Numerical methods in bridge structure analysis - Computational Analysis and Design of Bridge Structures - page 85

where ε e , ε c , and ε s denote the regular elastic strain, creep strain, and
shrinkage strain, respectively. Both creep and shrinkage strains depend on
the age of concrete and the observation time where the age for creep is
the duration after applying loads and the age for shrinkage is the duration
after concrete is allowed to dry. The creep strain also is proportional to the
elastic strain as

( )

t

ε = ε ϕ , τ (3.58)

c e

t , ( ) is the creep factor, which may be expressed by many different mathe-
matical models. The time origins of t and τ are the same as when the concrete
starts to cure. No matter what model is used to describe creep development,
the creep factor ϕ

ϕ τ

t , ( ) can be explained as at observation time t , the total
creep due to an elastic strain at τ divided by the elastic strain. ε

τ

s (

, τ )

is the

total shrinkage at time t , which is independent to the elastic strain ε e .
Given an external load acting on time τ , at time t the system is balanced
and the equilibrium equation is written as Equation 3.32. Considering a
small time increment dt , the variation of elastic strain can be obtained from
Equation 3.57 as

d d d d

ε = ε − ε ϕ − ε (3.59)

e e s

The internal stresses will have a change of d σ , and the incremental equilib-
rium equation can be obtained from Equation 3.32 as

∫

( ) =

d a B

T

d dv

ψ σ = 0 (3.60)

Substituting Equations 3.59 and 3.11 into Equation 3.60, the incremental
equilibrium equation of creep and shrinkage can be derived as

∫ ∫

K a d = B

T

σ d dv ϕ + B D

T

d dv ε (3.61)

s

where K is the global stiffness matrix as shown in Equation 3.30.
The physical meaning of Equation 3.61 is simple and clear: Incremental
creep and shrinkage will cause equivalent loads and will be balanced by
incremental displacements. By solving Equation 3.61, the incremental
displacements at the next time step due to creep and shrinkage can be
obtained. The total and elastic incremental strains can be computed from
Equations 3.28 and 3.59, respectively. The incremental stresses can further
be obtained. By accumulating all incremental values for each incremental
time, the total internal stresses and displacements at any time due to creep
and shrinkage can be solved.