Rotating Angle Shear Theories - Unified Theory of Concrete Structures - page 192

α l

Figure 5.23 m -coefficients for three pairs of S and

5.4.5.2 Strength of 2-D Elements

The basic equilibrium equations for PC 2-D elements subjected to proportional loadings can
be easily derived by substituting the definitions of the m -coefficients (Equations 5.115-5.117),
into the equilibrium equations 1 - 3 of rotating angle theory:

m σ 1 − ρ f − ρ p f p = σ d sin 2 α r (5.124)

m t σ 1 − ρ t f t − ρ tp f tp = σ d cos 2 α r (5.125)

m t σ 1 = ( − σ d )sin α r cos α r (5.126)

The strength of the element under proportional loadings can be represented by a single stress
σ 1 . This stress σ 1 can be solved from Equations (5.124)-(5.126) by eliminating the angle α r .

Multiplying Equation (5.124) by Equation (5.125) gives:

− σ d ) 2 sin 2 α r cos 2 ( m σ 1 − ρ f − ρ p f p )( m t σ 1 − ρ t f t − ρ tp f tp ) = ( α r (5.127)

Squaring Equation (5.126) gives:

( m t σ 1 ) 2

− σ d ) 2 sin 2 α r cos 2

= α r

( (5.128)

Equating the left-hand sides of Equations (5.127) and (5.128) and rearranging the terms
results in a quadratic equation for σ 1 as follows:

m 2

2
1

( m m t −

t )

σ − [ m ( ρ t f t + ρ tp f tp ) + m t ( ρ f + ρ p f p )] σ 1

+ ( ρ f + ρ p f p )( ρ t f t + ρ tp f tp ) = 0 (5.129)

= m m t −

t

m 2

Define S (5.130)

B = [ m ( ρ t f t + ρ tp f tp ) + m t ( ρ f + ρ p f p )] (5.131)

C = ( ρ f + ρ p f p )( ρ t f t + ρ tp f tp ) (5.132)