1. Reference problem#

1.1. Geometry#

We consider a beam with a rectangular cross section, with dimensions \(h={H}_{Z}=\mathrm{0,3}m\) and \({b}_{w}={H}_{Y}=\mathrm{0,5}m\).

_images/100002010000023E0000027F5875E08F68834DD3.png

1.2. Material properties#

Not applicable to mechanical resolution (see § 1.3). See paragraph 1.3.2 for the parameters for determining the limit state for the material.

1.3. Boundary conditions and loads#

There is no mechanical resolution operator called in this test.

A generalized analytical field of effort is applied as input to the operator CALC_FERRAILLAGE, corresponding to one of the following configurations:

1.3.1. Case of loading at ELU#

  1. compression force of \(1000000\text{N}\), and a shear force of \(100000\text{N}\) along the axis \(\text{Z}\)

  2. traction force of \(1000000\text{N}\), and a shear force of \(-100000\text{N}\) along the axis \(\text{Z}\)

  3. tensile force of \(1000000\text{N}\), a shear force of \(-100000\text{N}\) along the axis \(\text{Z}\) and a torsional moment of \(10000\text{Nm}\)

  4. bending moment of \(100000\text{Nm}\) around the \(\text{Z}\) axis.

  5. Bending moment of \(-100000\text{Nm}\) around the \(\text{Y}\) axis.

  6. bending moment of \(-100000\text{Nm}\) around the \(\text{Z}\) axis and pull force of \(100000\text{N}\)

  7. bending moment of \(-100000\text{Nm}\) around the \(\text{Y}\) axis and pull force of \(100000\text{N}\)

  8. bending moment of \(-100000\text{Nm}\) around the \(\text{Z}\) axis and pull force of \(2000000\text{N}\)

  9. bending moment of \(100000\text{Nm}\) around the \(\text{Z}\) axis and \(-150000\text{Nm}\) bending moment around the \(\text{Y}\) axis

  10. bending moment of \(100000\text{Nm}\) around axis \(\text{Z}\), moment of bending of \(-150000\text{Nm}\) around axis \(\text{Y}\), and compression force of \(3000000\text{N}\)

  11. bending moment of \(-150000\text{Nm}\) around the \(\text{Z}\) axis

  12. bending moment of \(-260000\text{Nm}\) around the \(\text{Z}\) axis

  13. bending moment of \(-380000\text{Nm}\) around the \(\text{Z}\) axis

  14. compression force of \(4500000\text{N}\), bending moment of \(380000\text{Nm}\) around the axis \(\text{Z}\) and a shear force of \(100000\text{N}\) following \(\text{Z}\)

1.3.2. Load cases with the ELS Feature#

  1. compression force of \(1000000\text{N}\), and a shear force of \(100000\text{N}\) along the axis \(\text{Z}\)

  2. traction force of \(1000000\text{N}\), and a shear force of \(-100000\text{N}\) along the axis \(\text{Z}\)

  3. tensile force of \(1000000\text{N}\), a shear force of \(-100000\text{N}\) along the axis \(\text{Z}\) and a torsional moment of \(10000\text{Nm}\)

  4. bending moment of \(100000\text{Nm}\) around the \(\text{Z}\) axis.

  5. Bending moment of \(-100000\text{Nm}\) around the \(\text{Y}\) axis.

  6. bending moment of \(-100000\text{Nm}\) around the \(\text{Z}\) axis and pull force of \(100000\text{N}\)

  7. bending moment of \(-100000\text{Nm}\) around the \(\text{Y}\) axis and pull force of \(100000\text{N}\)

  8. bending moment of \(-100000\text{Nm}\) around the \(\text{Z}\) axis and pull force of \(2000000\text{N}\)

  9. bending moment of \(100000\text{Nm}\) around the \(\text{Z}\) axis and \(-150000\text{Nm}\) bending moment around the \(\text{Y}\) axis

  10. bending moment of \(100000\text{Nm}\) around axis \(\text{Z}\), moment of bending of \(-150000\text{Nm}\) around axis \(\text{Y}\), and compression force of \(3000000\text{N}\)

  11. bending moment of \(-150000\text{Nm}\) around the \(\text{Z}\) axis

  12. bending moment of \(-260000\text{Nm}\) around the \(\text{Z}\) axis

  13. bending moment of \(-380000\text{Nm}\) around the \(\text{Z}\) axis

  14. compression force of \(4500000\text{N}\), bending moment of \(380000\text{Nm}\) around the axis \(\text{Z}\) and a shear force of \(100000\text{N}\) following \(\text{Z}\)

1.3.3. Case of loading at the ELS Quasi-Permanent#

  1. compression force of \(1000000\text{N}\), and a shear force of \(100000\text{N}\) along the axis \(\text{Z}\)

  2. traction force of \(1000000\text{N}\), and a shear force of \(-100000\text{N}\) along the axis \(\text{Z}\)

  3. tensile force of \(1000000\text{N}\), a shear force of \(-100000\text{N}\) along the axis \(\text{Z}\) and a torsional moment of \(10000\text{Nm}\)

  4. bending moment of \(100000\text{Nm}\) around the \(\text{Z}\) axis.

  5. Bending moment of \(-100000\text{Nm}\) around the \(\text{Y}\) axis.

  6. bending moment of \(-100000\text{Nm}\) around the \(\text{Z}\) axis and pull force of \(100000\text{N}\)

  7. bending moment of \(-100000\text{Nm}\) around the \(\text{Y}\) axis and pull force of \(100000\text{N}\)

  8. bending moment of \(-100000\text{Nm}\) around the \(\text{Z}\) axis and pull force of \(2000000\text{N}\)

  9. bending moment of \(100000\text{Nm}\) around the \(\text{Z}\) axis and \(-150000\text{Nm}\) bending moment around the \(\text{Y}\) axis

  10. bending moment of \(100000\text{Nm}\) around axis \(\text{Z}\), moment of bending of \(-150000\text{Nm}\) around axis \(\text{Y}\), and compression force of \(3000000\text{N}\)

  11. bending moment of \(-150000\text{Nm}\) around the \(\text{Z}\) axis

  12. bending moment of \(-260000\text{Nm}\) around the \(\text{Z}\) axis

  13. bending moment of \(-380000\text{Nm}\) around the \(\text{Z}\) axis

  14. compression force of \(4500000\text{N}\), bending moment of \(380000\text{Nm}\) around the axis \(\text{Z}\) and a shear force of \(100000\text{N}\) following \(\text{Z}\)

1.4. Other calculation parameters#

1.4.1. Settings at ELU#

In ELU, we will consider 9 configurations on which we will start the calculation of the 14 load cases presented in §1.3.1:

  • Setup 1- Basic calculation:

  • calculation at EC2

  • upper csup coating, Y = csup, Z =4 cm

  • lower cinf coating, Y = cinf, Z = 4 cm

  • elastic limit of steel fyk= 500 MPa

  • Young’s modulus of Eys steel = 210,000 MPa

  • type diagram (sant-901) for steel = “B2”

  • characteristic strength of concrete fck= 35 MPa

  • density of the steel [s] = 7800 Kg/m3

  • αcc= 1.0/γc= 1.5/γs= 1.15

  • Configuration 2- Taking into account compression reinforcement:

  • calculation at EC2

  • FERR_COMP = “OUI “

  • upper csup coating, Y = csup, Z =4 cm

  • lower cinf coating, Y = cinf, Z = 4 cm

  • elastic limit of steel fyk= 500 MPa

  • Young’s modulus of Eys steel = 210,000 MPa

  • type diagram (sant-901) for steel = “B2”

  • characteristic strength of concrete fck= 35 MPa

  • density of the steel [s] = 7800 Kg/m3

  • αcc= 1.0/γc= 1.5/γs= 1.15

  • Configuration 3- Taking into account the impact of compression on the shear force resistance and the impact of shear force and torsion on the longitudinal reinforcement:

  • calculation at EC2

  • FERR_COMP = “OUI “

  • UTIL_COMPR = “OUI “

  • EPURE_CISA = “OUI “

  • upper csup coating, Y = csup, Z =4 cm

  • lower cinf coating, Y = cinf, Z = 4 cm

  • elastic limit of steel fyk= 500 MPa

  • Young’s modulus of Eys steel = 210,000 MPa

  • type diagram (sant-901) for steel = “B2”

  • characteristic strength of concrete fck= 35 MPa

  • density of the steel [s] = 7800 Kg/m3

  • αcc= 1.0/γc= 1.5/γs= 1.15

  • Configuration 4- Search for a symmetric sizing reinforcement:

  • calculation at EC2

  • FERR_COMP = “OUI “

  • FERR_SYME = “OUI “

  • SEUIL_SYME = 1 cm2

  • upper csup coating, Y = csup, Z =4 cm

  • lower cinf coating, Y = cinf, Z = 4 cm

  • elastic limit of steel fyk= 500 MPa

  • Young’s modulus of Eys steel = 210,000 MPa

  • type diagram (sant-901) for steel = “B2”

  • characteristic strength of concrete fck= 35 MPa

  • density of the steel [s] = 7800 Kg/m3

  • αcc= 1.0/γc= 1.5/γs= 1.15

  • Configuration 5- Sizing at BAEL91:

  • calculation at BAEL91

  • FERR_COMP = “OUI “

  • upper csup coating, Y = csup, Z =4 cm

  • lower cinf coating, Y = cinf, Z = 4 cm

  • elastic limit of steel Fe= 500 MPa

  • Young’s modulus of Eys steel = 210,000 MPa

  • type diagram (sant-901) for steel = “B2”

  • characteristic strength of concrete fcj= 35 MPa

  • density of the steel [s] = 7800 Kg/m3

  • αcc= 1.0/γc= 1.5/γs= 1.15

1.4.2. Parameters with the ELS Feature#

In ELS Characteristic, we will consider 4 configurations on which we will start the calculation of the 10 load cases presented in §1.3.2:

  • Setup 1- Basic calculation:

  • calculation at EC2

  • upper csup coating, Y = csup, Z =4 cm

  • lower cinf coating, Y = cinf, Z = 4 cm

  • characteristic strength of concrete fck= 35 MPa

  • authorized limit stress in steel: μs, lim = 400 MPa

  • maximum compressive stress authorized at the level of the underside of concrete:

►c, inf, lim = 21 MPa

  • maximum compressive stress authorized at the level of the underside of concrete:

►c, sup, lim=21 MPa

  • steel-concrete equivalence coefficient: αE = 15

  • density of the steel [s] = 7800 Kg/m3

  • Configuration 2- Taking into account compression reinforcement:

  • calculation at EC2

  • FERR_COMP = “OUI “

  • upper csup coating, Y = csup, Z =4 cm

  • lower cinf coating, Y = cinf, Z = 4 cm

  • characteristic strength of concrete fck= 35 MPa

  • authorized limit stress in steel: μs, lim = 400 MPa

  • maximum compressive stress authorized at the level of the underside of concrete:

►c, inf, lim = 21 MPa

  • maximum compressive stress authorized at the level of the underside of concrete:

►c, sup, lim=21 MPa

  • steel-concrete equivalence coefficient: αE = 15

  • density of the steel [s] = 7800 Kg/m3

  • Configuration 3- Search for a symmetric sizing reinforcement:

  • calculation at EC2

  • FERR_COMP = “OUI “

  • FERR_SYME = “OUI “

  • SEUIL_SYME = 5 cm2

  • upper csup coating, Y = csup, Z =4 cm

  • lower cinf coating, Y = cinf, Z = 4 cm

  • characteristic strength of concrete fck= 35 MPa

  • authorized limit stress in steel: μs, lim = 400 MPa

  • maximum compressive stress authorized at the level of the underside of concrete:

►c, inf, lim = 21 MPa

  • maximum compressive stress authorized at the level of the underside of concrete:

►c, sup, lim=21 MPa

  • steel-concrete equivalence coefficient: αE = 15

  • density of the steel [s] = 7800 Kg/m3

  • Setup 4- Calculation at BAEL91

  • calculation at BAEL91

  • FERR_COMP = “OUI “

  • upper csup coating, Y = csup, Z =4 cm

  • lower cinf coating, Y = cinf, Z = 4 cm

  • characteristic strength of concrete FCj= 35 MPa

  • authorized limit stress in steel: μs, lim = 400 MPa

  • maximum compressive stress authorized at the level of the underside of concrete:

S-C, INF, LIM = 21 MPa

  • maximum compressive stress authorized at the level of the underside of concrete:

S-C, SUP, LIM = 21 MPa

  • steel-concrete equivalence coefficient: N = 15

  • density of the steel [s] = 7800 Kg/m3

1.4.3. Settings at the ELS Quasi-Permanent#

At ELS Permanent4 configurations will be considered on which we will start the calculation of the 10 load cases presented in §1.3.3:

  • Setup 1- Basic calculation:

  • calculation at EC2

  • upper csup coating, Y = csup, Z =4 cm

  • lower cinf coating, Y = cinf, Z = 4 cm

  • characteristic strength of concrete fck= 35 MPa

  • elastic limit of steel: fyk = 500 MPa

  • Young’s modulus of steel: Eys = 210,000 MPa

  • maximum crack opening allowed on the underside: wmax, inf = 0.15 mm

  • maximum crack opening allowed on the upper side: wmax, sup = 0.15 mm

  • loading time coefficient: kt= 0.4

  • maximum compressive stress allowed in concrete for the control of non-linear creep: μc, lim, NL= 15.75 MPa

  • estimated diameter of the bars on the lower face along the X axis: Φ INF, X = 25 mm

  • estimated diameter of the bars on the upper face along the X axis: Φ SUP, X= 25 mm

  • estimated diameter of the bars on the lower face along the Y axis: Φ INF, Y= 25 mm

  • estimated diameter of the bars on the upper face along the Y axis: Φ SUP, Y= 25 mm

  • steel-concrete equivalence coefficient: αE = 15

  • density of the steel [s] = 7800 Kg/m3

  • Configuration 2- Taking into account compression reinforcement:

  • calculation at EC2

  • FERR_COMP = “OUI “

  • upper csup coating, Y = csup, Z =4 cm

  • lower cinf coating, Y = cinf, Z = 4 cm

  • characteristic strength of concrete fck= 35 MPa

  • elastic limit of steel: fyk = 500 MPa

  • Young’s modulus of steel: Eys = 210,000 MPa

  • maximum crack opening allowed on the underside: wmax, inf = 0.15 mm

  • maximum crack opening allowed on the upper side: wmax, sup = 0.15 mm

  • loading time coefficient: kt= 0.4

  • maximum compressive stress allowed in concrete for the control of non-linear creep: μc, lim, NL= 15.75 MPa

  • estimated diameter of the bars on the lower face along the X axis: Φ INF, X = 25 mm

  • estimated diameter of the bars on the upper face along the X axis: Φ SUP, X= 25 mm

  • estimated diameter of the bars on the lower face along the Y axis: Φ INF, Y= 25 mm

  • estimated diameter of the bars on the upper face along the Y axis: Φ SUP, Y= 25 mm

  • steel-concrete equivalence coefficient: αE = 15

  • density of the steel [s] = 7800 Kg/m3

  • Configuration 3- Search for a symmetric sizing reinforcement:

  • calculation at EC2

  • FERR_COMP = “OUI “

  • FERR_SYME = “OUI “

  • SEUIL_SYME = 5 cm2

  • upper csup coating, Y = csup, Z =4 cm

  • lower cinf coating, Y = cinf, Z = 4 cm

  • characteristic strength of concrete fck= 35 MPa

  • elastic limit of steel: fyk = 500 MPa

  • Young’s modulus of steel: Eys = 210,000 MPa

  • maximum crack opening allowed on the underside: wmax, inf = 0.15 mm

  • maximum crack opening allowed on the upper side: wmax, sup = 0.15 mm

  • loading time coefficient: kt= 0.4

  • maximum compressive stress allowed in concrete for the control of non-linear creep: μc, lim, NL= 15.75 MPa

  • estimated diameter of the bars on the lower face along the X axis: Φ INF, X = 25 mm

  • estimated diameter of the bars on the upper face along the X axis: Φ SUP, X= 25 mm

  • estimated diameter of the bars on the lower face along the Y axis: Φ INF, Y= 25 mm

  • estimated diameter of the bars on the upper face along the Y axis: Φ SUP, Y= 25 mm

  • steel-concrete equivalence coefficient: αE = 15

  • density of the steel [s] = 7800 Kg/m3

  • Setup 4- Calculation at BAEL91

  • calculation at BAEL91

  • FERR_COMP = “OUI “

  • upper csup coating, Y = csup, Z =4 cm

  • lower cinf coating, Y = cinf, Z = 4 cm

  • characteristic strength of concrete fck= 35 MPa

  • elastic limit of steel: fyk = 500 MPa

  • Young’s modulus of steel: Eys = 210,000 MPa

  • maximum crack opening allowed on the underside: wmax, inf = 0.15 mm

  • maximum crack opening allowed on the upper side: wmax, sup = 0.15 mm

  • loading time coefficient: kt= 0.4

  • maximum compressive stress allowed in concrete for the control of non-linear creep: μc, lim, NL= 15.75 MPa

  • estimated diameter of the bars on the lower face along the X axis: Φ INF, X = 25 mm

  • estimated diameter of the bars on the upper face along the X axis: Φ SUP, X= 25 mm

  • estimated diameter of the bars on the lower face along the Y axis: Φ INF, Y= 25 mm

  • estimated diameter of the bars on the upper face along the Y axis: Φ SUP, Y= 25 mm

  • steel-concrete equivalence coefficient: αE = 15

  • density of the steel [s] = 7800 Kg/m3