Verification of stresses and crack openings for reinforcement plates according to Eurocode 2

Advance Design® BIM system is dedicated to structural engineers who require a comprehensive solution for simulating and optimizing all their projects. It includes a user-friendly structural modeler, automatic load and combination generators, a powerful FEM analysis engine (static, dynamic, time history, non linear, buckling, large displacement analysis, etc.), comprehensive wizards for designing concrete and steel members according to Eurocodes, efficient result post-processing, and automatic report generators.

Version 2012 of Advance Design focuses on the integration of new international codes and timber design EC5. It also provides many improvements in the implementation of EC1, 2 and 3.

Some of the new features of ADVANCE Design 2012 are a new design module for timber frames to Eurocode 5 (German, English, French, Romanian and Czech National Appendices), calculation of cracked inertia for linear and planar elements, implementation of the Baumann method for reinforcement plates to Eurocode 2, verification of stresses and crack openings as a function of the real reinforcement implemented in the element for Eurocode 2 (EN 1992-1-1).

With ADVANCE Design 2012 stresses and crack openings can be quickly checked, taking into consideration mainly the reinforcement solution used for linear and planar elements.

Main information regarding stresses and crack openings Checking the stresses and the crack openings is a requirement – they must be limited, otherwise it would affect the proper functioning of the entire structure. Stresses and the crack openings to elements made of reinforced concrete are determined either by the loads applied during construction, stresses from the subsidence of foundations, temperature changes and temperature gradient, contractions, slow flow, etc. Stresses and crack openings can occur in any of the following situations:

- due to plastic shrinkage; - due to slow flow of newly poured concrete; - due to stresses and crack openings caused by long-term loads.

SR EN 1992-1-1 (also referred to as Eurocode 2) according to Table 1, recommends the following limit values of the calculated crack openings (wmax), depending on the class of exposure of the reinforced concrete structures:

Table 1. Recommended values for wmax

Exposure Class Reinforced concrete elements Quasi-permanent loads

XD, XC1 wmax = 0.4 mm XC2, Xc3, XC4 wmax = 0.3 mm XD1, Xd2, XS1, XS2, XS3

Without some specific requirements (for example, water tightness) it is likely, for quasi-permanent combinations of loads, that the calculated crack opening can be limited to 0.3 mm, for all exposure classes. Without some appearance conditions, this limit may increase to 0.4 mm for exposure classes X0 and XC1. The theoretical value of the crack opening can be calculated according to Chapter 7.3.4 of SR EN 1992-1-1. Eurocode 2 offers structural engineers two methods for stresses and the crack openings to elements made of reinforced concrete:

1. Calculation of crack openings according to Chapter 7.3.4 of EN 1992-1-1, namely SR EN 1992-1-1 with the National Appendix.

The formula used is:

wk = Sr,max (εsm - εcm)

where:

Sr,max - maximum distance between cracks; εsm - average reinforcement strain, due to the combination of considered loads, including the

required deformation effect and taking into account the participation of tension stiffening; εcm - average strain of concrete between cracks.

The maximum distance between cracks can be calculated with the following formula (according to equation 7.11 of Eurocode 2):

effpr

kkkckS,

4213max, ρ

φ⋅⋅⋅+⋅=

where:

c - the concrete cover of the longitudinal reinforcements; k1 - the coefficient that takes into account the bond properties of the bonded reinforcement; k2 - the coefficient that takes into account the distribution of strain; φ - the diameter of the bars. If using more diameters in the same section, an equivalent

diameter shall be taken into account; ρp,eff - the effective reinforcement percentage, suitable to the effective concrete section, surrounding the tensioned reinforcement.

For k3 and k4, the recommended values from the National Appendix can be used.

„εsm - εcm” can be calculated with the following formula (according to equation 7.9 of Eurocode 2):

( )

s

s

s

effpeeffp

effctts

cmsm EE

fk

σρα

ρσ

εε ⋅≥⋅+⋅⋅−

=− 6.01 ,

,

,

where:

σs - stress in the tension reinforcement, considering cracked sections; αe - Es/Ecm ratio; kt - the factor depending on the duration of the load; fct,eff - average tensile stress of concrete, at the same time the first cracks occur; Es - design value of modulus of elasticity of reinforcing steel.

2. Control of crack openings with no direct calculation This is a simplified method, which uses rules derived from the formulas for calculating crack openings. The minimum area of reinforcement is determined (the reinforcement must not yield as soon as the first crack occurs) following the limitations required by Eurocode 2, for reinforcement bars and the distance between them (according to Table 2).

Table 2. Maximum diameter and distance between bars, stresses and the crack openings

Effort per unit in reinforcement

[MPa]

wmax = 0.4 mm wmax = 0.3 mm

Maximum diameter of bars

[mm]

or

Maximum distance between

bars [mm]

Maximum diameter of bars

[mm]

or

Maximum distance between

bars [mm] 160 40 300 32 300 200 32 300 25 250 240 20 250 16 200 280 16 200 12 150 320 12 150 10 100 360 10 100 8 50

Checking the stresses and the crack openings involves a large number of iterations, which is hard to manage by manual calculation. ADVANCE Designallows automatic control of stresses and the crack openings.

Checking stresses and crack openings with ADVANCE Design 2012 Since ADVANCE Design automatically performs the checking for the stresses and the crack openings, we will take as an example a circular tank made of reinforced concrete (Figure 1).

Figure 1. Circular tank made of reinforced concrete

The tank will be exposed to mechanical forces, as well as an alternating wet and dry environment (XC4 exposure class, in Table 4.1 of SR EN 1992-1-1). Therefore, we can use a certain type of concrete with a strength class C45/55. The concrete cover is 5 cm thick, for the reinforced bars.

The modeling of the tank was done with flat "shell" elements (the number of degrees of freedom of this element is 6) with a thickness of 35 cm. The mesh size used for the finite element analysis is 0.8 meters. The loads are determined by the weight of the elements and by the loading time of the fluid inside (water).

After the finite element analysis and the checking with the "Concrete Expert", the theoretical reinforcement areas which were obtained with . ADVANCE Design 2012 are analyzed (Figure 2), and the actual reinforcement is decided.

So, for the selected elements in Figure 3, we will select a reinforcement solution with individual bars on both directions, (Φ 16/14 cm, at the top and Φ 20/15 cm, at the bottom). Later, we can choose the reinforcement for the interior walls (Figure 4) and for the external walls (Figure 5). In addition to the global reinforcement, for the external walls we chose a local reinforcement with individual bars, which is required only for a segment of the wall (Figure 5).

Figure 2. Theoretical reinforcement areas along x and y local axes of the planar elements

Figure 3. Defining the global reinforcement for planar elements

Figure 4. Defining the reinforcement for interior walls Note: For planar elements, ADVANCE Design 2012 allows you to choose the reinforcement fabrics (either

for global or local reinforcement). This option is available in the program database.

Figure 5. Defining the reinforcement for external walls

To check the stresses and the crack openings, using the reinforcement previously selected, you have to resume the checking with the “Concrete Expert”. Note that the maximum opening of the crack does not exceed 0.3 mm (Figure 6 and Figure 7). This value is specified in Table 1.

Figure 6. Stresses and the crack openings for planar elements, along the x local axis (wk,x)

Figure 7. Stresses and the crack openings for planar elements, along the y local axis (wk,y)

Figure 8. The inferior tensioned reinforcement stress, along x and y local axes

Figure 9. The superior tensioned reinforcement stress, along x and y local axes

The selected reinforcement is compliant with the stresses and crack openings, therefore checking the stresses and the crack openings is achieved. Figures 8 and 9 show the reinforcement stress for quasi-permanent loads. The maximum value per unit for reinforcement is 192.67 N/mm2 and must not exceed the value of 22

3 /400/5008.0 mmNmmNfk yk =⋅=⋅ shown in SR EN 1992-1-1 Clause 7.2 (5).

Conclusions ADVANCE Design 2012 can achieve effective control for stresses and the crack openings under Eurocode 2 – National Appendix, allowing control of the distance between cracks, throughout the real reinforcement used (reinforced fabrics or individual reinforced bars), thereby avoiding a situation in which the crack opening exceeds certain acceptable limits.