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Concrete Stiffness – The General Method

Introduction to Concrete Stiffness

The stiffness of concrete sections has a significant impact on the capacity of columns. The higher the stiffness, the lower the second-order effect and thereby lower utilization.

EN1992-1-1 defines three different methods to determine stiffness

  • General method – EN1992 1-1 section 5.8.6
  • Nominal stiffness – EN1992-1-1 section 5.8.7
  • Nominal curvature – EN1992-1-1 section 5.8.8

Using the wrong method can significantly increase reinforcement demands (see comparison below). The Danish national annex specifies that if using the simplified methods (Nominal stiffness and -curvature) the method based on nominal stiffness must be used.

Comparison of Concrete stiffness

A 450x450mm C30 column with a normal force of 3000kN and a distributed load of 30kN/m is used to compare the two methods. Required reinforcement is calculated for increasing column heights.

The general method requires significantly less reinforcement than nominal stiffness. For a 5.5m column, nominal stiffness demands 300% more reinforcement for the same section.

The difference comes down to how stiffness is calculated. Nominal stiffness applies a fixed reduction regardless of the actual load. The general method calculates stiffness directly from the section response — and at low reinforcement ratios, that matters.

Concrete Section
Concrete Stiffness comparasion

Calculating Concrete Stiffness

To calculate the stiffness of a concrete section using the general method (EN1992-1-1 5.8.6) there are four parameters that need to be calculated:

  1. Uncracked stiffness: The stiffness before cracking of the concrete
  2. Cracking moment: The moment that causes initial cracking
  3. Cracked stiffness: The stiffness of the fully cracked section
  4. Effective stiffness: The stiffness used for design, taking tension stiffening into account

Calculation Example

In this example only the short-term case is considered. The section has the following properties:

  • C45
  • NEd = -2.500kN
  • MEd = 361kNm
  • Es = 200.000MPa
  • Ecd = 25.023MPa
  • fctd = 1,56MPa
  • fyd = 458MPa
Concrete Section

Uncracked Stiffness

The uncracked stiffness is found as the sum of the stiffness from the concrete and reinforcement

\text{EI}_{\text{y,uncracked}} = \text{E}_{\text{cd}} \cdot \text{I}_{\text{c,y,uncracked}} + \text{E}_{\text{cd}} \cdot \sum I_{\text{s,y}}

The stiffness from the concrete is found around the centroid. This stiffness is very sensitive to creep and will be much lower in the long-term case.

\text{E}_{\text{cd}} \cdot \text{I}_{\text{c,y,uncracked}} = 25.023 \,\text{MPa} \cdot \frac{1}{12} \cdot (500 \,\text{mm})^{4} = 130.328 \,\text{kN}\,\text{m}^{2}

The stiffness from the reinforcement is largely governed by the distance from the centroid, denoted z.

\text{E}_{\text{s}} \cdot \sum I_{\text{s,y}} = 200{,}000 \,\text{MPa} \cdot \sum \left( \frac{\pi}{64} \cdot (14 \,\text{mm})^{4} + \frac{\pi}{4} \cdot (14 \,\text{mm})^{2} \cdot z \right) = 10.930 \,\text{kN}\,\text{m}^{2}\text{EI}_{\text{y,uncracked}} = 130.328 \,\text{kN}\,\text{m}^{2} + 10.930 \,\text{kN}\,\text{m}^{2} = 141.258 \,\text{kN}\,\text{m}^{2}

The stiffness is largely governed by concrete in the short-term case.

Cracking Moment

The cracking moment is the moment at which the tensile stress exceeds the tensile capacity. It is found by rewriting Navier’s stress equation.

\text{M}_{\text{cr}} = \left( \text{f}_{\text{ctd}} + \frac{\text{N}_{\text{Ed}}}{\text{A}_{\text{uncracked}}} \right) \cdot \frac{\text{I}_{\text{y,uncracked}}}{y}

Two new parameters are introduced, an uncracked Area and moment of inertia. These can be found by normalizing the bending stiffness to the concrete model of elasticity.

\text{I}_{\text{y,uncracked}} = \frac{\text{EI}_{\text{y,uncracked}}}{\text{E}_{\text{cd}}} = \frac{141.258 \,\text{kN}\,\text{m}^{2}}{25.023 \,\text{MPa}} = 5.645 \cdot 10^{6} \,\text{mm}^{4}\text{A}_{\text{uncracked}} = (500 \,\text{mm})^{2} + \frac{200{,}000 \,\text{MPa}}{25.023 \,\text{MPa}} \cdot 12 \cdot \frac{\pi}{4} \cdot (14 \,\text{mm})^{2} = 264{,}765 \,\text{mm}^{2}

Entering these into the equation gives the cracking moment.

\text{M}_{\text{cr}} = \left( 1.56 \,\text{MPa} + \frac{2.500 \,\text{kN}}{264{,}765 \,\text{mm}^{2}} \right) \cdot \frac{5.645 \cdot 10^{6} \,\text{mm}^{4}}{250 \,\text{mm}} = 249 \,\text{kNm}

The cracking moment is dependent on the normal force. A large normal force will increase the cracking moment.

Cracked Stiffness

The cracked stiffness is found based on internal equilibrium. This is an iterative calculation that results in a neutral axis depth (x), a concrete strain and rotation of the neutral axis. The first step is to guess the state of internal equilibrium and afterwards verify it. In this example the parabolic stress-strain relation for concrete is used.

  • x = 331mm
  • εc = 0.0014
  • Neutral axis angle = 0°
Concrete stiffness

Through verification of the internal equilibrium the assumptions can be confirmed.

\sum \text{N} = \int \sigma_{\mathrm{c}} \cdot \mathrm{dA} + \sum \text{F}_{\mathrm{s}} + \text{N} = 0\sum \text{M} = \int \sigma_{\mathrm{c}} \cdot \text{z} \cdot \mathrm{dA} + \sum \text{F}_{\mathrm{s}} \cdot \text{z} + \text{M} = 0

Once equilibrium is established, the stiffness of the section can be determined based on the moment-curvature in the case of uniaxial bending. The next step is calculating the curvature.

\mathrm{\kappa} = \frac{\mathrm{\varepsilon}_{\mathrm{c}}}{\mathrm{x}} = \frac{0.0014}{331 \,\mathrm{mm}} = 0.0042

The cracked stiffness can now be found based on the moment and curvature.

\text{EI}_{\text{y}} = \frac{\text{M}_{\text{Ed}}}{\kappa} = \frac{361 \,\text{kNm}}{0.0042} = 85.351 \,\text{kNm}^{2}

The cracked stiffness is highly dependent on the external loads.

Effective stiffness

The general method allows the use of tension stiffening. This enables the use of effective stiffness instead of assuming either fully cracked or uncracked behavior.

The distribution coefficient can be found by comparing the bending moment to the cracking moment.

\zeta = 1 – \mathrm{\beta} \cdot \left( \frac{\mathrm{\sigma}_{\mathrm{sr}}}{\mathrm{\sigma}_{\mathrm{s}}} \right)^{2} = 1 – 0.5 \cdot \left( \frac{249 \,\mathrm{kNm}}{361 \,\mathrm{kNm}} \right)^{2} = 0.76

Equation 7.18 is used to determine the effective stiffness.

\frac{1}{\mathrm{EI}} = 0.76 \cdot \frac{1}{85.351 \,\mathrm{kNm}^{2}} + (1 – 0.76) \cdot \frac{1}{141.258 \,\mathrm{kNm}^{2}} = \frac{1}{95.605 \,\mathrm{kNm}^{2}}

Effective stiffness is somewhere in between the cracked- and the uncracked stiffness.

Key takeaways on Concrete Stiffness

The general method consistently requires less reinforcement than the simplified methods, particularly at low reinforcement ratios. For the 5.5m column in this example, the difference is 300%. The added computational complexity is manageable with modern tools such as PolyColumn.

How PolyStruc uses the General Method

PolyColumn implements the general method for both uniaxial and biaxial bending, giving you the most accurate stiffness estimate available under EN1992-1-1. For columns where the simplified methods demand excessive reinforcement, switching to the general method can reduce that demand by up to 300%, without compromising code compliance.

Download PolyColumn and get the most out of your concrete columns.

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