# PRE-Stress Theory Handbook

The Theory section gathers such information as does not directly pertain to the practical use of PRE-Stress. More specifically, it offers insight into the theory underpinning the calculations.

# Calculation of concrete beams

Unlike other ordinary construction materials, such as steel or timber, the characteristics of concrete render calculations – particularly in the case of statically undetermined structures – complicated. The main difficulty lies in cracking and the ensuing changes in stiffness. Non-linearity at higher strains, as well as long-time effects like creep, all contribute to the complexity.

## Calculation principles

The calculation uses member stiffness values calculated from a comprehensive and potentially non-linear section analysis. The method could be viewed as an iterative estimate of secant stiffness.

The calculation interprets non-linear behaviour from cracking into strain and curvature data, which influence the iteration as a load between the nodes corresponding to thermal load. Non-linear strain from creep and shrinkage are likewise converted to loads between nodes corresponding to thermal load. As the magnitude of creep and curvature depends on the distribution of section forces, the calculation of creep values also requires iteration.

Remaining deflection from earlier loadcases may also be added.

## Effective stiffness

## Inelastic strain at cracking

## Inelastic strain and curvature of creep and shrinkage

## Inelastic strain and curvature from previous loadcase

Creep and shrinkage mean that deflections remain in the beam after unloading. During the calculation, these deflections are stored for all relevant loadcases, and can be re-activated through Dependency. In other words, a current loadcase calculation may be influenced by a previous loadcase calculation. This way, the effect of a long-time load in one load combination can be made to influence another. When calculating the new loadcase, the inherited deflections will be treated as added deflections.

## Cracking in previous loadcase

Possible cracking in a load combination is stored during a calculation. Through the use of Dependency (more information can be found here: Guide to load combinations), the cracking from a previous load combination can be inherited, which may affect the analysis of the current load combination.

# Design theory

## Bending reinforcement

## Shear reinforcement

## Flange reinforcement

## End zone reinforcement

## Recess beam

# Control theory

## Section properties

## Stresses and strains at linear section analysis

## Stresses and strains at non-linear section analysis

## Elongation and curvature

## Ultimate moment capacity

## Shear capacity with shear reinforcement

## Shear capacity of the shear reinforcement

## Shear capacity of concrete

## Rules for shear reinforcement

## Resistance to spalling (hollowcore elements)

## Control of splitting

## Eigenfrequency calculations

# Fire theory

# References

**Contents**

- Calculation of concrete beams
- Design theory
- Control theory
- Section properties
- Stresses and strains at linear section analysis
- Stresses and strains at non-linear section analysis
- Elongation and curvature
- Ultimate moment capacity
- Shear capacity with shear reinforcement
- Shear capacity of the shear reinforcement
- Shear capacity of concrete
- Rules for shear reinforcement
- Resistance to spalling (hollowcore elements)
- Control of splitting
- Eigenfrequency calculations

- Fire theory
- References