# Stresses and strains at non-linear section analysis

**Contents**

Non-linear section analysis is used when stresses or strains are so large that the relation between stress and strain no longer is assumed to be linear. This is generally the case at Ultimate Limit State.

# Internal section forces from stresses and strains

In non-linear section analysis it is assumed that (see Stresses and strains at linear section analysis)

a) plane cross-sections remain plane at bending (Bernoulli’s hypothesis), and

b) only normal stresses occur in pure bending (Navier’s hypothesis).

In order to determine resultants to stresses in the section, numerical integration is generally needed. A simple method of integration is dividing the section into a number of concrete layers and reinforcement bars and then analyze each layer and bar separately. Strain and stress are determined in the mid-depth of each component and a numerical integration of the resulting moments and forces is performed:

with stresses according to

f_{ci} = f_{c} (ε_{ci}) where ε_{ci} is concrete strain at layer position, f_{cj} = f_{c} (ε_{cj}) where ε_{cj} is concrete strain at bar position, f_{sj} = f_{s} (ε_{sj}) where ε_{sj} is steel strain at bar position,

where

n, m = number of concrete layers and reinforcing bar elements,

b_{i}, h_{i} = width and depth of a concrete layer i,

A_{s} = cross-sectional area of reinforcing bar,

z_{cg} = z-coordinate for centroid of section in local coordinate system with y-axis parallel to neutral axis,

f_{c} = stresses in concrete,

f_{s} = stresses in reinforcement,

f_{c}(ε) = stress-strain relation (non-linear) for concrete,

f_{s}(ε) = stress-strain relation (non-linear) for steel.

If the section is divided into a sufficient number of layers the accuracy will be satisfactory even if large non-linearity in the stress distribution exists.

# Stress-strain relation for concrete

A rectangular stress distribution as shown below with height λ x may be assumed.

λ = 0.8 for f_{ck} ≤ 50 MPa

λ = 0.8 - (f_{ck} -50)/400 for 50 < f_{ck} ≤ 90 MPa

and

η = 1.0 for f_{ck} ≤ 50 MPa

η = 1.0 - (f_{ck} -50)/200 for 50 < f_{ck} ≤ 90 MPa

Note: If the width of the compression zone decreases in the direction of the extreme compression fibre, the value η f_{cd} should be reduced by 10%

The value of the design compressive strength is defined as

f_{cd} = α_{cc} f_{ck} / γ_{c} where

γ_{cc} = 1.0,

γ_{c} is the partial safety factor as above.

The value of the design tensile strength is defined as

f_{ctd} = α_{ct} f_{ctk,0.05} / γ_{c} where

α_{ct} = 1.0, γc is the partial safety factor as above.

## National Annex: United Kingdom

α_{cc} = 0.85

## National Annex: Finland

α_{cc} = 0.85

# Stress-strain relation for reinforcement steel

Strain limit ε_{ud} = 0.9 ε_{uk}

## National Annex: Denmark

ε_{ud} = f_{yd} / E_{s}

## National Annex: Finland

ε_{ud} = 0.02

# Calculation of stress and strains due to section forces

At non-linear section analysis, determination of the strain distribution on a section including position of the neutral axis is generally not possible to perform with a direct method. Instead an iterative method must be used. The method that is described in chapter Stresses and strain in cracked sections for cracked section is also suitable for use with non-linear calculation. The calculation of section forces out of assumed strains is then performed using the method described in chapter Internal section forces from stresses and strains.

The strain distribution is primarily determined by this method and after that, the stresses are calculated using a non-linear stress-strain relation.