Linear Principle of Creep Strain Superposition

If analysis is concerned only for stress ranges developed under service loads, creep of concrete may be assumed to be governed by linear principle of superposition in time. In this section, this principle is further explained.

Instantaneous (elastic) strain for a concrete member subjected to a load introduced at time

t_{o}

ε_{e} = \frac{σ_{c} (t_{o})}{E_{c} (t_{o})}

where

$σ_{c} (t_{o})$	=	stress caused by the applied load at $t_{o}$
$E_{c} (t_{o})$	=	concrete modulus of elasticity at $t_{o}$

If the load is kept constant on member, then, the creep strain calculated at time

\hat{t}

ε_{c r} (\hat{t}, t_{o}) = \frac{σ_{c} (t_{o})}{E_{c} (t_{o})} φ (\hat{t}, t_{o}) = ε_{e} φ (\hat{t}, t_{o})

where

φ (\hat{t}, t_{o})

a dimensionless factor known as the creep coefficient. It represents ratio of creep strain to instantaneous strain. Note that

φ (\hat{t}, t_{o})

is calculated at time

\hat{t}

for a member loaded at time

t_{o}

Combining the above two equations (i.e., superposition), total strain in member becomes

ε_{c} (\hat{t}) = \frac{σ_{c} (t_{o})}{E_{c} (t_{o})} (1 + φ (\hat{t}, t_{o}))

which assumes that total strain (instantaneous + creep) is proportional to applied stress. This linear relationship is acceptable for stresses developed under service loads.

If stress in member changes between the ages

t_{o}

and

\hat{t}

, the above equation is modified as follows:

ε_{c} (\hat{t}) = σ_{c} (t_{o}) (\frac{1 + φ (\hat{t}, t_{o})}{E_{c} (t_{o})}) + \int_{o}^{Δ σ_{c} (\hat{t})} (\frac{1 + φ (\hat{t}, τ)}{E_{c} (τ)}) d σ_{c} (τ)

The integral form in the above equation accounts for instantaneous plus creep strains between the period $t_{o}$ and $\hat{t}$ due to a stress change (i.e., $Δ σ_{c}$ ) introduced gradually within the same period.

In the current study, it is assumed that such stress increments are introduced at full magnitude at

t_{o}

and sustained to time

\hat{t}

. In this case, the above equation is simplified to the following form:

ε_{c} (\hat{t}) = σ_{c} (t_{o}) (\frac{1 + φ (\hat{t}, t_{o})}{E_{c} (t_{o})}) + Δ σ_{c} (\hat{t}) \frac{1 + χ (\hat{t}, t_{o}) φ (\hat{t}, t_{o})}{E_{c} (t_{o})}

where

χ (\hat{t}, t_{o})

the aging coefficient, a dimensionless multiplier (smaller than 1) applied to

φ (\hat{t}, t_{o})

With this approach, the integral term in the above equation, which accounts for history of instantaneous plus creep strains due to a stress change gradually introduced during the period

(\hat{t}, t_{o})

, is replaced with a term in which creep effects are reduced by the aging factor,

χ (\hat{t}, t_{o})

. The aging coefficient is intended to describe the effects of stress changes in creep-related deformations during the period of

(\hat{t}, t_{o})

Derivation of

χ (\hat{t}, t_{o})

can be found in literature [2]. It is repeated below

χ (\hat{t}, t_{o}) = \frac{E_{c} (t_{o})}{E_{c} (t_{o}) - R (\hat{t}, t_{o})} - \frac{1}{φ (\hat{t}, t_{o})}

where

R (\hat{t}, t_{o})

the relaxation function. In the present work, the relaxation function is implemented based on the study given in [3].

Finally, the following is obtained after rearranging the terms

ε_{c} (\hat{t}) = σ_{c} (t_{o}) (\frac{1 + φ (\hat{t}, t_{o})}{E_{c} (t_{o})}) + \frac{Δ σ_{c} (\hat{t})}{{\bar{E}}_{c} (\hat{t}, t_{o})}

where

{\bar{E}}_{c} (\hat{t}, t_{o})

\frac{E_{c} (t_{o})}{1 + χ (\hat{t}, t_{o}) φ (\hat{t}, t_{o})}

, which is referred to as age-adjusted elasticity modulus (AAEM).

The changes of stresses and deformations from time $t_{o}$ of the member loaded until time $\hat{t}$ are approximately calculated using age adjusted modulus. Under service level loads, stresses are generally low enough to be within linear range of concrete creep. In this case, analysis using age-adjusted elasticity modulus yield comparable results [3].