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

Maciej Pazdan
Maciej Pazdan
   | 30. Sept. 2021
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Article Category: Original Study
Online veröffentlicht: 30. Sept. 2021
Seitenbereich: 206 - 223
Eingereicht: 17. Feb. 2021
Akzeptiert: 26. Apr. 2021
DOI: https://doi.org/10.2478/sgem-2021-0012
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© 2021 Maciej Pazdan, published by Sciendo
This work is licensed under the Creative Commons Attribution 4.0 International License.

Figure 1

Graphical interpretation of Young's moduli of concrete.
Graphical interpretation of Young's moduli of concrete.

Figure 2

Shape of tested specimens and arrangement of electrical resistance strain gauges.
Shape of tested specimens and arrangement of electrical resistance strain gauges.

Figure 3

Investigation of stabilized secant elasticity modulus EC,SEC,S of concrete according to method I.
Investigation of stabilized secant elasticity modulus EC,SEC,S of concrete according to method I.

Figure 4

Investigation of stabilized elasticity modulus EC,S of concrete according to method II.
Investigation of stabilized elasticity modulus EC,S of concrete according to method II.

Figure 5

Concrete specimen placed in strength-testing machine.
Concrete specimen placed in strength-testing machine.

Figure 6

Failed concrete specimen.
Failed concrete specimen.

Figure 7

Diagram of specimen loading according to method I.
Diagram of specimen loading according to method I.

Figure 8

Stress–strain diagram for specimen tested according to method I.
Stress–strain diagram for specimen tested according to method I.

Figure 9

Young's modulus E versus compressive stress σ and stress intensity level in concrete (method I).
Young's modulus E versus compressive stress σ and stress intensity level in concrete (method I).

Figure 10

Diagram of specimen loading according to method II.
Diagram of specimen loading according to method II.

Figure 11

Stress–strain diagram for specimen tested according to method II.
Stress–strain diagram for specimen tested according to method II.

Figure 12

Young's modulus E versus compressive stress σ and stress intensity level in concrete (method II).
Young's modulus E versus compressive stress σ and stress intensity level in concrete (method II).

Figure 13

View and cross section of the considered beam (all dimensions in mm).
View and cross section of the considered beam (all dimensions in mm).

Figure 14

CDP stress–strain curve for concrete in compression (Abaqus Analysis User's Guide, 2014).
CDP stress–strain curve for concrete in compression (Abaqus Analysis User's Guide, 2014).

Figure 15

CDP stress–strain curve for concrete in tension (Abaqus Analysis User's Guide, 2014)
CDP stress–strain curve for concrete in tension (Abaqus Analysis User's Guide, 2014)

Figure 16

Stress–strain characteristic entered into ABAQUS.
Stress–strain characteristic entered into ABAQUS.

Figure 17

Beam model in ABAQUS.
Beam model in ABAQUS.

Figure 18

Beam deflection under load of 40 kN/m (deflection values in mm).
Beam deflection under load of 40 kN/m (deflection values in mm).

Figure 19

Image of beam cracking under load of 40 kN/m.
Image of beam cracking under load of 40 kN/m.

Figure 20

Structure and loading diagram – simply supported beam uniformly loaded along its whole length.
Structure and loading diagram – simply supported beam uniformly loaded along its whole length.

Figure 21

Stepped graph of bending moments with division of beam into fragments (dashed line shows actual shape of bending moments graph; dots on element's longitudinal axis denote points of division of beam into fragments).
Stepped graph of bending moments with division of beam into fragments (dashed line shows actual shape of bending moments graph; dots on element's longitudinal axis denote points of division of beam into fragments).

Figure 22

Normal strain, normal stress (for equivalent cross section) and Young's modulus of concrete in cross section of given beam fragment.
Normal strain, normal stress (for equivalent cross section) and Young's modulus of concrete in cross section of given beam fragment.

Figure 23

Visualization of way of calculating deflections of particular beam fragments.
Visualization of way of calculating deflections of particular beam fragments.

Figure 24

Maximum absolute deflection versus applied load for bar model taking into account constant and variable Young's modulus and for ABAQUS FEM model.
Maximum absolute deflection versus applied load for bar model taking into account constant and variable Young's modulus and for ABAQUS FEM model.

Figure 25

Values of effective moments of inertia according to, respectively, ACI Code 318-19 (2019) and EN 1992-1-1 (2004), depending on load in range of 15–65 kN/m.
Values of effective moments of inertia according to, respectively, ACI Code 318-19 (2019) and EN 1992-1-1 (2004), depending on load in range of 15–65 kN/m.

Design of concrete mix used to make tested cylindrical specimens.

No. Constituent Mass
[kg/m3]
1 Cement CEM I 42.5R 365.0
2 Aggregate 2–8 mm 650.0
3 Aggregate 8–16 mm 560.0
4 Sand 0–2 mm 650.0
5 Water 175.0
6 Superplasticizer (1.2% of cement mass) 4.5

Comparison of beam deflections for constant and variable Young's modulus of concrete.

No. Load Bending effort of the element Constant Young's modulus Variable Young's modulus Relative increment in deflection
α α’
[kN/m] [%] [mm] [mm] [%]
1 40 58 10.20 10.26 0.59
2 45 66 11.69 11.84 1.28
3 50 73 13.16 13.44 2.13
4 55 80 14.61 15.08 3.22
5 60 88 16.05 16.75 4.36
6 65 95 17.49 18.48 5.66

Comparison of deflections for bar model and FEM model.

No. Bar model FE model Relative difference between α〉 and αCDP
Load Constant Young's modulus Variable Young's modulus ABAQUS - CDP
α α’ αCDP
[kN/m] [mm] [mm] [mm] [%]
1 40 10.20 10.26 9.90 3.51
2 45 11.69 11.84 12.09 2.11
3 50 13.16 13.44 14.31 6.47
4 55 14.61 15.08 16.57 9.88
5 60 16.05 16.75 18.88 12.72
6 65 17.49 18.48 21.23 14.88
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