strain to the longitudinal strain (Poisson's ratio of concrete). This value is particularly important for calculations 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 compared to the concrete under long-term loading. E_s is the modulus of elasticity of the steel, E_c,l is the effective modulus of elasticity of the 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 cross-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 relation 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 loads.

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 effective modulus of elasticity E_c,eff of the concrete. This value shows that the steel stiffness 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 for Uncracked State

The distance to the center of gravity 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 surface and the reinforcement. The effect of the reinforcement is scaled with a conversion factor α_e,l, which represents the ratio between the modulus of elasticity of the steel and the effective modulus of elasticity of the concrete. This is particularly important, as 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 key parameter for the structural analysis.

The effective cross-section area in the uncracked state for long-term loading, A_I, represents the effective area that carries loads. In addition to the concrete area, the reinforcement area is also taken into account, and extended by the factor α_e,l. This provides a more realistic representation of the cross-section stiffness. This value is crucial for evaluating the load-bearing capacity and calculating 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, I_I describes the resistance of the cross-section to bending. It takes into account both the concrete surface and the reinforcement, with the latter generating additional moments due to its position relative to the center of gravity. 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 that arise in the cross-section, which has a direct effect on the deformations.

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

The distance of the center of gravity of the ideal cross-section in the uncracked state under short-term loading, z_I,st, describes the centroidal position under loads 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 loading. This centroidal distance is crucial for the distribution of loads and the determination of moments in short-term loading.

The effective cross-section area in the uncracked state under 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 results in a smaller area and has 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 bending resistance of the cross-section without the influence of long-term effects. It takes into account both the concrete and the reinforcement area and their distances from the center of gravity, which is crucial for calculating the deformation under short-term loads.

Geometric Parameters for Cracked State

The distance to the center of gravity of the ideal cross-section in the cracked state, z_II, takes into account the changed resistance of the cross-section, as the concrete tension zone no longer bears any loads after crack formation. The centroid location is recalculated, taking into account only the compression zone of the concrete and the reinforcement. This parameter is crucial for the analysis of the cross-section after cracking and influences the load-bearing capacity and deformation.

The effective cross-section area in the cracked state, A_II, represents the remaining area after the crack formation. Here, only the concrete compression zone and the reinforcement area are considered, which greatly reduces the cross-section stiffness. This value is crucial for the ultimate limit state design 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 center of gravity in the cracked state, I_II, describes the bending resistance after the cracking. Since the tension zone is no longer load-bearing, the moment of inertia is significantly reduced. This value is an essential factor for calculating the deformation and for evaluating the load-bearing capacity of cracked cross-sections.

The eccentricity of the ideal center of gravity of the cross-sectionin the cracked state, e_II, describes the displacement of the center of gravity due to the crack formation. This displacement affects the resulting moments and the cross-section deformation, 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 contribute to 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 reaction of the reinforcement. The calculated value shows how much the reinforcement is stressed by the shrinkage. At this point, the indirectly user-defined shrinkage strain ε_sh = -0.5‰ 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 the uncracked state M_sh,I, results from the shrinkage force N_sh and the eccentricity e_sh,I. It represents how the shrinkage force generates a moment due to 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 the 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 center of gravity influence the deformations of the structural component. This value is crucial in fully describing 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 divided 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, since 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 due to 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 that acts 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 carries 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,, the axial force N_Ed, and its eccentricity e_II are taken into account. The calculation sets the resulting moment in relation to the moment without shrinkage, and thus provides a measure of the influence of the shrinkage force.

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

Deformation of Cross-Section

The deformation of the cross-section describes the curvature of a structural component caused by external influences, taking into account its material and condition parameters.

The calculation of the cross-section deformation 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 effective modulus of elasticity, E_c,eff, and the moment of inertia of the uncracked cross-section, I_I, which determine the cross-section stiffness, are in the denominator.

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 cross-section in the cracked state, κ_II, shows the curvature of the cross-section after the crack formation, taking into account the shrinkage moment and the reduced resistance of the cracked cross-section. Here, the shrinkage moment M_y,Ed,def, the axial force N_Ed, 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. The effective modulus of elasticity of the concrete, E_c,eff, and the reduced moment of inertia of the cracked cross‑section, I_II, which reflect the lower stiffness of the cross-section, are in the denominator. 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 loading in order to ensure the ultimate and serviceability limit states 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 consists of two components:

• Contribution of the axial forces N_Ed and N_sh
• Contribution of the bending moments M_y,Ed,def, M_sh,I, and the moment arising from the eccentricity (z_I - h/2) of the axial force N_Ed.

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

The maximum stress in the uncracked state under short-term loading, σ_max,st, indicates the maximum stress in the cross-section under short-term loading. In contrast to long-term loading, only the axial force N_Ed and M_y,Ed,def are taken into account here, as no internal forces from shrinkage are present.

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 state. 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
where:
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.

The option selected for the crack state detection is important for the calculation of the distribution coefficient ζ_d. If 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 cross-section, κ_f, is calculated by interpolation between the cracked state (κ_II) and the uncracked state (κ_I), 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 center of gravity of the cross-section, based on the transition between the uncracked and the cracked state. 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, also takes into account the displacement of the center of gravity by the eccentricity, e_f, in addition to the ideal moment of inertia I_y,f, and the ideal cross-section area A_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. In this context, axial stiffnesses and flexural stiffnesses, as well as torsional and shear stiffnesses, are considered. These values are used as a basis for the analysis of the structural behavior and serviceability design of a structural component.
The tangent membrane stiffness, EA_f, describes the axial stiffness of the cross-section, taking into account the effective modulus of elasticity of the concrete, 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 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 resistance to bending about the local z-axis. It is defined by the effective modulus of elasticity of concrete, 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 about the y-axis GA_y,f takes into account the effective shear modulus of concrete, 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 to the torsional stiffness in the uncracked state, GI_T,I, in the considered case.

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 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 is corrected by the precamber 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), while the corresponding load combinations are always calculated with time-dependent properties (long-term). If there is more than one corresponding load, the deflection of the load with the highest value is used.
The limit deflection in z-direction, u_z,lim is calculated using the reference length in the 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, u_u, results from the difference of the total deflection, u_z,tot, and the precamber at the location x,u_z,c.

u_z = 19.4 mm - 0 = 19.4 mm

Design

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

To determine the efficient stiffnesses, the reinforced concrete cross-section is considered, whereby the reinforced concrete cross-section is classified as “cracked” or “uncracked” on the basis of the determined internal forces for the serviceability limit state. The effect of the concrete between the cracks is taken into account 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 the 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.