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Dipl.-Ing. Juliane Stopper-Akdag

2026-03-12

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Concrete Design Add-on

Theoretical Principles

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Tables for Concrete Design
Graphics for Concrete Design
- Design Checks
  
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Documentation

Confined Compression Elements

By applying constriction to the concrete cross-section with shear reinforcement, the lateral strain of the concrete is limited, thereby increasing its compressive strength and ductility. The effect is displayed schematically in the following image from [2].

Circular and Rectangular Compression Elements with Confining Reinforcement (Source: [2])

These effects are important for the seismic design of columns, as they ensure the availability of plastic deformation capacity and can improve the global load-bearing capacity of the structure in the event of an earthquake.

Design Model According to Eurocode 2

DIN EN 1992-1-1 does not provide an explicit design approach for constricted compression elements.

However, according to Section 3.1.9 of DIN EN 1992-1-1 [1], a multiaxial stress state may be taken into account in the design by applying a modified effective stress-strain relation with increased strengths and limit strains. It should be noted that these increased strength and strain values apply exclusively to the constricted concrete core, which is limited by the center lines of the confining reinforcement.

The load-bearing contribution of the concrete cover is generally not to be considered in the ULS, since the concrete cover is usually no longer effective due to spalling.

Unless more precise data is available, a parabolic-rectangular diagram according to [1], Figure 3.6, may be used for the constricted core concrete. It is necessary to use accordingly adjusted characteristic strength and strain values as a basis.

Diagram of the stress-strain relation for concrete under multiaxial compressive stress according to EC2.

Stress-Strain Behavior of Concrete Under Multiaxial Compression [1]

The strength of the constricted concrete core is described by the following formula.

f_{c k, c} = \{\begin{cases} f_{c k} \cdot (1, 000 + 5, 0 \cdot \frac{σ_{2}}{f_{c k}}) & f ü r σ_{2} \leq 0, 05 \cdot f_{c k} \\ f_{c k} \cdot (1, 125 + 2, 5 \cdot \frac{σ_{2}}{f_{c k}}) & f ü r σ_{2} > 0, 05 \cdot f_{c k} \end{cases}

σ2	Efficient transverse compression stress in the ULS due to transverse strain restriction

Characteristic Strength of Constricted Reinforced Concrete Core According to EC 2, (3.24) & (3.25)

The characteristic strains are determined using the following formulas.

ε_{c 2, c} = ε_{c 2} \cdot {(\frac{f_{c k, c}}{f_{c k}})}^{2}

εc2	Concrete elongation upon reaching f_ck according to EC2, Table 3.1

Concrete Strain Upon Reaching f_ck,c According to EC2, (3.26)

ε_{c u 2, c} = ε_{c u 2} + 0, 2 \cdot \frac{σ_{2}}{f_{c k}}

εcu2	Concrete ultimate strain under uniaxial compressive stress EC2, Table 3.1

Concrete Ultimate Strain Under Multiaxial Compression EN 1992-1-1, (3.27)

Eurocode 2 does not provide any guidelines for determining the transversal compression. Therefore, DAfStb Heft 630 ([2], 3.5.4) refers to the approaches of Mander et al. The equations for determining the transverse compression are described in the following section. σ₂ = σ'_l

Analysis Model According to Mander et al.

The model developed by Mander et al. explicitly describes the stress-strain behavior of constricted concrete as a function of the effective confining pressure. From this, an increased effective compressive strength and an increased peak strain of the constricted concrete core are derived, as well as a ductile, gradually decreasing stress-strain curve. Among others, the input parameters include the reinforcement content and arrangement of the stirrups/spirals, their steel strength, and the geometry of the enclosed core cross-section.

Comparison of the stress-strain relation of concrete and constricted concrete according to Mander [3], displayed in blue and red.

Stress-Strain Relation for Concrete and Constricted Concrete According to Mander [3]

The stress-strain relation is described by the following equation, [3] (3), (4), (6), (7), and (8).

σ_{c} = \frac{f_{c c} \cdot x \cdot r}{r - 1 + x^{r}}, m i t : x = \frac{ε_{c}}{ε_{c c}} u n d r = \frac{E_{c}}{E_{c} - E_{s e c}}

E_sec	=f_cc/εcc
E_c	=5,000·fc

Stress Equation According to Mander or Popovics

The corresponding strain can be described by the following formula, [3] (5).

ε_{c c} = ε_{c} \cdot (1 + 5 \cdot (\frac{f_{c c}}{f_{c}} - 1))

Strain Upon Reaching Compressive Strength of Constricted Core Concrete According to Mander et. al.

The compressive strength of the constricted concrete is given according to [3] (29).

f_{c c} = f_{c} \cdot (- 1, 254 + 2, 254 \cdot \sqrt{1 + \frac{7, 94 \cdot σ'_{l}}{f_{c}}} - 2 \cdot \frac{σ'_{l}}{f_{c}})

Compressive Strength of Constricted Concrete

The effective transverse compression stress for this can be determined from the equation below.

σ'_{l} = \frac{1}{2} \cdot ω_{v} \cdot f_{y w} \cdot k_{e}, m i t : ω_{v} = \frac{4 \cdot A_{s w}}{d_{c} \cdot s_{c}}

k_e	Effective Constriction Coefficient
ωv	Volume ratio between confining reinforcement and confined core concrete
f_yw	Yield stress of the transverse reinforcement

Effective Transverse Compression Stress According to Mander et. al.

The effectivity coefficient of the constriction is given according to the following expression.

k_{e} = \frac{A_{e c}}{A'_{c c}}

A_ec	Surface of the efficiently constricted concrete core

Effectiveness Coefficient of Constriction

Where the net core concrete surface can be determined from:

A'_{c c} = A_{c c} \cdot (1 - ρ_{c c}) = A_{c c} - A_{s l}

ρcc	=A_sl/A_cc longitudinal reinforcement ratio of the constricted concrete core

Cross-Sectional Area of Constricted Concrete Core

The effective constricted concrete surface can be determined from

A_{e} = \frac{π}{4} \cdot {d_{c}}^{2} \cdot {(1 - \frac{s'}{2 \cdot d_{c}})}^{2} f ü r K r e i s q u e r s c h n i t t e

A_{e} = (b_{c} \cdot a_{c} - \sum_{i = 1}^{n} \frac{(w'_{i}) ²}{6}) \cdot (1 - \frac{s'}{2 \cdot b_{c}}) \cdot (1 - \frac{s'}{2 \cdot a_{c}}) f ü r R e c h t e c k q u e r s c h n i t t e

d_c	Diameter of the constricted core concrete
a_c, b_c	Horizontal dimensions of the constricted core concrete
s'	Light spacing of the confining reinforcement in the direction of the compressive stress
w'_i	Clear distance between the longitudinal rebars

Efficiently Constricted Concrete Surface According to Mander et al.

Application

The design model according to EC2, 3.1.9 is used for standard-compliant ultimate limit state design checks that allow for an increase in compressive strength due to constriction.

The Mander model (Mander et al., 1988) is used as a physically based material model when the full σ < sub > c < /sub > - ε_c behavior of constricted core concrete (increased f_cc, ε_cc, post-peak behavior) needs to be taken into account for nonlinear analyses.

References

European Committee for Standardization (CEN). (2010). Eurocode 2: Design of Concrete Structures – Part 1‑1: General Rules and Rules for Buildings, EN 1992‑1‑1:2004 + AC:2010. Brussels, CEN.
German Committee for Structural Concrete e.V. (2018). Heft 630: Bemessung nach DIN EN 1992 in den Grenzzuständen der Tragfähigkeit und der Gebrauchstauglichkeit. Berlin, DAfStb.
Mander, J. B., Priestley, M. J., & Park, R. (1988). Theoretical Stress‐Strain Model for Confined Concrete. Journal of Structural Engineering, 114(8), 1804–1826.

Parent Chapter

Design in Seismic Situation