FEM modelling of the static behaviour of reinforced concrete beams considering the nonlinear behaviour of the concrete

Tests were carried out on 12 cylindrical concrete specimens 300 mm high and 150 mm in diameter. The concrete mix design is presented in Table 1.

Concrete mix consistency was controlled by gradually adding superplasticizer Sikament FM-6. Ultimately, the superplasticizer mass was assumed to be equal to 1.2% of the cement mass.

Investigations of the Young's modulus of concrete were carried out using two methods (I and II) based on Code method B (in accordance with EN 12390-13, 2013) for determining the stabilized Young's modulus of concrete. The method was modified to take into account the influence of different concrete stress intensity levels on the Young's modulus.

Three electrical resistance strain gauges RL=350/50 were stuck at every 120° on each cylindrical specimen at half of its height along three lines on the cylinder side wall. The shape of the specimens and the arrangement of the electrical strain gauges are shown in Fig. 2.

Figure 2

In the case of method I the load was applied according to the loading diagram shown in Fig. 3.

Figure 3

The symbols used in the description below should be interpreted in accordance with Fig. 3.

The numbers in brackets (1)–(6) represent the particular stops, i.e. pauses no longer than 20 s during which the specified stress level in the specimen should be maintained.

Method I differs from the standard code method in this that mean stress and strain in the specimen in the final load cycle should be read also halfway between lower stress level σ_p and upper stress level σ_a and above stress level σ_a at every 1/6 f_cm.

The secant Young's modulus of concrete for a given stress interval is calculated as the ratio of the difference between the stress indicated by two neighbouring readings and the difference between the mean strain corresponding to the readings.

In the case of method II, the load was applied in accordance with the diagram shown in Fig. 4.

Figure 4

The symbols in the description below should be interpreted consistently with Fig. 4. The numbers in brackets (1)–(12) represent the particular pauses.

In the case of method II, in the first two cycles the specimen is loaded up to the stress level of 2/3 f_cm at a stop at 1/3 f_cm. In the final load cycle, the mean strain and stress in the specimen are read at the end of each stop. The readings should be taken for stress halfway between stress levels σ_p and σ_a and between stress levels σ_a and σ_a‘, whereas after stress σ_a‘ is reached and a stop is made the specimen is loaded at stops at every 1/6 f_cm.

In the case of method II, the secant elasticity modulus of concrete for a given stress range is calculated similarly as in method I.

The subsequent photographs show the specimen in the strength-testing machine prior to the test (Fig. 5) and after failure (Fig. 6). The visible characteristic cones in the core of the failed specimen indicate that the specimen was compressed axially.

Figure 5

Figure 6

The test results for an exemplary specimen tested according to method I are presented in Figs. 7–9. The specimen loading diagram is shown in Fig. 7. The stress–strain curve drawn on the basis of the test results is shown in Fig. 8. The dependence between the Young's modulus of the concrete and the compressive stress and the stress intensity level in the specimen is illustrated in Fig. 9.

Figure 7

Figure 8

Figure 9

In accordance with EN 1992-1-1 (2004), the secant elastic modulus of concrete for the specimen represented by Figs. 7–9 amounts to E_cm=30.28 GPa.

The test results for the exemplary specimen tested according to method II are presented in Figs. 10–12. The specimen loading diagram is shown in Fig. 10. The stress–strain curve is shown in Fig. 11. A diagram of the Young's modulus of concrete versus the compressive stress and stress intensity level in the specimen is shown in Fig. 12.

Figure 10

Figure 11

Figure 12

In accordance with EN 1992-1-1 (2004), the secant elasticity modulus of concrete for the specimen represented by Figs. 10–12 amounts to E_cm=28.24 GPa.

It appears that method II yields an almost constant value of the Young's modulus for the stress intensity level of about 60%. Whereas the value of the Young's modulus investigated using method I begins to decrease at lower stress intensity levels. According to Neville (1977), the multiple loading of concrete eliminates creep (which occurs already as the concrete is being loaded). Subsequently, the value of the Young's modulus stabilizes and the curvature of the stress–strain curve decreases. This can be observed for testing according to method II, where in the first two cycles the concrete is loaded not to the stress level of 1/3 f_cm as in method I, but to 2/3 f_cm. As a result, the material adapts to higher stress intensity levels, as shown in Fig. 12.

Thus for the same concrete not subjected to loading yet versus already loaded one gets different elastic modulus curves. Therefore in a precise analysis it is important whether the considered element is a debuting element or whether it has already been subjected to loading.

An analysis of the static behaviour of the beam in ABAQUS was carried out for the concrete tested using method I (the specimen represented by Figs. 7–9). Reinforcement ratio ρ=1% (commonly used and economically viable) was assumed for the beam.

The geometry of the considered element is shown in Fig. 13.

Figure 13

The following nominal loading steps with the load uniformly distributed along the whole length of the beam were considered: 40, 45, 50, 55, 60 and 65 kN/m.

The beam was modelled in ABAQUS, taking into account the CDP model. The stress–strain relationship obtained from the tests was entered into program. Moreover the actual arrangement of reinforcement was taken into account in the model.

The beam alone was defined as a three-dimensional (3D) solid (element type: solid), while the reinforcement was built of one-dimensional elements in 3D space (element type: wire). Moreover, steel support pads (element type: solid) were designed in order for the FE analysis to map well the actual support.

The rebars and the steel support pads were assigned linearly elastic (elastic) materials. The rebars were assigned reinforcement steel with E = 200 GPa and Poisson's ratio ν = 0.3, while the steel support pads were assigned structural steel with E = 210 GPa and ν = 0.3.

A definition of a material such as concrete, to which a nonlinear stress–strain curve is to be assigned and which is to take element cracking into account, is more complex and requires a special material model, i.e. the CDP model. Moreover, one should endow the concrete with the elastic material characteristic since the stress–strain relationship in CDP is a polyline and the Young's modulus assigned to the elastic characteristic corresponds to the slope of the first (counting from the origin of the coordinate system) segment of the stress–strain polyline. In addition, the elastic characteristic includes a Poisson's ratio which for the concrete was assumed to amount to ν = 0.2.

The following concrete parameters were used in CDP: –

a dilation angle (Szczecina and Winnicki, 2015a,b, 2016): ψ=5°,

Stress values and the corresponding inelastic strain values were entered in order to implement the stress–strain curve for concrete under compression into the program. The stress–strain curve for concrete (taken from the Abaqus Analysis User's Guide (2014)) is shown in Fig. 14.

Figure 14

The inelastic strain to be entered into the program for a given stress level is calculated from the eq. (1): (1) ε˜cin=εc−ε0cel, \tilde \varepsilon _{\rm{c}}^{{\rm{in}}} = {\varepsilon _{\rm{c}}} - \varepsilon _{0{\rm{c}}}^{{\rm{el}}}, where: ɛ_c – strain due to compression for a given point on the stress–strain curve, ɛ^0c_el – a part of the strain due to compression, normalized to the initial Young's modulus, expressed by the eq. (2): (2) ε0cel=σc/E0, \varepsilon _{{\rm{0c}}}^{{\rm{el}}} = {\sigma _{\rm{c}}}/{E_0}, where: σ_c – compression stress for the given point on the stress–strain curve, E₀ – the initial Young's modulus.

The notations in eqs. (1) and (2) should be interpreted in accordance with Fig. 14.

When characterizing concrete in tension, taking into account tension stiffening, one should specify stress and the corresponding cracking strain values. The stress–strain curve for concrete in tension is shown in Fig. 15.

Figure 15

The tensile strength of concrete was assumed on the basis of the following eqs. (3)–(5) according to EN 1992-1-1 (2004): (3) fcm=fck+8[ MPa ]⇒fck=fcm−8[ MPa ], {f_{{\rm{cm}}}} = {f_{{\rm{ck}}}} + 8\left[ {{\rm{MPa}}} \right] \Rightarrow {f_{{\rm{ck}}}} = {f_{{\rm{cm}}}} - 8\left[ {{\rm{MPa}}} \right], (4) fctm=0.30fck2/3,fctm=2.96 MPa \matrix{ {{f_{{\rm{ctm}}}} = 0.30f_{{\rm{ck}}}^{2/3},} \hfill \cr {{f_{{\rm{ctm}}}} = 2.96\,{\rm{MPa}}} \hfill \cr } Cracking strain is calculated from eq. (5): (5) ε˜tck=εt−ε0tel, \tilde \varepsilon _{\rm{t}}^{{\rm{ck}}} = {\varepsilon _{\rm{t}}} - \varepsilon _{0{\rm{t}}}^{{\rm{el}}}, where: ɛ_t – strain due to tension for a given point on the stress–strain curve, ɛ^0t_el – a part of the strain due to tension, normalized to the initial Young's modulus, expressed by the eq. (6): (6) ε0tel=σt/E0, \varepsilon _{{\rm{0t}}}^{{\rm{el}}} = {\sigma _{\rm{t}}}/{E_0}, where: σ_t – tensile stress for a given point on the stress–strain curve, E₀ – the initial Young's modulus.

Moreover, data relating to the failure parameter and the corresponding cracking strain were entered in order to obtain results for the failure of concrete in tension. The failure parameter can be estimated using the eq. (7): (7) dt=1−σtσt0, {d_{\rm{t}}} = 1 - {{{\sigma _{\rm{t}}}} \over {{\sigma _{{\rm{t}}0}}}}, where: σ_t0 – the tensile strength of concrete in a uniaxial stress state.

The notations in eqs. (5)–(7) should be interpreted in accordance with Fig. 15.

Taking into account the above, the stress–strain characteristic (based on the author's own research) shown in Fig. 16 was entered into the program.

Figure 16

Material (bulk) density was assigned to both the steel and the concrete since in the considered case a dynamic explicit analysis which requires this parameter is carried out.

The object model consists of the beam's concrete, the rebars and the steel support pads. Beam–rebars interactions and beam concrete–steel pads interactions were introduced in the next step. The rebars were connected with the beam's concrete through an embedded region constraint while steel pads–concrete interactions were effected through a tie constraint. The model is shown in Fig. 17.

Figure 17

Then boundary conditions and loads are entered. The supports were introduced as linear pinned supports with the possibility of shifting one beam end along the beam. The supports were located at half the width of the steel pads. The load was defined as a pressure-type load applied to the beam's top surface.

Subsequently, all the model's parts were divided into FEs. C3D8R FEs (8-node reduced integration-type cuboid FEs) were assigned to the beam and the steel pads. The element size amounted to approximately 50 mm for the beam and to 25 mm for the steel pads. T3D2 FEs (two-node truss elements in 3D space) with the element size of 50 mm were assumed for the rebars.

After computations, the results for vertical displacements (an exemplary map is shown in Fig. 18) and the images of cracking (see Fig. 19) were analysed.

Figure 18

Figure 19

The deflections of the beam analysed in ABAQUS were also calculated using a numerical bar model developed by the author on the basis of the Euler–Bernoulli beam theory. The procedure was carried out for the same concrete which had been assigned to the beam in the FE analysis. Figure 20 shows the beam structure and loading diagram.

Figure 20

The proposed method requires knowledge of beam deflections according to EN 1992-1-1 (2004). For example, the considered beam's deflection calculated in accordance with EN 1992-1-1 (2004) for the load of 40 kN/m amounts to 10.20 mm.

In order to calculate the beam's deflection, taking into account the changes in the Young's modulus, an intermediate cross section (between cross section behaviour in phases I and II) was determined. Such a cross section with a certain concrete tension zone is determined to take into account tension stiffening. When the obtained moment of inertia is substituted into the formula for the deflection of a homogenous prismatic bar, one gets the deflection value calculated according to EN 1992-1-1 (2004). Then the beam was divided into n=40 equal fragments. The bending moment along the length of each of them is constant and equal to the calculated bending moment in the middle of each of the fragments. Consequently, the diagram of bending moments for the whole beam has a stepped shape as shown in Fig. 21.

Figure 21

Each of the fragments has an identical moment of inertia (for the cross section mentioned above), but the Young's modulus for each of them was calculated independently on the basis of the compressive stress in the cross section. The Young's modulus for a given fragment is an integral mean of the Young's modulus of the concrete in the compressed part of the cross section of the considered fragment. The graphical interpretation of the determination of the Young's modulus for the given fragment of the beam is shown in Fig. 22., where: x – the extent of the compression zone, t – the extent of the tension zone in concrete, ɛ_cg – normal strain in top concrete fibres, ɛ_cd – normal strain in bottom concrete fibres, ɛ_s1 – normal strain in bottom reinforcement steel, ɛ_s2 – normal strain in top reinforcement steel, σ_c,g – normal stress in top concrete fibres, σ_c,d – normal stress in bottom concrete fibres, σ_s1 – normal stress in bottom reinforcement steel, σ_s2 – normal stress in top reinforcement steel, α_e – a ratio of the Young's modulus of the steel to the secant Young's modulus of the concrete, acc. to EN 1992-1-1 (2004), E_i – the Young's modulus for the i-th fragment of the beam, E₁ – the Young's modulus of the concrete along segment x₁, E_2,i – the Young's modulus of the concrete at the end of segment x₂ in the i-th fragment of the beam, x₁ – the extent of the compression zone in which the Young's modulus of concrete is constant, x₂ – the extent of the compression zone in which the Young's modulus of concrete is variable.

Figure 22

The Young's modulus of the concrete for a given fragment of the beam was calculated from the eq. (8): (8) Ei=x1E1+0.5(E1+E2,i)x2x, {E_{\rm{i}}} = {{{x_1}{E_1} + 0.5\left( {{E_1} + {E_{2,{\rm{i}}}}} \right){x_2}} \over x}, After moduli E_i for each fragment of the beam had been calculated, the deflection was calculated by adding up the displacements of the ends of all the fragments in one half of the beam. The displacements were calculated as for a cantilever loaded on its free end with a point moment (the bending moment is constant along the length of the fragment), taking into account the angle of rotation of the cantilever beginning (the cantilever support). For each fragment this angle is equal to the angle of rotation of the end of the preceding fragment. The starting point for the calculations is the first beam fragment (numbering as in Fig. 21) for which the angle of rotation of its beginning is equal to zero (ϕ_0,1=0). It is the consequence that the angle of rotation of the axis of a simply supported beam at half of its span for a load uniformly distributed along the beam's length is equal to zero.

The formula for the deflection of the beam with Young's modulus variation taken into account is eq. (9): (9) α′=∑i=0nwi=∑i=1n(0.5Milf2EiIy+φ0,ilf), \alpha ' = \sum\limits_{{\rm{i}} = 0}^{\rm{n}} {{{{w}}_{\rm{i}}} = \sum\limits_{{\rm{i}} = 1}^{\rm{n}} {\left( {0.5{{{M_{\rm{i}}}l_{\rm{f}}^2} \over {{E_{\rm{i}}}{I_{\rm{y}}}}} + {\varphi _{0,{\rm{i}}}}{l_{\rm{f}}}} \right),} } where: α‘ - the beam's deflection with the variation of the Young's modulus of the concrete taken into account, w_i – the deflection of the particular beam fragments, M_i – the bending moment in the i-th fragment of the beam, l_f – the length of a single fragment of the beam, I_y – the moment of inertia of the cross-section intermediate between phases I and II, ϕ_0,i – the angle of rotation of the beginning of the i-th element, defined by the eq. (10): (10) φ0,i=φ0,i−1+Mi−1lfEi−1Iy,φ0,1=0. \matrix{ {{\varphi _{0,{\rm{i}}}} = {\varphi _{0,{\rm{i}} - 1}} + {{{M_{{\rm{i}} - 1}}{l_{\rm{f}}}} \over {{E_{{\rm{i}} - 1}}{I_{\rm{y}}}}},} \cr {{\varphi _{0,1}} = 0.} \cr } The graphical interpretation of the determination of the beam's deflection is shown in Fig. 23.

Figure 23

The deflection of the considered beam with the variable Young's modulus of the concrete taken into account amounted to α‘=10.26 mm. Similar calculations were made for the loads of 45, 50, 55, 60 and 65 kN/m. The results are compiled in Table 2.