Nominal stiffness method (§5.8.7.2) - Example

 

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:

 

Top loads

 

 

Bottom loads

 

 

Combinations

 

 

 

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.

 

 

minimum reinforcement

 

 

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.

 

 

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

 

 

Diameters

 

 

 

 

 

Total nominal stiffness (Ax reinforcement)

 

 

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

 

 

 

concrete's modulus of elasticity

 

 

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.

 

 

Ecm

 

 

 

 

 

Ic

 

 

 

Kc

 

 

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

 

 

K

 

where n is the relative axial force

 

n

 

 

The value for λx is automatically calculated and it is displayed in the Buckling Length dialog and in the Geometry chapter of the report.

 

 

Lamda x

 

 

λx = 25.06

 

 

k2

 

 

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:

 

ei

 

 

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)

 

 

  M0Ed

 

 

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):

 

 

Es

 

 

Area of reinforcement's second moment, about the centre area of concrete:

 

 

Area of reinforcement's second moment

 

 

Section B

 

 

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:

 

 

concrete cover

 

 

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.

 

 

diameters

 

 

In the final stage, the calculation is done again with the real d’ obtained as:

 

d

 

The report will reflect the final stage, with the real value of d’.

 

the real value of d

 

 

Nominal stiffness value for concrete:

 

Kc Ecd Ic formula

 

Nominal stiffness value for steel:

 

Ks Es Is formula

 

Total nominal stiffness value:

 

EI formula

 

 

Bending moment magnification (§5.8.7.3)

 

The total bending moment, including the 2nd order effects is designed with the following formula increasing the 1st order bending moment:

 

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

 

 

β

 

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.).

 

β formula

 

 

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.

 

 

buckling length

 

 

 

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

 

 

formula

 

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.

 

 

the slenderness λ

 

 

φef is the effective creep ratio detailed here.

 

φef = 1.931

 

A

 

 

ω is the mechanical reinforcement ratio

 

ω

 

 

As is the total area of longitudinal reinforcement

 

As

 

 

The relative normal force n is detailed above.

 

n = 0.42

 

 

rm is the moment ratio

 

rm

 

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)

 

rm formula

 

 

=> The second order effects are considered.