Analysis of second order effects with axial load: the nominal stiffness EI
Eurocode 2 part 1-1: Design of concrete structures 5.8.7.2 (1)
The nominal stiffness of slender compression members with arbitrary cross section may be estimated as follows:
EI = K_{c} E_{cd} I_{c} + K_{s} E_{s} I_{s} | (5.21) |
where:
- E_{cd}
- is the design value of the modulus of elasticity of concrete, see § 5.8.6 (3). Cf. also 5.8.7.2 (4) to know when to use E_{cd,eff} = E_{cd}/(1 + φ_{ef})
- I_{c}
- is the moment of inertia of concrete cross section
- E_{s}
- is the design value of the modulus of elasticity of reinforcement, see § 3.2.7 (4)
- I_{s}
- is the second moment of area of reinforcement, about the centre of area of the concrete
- K_{c}
- is a factor for effects of cracking, creep etc., see options below.
- K_{s}
- is a factor for contribution of reinforcement, see options below
• ρ ≥ 0,002 and with an accurate method:
K_{s} = 1 | |
K_{c} = k_{1} k_{2} / (1 + φ_{ef}) | (5.22) |
where:
- ρ
- is the geometric reinforcement ratio, = A_{s}/A_{c}
with
- A_{s}
- the total area of reinforcement
- A_{c}
- the area of concrete section
- φ_{ef}
- is the effective creep ratio, cf. 5.8.4
- k_{1}
- is a factor which depends on concrete strength class:
k_{1} = (f_{ck}/20)^{1/2} (5.23) - k_{2}
- is a factor which depends on axial force and slenderness:
k_{2} = n⋅λ/170 ≤ 0,20 (5.24) with
- n
- the relative axial force, n = N_{Ed}/(A_{c}⋅f_{cd})
where:
N_{Ed} is the design value of the applied axial force,
f_{cd} is the design compressive strength of concrete, see § 3.1.6 (1)P - λ
- the slenderness ratio, cf 5.8.3
If the slenderness ratio λ is not defined, k_{2} may be taken as:
k_{2} = n⋅0,30 ≤ 0,20 (5.25)
• ρ ≥ 0,01 and with a simplified method:
K_{s} = 0 | |
K_{c} = 0,3 / (1 + 0,5φ_{ef}) | (5.26) |
This simplified method may be suitable as a preliminary step, followed by a more accurate calculation above.
This application calculates the nominal stiffness EI from your inputs. Intermediate results will also be given.
First, change the following option if necessary:(*) If λ is not defined, you can put the value of zero (0) for λ, k_{2} will be taken following the Expression (5.25).