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 reexamination 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 flexurecompressive 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.
The bottom longitudinal reinforcement is extended outside the beam and welded to oneinch thick plates. It is assumed that the dimensions of these plates are 0.192×0.350×0.025 m3.
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  
ƒ_{cm}
(N/mm^{2}) 
d_{max}
(mm) 

43.5  20  
Reinforcement properties  
Bar 
ϕ
(mm) 
A_{s}
(mm^{2}) 
E_{s}
(N/mm^{2}) 
ƒ_{ym}
(N/mm^{2}) 
ƒ_{tm}
(N/mm^{2}) 
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 outofplane dimensions of the loading and support plates are larger than the outofplane thickness of the beam.
Experimental Results
The beam exhibited a flexuralcompressive failure mode with a clear maximum in the loaddeflection response, Figure 4 and Figure 5. The experimental ultimate value of applied load was equal to P_{Exp}
= 265kN at the deflection of 44.3 mm.
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 / m^{3} = 2.098kN / 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
Load case 2
The experimental ultimate value of the applied load is equal to P_{E,max} = 265kN.
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 V_{E,max} = 1325kN.
Load 1 + Load 2
At failure the maximum value of applied moment is
M_{E,max }
= 10.69kNm + 424kNm = 434.69kNm and the value of applied shear force is equal
to V_{E,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 parabolarectangle relation,
 he stresses in the reinforcing steel are derived from an elastic–plastic stressstrain 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 ϓ_{c}=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.
Assuming that reinforcement yields at strains:
which is smaller than x_{lim}
= 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 V_{Rd }= min (V_{Rd,c},V_{Rd,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 P_{Ed}
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 P_{Rd}
P_{Rd} (EC2 – MC2010) 
(kN) 
181.232 
In Table 2 the design value of beam resistance expressed in terms of applied load P_{Rd} 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 stressstrain curve is shown in Figure 10. In the input file of the analysis, the G_{F} 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, G_{F,reduced} = 0.144 / √2 = 0.102.
Table 3: Constitutive model parameters for concrete
f_{cm}
(N/mm^{2}) 
f^{ctm}
(N/mm^{2}) 
E_{c}
(N/mm^{2}) 
ν 
G_{F}
(Nmm/mm^{2}) 
G_{C}
(Nmm/mm^{2}) 

Mean measured values  43.5  3.24*  34925*  0.15  0.144*  35.99* 
* Not specified in reference; estimated according to MC2010 (fib, 2013). 
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.
For the steel plates a linear elastic behavior is assumed, see Table 4.
Table 4: Steel plates properties
E(N/mm^{2})  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. Stressstrain 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.
Table 5: Interface properties
K_{nn in tension}
(N/mm^{3}) 
K^{nn in compression}
(N/mm^{3}) 
K_{t}
(N/mm^{3}) 
3.42×10^{8}  3.42×10^{+4}  3.42×10^{8} 
Element types and finite element mesh
For meshing the concrete, 8node membrane elements (CQ16M) with a full Gauss integration scheme (3×3) are used. The average element size is 44×36 mm^{2}
. 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 8node membrane elements (CQ16M) with a full Gauss integration scheme (3×3) are used.
The 6node 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.
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 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 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.
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.
Load increments and convergence criteria
Load case 1 is applied in a single step. The regular NewtonRaphson 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.
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. Arclength control was applied based on translation along y axis of node 5514 at midspan (“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 loaddeflection 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.
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 f_{ctm}
. 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.
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 ).
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 ).
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 ).
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 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 P_{u} = 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  
0.24945D09  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  
0.63461D09  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  
0.16393D08  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  
0.67223D10  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  
0.79772D12  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  
0.72739D10  0.28331D+06 
5 Application of Safety Formats Model Code 2010
ƒ_{c}
(N/mm^{2}) 
ƒ_{ct}
(N/mm^{2}) 
E
_{c}
(N/mm^{2}) 
v 
G_{F}
(Nmm/mm^{2}) 
G_{C}
(Nmm/mm^{2}) 

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 
ϕ (mm) 
A_{s}
(mm^{2}) 
ƒ_{y}
(N/mm^{2}) 
ƒ_{t}
(N/mm^{2}) 
E_{s}
(Nmm/mm^{2}) 
ε^{sy}


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 
ϕ (mm) 
A_{s}
(mm^{2}) 
ƒ_{y}
(N/mm^{2}) 
ƒ_{t}
(N/mm^{2}) 
E_{s}
(Nmm/mm^{2}) 
ε^{sy}


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 
ϕ (mm) 
A_{s}
(mm^{2}) 
ƒ_{y}
(N/mm^{2}) 
ƒ_{t}
(N/mm^{2}) 
E_{s}
(Nmm/mm^{2}) 
ε^{sy}


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 
ϕ (mm) 
A_{s}
(mm^{2}) 
ƒ_{y}
(N/mm^{2}) 
ƒ_{t}
(N/mm^{2}) 
E_{s}
(Nmm/mm^{2}) 
ε^{sy}


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 nonlinear 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 nonlinear analyses. In Table 7  Table 11 the mechanical properties used in the nonlinear analyses are summarized.
In Figure 31 the loaddeflection 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 1×103 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.
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.
In Table 12 the design values of beam resistances, expressed in terms of applied load P_{Rd} , 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).
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 elastoplastic law with hardening was adopted for steel. The analyses were carried out in load control with arclength 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 (CEBFIP, 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 d_{aggr}
, the crack normal strain ε_{n} and the crack bandwidth value h is adopted:
In Figure 33 the loaddeflection 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.
Analysis  Total strain crack model  Limit to β_{Ф}  GF  GC  Peak load value (kN) 
Analysis 1  rotating  0.6  MC2010  250 G_{F}  268 
Analysis 2  rotating  0.6  MC1990  250 G_{F}  264 
Analysis 3  fixed  0.6  MC2010  250 G_{F}  289 
In Figure 33 the loaddeflection curves resulting from the parametric study are plotted.
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 postpeak 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 G_{F}/^{h}
, and (ii) on postanalysis 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 G_{F}
. 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 G_{F} divided by the crack bandwidth h is:
The ultimate crack strain results to be evaluated with:
The maximum crack strain value ε_{knn,max}
is defined as the crack strain corresponding to a residual stress equal to 1% of ƒ_{t}
:
ε_{knn,max}
 ε_{u}
ln 0.001 = 4.6ε_{u}
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 P_{u}
.
Tension (mm) 
Compression 
G_{F}
(Nmm/mm^{2}) 
G_{C}
(Nmm/mm^{2}) 
ε_{knn,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} 
P_{Exp} (kN)  P_{u} (kN)  
0.5h  h  2h  
265  271  268  255.98 
Figure 34 shows the loaddeflection curves for this beam obtained with different crack bandwidth values. The three ascending branches of the loaddeflection 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 arclength, 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) stressstrain relations: red denotes strains beyond the ultimate crack strain of the softening stressstrain 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.
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.
The total dissipated fracture energy can be calculated with:
The total dissipated fracture energy in the ith 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).
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, ε_{knn,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 parametersmaterial models and parameters as explained in section 3.
Case study  Flexural beam 
Compression model  Parabolic, low G_{c}
— Parabolic, medium G_{c}
— Parabolic, high G_{c} 
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 G_{C}/G_{F} equals 100, 250 and 500 respectively. The fracture energy of concrete in tension G_{F} was calculated with Model Code 2010 (fib, 2013).
low G_{C}
(Nmm/mm^{2}) 
medium G_{C}
(Nmm/mm^{2}) 
high G_{C}
(Nmm/mm^{2}) 
G_{F}
(Nmm/mm^{2}) 
14.39  35.99  71.98  0.144 
For analyses carried out in load control with arclength 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 G_{C} , 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 P_{u} . 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.
Comp. model  criterion  Analysis label 
tolerance strict 
P_{u}
(kN) 
Analysis label 
tolerance loose 
P_{u} (kN) 
Low G_{C}  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 G_{C}  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 G_{C}  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 40 presents the loaddeflection 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.
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 P_{u}
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.
The characteristic crack width w_{k} was calculated according to the Model Code 2010 (fib, 2013) as follows:
w_{k} = 2l_{rsmax} (ε_{sm}  ε_{cm} )  (31) 
where:
l_{s,max}  length over which slip between concrete and steel occurs, 
ε_{sm}  average steel strain over the length l_{s,max} , 
ε_{cm}  average concrete strain over the length l_{s,max} , 
The relative mean strain in equation (3 1) follows from:
(32) 
Where:
σ_{s}  the steel stress in a crack, 
σ_{sy}  maximum the steel stress in a crack in the crack formation stage, which for pure tension is 
Where:
and σ_{sp} can be easily calculated from sectional analysis for bending.
For the length l_{s,max} the following expression applies:
(33) 
k = 1
For a stabilized cracking stage and long term loading: ͳ_{bms}
= 1.8 x ƒ_{ctm}, 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).
w_{d} = l_{s,max} ε _{s}  (34) 
In Figure 42 the crack strain values of the beam at the SLS (step 43, P = 157kN ) are shown.
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.
2
_{ls,max}
(mm) 
w_{d}
(mm) 
w_{d}
(mm) 
236  0.297  0.318 
10 Concluding remarks
This benchmark beam tested in the experimental program of Vecchio & Shim (2004) exhibited a flexuralcompressive 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 P_{Rd}
= 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 threedimensional 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 flexuralcompressive 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 nonlinear 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 P_{Rd}
= 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 postpeak behavior appears to be sensitive to the stressstrain 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 l_{s,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 11: General rules and rules for buildings, EN 199211, Brussels: CEN.
fib (2013), fib Model Code for Concrete Structures 2010, Ernst & Sohn.
Selby R.G., Vecchio F.J. (1993). “Threedimensional Constitutive Relations for Reinforced Concrete”, Tech. Rep. 9302, 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), 460469.
Vecchio F. J. & Collins M. P. (1986), "The modified compressionfield theory for reinforced concrete elements subjected to shear", ACI Journal 83, 219231
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., 471487, Benson P. Shing.
CEBFIP Model Code 1990. (1993), Bullettin d’Information n° 213/214. Thomas Telford.