General data and short workflow description
We consider a column with a 0.3 x 0.4 m rectangular section and a 3.10 m height. The concrete class is 25/30. The loads on column are the following:
The workflow followed by the program to establish the reinforcement area is the following:
This calculation is done individually for X and Y direction.
The program starts from a minimum reinforcement area and for the defined concrete section generates the interaction curve.
The interaction curve used for verification is a simplified curve obtained by considering only the reinforcement on the direction for which the calculation is done (X or Y).
The real interaction curve is obtained considering the entire area reinforcement of the column (on both directions), and it is the one displayed when launching the Interaction Curves tool.
At this stage, MRd and NRd are known.
The next step is to calculate the design bending moment (including the second order effects or not) based on the stiffness EI calculated below, and on the initial reinforcement.
Then, the program verifies that the point defined by the pair (NEd, MEd) is inside the simplified interaction curve. If it is not, the calculation is iterated with a bigger reinforcement area, until the point is inside the curve.
The increment step of the iterative calculation of reinforcement bars can be defined in the Design Assumptions dialog, as Reinforcement precision.
The detailed report gives the final theoretical reinforcement that verifies the condition. The following example follows will use the reinforcement areas from the report to exemplify the calculation for nominal stiffness and the design bending moment calculation.
As,x = 9.2 cm2
As,y = 3.2 cm2
Total nominal stiffness (Ax reinforcement)
The Ax reinforcement is considered the reinforcement placed along the direction perpendicular to the X axis.
Combination 104: +1.35x[1 G]+1.5x[2 Q]
The detailed calculation below is given for combination 104, giving the maximum Ax.
The nominal stiffness is calculated using the following formula:
EI = Kc × Ecd × Ic + Ks × Es × Is
where:
Ecd is the design value of the concrete's modulus of elasticity, see 5.8.6 (3).
Ic is the concrete cross section's moment of inertia.
Es is the design value of the reinforcement's modulus of elasticity, see 5.8.6 (3).
Is is the second moment of the reinforcement area, about the centre of area of the concrete.
Kc is a factor for effects of cracking, creep etc, see 5.8.7.2 (2) or (3).
Ks is a factor for contribution of reinforcement, see 5.8.7.2 (2) or (3).
Concrete stiffness
Design value of the concrete's modulus of elasticity (5.8.6(3)):
Note: The value of ɣcE to be used in a certain country may be found in its National Annex. The recommended value is 1.2.
Ecm is displayed in the Materials chapter of the report.
Concrete cross section's moment of inertia:
Factor for the effects of cracking, creep, etc. (5.8.7.2(2)):
where:
φef is the effective creep ratio
k1 is a factor which depends on the concrete strength class
k2 is a factor which depends on the axial force and slenderness
where n is the relative axial force
The value for λx is automatically calculated and it is displayed in the Buckling Length dialog and in the Geometry chapter of the report.
λx = 25.06
φ(∞,t0) is the final creep coefficient.
This value is displayed in the report in the Creep Coefficient chapter.
φ(∞,t0) = 2.71
M0Eqp is the first order bending moment in quasi-permanent load combination (SLS).
M0Ed is the first order bending moment in design load combination (ULS).
Regardless of the method used for element calculation, the geometrical imperfections should be considered only at ULS.
We consider an initial eccentricity defined by:
M0Ed = M0Ed,input + NEd × ei = 40.5 + 837.32 × 0.02 = 57.246 kNm (in design load combination 105)
M0Eqp = M0Eqp,input + NEqp × ei = 30 + 539.12 × 0.02 = 40.782 kNm (in quasi-permanent load combination 114)
Steel stiffness
Factor for the contribution of reinforcement (5.8.7.2(2)): Ks = 1
Design value of the reinforcement modulus of elasticity, 5.8.6 (3):
Area of reinforcement's second moment, about the centre area of concrete:
Since, in the design phase, the real d’ is not known, the program will use in calculation the concrete cover from the Concrete cover dialog:
In the initial stage, the calculation will be done using the value from the dialog. After a certain number of iterations, a longitudinal reinforcement area and a transversal reinforcement area with specific diameters will result.
In the final stage, the calculation is done again with the real d’ obtained as:
The report will reflect the final stage, with the real value of d’.
Nominal stiffness value for concrete:
Nominal stiffness value for steel:
Total nominal stiffness value:
Bending moment magnification (§5.8.7.3)
The total bending moment, including the 2^{nd} order effects is designed with the following formula increasing the 1^{st }order bending moment:
M0Ed is the first order bending moment in design load combination (ULS) considering the geometrical imperfections.
β is a factor which depends on the distribution of 1st and 2nd order moments, see 5.8.7.3 (2)-(3).
NEd is the axial load design value.
NB is the buckling load based on nominal stiffness (critical load).
M0Ed = 57.246 kNm
c0 is a coefficient which depends on the distribution of first order moment (for instance, c0 = 8 for a constant first order moment, c0 = 9,6 for a parabolic, and c0 = 12 for a symmetric triangular distribution etc.).
The effective length is displayed in the Geometry chapter of the report and in the Buckling Length dialog. The effective length of a member will depend on its end conditions.
In our example, the column is fixed at the bottom end and pinned at the top end:
l0= Lfx =0.7 × L 0.7 × 3100 = 2170 mm
Eurocode 2 indicates certain cases in which it is not necessary to take into account the effects of the second order moment:
According to 5.8.3.1(1)) from EN 1992-1-1, […] second order effects may be ignored if the slenderness λ (as defined in 5.8.3.2) is below a certain value λlim.
φef is the effective creep ratio detailed here.
φef = 1.931
ω is the mechanical reinforcement ratio
As is the total area of longitudinal reinforcement
The relative normal force n is detailed above.
n = 0.42
rm is the moment ratio
M01 and M02 are the first order end moments, | M02| ≥ | M01 |
If the end moments M01 and M02 give tension on the same side, rm should be considered positive (C ≤ 1.7); otherwise, it should be considered negative (C ≥ 1.7).
M01 = 0.00 (top end)
M02 = My = 40.5 kN × m (bottom end)
=> The second order effects are considered.