Materials

Basic functionality, concept and integration

In steel-concrete composite structures Concrete, Steel and Reinforcement are relevant materials. For these materials sophisticated material models are available to describe their behaviour under loading (see for example formula (3)).

For integration into the program these – in part discrete – formulas need to be abstracted to multi-linear stress-strain relationships.

Concrete

General

Concrete is in general characterised by it class. The classes describe the characteristic concrete compressive strength \(f_\mathrm{ck}\). Following EN 1992-1-1 [1] most of the mechanical characteristic are derived from \(f_\mathrm{ck}\).

The mean concrete compressive strength \(f_\mathrm{cm}\) according to EN 1992-1-1 [1] is given in formula (1).

(1)\[f_\mathrm{cm} = f_\mathrm{ck} + 8 \text{ in N/mm²}\]

The modulus of elasticity \(E_\mathrm{cm}\) by EN 1992-1-1 [1] is given in formula (2).

(2)\[E_\mathrm{cm} = 22000 \cdot \left(\frac{f_\mathrm{cm}}{10} \right)^{0.3}\]

Compression

Introduction

EN 1992-1-1 [1] provides three material models to define the stress-strain-relationship of concrete in compression. These are Stress-strain-relationship for non-linear determination of stress-resultants and deformations, Parabola-rectangle stress-strain-relationship for section-design and Bi-linear stress-strain-relationship for section-design. Every of these three stress-strain-relationships of the concrete according to EN 1992-1-1 [1] is implemented in Concrete and may be chosen argument compression_stress_strain_type.

Stress-strain-relationship for non-linear determination of stress-resultants and deformations

Non-linear stress-strain-relationship of concrete under compression acc. EN 1992-1-1 [1]

The stresses according to the non-linear determination of stress-resultants and deformations are computed by formula (3) in the range \(0 < | \varepsilon_\mathrm{c1} | < | \varepsilon_\mathrm{cu1} |\).

(3)\[\sigma_\mathrm{c} = \frac{k \cdot \eta - \eta^{2}}{1 + (k - 2) \cdot \eta} \cdot f_\mathrm{cm}\]

Where:

\[ \begin{align}\begin{aligned}\eta & = \varepsilon_\mathrm{c} / \varepsilon_\mathrm{c1}\\\varepsilon_\mathrm{c1} & = 0.7 \cdot f_\mathrm{cm}^{0.31} \leq 2.8\\k & = 1.05 \cdot E_\mathrm{cm} \cdot | \varepsilon_\mathrm{c1} | / f_\mathrm{cm}\\\varepsilon_\mathrm{cu1} & = 2.8 + 27 \cdot \left[\frac{98-f_\mathrm{cm}}{100}\right]^{4}\end{aligned}\end{align} \]

\(\varepsilon_\mathrm{c1}\) is the strain at maximum stress, whereas \(\varepsilon_\mathrm{cu1}\) is the strain at failure.

The above given nonlinear stress-strain-relationship is implemented by passing compression_stress_strain_type='Nonlinear' to Concrete. Formula (3) is approximated by a multi-linear curve in Concrete.

Parabola-rectangle stress-strain-relationship for section-design

Parabola-Rectangle stress-strain-relationship of concrete under compression acc. EN 1992-1-1 [1]

(4)\[ \begin{align}\begin{aligned}\sigma_\mathrm{c} & = f_\mathrm{ck} \cdot \left[1 - \left(1 - \frac{\varepsilon_\mathrm{c}}{\varepsilon_\mathrm{c2}} \right)^{n} \right] & & \text{ for } 0 \leq \varepsilon_\mathrm{c} \leq \varepsilon_\mathrm{c2}\\\sigma_\mathrm{c} & = f_\mathrm{ck} & & \text{ for } \varepsilon_\mathrm{c2} \leq \varepsilon_\mathrm{c} \leq \varepsilon_\mathrm{cu2}\end{aligned}\end{align} \]

where

\[ \begin{align}\begin{aligned}\varepsilon_\mathrm{c2} & = 2.0 + 0.085 \cdot (f_\mathrm{ck} - 50)^{0.53}\\\varepsilon_\mathrm{cu2} & = 2.6 + 35 \cdot \left[\frac{90 - f_\mathrm{ck}}{100}\right]^{4}\\n & = 1.4 + 23.4 \cdot \left[\frac{90 - f_\mathrm{ck}}{100}\right]^{4}\end{aligned}\end{align} \]

\(\varepsilon_\mathrm{c2}\) is the strain at maximum stress and \(\varepsilon_\mathrm{cu2}\) is the strain at failure.

This stress-strain-relationship is applied by passing compression_stress_strain_type='Parabola' to Concrete.

Bi-linear stress-strain-relationship for section-design

Bi-linear stress-strain-relationship of concrete under compression acc. EN 1992-1-1 [1]

(5)\[ \begin{align}\begin{aligned}\sigma_\mathrm{c} & = f_\mathrm{ck} \cdot \frac{\varepsilon_\mathrm{c}}{\varepsilon_\mathrm{c2}} & & \text{ for } 0 \leq \varepsilon_\mathrm{c} \leq \varepsilon_\mathrm{c3}\\\sigma_\mathrm{c} & = f_\mathrm{ck} & & \text{ for } \varepsilon_\mathrm{c3} \leq \varepsilon_\mathrm{c} \leq \varepsilon_\mathrm{cu3}\end{aligned}\end{align} \]

where

\[ \begin{align}\begin{aligned}\varepsilon_\mathrm{c3} & = 1.75 + 0.55 \cdot (\frac{f_\mathrm{ck} - 50}{40})\\\varepsilon_\mathrm{cu3} & = \varepsilon_\mathrm{cu2}\end{aligned}\end{align} \]

The bi-linear stress-strain-relationship is applied by passing compression_stress_strain_type='Bilinear' to Concrete.

Tension

Stress-strain-relationship of concrete under tension

For a realistic load-carrying behaviour of the concrete the behaviour under tension is crucial.

If the tensile strength of the concrete \(f_\mathrm{ctm}\) is not given, it may be computed by formula (6).

(6)\[ \begin{align}\begin{aligned}f_\mathrm{ctm} & = 0.3 \cdot f_\mathrm{ck}^{2/3} & & \leq \text{ C50/60}\\f_\mathrm{ctm} & = 2.12 \cdot \ln\left[1 + \frac{f_\mathrm{cm}}{10}\right] & & > \text{ C50/60}\end{aligned}\end{align} \]

The strain when \(f_\mathrm{ctm}\) is reached may than be computed by formula (7).

(7)\[\varepsilon_\mathrm{ct} = \frac{f_\mathrm{ctm}}{E_\mathrm{cm}}\]

where \(E_\mathrm{cm}\) is the modulus of elasticity according to formula (2).

As soon as the strain reaches \(\varepsilon_\mathrm{ctm}\) the concrete starts to break. Different post-failure behaviours are possible in Concrete if \(\varepsilon_\mathrm{c} > \varepsilon_\mathrm{ct}\).

1. The resisting stresses drop immediately to \(\sigma_\mathrm{c} = 0\).

2. The crack-opening behaviour follows the recommendations by fib Model Code (2010) [2].

fib Model Code (2010) [2] defines the crack-opening behaviour as described in formula (9).

(8)\[ \begin{align}\begin{aligned}\sigma_\mathrm{ct} & = f_\mathrm{ctm} \cdot \left(1.0 - 0.8 \cdot \frac{w}{w_1}\right) & & \text{ for } w \leq w_1\\\sigma_\mathrm{ct} & = f_\mathrm{ctm} \cdot \left(0.25 - 0.05 \cdot \frac{w}{w_1}\right) & & \text{ for } w_1 < w \leq w_\mathrm{c}\end{aligned}\end{align} \]

where \(w\) is the crack opening in mm and \(w_1\) and \(w_\mathrm{c}\) are defined in theory.materials.concrete_crack_opening_values.

(9)\[ \begin{align}\begin{aligned}w_1 & = \frac{G_\mathrm{f}}{f_\mathrm{ctm}} & & \text{ if } \sigma_\mathrm{ct} = 0.2 \cdot f_\mathrm{ctm}\\w_\mathrm{c} & = 5 \cdot \frac{G_\mathrm{f}}{f_\mathrm{ctm}} & & \text{ if } \sigma_\mathrm{ct} = 0\end{aligned}\end{align} \]

The fracture energy \(G_\mathrm{F}\) is computed by (10).

(10)\[G_\mathrm{F} = 73 \cdot f_\mathrm{cm}^{0.18}\]

where \(f_\mathrm{cm}\) is the mean concrete compressive strength in N/mm².

The crack opening is considered by passing tension_stress_strain_type='consider opening behaviour' to py:class:~m_n_kappa.Concrete.

Steel

The stress-strain-relationship of structural steel is assumed to be point-symmetric around the origin. It may may be determined by one of following three ways:

1. Linear-elastic behaviour \(\sigma_\mathrm{a} = \varepsilon_\mathrm{a} \cdot E_\mathrm{a}\).

   Achieved if f_u = None and epsilon_u = None are passed to Steel.

2. Bi-linear behaviour where \(f_\mathrm{y} = f_\mathrm{u}\)

   (11)\[ \begin{align}\begin{aligned}\sigma_\mathrm{a} & = \varepsilon_\mathrm{a} \cdot E_\mathrm{a} & & \text{ if } 0 < | \varepsilon_\mathrm{a} | \leq | \varepsilon_\mathrm{y} |\\\sigma_\mathrm{a} & = f_\mathrm{y} & & \text{ if } | \frac{f_\mathrm{y}}{E_\mathrm{a}} | < | \varepsilon_\mathrm{a} | < | \varepsilon_\mathrm{u} |\end{aligned}\end{align} \]

   where \(f_\mathrm{y}\) is the yield strength of the steel and \(\varepsilon_\mathrm{y} = \frac{f_\mathrm{y}}{E_\mathrm{a}}\) is the strain at yielding and \(\varepsilon_\mathrm{u}\) is the strain at failure.

   Achieved if and epsilon_u != None is passed to Steel.

3. Bi-linear behaviour where \(f_\mathrm{y} < f_\mathrm{u}\)

   (12)\[ \begin{align}\begin{aligned}\sigma_\mathrm{a} & = \varepsilon_\mathrm{a} \cdot E_\mathrm{a} & & \text{ if } 0 < | \varepsilon_\mathrm{a} | \leq | \varepsilon_\mathrm{y} |\\\sigma_\mathrm{a} & = f_\mathrm{y} + (f_\mathrm{u} - f_\mathrm{y}) \cdot \frac{\varepsilon_\mathrm{a} - \varepsilon_\mathrm{y}}{\varepsilon_\mathrm{u} - \varepsilon_\mathrm{y}} & & \text{ if } | \varepsilon_\mathrm{y} | < | \varepsilon_\mathrm{a} | < | \varepsilon_\mathrm{u} |\end{aligned}\end{align} \]

   where \(f_\mathrm{y}\) is the yield strength of the steel, \(\varepsilon_\mathrm{y} = \frac{f_\mathrm{y}}{E_\mathrm{a}}\) is the strain at yielding, \(\varepsilon_\mathrm{u}\) is the strain at failure and \(f_\mathrm{u}\) is the stress at failure.

Elastic stress-strain-relationship of steel

Bi-linear stress-strain-relationship of steel

Bi-linear stress-strain-relationship with hardening of steel

The above given three ways are implemented in Steel.

Reinforcement

The characteristics of the stress-strain-relationship of reinforcement steel is similar to those of Steel. Solely the input-parameters change in Reinforcement as follows:

• Yield strength \(f_\mathrm{s}\): f_s (eqivalent to f_y in Steel)

• Failure strain \(\varepsilon_\mathrm{su}\): epsilon_su (eqivalent to epsilon_u in Steel)

• Failure strength \(f_\mathrm{su}\): f_su (eqivalent to f_u in Steel)

References

[1] (1,2,3,4,5,6,7,8)

EN 1992-1-1: Eurocode 2: Design of concrete structures - Part 1-1: General rules and rules for buildings, European Committee of Standardization (CEN), 2004

[2] (1,2)

fib Model Code 2010: fib Model Code for Concrete Structures 2010, International Federation for Structural Concrete, 2013, Ernst & Sohn