CAPÍTULO II. DE LA UTILIZACIÓN Y MANTENIMIENTO DE LAS REDES
27. ACTIVACIÓN O REACTIVACIÓN TEMPORAL DEL SERVICIO DE GAS DURANTE EL
4.2.1 Formulation
The simplest and oldest technique for including creep in structural analysis is Faber’s effective modulus method (EMM) (Ref. 1). In Section 1.2.4, the instantaneous and creep components of strain were combined and a reduced or effective modulus for concrete, Ee(t,τ0), was defined in Eqs 1.11 and 1.12. In Section 1.2.5, the integral- type creep law was presented in Eq. 1.17 to describe the time-dependent deformation of a continuously varying stress history. In the EMM, Eq. 1.17 is approximated by assuming that the stress-dependent deformations are produced only by a sustained stress equal to the final value of the stress history, that is:
ε(t) = t τ0 1+ ϕ (t,τ) Ec(τ) dσc(τ) + εsh(t)≈ 1+ ϕ (t,τ0) Ec(τ0) σc(t)+ εsh(t) = σc(t) Ee(t,τ0)+ εsh (t) (4.2)
where Ee(t,τ0) is the effective modulus for concrete defined in Eq. 1.12 (which for convenience is repeated here) as:
Ee(t,τ0) = Ec(τ0) 1+ ϕ(t,τ0)
(4.3)
Creep is treated as a delayed elastic strain and is taken into account simply by reducing the elastic modulus of concrete with time. A time analysis using the effective modulus method is nothing more than an elastic analysis in which Ee(t,τ0) is used instead of Ec(τ0). Shrinkage may be included in this elastic time analysis in a similar way as a sudden temperature change in the concrete would be included in a short-term elastic analysis.
According to the EMM, the creep strain at time t (in Eq. 4.2) depends only on the current stress in the concreteσc(t) and is therefore independent of the previous stress
history. This, of course, is not the case. The ageing of the concrete has been ignored. For an increasing stress history, the EMM overestimates creep, while for a decreasing
stress history, creep is underestimated. If the stress is entirely removed, the creep strains disappear. The EMM therefore predicts complete creep recovery, which is not correct.
Eq. 4.2 is valid only when the concrete stress is constant in time. In such cases, the EMM gives excellent results. Good results are also obtained if the concrete is old when first loaded and the effect of ageing is not great. Despite its shortcomings, the EMM is the simplest of all of the methods for the time analysis of concrete structures and its simplicity recommends it, particularly under the conditions mentioned above. In many practical problems, the method is sufficiently accurate for design purposes.
However, many practical situations involve rapidly changing stress histories in young concrete. In such cases, the EMM may be unsuitable and potentially misleading, and a more sophisticated method of analysis is required.
4.2.2 Example application (EMM)
Consider the short, axially-loaded, symmetrically reinforced column shown in Fig. 4.1. The column is subjected to a constant, sustained axial force P as shown. Creep and shrinkage cause a gradually decreasing stress history in the concrete similar to that shown in Fig. 1.15.
The redistribution of internal forces due to the gradual development of creep and shrinkage strains is to be examined and the time-dependent stresses and strains in both the concrete and the steel are to be calculated using the EMM.
The problem is solved by enforcing the three basic requirements of any time analysis: namely, equilibrium of forces, compatibility of strains and satisfaction of the constitutive relationships of the concrete and the steel.
The external compressive load P is applied at timeτ0and is resisted by the internal forces in the concrete and steel Nc(t) and Ns(t), where at any time t after loading Nc(t)= σc(t)Acand Ns(t)= σs(t)As.
Equilibrium requires that the sum of the internal forces at time t equals the external load, that is:
P= Nc(t)+ Ns(t)= σc(t)Ac+ σs(t)As (4.4) P P P = constant Ac As Nc(t) Ns(t) (a) Elevation L (b) Section Figure 4.1 Axially-loaded short column.
Compatibility requires that the total concrete strainε(t) and the steel strain εs(t) are
identical at all times:
ε(t) = εs(t) (4.5)
The concrete constitutive relationship used in the EMM is given by Eq. 4.2 and the steel is assumed to be linear and elastic, with modulus Es. Therefore, at any time t:
ε(t) = σc(t) Ee(t,τ0)+ εsh(t) (4.6a) and εs(t)=σs(t) Es (4.6b)
Re-arranging Eqs 4.6 in terms ofσc(t) andσs(t) gives:
σc(t)= ε(t)Ee(t,τ0)− εsh(t)Ee(t,τ0) (4.7a) and
σs(t)= εs(t)Es (4.7b)
Inserting these equations into the equilibrium equation (Eq. 4.4) and enforcing compatibility (Eq. 4.5) yields:
P= Ee(t,τ0)ε(t)Ac− Ee(t,τ0)εsh(t)Ac+ ε(t)EsAs (4.8)
which represents the governing equation of the problem. Solving Eq. 4.8 for the unknown total concrete strainε(t) gives:
ε(t) = P AcEe(t,τ0)+ AsEs+ AcEe(t,τ0)εsh(t) AcEe(t,τ0)+ AsEs = P AcEe(t,τ0)(1+ neρ)+ εsh(t) 1+ neρ (4.9)
where ne= the effective modular ratio = Es/Ee(t,τ0) andρ = the reinforcement ratio = As/Ac.
By substituting Eq. 4.9 into Eq. 4.7a, the concrete stress at time t is obtained:
σc(t)= P
Ac(1+ neρ)−
Esρ εsh(t)
1+ neρ
(4.10)
The steel strainεs(t) is identical to the total concrete strainε(t) (Eq. 4.5) and therefore
the steel stress may be obtained by substituting Eq. 4.9 into Eq. 4.7b: σs(t)=
neP Ac(1+ neρ)+
Esεsh(t)
The steel stress may also be obtained from the equilibrium equation (Eq. 4.4):
σs(t)=
P− σc(t)Ac
As (4.11b)
The elastic component of concrete strain at time t is usually taken to be εe(t)= σc(t)/Ec(τ0) and the creep strain is thereforeεcr(t)= ε(t) − εe(t)− εsh(t).
Example 4.1
Consider the short, symmetrically reinforced column shown in Fig. 4.1. The column is subjected to a constant sustained axial force P= −1000 kN first applied at ageτ0= 14 days. The cross-sectional areas of the concrete and the steel reinforcement are Ac= 90,000 mm2and As= 1800 mm2, respectively, and
the reinforcement ratio is thereforeρ = As/Ac= 0.02. The column is located
in a temperate environment and shrinkage is assumed to commence at the age of first loading. The mean in-situ compressive strength of concrete at the time of first loading is taken to be fcmi(τ0)= 28 MPa and the characteristic 28-day compressive strength is fc= 40 MPa. The elastic modulus of concrete at first loading is Ec(τ0)= 26.7 GPa and the elastic modulus of steel is Es= 200 GPa.
The creep coefficient is obtained from Eq. 2.3 and shrinkage strain is obtained from Eq. 2.6, with final (long-term) values:ϕ∗(τ0)= 2.39 and ε∗sh= −510×10−6. The variations of the creep coefficient and shrinkage with time are given in the following table and shown in Fig. 4.2. For illustrative purposes it has been assumed that shrinkage begins at the time of loading.
(t− τ0) in days 0 10 30 70 200 500 10,000
ϕ(t,τ0) 0 0.53 0.98 1.38 1.83 2.10 2.39
εsh(t− τ0)× 10−6 0 −142 −246 −325 −407 −456 −510
The redistribution of internal forces due to the gradual development of creep and shrinkage strains is to be examined and the time-dependent changes in stresses and strains in both the concrete and the steel are to be calculated. Sample calculations are provided below.
0 0.5 1 1.5 2 2.5 0 Creep coefficient
Time, t (days) Time, t (days)
0 100 200 300 400 500 600 Shrinkage strain ( × − 10 − 6) 200 400 600 800 1000 0 200 400 600 800 1000
At first loadingτ
0, (t− τ
0)= 0:
ϕ(t,τ0)= 0 and εsh(t− τ0)= 0.With Ee(t,τ0)= Ec(τ0)= 26.7 GPa; and ne= n = Es/Ec(τ0)= 7.48.
Eq. 4.9: ε(τ0)= −1000 × 10 3
90,000 × 26,700 × (1 + 7.48 × 0.02)+ 0 = −361 × 10 −6;
Eq. 4.5: εs(τ0)= −361 × 10−6;
Eq. 4.7a: σc(τ0)= −361 × 10−6× 26,700 − 0 = −9.67 MPa; Eq. 4.7b: σs(τ0)= −361 × 10−6× 200,000 = −72.3 MPa.
The stresses in the concrete and steel could have also been calculated directly from Eqs 4.10 and 4.11:
Eq. 4.10: σc(τ0)= −1000 × 10 3 90,000(1 + 7.48 × 0.02)− 0 = −9.67 MPa; Eq. 4.11b: σs(τ0)=−1000 × 10 3+ 9.67 × 90,000 1800 = −72.3 MPa.
With the creep and shrinkage components of the concrete strain both equal to zero at first loading, the concrete strainε(τ0) is entirely made up of the elastic or instantaneous component of strainεe(τ0).
At (t− τ
0)= 10 days:
ϕ(t,τ0)= 0.53 and εsh(t)= −142 × 10−6. Ee(t,τ0) = Ec(τ0)/(1 + ϕ(t,τ0)) = 17.48 GPa; and ne = Es/Ee(t,τ0) = 11.44; and Eq. 4.9: ε(t) = −1000 × 103 90,000 × 17,480 × (1 + 11.44 × 0.02)+ −142 × 10−6 1+ 11.44 × 0.02 = −633 × 10−6; Eq. 4.5: εs(t)= −633 × 10−6; Eq. 4.7a: σc(t)= −633 × 10−6× 17,480 − (−142 × 10−6)× 17,480 = −8.58 MPa; Eq. 4.7b: σs(t)= −633 × 10−6× 200,000 = −126.6 MPa.The elastic and shrinkage components of the total concrete strain are εe(t)= σc(t)/Ec(τ0) = −321 × 10−6; the shrinkage component of strain is εsh(t)=
−142 × 10−6; and therefore the creep component of strain is εcr(t) = ε(t) − εe(t)− εsh(t)= −170 × 10−6.
The results of the analyses at first loading and throughout the period of sustained loading are provided in the following table.
Time t Duration Concrete Steel Total Elastic Creep Shrinkage
(days) of load stress stress strain concrete strain strain
t− τ0 σc(t) σs(t) ε(t) strainεe(t) εcr(t) εsh(t)
(days) (MPa) (MPa) (×10−6) (×10−6) (×10−6) (×10−6)
14 0 −9.67 −72.3 −361 −361 0 0 24 10 −8.58 −127 −633 −321 −170 −142 44 30 −7.82 −165 −824 −292 −285 −246 84 70 −7.23 −194 −969 −270 −374 −325 214 200 −6.66 −222 −1112 −249 −456 −407 514 500 −6.35 −238 −1191 −237 −497 −456 10,014 10,000 −6.02 −255 −1273 −225 −538 −510