576x
001025
2024-02-28

VE 1025 | Reinforced Concrete Design of a Two-span Continuous Beam with Tapered Cantilever

Description

A reinforced concrete beam is designed as a two-span beam with a cantilever. The cross-section varies along the length of the cantilever (tapered cross-section). The internal forces, the required longitudinal and shear reinforcement for the ultimate limit state are calculated and compared to the results in [1].


Material Concrete C25/30 Modulus of Elasticity E 31000 N/mm2
Design value of concrete compressive strength fcd 14.167 N/mm2
Reinforcing Steel B500S(B) Characteristic yield strength fyk 500.000 N/mm2
Design yield strength fyd 434.783 N/mm2
Geometry Structure Cantilever length leff,cantilever 4.000 m
Span 1 length leff,1 8.000 m
Span 2 length leff,2 8.000 m
Cross-section Height h 1500 mm
Width b 2620 mm
Flange height hf 150 mm
Web width bw 380 mm
Concrete cover cnom 35 mm
Loads Permanant loads LC1 gk,1 10.500 - 90.000 (trapezoidal) kN/m
LC2 Gk,2 216.000 kN
LC3 Gk,3 416.000 kN
Imposed loads LC4 qk,1,1 40.000 kN/m
LC5 qk,1,2 40.000 kN/m
LC6 qk,1,3 30.000 kN/m
LC7 Qk,2 284.000 kN


RFEM Settings

  • Consideration of limited moment redistribution of the supporting moment acc. to 5.5
  • Reduction of the moments or dimensioning for the momentsat the face of a monolithic support acc. to 5.3.2.2
  • Reduction of shear forces in the support face and distance d acc. to 6.2.1(8)
  • The distribution type of the section used is tapered at start of member, to consider the height change of the cross-section.

Results

  1. Bending Moment and Shear force from Permanant and Imposed Loads

    Bending Moment and Shear Force due to gk,1
    Inner Force Unit RFEM / Analytical Solution Span 1 Span 2 Axis A Axis B Axis C
    Bending Moment [kNm] RFEM 248.890 432.840 -296.460 -645.760 0
    Analytical Solution 249.000 433.000 -296.000 -646.000 0
    Shear Force [kN] RFEM -43.330 80.830 -201.000/316.340 -403.660/440.720 -279.280
    Analytical Solution -44.000 81.000 -201.000/316.000 -404.000/441.000 -279.000
    Bending Moment and Shear Force due to Gk,2
    Inner Force Unit RFEM / Analytical Solution Span 1 Span 2 Axis A Axis B Axis C
    Bending Moment [kNm] RFEM -305.850 101.850 -815.400 203.720 0
    Analytical Solution -306.000 102.000 -815.000 204.000 0
    Shear Force [kN] RFEM 127.390 -25.460 -215.670/127.390 -127.390/-25.460 -25.460
    Analytical Solution 127.000 -25.500 -216.000/127.000 -127.000/-25.500 -25.500
    Bending Moment and Shear Force due to Gk,3
    Inner Force Unit RFEM / Analytical Solution Span 1 Span 2 Axis A Axis B Axis C
    Bending Moment [kNm] RFEM 676.040 -155.960 0 -311.920 0
    Analytical Solution 676.000 156.000 0 -312.000 0
    Shear Force [kN] RFEM 169.010/-246.990 -38.990 169.010 -246.990/38.990 38.990
    Analytical Solution 169.000/247.000 39.000 169.000 -247.000/39.000 39.000
    Bending Moment and Shear Force due to qk,1,1
    Inner Force Unit RFEM / Analytical Solution Span 1 Span 2 Axis A Axis B Axis C
    Bending Moment [kNm] RFEM -120.100 40.000 -320.200 79.950 0
    Analytical Solution -120.220 40.030 -320.490 80.060 0
    Shear Force [kN] RFEM 50.070 -10.000 -160.000/50.020 50.020/-10.000 -10.000
    Analytical Solution 50.000 -10.010 -160.000/50.070 50.070/-10.010 -10.010
    Bending Moment and Shear Force due to qk,1,2
    Inner Force Unit RFEM / Analytical Solution Span 1 Span 2 Axis A Axis B Axis C
    Bending Moment [kNm] RFEM 240.020 -79.980 0 -159.960 0
    Analytical Solution 240.000 -80.000 0 -160.000 0
    Shear Force [kN] RFEM -19.990 19.990 140.010 -179.990/19.999 19.999
    Analytical Solution -20.000 20.000 140.000 -180.000/20.000 20.000
    Bending Moment and Shear Force due to qk,1,3
    Inner Force Unit RFEM / Analytical Solution Span 1 Span 2 Axis A Axis B Axis C
    Bending Moment [kNm] RFEM -59.980 180.010 0 -119.970 0
    Analytical Solution -60.000 184.000 0 -120.000 0
    Shear Force [kN] RFEM -15.000 15.000 -15.000 -15.000/135.000 -105.000
    Analytical Solution -15.000 15.000 -15.000 -15.000/135.000 -105.000
    Bending Moment and Shear Force due to Qk,2
    Inner Force Unit RFEM / Analytical Solution Span 1 Span 2 Axis A Axis B Axis C
    Bending Moment [kNm] RFEM 461.530 -106.470 0 -212.950 0
    Analytical Solution 462.000 -106.500 0 -213.000 0
    Shear Force [kN] RFEM 115.380/-168.620 26.620 115.380 -168.620/26.620 26.620
    Analytical Solution -169.000/115.000 26.600 115.000 -15.000/135.000 -169.000/26.600


  2. Internal Forces
    The table below contains all the load combinations of the ultimate limit state:

    Load Comibation Assigned Load Cases
    CO1 1.00·LC1 + 1.00·LC2 + 1.00·LC3
    CO2 1.35·LC1 + 1.35·LC2 + 1.35·LC3 + 1.50·LC4 + 1.50·LC5 + 1.50·LC6 + (1.50·0.80)·LC7
    CO3 1.35·LC1 + 1.35·LC2 + 1.35·LC3 + (1.50·0.70)·LC4 + (1.50·0.70)·LC5 + (1.50·0.70)·LC6 + 1.50·LC7
    CO4 1.35·LC1 + 1.00·LC2 + 1.35·LC3 + 1.50·LC5 + 1.50·LC6 + (1.50·0.80)·LC7
    CO5 1.35·LC1 + 1.00·LC2 + 1.35·LC3 + (1.50·0.70)·LC5 + 1.50·LC7
    CO6 1.00·LC1 + 1.35·LC2 + 1.35·LC3 + (1.50·0.70)·LC4 + 1.50·LC7
    CO7 1.35·LC1 + 1.00·LC2 + 1.35·LC3 + (1.50·0.70)·LC5 + (1.50·0.70)·LC6+ 1.50·LC7
    CO8 1.35·LC1 + 1.35·LC2 + 1.00·LC3 + 1.50·LC4 + 1.50·LC6
    CO9 1.35·LC1 + 1.35·LC2 + 1.35·LC3 + 1.50·LC4 + 1.50·LC5 + (1.50·0.80)·LC7

    Action Unit Load Combination RFEM Result Reference Result Ratio
    MEd,A kNm CO8 -1981.830 -1980.000 1.00
    MEd,B kNm CO4 -1764.600 -1765.000 0.99
    MEd,1 kNm CO5 1887.120 1887.000 1.00
    MEd,2 kNm CO8 885.540 895.000 0.99
    VEd,A,li kN CO2 -802.500 -803.000 0.99
    VEd,A,re kN CO9 1250.770 1250.000 1.00
    VEd,1,li kN CO6 582.090 581.000 1.00
    VEd,1,re kN CO7 -554.660 -555.000 0.99
    VEd,B,li kN CO4 -1245.820 -1246.000 0.99
    VEd,B,re kN CO4 -886.580 -887.000 0.99
    VEd,C kN CO8 -544.930 -545.000 0.99

  3. Required Longitudinal Reinforcement
    In the literature, a 15% moment redistribution was considered at support B within load combinations 4, and a 12% moment redistribution was considered within load combination 7. In contrast, RFEM applies the same moment redistribution across all load combinations. To facilitate a meaningful comparison with the literature, adjustments will be made to the RFEM model. Subsequently, the actual solution provided by RFEM will be presented.
    Comparing RFEM results to literature results:
    Support A:
    The beam is monolithically connected to the support, and therefore, the critical design moment is at the face of the support. However, the literature neglects the influence of the load when calculating the moment at the face of the support. To enable a meaningful comparison with the results in RFEM, it is necessary to recalculate it while considering the influence of the load. The design moment at the face of the support without load influence consideration, MEd, is -1819.0 kNm. Considering the effect of the loads, MEd increases to -1823.0 kNm.
    RFEM Analytical Solution Ratio
    Load Case Design Bending Moment MEd Required Reinforcement As,stat,tot Design Bending Moment MEd Required Reinforcement As,stat,tot MEd As,stat,tot
    [kNm] [cm2] [kNm] [cm2] [kNm] [cm2]
    CO8 -1824.790 32.50 -1823.000 31.60 1.00 1.02

    In the literature, it is assumed that the cross-section height at the edge of the support is equal to the cross-section height at the middle of the support. However, in RFEM, the actual cross-section height is considered due to the tapered cross-section. As a result, this leads to higher reinforcement requirements in RFEM.
    Support B:
    The critical load combination in this case is the load combination 4. To match the literature, the ratio of moment redistribution in the support B is set to 0.850.
    Support B
    RFEM Analytical Solution Ratio
    Load Case Design Bending Moment MEd Required Reinforcement As,stat,tot Design Bending Moment MEd Required Reinforcement As,stat,tot MEd As,stat,tot
    [kNm] [cm2] [kNm] [cm2] [kNm] [cm2]
    CO4 -1345.870 22.40 -1360.000 22.80 0.99 0.98

    When calculating the design moment, the literature takes into consideration that the moment at the face of the support should not be less than 0.65 of the full fixed end moment (DIN EN 1992-1-1, 5.3.2.2). This condition is not implemented in RFEM. This explains the difference in the design moment.

    Span 1:
    Since the beam is defined as a continuous member in RFEM, it is not possible to set an effective width beff to each span. The smallest value from the two effective widths from span 1 and 2 is used for simplification. beff is than set to 2.620 m. The literature considers a 12% moment redistrubiton for the load combination 7, the moment redistribution ratio in the middle support is therefore now set to 0.880.
    Span 1
    RFEM Analytical Solution Ratio
    Load Case Design Bending Moment MEd Required Reinforcement As,stat,tot Design Bending Moment MEd Required Reinforcement As,stat,tot MEd As,stat,tot
    [kNm] [cm2] [kNm] [cm2] [kNm] [cm2]
    CO7 1926.280 30.13 1927.000 33.10 0.99 0.91

    Span 2:
    In this case, no moment redistribution is considered. The ratio of moment redistribution is set to 1.000.
    Span 2
    RFEM Analytical Solution Ratio
    Load Case Design Bending Moment MEd Required Reinforcement As,stat,tot Design Bending Moment MEd Required Reinforcement As,stat,tot MEd As,stat,tot
    [kNm] [cm2] [kNm] [cm2] [kNm] [cm2]
    CO8 885.520 13.79 895.000 15.10 0.99 0.91


    In the literature, the required longitudinal reinforcement is determined using approximation methods for T-beams according to DAstb-heft 425. Using this method, the compressive force in the flange is assumed to be in the center of the flange (hf/2). in RFEM, the required reinforcement is determined with a cross-section analysis. This results in a required reinforcement lower than in the literature.
    RFEM provided solution
    Now, the moment redistribution in the middle support is set to 15% across all load combinations. The results are summarized in the tables below.
    Support A:
    Load case 8 delivers the highest bending moment, and is therefore decisive.
    Support A
    Load Case Design Bending Moment MEd Required Reinforcement As,stat,tot
    [kNm] [cm2]
    CO8 -1824.840 32.32

    Support B:
    Support B:
    Load Case Design Bending Moment MEd Required Reinforcement As,stat,tot
    [kNm] [cm2]
    CO4 -1345.890 22.40

    Span 1:
    When moment redistribution is taken into consideration in all load combinations, CO5 has the highest design bending moment in Span 1.
    Span 1:
    Load Case Design Bending Moment MEd Required Reinforcement As,stat,tot
    [kNm] [cm2]
    CO5 2005.410 31.44

    Span 2:
    CO8 has a design moment after moment redistibution MEd of 940 kNm.
    Span 2:
    Load Case Design Bending Moment MEd Required Reinforcement As,stat,tot
    [kNm] [cm2]
    CO8 940.000 14.73

  4. Shear Reinforcement
    Shear reinforcement in the cantilever:
    To determine the required stirrups in the cantilever, 3 locations are examined. The results are summarized in the table below:
    Cantilever
    Location x Parameter Symbol Unit RFEM Analytical Solution Ratio
    x = 0.45m Effective depth d [m] 0.940 0.920 1.02
    Inner lever arm z [m] 0.848 0.828 1.02
    Shear force VEd [kN] -327.190 -328.000 0.99
    Design bending moment MEd [kNm] -73.320 -74.000 0.99
    Design shear component of the force in compression area Vccd [kN] 12.550 13.000 0.99
    Design shear force VEd,red [kN] 314.640 314.000 1.0
    Shear capacity without reinforcement Vrd,cc [kN] 219.420 221.00 0.99
    Inclination of the compression strut cot Θ [-] 3.0 3.0 1.0
    Capacity of compression strut Vrd,max [kN] 996.230 1003.000 0.99
    Required reinforcement asw,req [cm2/m] 2.84 2.91 0.98
    x = 1.37m Effective depth d [m] 1.070 1.050 1.02
    Inner lever arm z [m] 0.965 0.945 1.02
    Shear force VEd [kN] -417.720 -418.000 1.00
    Design bending moment MEd [kNm] -414.250 -415.000 1.00
    Design shear component of the force in compression area Vccd [kN] 62.210 66.000 0.94
    Design shear force VEd,red [kN] 355.510 353.000 1.01
    Shear capacity without reinforcement Vrd,cc [kN] 250.070 252.000 0.99
    Inclination of the compression strut cot Θ [-] 3.0 3.0 1.0
    Capacity of compression strut Vrd,max [kN] 1135.860 1144.000 0.99
    Required reinforcement asw,req [cm2/m] 2.83 2.86 0.99
    x = 2.37m Effective depth d [m] 1.210 1.190 1.02
    Inner lever arm z [m] 1.090 1.070 1.02
    Shear force VEd [kN] -541.800 -543.000 1.0
    Design bending moment MEd [kNm] -891.790 -893.00 1.00
    Design shear component of the force in compression area Vccd [kN] 118.250 125.000 0.95
    Design shear force VEd,red [kN] 423.550 418.000 1.01
    Shear capacity without reinforcement Vrd,cc [kN] 283.220 285.000 0.99
    Inclination of the compression strut cot Θ [-] 3.0 3.0 1.0
    Capacity of compression strut Vrd,max [kN] 1286.410 1298.000 0.99
    Required reinforcement asw,req [cm2/m] 2.98 2.99 1.0


    Span 1:
    The decisive member location for the calculation of the stirrups in field 1 is at a distance d from the right edge of the support A.
    Span 1
    Parameter Symbol Unit RFEM Analytical Solution Ratio
    Effective depth d [m] 1.440 1.430 1.00
    Shear force at the support A VEd,A [kN] 1250.770 1250.000 1.00
    Design shear force VEd,A,re [kN] 952.430 954.000 1.00
    Shear capacity without reinforcement VRd,cc [kN] 346.210 343.000 1.00
    Inclination of compression strut cot Θ [-] 1.88 1.87 1.00
    Required shear reinforcment asw,req [cm2/m] 8.95 9.11 0.98


    Span 2:

    The calculation of the stirrups is done analog to span 1.
    Span 2
    Parameter Symbol Unit RFEM Analytical Solution Ratio
    Effective depth d [m] 1.440 1.440 1.02
    Shear force at the support B VEd,B [kN] 886.580 855.000 1.03
    Design shear force VEd,B,re [kN] 613.100 584.000 1.05
    Shear capacity without reinforcement VRd,cc [kN] 346.210 343.000 1.00
    Inclination of compression strut cot Θ [-] 2.75 2.91 0.95
    Required shear reinforcment asw,req [cm2/m] 3.94 3.58 1.10


    The differences in the results for span 2 are due to the fact that the literature considered the shear force at support B after moment redistribution. However, moment redistribution does not influence the shear force design in RFEM.


References
  1. German Concrete and Structural Engineering Association E. V, examples for design according to Eurocode 2. Volume 1: Building construction, Berlin: Ernst & Sohn 2012, 1st prize corrected reprint of 1. Edition, 978-3-433-01877-4