Numerical methods in bridge structure analysis - Computational Analysis and Design of Bridge Structures - page 86

It should be noted that the most available creep and shrinkage models are
based on experiments on axial compression components. However, creep
and shrinkage factors can be treated the same in all directions, includ-
ing shear strains. Therefore, when computing the equivalent loads as
Equation 3.61, the incremental creep and shrinkage factors can be isolated
from matrix operations.

3.3.2 calculation of equivalent loads due to
incremental creep and shrinkage
The development of concrete shrinkage at a given observation time depends
only on the concrete age when it is allowed to dry and is independent to
stresses. Thus, the equivalent loads due to shrinkage (the second term on
the right side of Equation 3.61) are straightforward. The computation of
creep equivalent load, however, is complicated because it depends on both
stresses and the concrete age when stresses are loaded. Figure 3.12 shows
generic stress changes of one component at different time steps. Each stress
change could be caused by external loads or creep/shrinkage redistribu-
tion. As time and the concrete age are considered when each stress change
applies, this diagram represents a typical loading history. Assuming stress
change at each time step is ∆σ i , the time ordinate at each time step is t i , and
the concrete age is τ 0 when the first stress change ∆σ 0 is loaded, the total
stress at any time step t i is

i

∑ 0

σ = ∆σ (3.62)

i j

=

j

Δσ

Δσ 2

Δσ i + 1

Δσ 1

Δσ 3

Δσ i

Δσ 0

Δσ i − 1

. . . . . . . . . . . .

t
Loading time
Age when loading

0

t 0 t 1 t 3

t i − 1 t i t i + 1

τ 0 τ 0 + t 1 − t 0 τ 0 + t 3 − t 0 τ 0 + t i − t 0

Figure 3.12 Stress changes and loading history of a concrete component.