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publicatie: Finite Element Modelling

1 Reinforced beam - Flexure compressive failure

1 Reinforces beam - Flexure compressive failure

The experimental program of Vecchio & Shim (2004) is a re-examination of the classical experiments of Bresler & Scordelis (1963). The complete experimental program consisted of twelve beams with different ratios of shear and longitudinal reinforcement. Beam C3 is selected, as this beam has the longest span (6400 mm) and featured a flexure-compressive failure mechanism.

1 Experimental setup and results

Geometry
The geometry of the beam and reinforcement is shown in Figure 1 and Figure 2. The beam has a total length of 6.840 m, a depth of 0.552 m, and a width of 0.152 m.

Figure 1: Dimensions (in mm), reinforcements and loading (Vecchio & Shim 2004)

The bottom longitudinal reinforcement is extended outside the beam and welded to one-inch thick plates. It is assumed that the dimensions of these plates are 0.192×0.350×0.025 m3.

Figure 2: Cross section details (in mm), (Vecchio & Shim 2004)

Material Properties
The concrete and reinforcement properties, given in Vecchio & Shim (2004), are listed in Table 1

Table 1: Concrete and reinforcement properties

Concrete properties
ƒcm
(N/mm2)
dmax
(mm)
43.5 20
Reinforcement properties
Bar ϕ
(mm)
As
(mm2)
Es
(N/mm2)
ƒym
(N/mm2)
ƒtm
(N/mm2)
M10 11.3 100 200000 315 460
M25 25.2 500 220000 445 680
M30 29.9 700 200000 436 700
D4 3.7 25.7 200000 600 651

Loading and Boundary Conditions
The loading and boundary conditions in the experimental setup are shown in Figure 3.
The out-of-plane dimensions of the loading and support plates are larger than the out-of-plane thickness of the beam.

Figure 3: Loading and boundary conditions (Vecchio & Shim 2004)

Experimental Results
The beam exhibited a flexural-compressive failure mode with a clear maximum in the load-deflection response, Figure 4 and Figure 5. The experimental ultimate value of applied load was equal to PExp = 265kN at the deflection of 44.3 mm.

Figure 4: Failure mechanisms at experimental ultimate value of applied load (Vecchio & Shim 2004)

Figure 5: Experimental load-deflection at midspan

2 Analytical analysis

In Figure 6 the load configuration at failure is reported. The distributed load representing the beam weight is equal to q = 0.152m x 0.552m x 25kN / m3 = 2.098kN / m .

Figure 6: Load configurations (dimension in m)

Load case 1
Figure 7 shows that the maximum value of applied moment at the midspan is equal to
and the maximum value of applied shear force at the supports is equal to

Figure 7: Load 1: Internal forces

Load case 2
The experimental ultimate value of the applied load is equal to PE,max = 265kN.

Figure 8: Load 2: Internal forces

Figure 8 shows that the maximum value of applied moment at midspan is equal to and the value of applied shear force is equal to VE,max = 1325kN.

Load 1 + Load 2
At failure the maximum value of applied moment is
ME,max = 10.69kNm + 424kNm = 434.69kNm and the value of applied shear force is equal
to VE,max = 6.71kN + 1325kN + 139.21kN.

The design value of resistance moment is evaluated with sectional analysis by assuming:

  • the tensile strength of concrete is ignored,
  • the compressive stresses in concrete are derived from a parabola-rectangle relation,
  • he stresses in the reinforcing steel are derived from an elastic–plastic stress-strain relation with hardening,
  • the partial safety factor for the mechanical properties of reinforcing steel equals ϓs=1.15, the factor for concrete material properties equals ϓc=1.5.

The failure mechanism of the specimen is caused due to crushing of concrete after yielding of reinforcing steel. The design bending moment resistance is calculated below.

Figure 9: Stress block for determination of the design moment resistance

Assuming that reinforcement yields at strains:
which is smaller than xlim = 327mm.

Verification of the assumptions for the calculated value of x:

thus the assumptions that reinforcement yields is fulfilled.
The design value of the moment resistance in calculated around the centre of the compression zone 0.416x :

The value of applied moment equals to

which results in the value of the applied load

The design value of shear resistance is VRd = min (VRd,c,VRd,s) = 149.96kN.
The design shear resistance attributed to concrete is evaluated with:

whereas the design shear resistance provided by stirrups is calculated as:

The value of shear resistance equals to which solved for unknown PEd yields the value of the applied load of 286.5kN.
From the comparison of the calculated values of the applied load related to the design moment resistance – 181.23kN and dictated by the shear resistance – 286.5kN, it can be concluded that because the former force is lower – thus governing, the beam fails in bending.

Table 2: Design value of beam resistance expressed in terms of applied load PRd

PRd (EC2 – MC2010)
(kN)
181.232

In Table 2 the design value of beam resistance expressed in terms of applied load PRd obtained with Eurocode 2 (CEN, 2005) and Model Code 2010 (fib, 2013) is summarized.

3 Finite element model

Units
Units are N, m.

Material models and parameters
The concrete model is based on a total strain rotating crack model with

  • exponential softening in tension and parabolic behavior in compression,
  • variable Poisson’s ratio of concrete dependent on crack strain values,
  • reduction of compressive strength of concrete due to lateral cracking with a lower limit of 0.6 according to (Vecchio, 1986),
  • increase in compressive strength due to lateral confinement according to the model proposed by Selby and Vecchio (Selby and Vecchio, 1993).

The mechanical properties for concrete are summarized in Table 3. The uniaxial stress-strain curve is shown in Figure 10. In the input file of the analysis, the GF value has been decreased with a factor 2 in order to compensate for an underestimation of the crack band width for cracks with an inclination angle of 45 degrees, GF,reduced = 0.144 / 2 = 0.102.

Table 3: Constitutive model parameters for concrete

fcm
(N/mm2)
fctm
(N/mm2)
Ec
(N/mm2)
ν GF
(Nmm/mm2)
GC
(Nmm/mm2)
Mean measured values 43.5 3.24* 34925* 0.15 0.144* 35.99*
* Not specified in reference; estimated according to MC2010 (fib, 2013).

Figure 10: Stress-strain curve for concrete

The model for the reinforcement bars and stirrups is based on hardening plasticity. Geometrical and mechanical features of reinforcing bars are summarized in Table 1.
The stress- strain curve of the M30 reinforcing bars is plotted in Figure 11.

Figure 11: Stress-strain curve for M30 reinforcing bars

For the steel plates a linear elastic behavior is assumed, see Table 4.

Table 4: Steel plates properties

E(N/mm2) v
200000 0.3

Interfaceelements were used between the steel plates and the concrete beam at the supports and loading positions. The thickness of interface elements equals 10 mm. Stress-strain relation in compression was derived by assuming a stiffness equivalent to the stiffness of a layer of mortar 1 mm thick having a Young’s modulus derived from the mean measured of compressive strength of concrete as reported in Table 3.

A bilinear behavior is assumed in normal direction (see Figure 12) and a linear elastic relation is assumed in shear direction. The normal stiffness in tension and the stiffness in shear direction were assumed almost equal to zero. For stableness of the analysis horizontal displacements of one pair of nodes across the interface elements of support plates and loading plate were tied. The mechanical features of the interface elements are summarized in Table 5.

Figure 12: Traction-displacement diagram in normal direction for interfaces (not to scale)

Table 5: Interface properties

Knn in tension
(N/mm3)
Knn in compression
(N/mm3)
Kt
(N/mm3)
3.42×10-8 3.42×10+4 3.42×10-8

Element types and finite element mesh
For meshing the concrete, 8-node membrane elements (CQ16M) with a full Gauss integration scheme (3×3) are used. The average element size is 44×36 mm2 . The reinforcement bars and stirrups are modelled with embedded truss elements with two Gauss integration points along the axis of the element. A perfect bond is assumed. For the steel plates 8-node membrane elements (CQ16M) with a full Gauss integration scheme (3×3) are used. The 6-node interfaces element have three Lobatto integration points. The adopted dimensions for the beam and for the transversal cross section of the beam are given in Figure 13 and Figure 14, respectively.

Figure 13: Dimensions adopted for the beam (in mm)

Figure 14: Dimensions adopted for the transversal cross section of the beam (in mm)

The mesh of the beam is presented in Figure 15(a). The different materials are indicated with different colors in Figure 15(b).
Different groups of elements were defined to distinguish the concrete elements that can be subjected to crushing or cracking during the analyses and the steel elements that can yield during the analysis.
These groups will be used in section 4 to monitor the failure mode during the analysis.
For monitoring steel yielding the groups RETOPM10 (red), REBOTM25 (blue), REBOTM30 (green) and STIRRUPS (black) refer to reinforcing bars and stirrups of the beam, see Figure 16

Figure 15: (a) Mesh and (b) material sets.

Figure 16: Groups of steel elements monitoring yielding

Figure 17 shows the group of elements named CRUSHING, used for monitoring the inelastic behavior of concrete in compression. This group of elements has a length equal to 5 times the length of the loading steel plate and a depth equal to the length of the loading steel plate, Figure 17.

Figure 17: Group of concrete elements monitoring crushing due to bending

Figure 18 shows the group of elements named SHEAR, where the inelastic behavior of concrete due to shear was monitored. The group SHEAR has a length equal to the space between the end of the loading plate and the end of the support plate and a depth equal to the space between upper and lower reinforcement. Group shear is lying between the RETOPM10 and the group REBOTM25 and between the edges of the load and support steel plate.

Figure 18: Group of concrete elements monitoring inelastic behavior due to shear

Boundary conditions and loading
The translations along x and y axes at a single node of the left steel plate (support 1) are constrained as well as the translation alongy axis at a single node of the right steel plate (support 2), Figure 19. Dead load is applied in load case 1; load P as a unit load of 1 x 10-3 N is added at the load case 2 as a concentrated load applied at the mid node of the loading plate, Figure 19.

Figure 19: Boundary conditions and load case 2

Load increments and convergence criteria
Load case 1 is applied in a single step. The regular Newton-Raphson method with a maximum of 25 iterations is used. The selected convergence norms are according to both force and energy norms. The analysis continues even if the convergence criteria are not satisfied. The convergence tolerance is equal to 5×10-2 for force norm and equal to 1×10-2 the energy norm. A Line Search algorithm is used to improve the convergence performance.

Figure 20: `Indirect Displacement control' technique applied referring to node 5514

Load case 2 is applied with automatic adaptive load increments based on energy. The initial load factor equals 5, the upper limit of the incremental load factor equals 10 and the lower limit of the incremental load factor equals 5. The maximum number of steps is 150. Arc-length control was applied based on translation along y axis of node 5514 at mid-span (“indirect displacement control”), Figure 20. The analysis continues even if the convergence criteria are not satisfied. The convergence tolerances are equal to 1×10-3 and 1×10-2 for energy and forces, respectively. A maximum of 25 iterations is used. A line search algorithm is used to improve the convergence performance.

4 Nonlinear finite element analysis

Load deflection
The load-deflection curve is presented in Figure 21 where the applied load values corresponding to the onset of yielding of the M30 and M25 longitudinal bars, yielding of the M10 longitudinal bars, yielding of the stirrups D4 are indicated. The step in which the first integration point reaches a minimum principal strain value lower than -3.5×10-3 is defined crushing of concrete.

Convergence behavior
For most steps convergence is till reaching the peak load achieved on the basis of the energy criterion, Figure 22 and Figure 23.
For load case 2 the peak load is defined as the highest load step for which the energy norm ratio satisfies the fixed tolerance of 1×10-3 and it is marked in Figure 22 and Figure 23 with a red dot.
The convergence behavior is quite poor after reaching the peak load. After step 82, the analysis continues even if the energy convergence criteria are not satisfied within the maximum number of iterations equal to 25. The post peak branch of the load – deflection curve is for this reason plotted with a dot line. The force norm ratio is higher than the fixed tolerance of 1×10-2 for most of the steps.

Figure 21: Load-deflection curves

Figure 22: Evolution of the energy norm (blue lines indicate steps, red line indicates tolerance, green points indicate iterative results)

Figure 23: Evolution of the force norm (blue lines indicate steps, red line indicates tolerance, green points indicate iterative results)

Strains
Figure 24 shows the crack strain values (which are the plastic part of the maximum principal strain values) at step 65 (load P = 238 kN). The first crack strain value plotted in Figure 24, equal to 7.86×10-4, corresponds to the ultimate crack strain value calculated as , while the third crack strain value, equal to 3.62×10-3, is the crack strain value corresponding to a stress value equal to 1% of fctm . An intermediate crack strain value was added in the contour plot. The crack pattern, which can be derived from the contour of the principal strain value, shows that the failure is mainly due to bending.

Figure 24: Crack strain values at step 65 ( P = 238kN)

Figure 25: Experimental crack pattern at failure (load PExp = 277kN) (Vecchio & Shim 2004): a) north side, (b) south side

The yielding strain for the M10 reinforcing bars is equal to . The M10 reinforcing bars start to yield in compression at the load equal to 213kN (step 58). Figure 26 shows yielding of M10 reinforcing bars at step 70 (load P = 254kN ).

Figure 26: Yielding of reinforcing bars M10 at step 70 (load P = 254kN ).

The yielding strain of stirrups is equal to. Stirrups start to yield at the load equal to 231kN (step 63). Figure 27 shows yielding of stirrups at step 80 (load P = 267kN ).

Figure 27: Yielding of stirrups at step 80 (load P = 267kN )

The yielding strain of the M30 reinforcing bars is equal to The longitudinal M30 reinforcing bars start to yield in tension at the load equal to 251 kN (step 69). Figure 28 shows yielding of M30 reinforcing bars at step 75 (load P = 263kN ).

Figure 28: Yielding of reinforcing bars M30 at step 75 (load P = 263kN )

The yielding strain of the M25 reinforcing bars is equal to. The longitudinal M25 reinforcing bars start to yield at a load equal to 257 kN (step 71). Figure 29 shows yielding of M25 reinforcing bars at step 75 (load P = 263kN ).

Figure 29: Yielding of M25 reinforcing bars at step 75 (load P = 263kN )

Figure 30shows principal strain state at the step 82 (peak load). The values of the minimum principal strain at the location of applied load are lower than -3.5×10-3 which indicates crushing of concrete in this area. The peak value of applied load obtained from the NLFEA is equal to Pu = 268kN .

Figure 30: Minimum principal strain at step 82 (load Pu = 268kN )

Gauss point statistics
In Table 6 lists the number of cracking points, crushing points and yield points at step 58 (beginning of yielding of M10 reinforcing bars), at step 63 (beginning of yielding of stirrups), at step 67 (when the first element reaches minimum principal strains lower than -3.5×10-3), at step 69 (beginning of yielding of M30 reinforcing bars), at step 71 (beginning of yielding of M25 reinforcing bars) and at step 82 (peak load). Crushing is defined as soon as the softening branch in compression is reached. In the current case, it is at the minimum principal strain of -2.1×10-3.

Table 6: Number of cracking points, crushing points, and yield points

YIELDING OF REINFORCING BARS M10
STEP
58 ITERATIONS 17
GROUP NAME PLAST PRV. PL CRITIC PLAST NEW PRV. PL NEW CRITIC NEW
BEAM 278 2 0 22 0 0
RETOPM10 18 0 0 18 0 0
CRUSHING 252 2 0 16 0 0
TOTAL MODEL 296 2 0 40 0 0
CRACKING LOGGING SUMMARY
GROUP NAME CRACK OPEN CLOSED ACTIVE INACTI ARIRES
BEAM 9725 9725 0 5725 4000 93
SHEAR 1799 1799 0 1031 768 34
TOTAL MODEL 9725 9725 0 5725 4000 93
CUMULATIVE REACTION:
FORCE X FORCE Y
-0.24945D-09 -0.22759D+06
YIELDING STIRRUPS
STEP 63 ITERATIONS 12
GROUP NAME PLAST PRV. PL CRITIC PLAST NEW PRV. PL NEW CRITIC NEW
BEAM 338 0 0 28 0 0
RETOPM10 27 0 0 12 0 0
STIRRUPS 2 2
CRUSHING 280 0 0 18 0 0
SHEAR 3 0 0 3 0 0
TOTAL MODEL 367 0 0 42 0 0
CRACKING LOGGING SUMMARY
GROUP NAME CRACK OPEN CLOSED ACTIVE INACTI ARISES
BEAM 9774 9768 6 6184 3590 93
SHEAR 1731 1791 0 1059 732 39
TOTAL MODEL 9774 9768 6 6184 3590 93
CUMULATIVE REACTION:
FORCE X FORCE Y
-0.63461D-09 -0.24465D+06
CRUSHING OF CONCRETE
STEP 67 ITERATIONS 7
GROUP NAME PLAST PRV. PL CRITIC PLAST NEW PRV. PL NEW CRITIC NEW
BEAM 505 8 0 25 0 0
RETOMP10 54 0 0 6 0 0
STIRRUPS 20 0 0 4 0 0
CRUSHING 379 8 0 11 0 0
SHEAR 16 0 0 1 0 0
TOTAL MODEL 579 8 0 35 0 0
CRACKING LOGGING SUMMARY
GROUP NAME CRACK OPEN CLOSED ACTIVE INACTI ARISES
BEAM 10368 10368 0 7047 3321 86
SHEAR 2007 2007 0 1351 656 31
TOTAL MODEL 10368 10368 0 7047 3321 86
CUMULATIVE REACTION:
FORCE X FORCE Y
-0.16393D-08 -0.26027D+06
YIELDING OF REINFORCING BARS M30
STEP 69 ITERATIONS 13
GROUP NAME PLAST PRV. PL CRITIC PLAST NEW PRV. PL NEW CRITIC NEW
BEAM 555 10 0 21 0 0
RETOPM10 60 0 0 0 0 0
RETOPM30 14 0 0 14 0 0
STIRRUPS 26 0 0 4 0 0
CRUSHING 407 10 0 13 2 0
SHEAR 23 0 0 2 0 0
TOTAL MODEL 655 10 0 39 2 0
CRACKING LOGGING SUMMARY
GROUP NAME CRACK OPEN CLOSED ACTIVE INACTI ARISES
BEAM 10558 10558 0 6779 3779 111
SHEAR 2067 2067 0 1321 746 35
TOTAL MODEL 10558 10558 0 6779 3779 111
CUMULATIVE REACTION:
FORCE X FORCE Y
-0.67223D-10 -0.26644D+06
YIELDING OF REINFORCING BARS M25
STEP 71 ITERATIONS 8
GROUP NAME PLAST PRV. PL CRITIC PLAST NEW PRV. PL NEW CRITIC NEW
BEAM 607 8 0 25 0 0
RETOPM10 66 0 0 6 0 0
REBOTM30 50 0 0 24 0 0
REBOTM25 6 0 0 6 0 0
STIRRUPS 38 0 0 9 0 0
CRUSHING 435 8 0 15 0 0
SHEAR 31 0 0 4 0 0
TOTAL MODEL 767 8 0 70 0 0
CRACKING LOGGING SUMMARY
GROUP NAME CRACK OPEN CLOSED ACTIVE INACTI ARISES
BEAM 10738 10738 0 6978 3760 93
SHEAR 2129 2129 0 1384 745 31
TOTAL MODEL 10738 10738 0 6978 3760 93
CUMULATIVE REACTION:
FORCE X FORCE Y
-0.79772D-12 -0.27184D+06
PEAK LOAD
STEP 82 ITERATIONS 12
GROUP NAME PLAST PRV. PL CRITIC PLAST NEW PRV. PL NEW CRITIC NEW
BEAM 302 446 0 5 210 0
RETOPM10 36 48 0 0 12 0
REBOTM30 104 0 0 4 0 0
REBOTM25 84 0 0 2 0 0
STIRRUPS 68 14 0 4 4 0
CRUSHING 220 298 0 3 62 0
SHEAR 18 30 0 0 6 0
TOTAL MODEL 594 509 0 9 226 0
CRACKING LOGGING SUMMARY
GROUP NAME CRACK OPEN CLOSED ACTIVE INACTI ARISES
BEAM 11378 11378 0 6294 5084 62
SHEAR 2336 2336 0 1215 1121 22
TOTAL MODEL 11378 11378 0 6294 5084 62
CUMULATIVE REACTION:
FORCE X FORCE Y
-0.72739D-10 -0.28331D+06

5 Application of Safety Formats Model Code 2010

Table 7: Constitutive model parameters for concrete

ƒc
(N/mm2)
ƒct
(N/mm2)
E c
(N/mm2)
v GF
(Nmm/mm2)
GC
(Nmm/mm2)
Mean measured 43.50 3.24 34925 Var. 0.144 35.99
Characteristic 35.50 2.27 32659 Var. 0.139 34.70
Mean GRF 30.17 2.91 30954 Var. 0.135 33.70
Design 23.67 1.51 28569 Var. 0.129 32.26

Table 8: Constitutive model parameters for reinforcing bars M10

ϕ
(mm)
As
(mm2)
ƒy
(N/mm2)
ƒt
(N/mm2)
Es
(Nmm/mm2)
εsy
Mean measured 11.3 100 315.00 460.00 200000 0.00158
Characteristic 11.3 100 285.31 416.64 200000 0.00143
Mean GRF 11.3 100 313.84 458.31 200000 0.00157
Design 11.3 100 248.09 362.30 200000 0.00124

Table 9: Constitutive model parameters for reinforcing bars M25

ϕ
(mm)
As
(mm2)
ƒy
(N/mm2)
ƒt
(N/mm2)
Es
(Nmm/mm2)
εsy
Mean measured 25.2 500 445.00 680.00 220000 0.00202
Characteristic 25.2 500 403.06 615.91 220000 0.00183
Mean GRF 25.2 500 443.36 677.50 220000 0.00202
Design 25.2 500 350.48 535.57 220000 0.00159

Table 10: Constitutive model parameters for reinforcing bars M30

ϕ
(mm)
As
(mm2)
ƒy
(N/mm2)
ƒt
(N/mm2)
Es
(Nmm/mm2)
εsy
Mean measured 29.9 700 436.00 700.00 200000 0.00218
Characteristic 29.9 700 394.90 634.02 200000 0.00197
Mean GRF 29.9 700 434.39 697.42 200000 0.00217
Design 29.9 700 343.39 551.32 200000 0.00172

Table 11: Constitutive model parameters for reinforcing bars D4

ϕ
(mm)
As
(mm2)
ƒy
(N/mm2)
ƒt
(N/mm2)
Es
(Nmm/mm2)
εsy
Mean measured 5.7 25.7 600.00 651.00 200000 0.00300
Characteristic 5.7 25.7 543.45 589.64 200000 0.00272
Mean GRF 5.7 25.7 597.79 648.60 200000 0.00299
Design 5.7 25.7 472.56 512.73 200000 0.00236

As proposed by the Model Code 2010 (fib, 2013) safety formats for non-linear finite element analyses include three numerical methods denoted as GRF (Global Resistance Factor method), PF (Partial Factor method) and ECOV (Method of Estimation of a Coefficient of Variation of resistance). Application of the safety formats requires in total 4 non-linear analyses. In Table 7 - Table 11 the mechanical properties used in the non-linear analyses are summarized.

In Figure 31 the load-deflection curves obtained with mean measured, characteristic, mean GRF and design values of material strengths, calculated according to the Model Code 2010 (fib, 2013) are shown. In Figure 31 the peak loads of the analyses are presented. The aforementioned loads were defined as the highest load step which satisfied the imposed energy convergence tolerance of 1×10-3 or as the highest load value when the energy convergence norm was met in the subsequent steps. The peak loads are indicated with dots in Figure 31.

Figure 31: Load-deflection curves obtained with mean measured, characteristic, mean (GRF) and design values of material strengths calculated according to Model Code 2010 (fib, 2013)

This beam was analyzed with the analytical procedures proposed for sectional analysis an as well as numerically, with application of the safety formats for NLFEA as proposed by the Model Code 2010. Figure 32 shows the comparison of the analytical and numerical design values of the beam resistance expressed in terms of a percentage of the experimental ultimate value of applied load.

Figure 32: Analytical and numerical design values of beam resistance expressed in terms of a percentage of the experimental ultimate value of applied load, PExp = 265kN

In Table 12 the design values of beam resistances, expressed in terms of applied load PRd , obtained from numerical and analytical procedures are reported. The analytical beam resistance was obtained in section 2 with application of the sectional analysis according to Eurocode 2 (CEN, 2005) and Model Code 2010 (fib, 2013).

Table 12: Values of beam resistances, expressed in terms of applied load PRd

Experimental
(kN)
EC2,MC2010
(kN)
GRF
(kN)
PF
(kN)
ECOV
(kN)
No Safety Formats
(kN)
265 181 190 193 203 268

6 Parametric study on crack models

A parametric study was carried out by varying sensitive input parameters of the concrete constitutive model, such as the crack model and the fracture energy of concrete in tension.
In Table 13 the material parameters applied in NLFE analyses performed in the parametric study are reported. Analysis 1 to Analysis 3 refer to the three analyses carried out by varying the aforementioned material parameters. All the analyses were performed considering mean measured values of material strengths. Parabolic law in compression and exponential law in tension were used for concrete, while an elasto-plastic law with hardening was adopted for steel. The analyses were carried out in load- control with arc-length control. A variable Poisson ratio was adopted for all analyses.
For all analyses a limit value of the reduction of the compressive strength of concrete due to lateral cracking was adopted:

The effect of the used values of the fracture energy of concrete in tension on the beam response was investigated by adopting the formulation proposed by Model Code 1990 (CEB-FIP, 1993) and the formulation proposed by Model Code 2010 (fib, 2013). The fracture energy of concrete in compression was considered for all analyses equal to 250GF (Nakamura et al. 2001).
Within the fixed crack model a variable shear retention factor, which depends on the mean aggregate size daggr , the crack normal strain εn and the crack bandwidth value h is adopted:

In Figure 33 the load-deflection curves obtained from the parametric study are plotted and the peak load of each analysis is indicated with a circular indicator. The peak load is defined as the highest load step where the energy norm ratio satisfies the fixed tolerance of 1×10-3.
The peak load values are reported in Table 13.

Table 13: Data used for the parametric study

Analysis Total strain crack model Limit to βФ GF GC Peak load value (kN)
Analysis 1 rotating 0.6 MC2010 250 GF 268
Analysis 2 rotating 0.6 MC1990 250 GF 264
Analysis 3 fixed 0.6 MC2010 250 GF 289

In Figure 33 the load-deflection curves resulting from the parametric study are plotted.

Figure 33: Load-deflection curves (Analysis 1 to 3)

The crack model and mechanical properties used in Analysis 1 were the same as those used to predict the design value of beam resistance from NLFE analyses. From the comparison of analyses 1 and 2, the influence of the adapted values of the fracture energy of the concrete in tension ( G F,MC1990 = 0.106N/mm and G F,MC2010 = 0.144N/mm ) and corresponding compressive fracture energy ( G C,MC1990 = 26.58N/mm and G C,MC2010 = 35.99N/mm ) can be observed. Because of the fact that the beam failed in bending with crushing of concrete, the fracture energy of concrete in compression plays an important role on the ductility of the beam – especially on the peak and post-peak deformation. Comparing analyses 1 till 3, it is clear that for this beam, the adopted crack model (total strain rotating or fixed crack model) has a moderate influence on the beam response, both in terms of peak load and in terms of peak deformation.

7 Parametric study on crack bandwidth

This section reports on (i) the sensitivity of analyses results on h, or actually GF/h , and (ii) on post-analysis checks on the correctness of the a priori estimates for h. Material models and parameters, element types and finite element mesh, boundary conditions and loading, load increments and convergence criteria are the same as those used for the analysis carried out with mean measured material strength (please refer to section 1).
Table 14 lists the a priori estimates for the crack bandwidth that are used in this study. Note that the compressive bandwidths hc are unaltered. For practical reasons the variations of the crack bandwidths h were applied in the finite element models by variations of GF . The exponential softening adopted for total strain crack models and exercised in the analysis can be formulated as given below:

The fracture energy of concrete in tension GF divided by the crack bandwidth h is:

The ultimate crack strain results to be evaluated with:

The maximum crack strain value εknn,max is defined as the crack strain corresponding to a residual stress equal to 1% of ƒt :
εknn,max - εu ln 0.001 = 4.6εu
The values of fracture energy of concrete in tension used as input data for the analyses and maximum crack strain values used in the contour plot are reported in Table 14.
Table 15 gives an overview of the obtained peak loads Pu .

Table 14: Estimates for the crack bandwidth h for quadratic plane stress quadrilaterals with 3×3 Gaussian integration

Tension
(mm)
Compression GF
(Nmm/mm2)
GC
(Nmm/mm2)

εknn,max

Short-
width
h = ½√A√2 = 20√2 hc = √A = 40 0.144 35.99 7.24×10-2
Default h = √A√2 = 40√2 hc = √A = 40 0.144 35.99 3.62×10-3
Long-
width
h = 2√A√2 = 80√2 hc = √A = 40 0.144 35.99 1.81×10-3

Figure 34: a) Load-deflection curves obtained with different crack bandwidth values

Table 15: Case studies and modifications

PExp (kN) Pu (kN)
0.5h h 2h
265 271 268 255.98

Figure 34 shows the load-deflection curves for this beam obtained with different crack bandwidth values. The three ascending branches of the load-deflection graphs in show a comprehensible trend, with the “0.5h” branch above the “h” branch, and with the “h” branch above the “2h” branch. This does not hold true for the three descending branches in the Figure 32. A possible explanation is the sensitivity of the descending branches with respect to convergence criteria, especially for load controlled analysis with arc-length, see Section 8. A main observation is that the peak loads are hardly influenced by the choice of the crack bandwidth.

Figure 35 shows contour plots of the crack strain values for the corresponding peak values of the applied loads, obtained with different crack bandwidth values. In the contour plots the color ranges are adjusted to the (tensile) stress-strain relations: red denotes strains beyond the ultimate crack strain of the softening stress-strain diagram.

For a better assessment of the strain localization, mesh extractions from the marked regions in Figure 35 with indicated a priori estimates of h are illustrated in Figure 36.

Figure 35: Maximum principal strain values obtained with different crack bandwidth values

Figure 36: (a) Maximum principal strains for the area indicated in Figure 35. (b) Maximum principal strain-distance between integration points along the lines indicated above. The ultimate strain values are indicated by a red dashed line. (c) fracture energy over crack bandwidth-distance between integration points.

The dissipated fracture energy divided by the crack bandwidth (g Fp ) is plotted versus the distance between different integration points along a line considered perpendicular to the crack. In all graphs 9 integration points, as shown in Figure 37, are taken into account.

Figure 37: Example of integration points considered

The total dissipated fracture energy can be calculated with:

The total dissipated fracture energy in the i-th integration point is illustrated in Figure 38, while the values of dissipated fracture energy in tension are calculated for different values of crack bandwidth in Figure 36 (c).

Figure 38: Fracture energy dissipated in the i- th integration point

From Figure 36 (c) the a posteriori crack bandwidths can be evaluated as the length characterized by maximum dissipated fracture energy of concrete in tension; this means the length characterized by principal strains higher than the ultimate crack strain in tension, εknn,max .
Remarkably, Figure 36 shows that by comparing the a priori crack bandwidths with the obtained a posteriori crack bandwidths, the results are quite close. None of the threea prioricrack bandwidths is clearly superior to the two remaining crack bandwidths.

8 Parametric study of convergence criteria

A sensitivity study were carried out with respect to (i) the fracture energy of concrete in compression, (ii) the convergence method, (iii) the convergence criteria and (iv) the maximum number of iterations, Table 16.
The current sensitivity study employs in the models the mean measured material parameters-material models and parameters as explained in section 3.

Table 16: Case studies and modifications

Case study Flexural beam
Compression model Parabolic, low Gc — Parabolic, medium Gc
Parabolic, high Gc
Control Load control with arc length
Max. number of iterat. 25 - 50

Table 17 presents the values for the fracture energy of concrete in compression. For “low”, “medium” and “high” values of fracture energy of concrete in compression the ratio GC/GF equals 100, 250 and 500 respectively. The fracture energy of concrete in tension GF was calculated with Model Code 2010 (fib, 2013).

Table 17: Values for the fracture energy of concrete in compression GC

low GC
(Nmm/mm2)
medium GC
(Nmm/mm2)
high GC
(Nmm/mm2)
GF
(Nmm/mm2)
14.39 35.99 71.98 0.144

For analyses carried out in load control with arc-length method, Table 18 gives an overview of all analyses which were performed to determine the sensitivity of the analyses results to variations in the convergence criteria.

For low, medium and high values of fracture energy of concrete in compression GC , 6 analyses were carried out:

  • 3 different type of criteria: displacement (D), energy (E) and force (F)
  • 2 levels of convergence tolerances ε': strict (1) and relatively loose (2)

As a main NLFEA result, the Table 18 presents the peak value of applied load Pu . Peak load values are identified in correspondence of the last step load values in which the convergence criterion is satisfied within the maximum number of iterations.

Table 18: Overview of NLFE analyses using load control and obtained peak values of the applied load Pu , (PExp = 265 kN )

Comp. model criterion Analysis
label
tolerance
strict
Pu
(kN)
Analysis
label
tolerance
loose
Pu (kN)
Low GC disp. DA1L 1×10-2 263 DA2L 5×10-2 279
energy EA1L 1×10-3 251 EA2L 5×10-3 263
force FA1L 1×10-2 27 FA2L 5×10-2 42.3
Medium GC disp. DA1M 1×10-2 277 DA2M 5×10-2 303
energy EA1M 1×10-3 268 EA2M 5×10-3 276
energy (50 ite) EA1M50 1×10-3 267
force FA1M 1×10-2 27 FA2M 5×10-2 39.4
High GC disp. DA1H 1×10-2 279 DA2H 5×10-2 320
energy EA1H 1×10-3 280 EA2H 5×10-3 294
force FA1H 1×10-2 268 FA2H 5×10-2 259

Typical results for this beam are given in Figure 39 where the convergence criteria was kept constant and the fracture energy of concrete in compression was varied. Because of the fact the beam fails due to bending after crushing of concrete, the ductility of the beam is increasing as the value of the fracture energy of concrete in compression is increased. In Figure 39 the peak load values are marked with the circular markers. The peak load values are identified in correspondence of the last step load values in which the energy convergence criterion is satisfied within the maximum number of iterations.

Figure 39: Load deflection curves for this beam with varying the fracture energy of concrete in compression

Figure 40 presents the load-deflection curves obtained with the mean values of the fracture energy of concrete in compression and the varying convergence criteria. It can be observed that for the case of the analysis with the governing energy criterion and alternating number of iterations (25 and 50 iterations), no differences in the peak load value takes place. Moreover, from Figure 40 it is noted that for the analysis with the force convergence norm, the convergence criterion was satisfied within the maximum number of iterations for only few load steps.

Figure 40: Load-deflection curves with varying the convergence criteria

Based on the recorded results in Table 18, a number of conclusions can be drawn. The “loose” displacement norm is inappropriate for the estimation of load carrying capacity for the case under consideration. All analyses with the displacement criterion and the convergence tolerance of 5x10-2 resulted in the overestimated peak value when compared to the experimental failure load.
Good and consistent results were obtained by the “strict” energy norm. Based on the results of the analysis with an increased number of iterations per step as well as interpretations of the convergence graphs, a “strict” energy tolerance of 10-3 is recommended.

9 Estimating crack widths

In the current section, a way to estimate the crack width from the nonlinear finite element analyses results is proposed. The crack width was estimated from NLFE results in combination with the Model Code 2010 (fib, 2013).
The crack width is calculated at the Serviceability Limit State. The load at the Serviceability Limit State (PSLS) was defined as:

where Pu is the peak value of applied load obtained from NLFEA. In this way the step corresponding to the Serviceability Limit State is identified.
In the following, the procedure for the determination of the crack width in case of development of bending and shear cracks is explained.

Bending cracks

In order to better evaluate the crack width of the beam, an equivalent tie with the same ratio of longitudinal reinforcement and the same effective area in tension as this beam was modeled. Crack opening width values and crack spacing were evaluated both for the equivalent tie and for the beam in correspondence of the same load level, equal to 268kN/1.7=157 kN.

In Figure 41(a) the cross section of the tie compared with the cross section of the beam and the mechanical model of the equivalent tie (b) is shown.

Figure 41: (a) Cross section and of the equivalent tie (b) Mechanical model of the tie

The characteristic crack width wk was calculated according to the Model Code 2010 (fib, 2013) as follows:

wk = 2lrsmax (εsm - εcm ) (3-1)

where:

ls,max length over which slip between concrete and steel occurs,
εsm average steel strain over the length ls,max ,
εcm average concrete strain over the length ls,max ,

The relative mean strain in equation (3 1) follows from:

(3-2)

Where:

σs the steel stress in a crack,
σsy maximum the steel stress in a crack in the crack formation stage, which for pure tension is

Where:

and σsp can be easily calculated from sectional analysis for bending.

For the length ls,max the following expression applies:

(3-3)

k = 1

For a stabilized cracking stage and long term loading: ͳbms = 1.8 x ƒctm, β = 0.4
Because of the beam fails due to bending, the crack opening width values obtained from NLFE analyses were calculated from the average value of strain of reinforcing steel M30, ε s , over the length 2 ls,max at midspan. The average value of strain of reinforcing steel M30 obtained from the NLFEA was multiplied by the crack spacing 2 ls,max obtained through Model Code 2010 (fib, 2013) formulation, equation.(3 3).

wd = ls,max ε s (3-4)

In Figure 42 the crack strain values of the beam at the SLS (step 43, P = 157kN ) are shown.

Figure 42: Crack strain values at step 43 ( P = 157kN )

In Table 19 the crack spacing obtained from equation (3 3) and the crack width obtained from Model Code 2010 (fib, 2013) (equation (3 1)) and equation (3 4) are summarized.
It can be noted that the crack opening width values mainly due to bending are well predicted with equation. (3 4) and on the safe side.

Table 19: Crack width according to Model Code 2010 (fib, 2013) formulation and equation (3 4))

2 ls,max
(mm)
wd
(mm)
wd
(mm)
236 0.297 0.318

10 Concluding remarks

This benchmark beam tested in the experimental program of Vecchio & Shim (2004) exhibited a flexural-compressive failure mode at a load equal to . The analytical calculation based on sectional analysis demonstrated that the beam fails due to bending. The shear force corresponding to the design value of moment resistance is lower than the design value of shear resistance. The design beam resistance evaluated with sectional analysis equals to PRd = 176kN.
The behavior of the beam is highly influenced by crushing of concrete beneath and adjacent to the loading plates. The disturbances around the loading plates introduce complex three-dimensional effects, making the modelling of interface elements placed between loading or supporting steel plates and the RC beam a fundamental aspect to be considered.
A flexural-compressive failure mode was achieved from NLFEA carried out with mean measured value of material strengths. The peak value of applied load obtained from NLFEA is equal to 268 kN and the failure mode is characterized by crushing of concrete, yielding of M10 top bars, yielding of M30 and M25 bottom bars and yielding of stirrups.
Safety formats for non-linear finite element analyses as proposed by the Model Code 2010 (fib, 2013), were used to derive the design value of beam resistance. The design value of beam resistance obtained from application of safety formats is higher than the design value of beam resistance obtained from analytical methods based on the sectional analysis. The maximum value was obtained with ECOV method and equals to PRd = 203kN .
Since the beam fails in bending, the failure mode is not heavily dependent on the crack model and tensile strengths adopted for concrete in tension, whereas the post-peak behavior appears to be sensitive to the stress-strain relation adopted for concrete in compression.
The first sensitivity study investigated the influence of the varying crack bandwidth (0.5h, h, 2h). A priori crack bandwidths (based on integration schemes) were compared with a posteriori crack bandwidths derived on the calculation of the dissipated fracture energy of concrete in tension. It was concluded that none of the three a priori crack bandwidths is clearly superior to the two remaining crack bandwidths
The second sensitivity study was carried out by varying the compression model for concrete, the convergence method, the convergence criteria and the maximum number of iterations. From the results, it was concluded that good and consistent results are obtained with the “strict” force and the “strict” energy convergence norms. Moreover, based on the above conclusions an energy norm with a tolerance of 10-3 is recommended.
Finally, the method to estimate the crack width from the results of nonlinear finite element analyses is proposed to satisfy performance requirements of serviceability limit states (SLS). It was shown that for this beam the crack width values, mainly due to bending, can be evaluated with good accuracy by multiplying the average value of strain of reinforcing steel by the crack spacing ls,max calculated with expressions of the MC2010.

11 References

Bresler, B. & Scordelis, A.C. (1963), "Shear strength of reinforced concrete beams", J. Am. Concr. Inst. 60(1), 51–72.

CEN (2005), Eurocode 2 - Design of concrete structures - Part 1-1: General rules and rules for buildings, EN 1992-1-1, Brussels: CEN.

fib (2013), fib Model Code for Concrete Structures 2010, Ernst & Sohn.

Selby R.G., Vecchio F.J. (1993). “Three-dimensional Constitutive Relations for Reinforced Concrete”, Tech. Rep. 93-02, Univ. Toronto, dept. Civil Eng., Toronto, Canada.

Vecchio, F.J. & Shim W. (2004), "Experimental and Analytical Reexamination of Classic Concrete Beam Tests", J. Struct. Engrg. ASCE 130(3), 460-469.

Vecchio F. J. & Collins M. P. (1986), "The modified compression-field theory for reinforced concrete elements subjected to shear", ACI Journal 83, 219-231

Nakamura, H. & Higai T. (2001), "Compressive Fracture Energy and Fracture Zone Length of Concrete" in "Modeling of Inelastic Behavior of RC Structures under Seismic Loads", Benson P. Shing (editor), ASCE J. Str. Eng., 471-487, Benson P. Shing.

CEB-FIP Model Code 1990. (1993), Bullettin d’Information n° 213/214. Thomas Telford.