Torsion - Unified Theory of Concrete Structures - page 302

The average strain ε d can be simply related to the maximum strain ε ds by:

ε d = ε ds
2

(7.24) or [11]

Equations [8], [9], [10] and [11] are the four additional compatibility equations. They
introduce four additional variables, θ , ψ , t d and ε ds .

7.1.3 Constitutive Relationships of Concrete
7.1.3.1 Softened Compression Stress Block

When the strain distribution in the concrete struts is assumed to be linear, the stress distribution
is represented by a curve as shown in Figure 7.4(c). This softened compression stress block
has a peak stress

σ p defined by

f c σ p = ζ (7.25)

where the softening of the concrete struts is taken into account by the coefficient ζ . The average

stress of the concrete struts σ d is defined as

f c

σ d = k 1 σ p = k 1 ζ (7.26) or [12]

where the nondimensional coefficient k 1 is defined as the ratio of the average stress to the
peak stress. The symbol σ d in Equation [12] has been generalized to represent the average

compression stress in a concrete strut subjected to bending and compression, rather than
the compression stress of axially loaded concrete struts in a membrane element as defined
for the symbol

σ d in the equilibrium equations [1]-[3]. The generalization of the symbol
σ d implies two assumptions. First, the relationship between the average stress σ d defined by

Equation [12] and average strain ε d defined by Equation [11] is identical to the stress-strain

relationship of an axially loaded concrete strut. Second, the softening coefficient , which has

been determined from the tests of 2-D elements with concrete struts under axial compression,
is assumed to be applicable to the concrete struts under combined bending and compression.
The resultant C of the softened compression stress block has a magnitude of

ζ

f c t d

C = σ d t d = k 1 σ p t d = k 1 ζ (7.27)

This resultant C is located at a distance k 2 t d from the extreme compression fiber, Figure
7.4(c). The nondimensional coefficient k 2 is defined as the ratio of the distance between
the resultant C and the extreme compression fiber to the depth of the compression zone t d .
The compression stress block is statically defined, when the two coefficients k 1 and k 2 are
determined.
The coefficient k 1 can be determined from the equilibrium of forces by integrating the
compression stress block. Designating the stress at a distance x from the neutral axis as

σ

( x )

(Figure 7.4c), gives

t d

C = k 1 σ p t d = σ ( x )d x (7.28)

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