Finite Element Modeling of Frames and Walls - Unified Theory of Concrete Structures - page 395

of tangent formulations. The details of the derivation of the secant and tangent material con-
stitutive matrices are presented in (Zhong, 2005). The tangent material constitutive matrix is
implemented in the FEA in this topic since OpenSees adopts the tangent stiffness formulation.
A tangent material constitutive matrix [ D ] for a reinforced concrete plane stress element is
formulated as:

⎧
⎨
⎩

⎫
⎬
⎭

σ x
σ y
τ xy

d

⎧
⎨

⎫
⎬

[ D ] = (9.33)

ε x
ε y d

⎩ ⎭

1

2 γ xy

[ D ] is evaluated by

[ D ] = [ T ( α x 1 )][ D c ][ V ][ T ( α 1 x )] + [ T ( α xi )][ D si ][ T ( α i 1 )][ V ][ T ( α 1 x )] (9.34)

i

In Equation (9.34), [ D c ] is the uniaxial tangent constitutive matrix of concrete, [ D si ] is
the uniaxial tangent constitutive matrix of steel, and [ V ] is the matrix defined previously in
Equation (6.37), which converts the biaxial strains into uniaxial strains using the Hsu/Zhu
ratios. [ T (

α 1 x )] is the transformation matrix from the x-y coordinate to the 1 - 2 coordinate;

[ T ( α x 1 )] is the transformation matrix from the 1 - 2 coordinate to the x-y coordinate; [ T ( α i 1 )]

is the transformation matrix from the 1 - 2 coordinate to the si - ti coordinate; [ T ( α xi )] is the

transformation matrix from the si - ti coordinate to the x-y coordinate;
The uniaxial constitutive matrix of concrete [ D c ] is given by

⎡ ⎤

c
1
∂ε 2

∂σ

E c
1

0

⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

2
∂ε 1

∂σ

[ D c ] =

c
2

(9.35)

E 0

G 12 0 0

c
2 are the uniaxial tangent moduli of concrete in the 1 and 2
directions, respectively, evaluated at a certain stress/strain state. The off-diagonal terms

c
1

In Equation (9.35), E and E

c
1
∂ε 2

c
2
∂ε 1

∂σ

and ∂σ

are obtained by using the uniaxial constitutive relationships and taking into account the states
of the concrete stresses and uniaxial strains in the 1-2 directions, which are not zero because
the stress and strain of the concrete in compression is softened by the orthogonal tensile strains.

2
ε 1 − ε 2

1

G 12 = σ

− σ

is the shear modulus of concrete as described previously.