Tensile Strength and degree of restraint

Since concrete will crack once the tensile stress in the structure exceeds the tensile strength of the concrete, which then affects the cross section, the tensile strength of the concrete is a significant factor in the long-term performance of the structure.

While the British Standard used a fixed tensile strength based on the concrete grade, the Eurocode allows for some modification for time with an increase in the tensile strength as the concrete compressive strength increases with time.

The tensile stress in the concrete is influenced by the restraint to the structure, since the restraint of the structure against shrinkage will induce tensile stresses in the structure which will then have an influence on the formation of cracking.

The restraint to the slab will have an influence on the effective tensile strength of the concrete. Where slabs are highly restrained, the shrinkage of the slab will induce tensile stresses which effectively reduces the tensile strength of the concrete. For high restraint situations, the concrete tensile strength fctm should be used, while for no restraint the tensile strength fctm,fl can be used. The Concrete Centre publication “How to design concrete structures using Eurocode 2” recommends that for low restraint, the average value of fctm and fctm,fl should be used, to take account of any unintentional restraint.

Cracking

The deflection of the structure is very closely linked with the formation of cracking, which is itself closely linked to the tensile stress and tensile strength of the concrete. The tensile stress in the structure is linked to the loading while the tensile strength is dependent on the material properties of the concrete and so both are time dependent properties.

Once cracking has occurred, the cross section in the vicinity of the crack is permanently altered, resulting in a reduced effective cross section and hence a reduction of stiffness. The presence of cracking can have a significant effect on the long-term subsequent distribution of forces in the structure in later loading events and so influence the performance and deflections of the structure. It is, therefore, important to identify the stages at which cracking occurs. Since cracking is dependent on loading and material properties, both of which are time dependent, crack formation and its effects are then also time dependent.

Reinforcement

The amount of reinforcement in a cross section, along with the relative quantities of reinforcement in the compression and tension zones can induce additional curvatures in the concrete as an result of the shrinkage of the concrete. In zones where the reinforcement is symmetrical, the strain due to curvature is uniform and so no curvature will be induced, but once the reinforcement becomes non-symmetrical, then an imbalance in the strains occurs which leads to an imbalance in the forces in the cross section which in turn leads to an induced curvature, with the increased curvature occurring on the side with the largest area of reinforcement. As a consequence of this, the larger amount of reinforcement on the tension side of the section will induce a curvature which acts to increase the deflection of the slab or beam.

The shrinkage curvature is calculated from:

                                (4)

Where

•    the curvature due to shrinkage
•    the free shrinkage strain
•    the effective modular ratio                    
•    the first moment of area of the reinforcement about the centroid of the section
• I    the second moment of area of the section

The effective modular ratio is calculated from:

                            (5)

Where  is calculated from Eq (3)