Nonlinear Finite Element Analysis of Concrete Structures

e.. = e e . + eP (3-15)
ID i] i]
From Hooke's law follows
e
e ij
= fil
2 G
where G is the shear modulus. Inserting this equation in eq. (15)
and eliminating s.. by means of eq. (5) gives
e.. = (-^- + l) e?. (3-16,
ID v 3Ge p ; ID
Multiplication of eq. (16) with itself yields
FP = e _ e(e y ) (3-17)
e e et 3G
where eq. (6) has been used and where the equivalent total
strain e . is defined by
e et
= (I „ e .Y«
V3
.. e..)
i] ID/
which using the definition of deviatoric total strain can be
written
e et = -2 ^Ul
~ £ 2 )2 + (£ 1 " e 3 )2 + (c 2 " e 3 )
Moreover, as the stress-plastic strain curve obtained from uniaxial
loading and derived from fig. 3-1 a) determines a as a
p
unique function of e , equation (17) is the expression sought,
as it determines the equivalent plastic strain e p as a function
of e . determined by the total strains. The iteration sequence
is then as follows:
From the present values of the total strains e, and e„ and from
the values of o and e p from the previous loading stage a e,-
value is determined through eq. (8). The equivalent total strain
e . is then evaluated by means of eq. (10) . Knowing e . and a ,
eq. (17) determines a new value of e p and thereby also a new
value of a . This iteration loop is continued until values for
e p and o that are in suffficiently close agreement with the pre-