The degree of the formulas needed to compute the compression zone in phase III are strongly dependent on both the shape of the cross section and  constitutive relation assumed in the analysis. For example in the simplest case, rectangular cross section and rectangular distribution of stress in concrete followed by some simplifications needed for producing analytical formulas (according to Eurocode 2) that can be used for the design purposes produces equations in the attachment. The solution will always be given in specific intervals.

For sections made of multiple rectangles (+ rectangle relation for concrete), solution is made of similar functions but is far more complicated - too long to be written explicitly in one equation.  

For sections like circular or circular hollow (+ rectangular consitutive relation for concrete) the solution is made of trigonomic and inverse trigonometric functions (plus many others) and it is impossible to obtain explicit analitical formulas for x(N,M) - only numerical methods remain but it is possible to write down these equations at least.

If we assume the parabolic constitutive relation for concrete the solution complicates itself considerably even for rectangular cross section let alone for other sections - in my opinion only numerical methods remain but for rectangular sections we are of course able to write down these equations. 

Best regards,

Bartosz Grzeszykowski

For RC structures? In phase 2 or phase 3? For the rectangular cross section?

At failure.

For any section.

The degree of the formulas needed to compute the compression zone in phase III are strongly dependent on both the shape of the cross section and  constitutive relation assumed in the analysis. For example in the simplest case, rectangular cross section and rectangular distribution of stress in concrete followed by some simplifications needed for producing analytical formulas (according to Eurocode 2) that can be used for the design purposes produces equations in the attachment. The solution will always be given in specific intervals.

For sections made of multiple rectangles (+ rectangle relation for concrete), solution is made of similar functions but is far more complicated - too long to be written explicitly in one equation.  

For sections like circular or circular hollow (+ rectangular consitutive relation for concrete) the solution is made of trigonomic and inverse trigonometric functions (plus many others) and it is impossible to obtain explicit analitical formulas for x(N,M) - only numerical methods remain but it is possible to write down these equations at least.

If we assume the parabolic constitutive relation for concrete the solution complicates itself considerably even for rectangular cross section let alone for other sections - in my opinion only numerical methods remain but for rectangular sections we are of course able to write down these equations. 

Best regards,

Bartosz Grzeszykowski

Thank you my friend

Good answer Bartosz Grzeszykowski

Usually third order for this case

I agree with  Bartosz Grzeszykowski

I mean at serviceability stage not at failure Mr Bartoz !!!!

At serviceability stage the stress diagram is not parabolic.

In SLS we usually assume that the constitutive relation for concrete is linear (trianlgle distribution along the height of a compressive zone without any stress and strain limit), the steel is not yielding (Hooke's law) and concrete does not bear tensile stress. Under that specific conditions the solutions for different cross sections are as follows:

1. rectangular cross section, the compressive zone is the solution of a 3-rd degree polynomial:

f = f(x, N, M, As, b, d) = a x^3 + b x^2 +c x +d = 0             (1)

where a, b, c, d - coefficients of a 3-rd degree polynomial

and it is relatively easy to write down this equation in an explicit form.

2. cross section that consists of traingles (I- beams, T-beams, square hollow sections etc., but for uniaxiall bending + axial force only):

f is in the same form but is given in intervals and is therefore more complex.

3. Circular and circular hollow sections: f is given in intervals and it is impossible to write it down in reasonable number of lines on a peace of paper. Moreover it consists of functions: trigonomic, inverse trigonomic and others. Only numerical methods remain.

4. The general equation for any cross section and distribution of reinforcement along the height of a cross section is the solution of the following equation :

N / AII + MII (vII - x) / JII = 0                       (2)

where:

AI = area of an uncracked concrete section  (without the influence of reinforcement)

AII = area of a cracked concrete section + the influence of reinforcement

JII = second moment of area for cracked concrete + the influence of reinforcement

vI = distance from the compressed edge to the centre of gravity of the AI

vII = distance from the compressed edge to the centre of gravity of the AII

N = axial force

M = bending moment given in relation to vI

H = height of a cross section

MII = bending moment given in relation to vII; MII = M - N (vI - vII)

The solution of eq. (2) is very often complex, because most functions from eq. (2) depend on x:

AII = AII(x), MII = MII(x), vII = vII(x), JII = JII(x)

For rectangular cross sections: If we substitute all the above  functions into eq. (2) and expand it with respect to x, we get eq. (1).

Best Regards,

Bartosz Grzeszykowski

Many thanks