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Article by Lucy

The strength of concrete in compression is measured by crushing standard samples. In the UK these samples are in the form of 150 mm cubes, while on the continent of Europe and in theUSA they are in the form of cylinders of 150 mm diameter and 300 mm height. In general, the stressat which a cylinder fails is approximately 85 per cent of the strength of a cube made of thesame material, although the difference is less the stronger the concrete [2].

As the strength ofconcrete increases with time, it is conventionally determined at an age of 28 days aftercasting,although the 90 day strength is also relevant. The degree to which the strengthincreases with time depends on the nature of the cement. For instance, concrete madewith blended cement including a proportion of pulverised fuel ash (PFA), increases instrength more than that of unblended Portland cement. This enhanced strength is signi? cantin establishing the true factor of safety of a structure.

The strength of a concrete test specimen depends on the rate at which it is loaded, ‘the faster the application of the load the higher the strength. Thus tests for concrete need to specify a rateof loading [3].Concrete used in bridges typically has a 28 day cube strength of between 40MPa and 60 MPa. Strengths up to 100 MPa have been used exceptionally while even strongerconcretes, up to 150 MPa and beyond, are still the subject of research involving theconstruction of some trial structures.

Concrete is not an elastic material. As the stress/strain curve does not have a straight portion, one cannot de ne a unique Young’s modulus forconcrete and different codes of practice use different de nitions. The most common de?nitions are the tangent modulus, which is the slope of the curve at one point and the secantmodulus which is the slope of the line that connects the origin to a point on the curve,Figure 3.2 (a) [4]. If the concrete is loaded by a pulse, it will exhibit a dynamic modulusthat is similar to the initial tangent modulus [5]. The Young’s modulus of concreteincreases with its strength, but is typically, as de ned by the British code, of the order of 34,000 MPa for oads applied for a short time on bridge quality concrete.

When concrete fails in compression it exhibits a degree of ductility. The strain of plain concrete at failure in compression is conventionally assumed to be of the order of 2-4 × 1in the UK code of practice), Figure 3.2 (b) [6]. This ductility in compression may be greatlyincreased by the provision of reinforcement speci cally designed for that purpose.When concrete is loaded in compression and the load is maintained, the instantaneous deformation isfollowed by a deferred deformation known as creep. This deferred deformation is frequently upto twice the magnitude of the instantaneous value, and in a large structure such as a bridgedeck, may take several years to complete.

The corollary of this behaviour is that if a xed deformation is imposed on a concrete member,the force required to maintain that deformation would reduce with time. This is knownas relaxation or load decay. Creep and relaxation of concrete are discussed in more detail in 3.9.The effect of creep on the bending moments, de ections and stresses in bridge decks isdiscussed in 6.21 and 6.22.Concrete has a tensile strength that is of the order of 10 per centof its compressive strength. This tensile capability is essential to its strength in shear and to createbond with reinforcement. However, as concrete fails in tension in a brittle manner, andmay be cracked before loading due to internal stresses, this tensile strength is usuallyignored in calculating the bending strength of beams and slabs.

The steel generally used as reinforcement has a yield strength which lies between 400 and 500 MPa and a Young’s modulus that lies between 190,000 and 210,000 MPa. The typicalstress/strain curve is shown in Figure 3.3, and demonstrates a high degree of ductility;usually the strain at rupture is in excess of 12 per cent. The reinforcing bars aredeformed by rolling ribs onto the bar during manufacture. These deformations improve thebond of the bars with the concrete. Higher strength steel is available, but cannot in general beused effectively as it causes excessive cracking of the concrete. Undeformed mild steel, witha yield stress of 250 MPa is used occasionally, generally as starter bars where its greater ductilityallows it to be bent and re-bent.

The ratio of the Young’s moduli of steel to concrete is called the modular ratio. This ratio varies between about 6 for short-term loads, to 15 or over for sustained loads, and is important, as itdetermines how the steel and the concrete in a structural member share the applied loadsbetween them.

For instance, consider a concrete column reinforced with steel bars that represent 3 per cent ofthe total cross-section area. When subjected to an axial compressive load, in the short termthe stress in the steel will be six times higher than the stress in the concrete, and the steel wouldcarry some 16 per cent of the total load. If the load is maintained, the modular ratioincreases as the concrete creeps, and load is shed from the concrete onto the steel, whicheventually will carry nearly one-third of the load.

A reinforced concrete member has to satisfy both the Serviceability Limit State and the Ultimate Limit State. The SLS criteria include considerations of de ection, vibration and of control of thewidth of cracks (3.5), while the ULS is principally the consideration of the collapse of themember in bending, by yield of the steel or crushing of the concrete, or in shear. Althoughbeams are usually designed at the ULS, the beam actually spends its life at workingloads, and its performance at the SLS is just as important.

Most correctly designed beams are ‘under-reinforced’ in bending. This means that the steelreinforcement yields before the concrete crushes, and the mode of failure is ductile. Thesimple methods described below are aimed at demystifying the preliminary sizing of underreinforced concrete members at the ULS. Clearly the detailed design and veri cation of abeam involve much more extensive and careful calculations, at both the SLS and the ULS.

The strength of concrete in compression is measured by crushing standard samples. In the UK these samples are in the form of 150 mm cubes, while on the continent of Europe and in theUSA they are in the form of cylinders of 150 mm diameter and 300 mm height. In general, the stressat which a cylinder fails is approximately 85 per cent of the strength of a cube made of thesame material, although the difference is less the stronger the concrete [2].

As the strength ofconcrete increases with time, it is conventionally determined at an age of 28 days aftercasting,although the 90 day strength is also relevant. The degree to which the strengthincreases with time depends on the nature of the cement. For instance, concrete madewith blended cement including a proportion of pulverised fuel ash (PFA), increases instrength more than that of unblended Portland cement. This enhanced strength is signi? cantin establishing the true factor of safety of a structure.

The strength of a concrete test specimen depends on the rate at which it is loaded, ‘the faster the application of the load the higher the strength. Thus tests for concrete need to specify a rateof loading [3].Concrete used in bridges typically has a 28 day cube strength of between 40MPa and 60 MPa. Strengths up to 100 MPa have been used exceptionally while even strongerconcretes, up to 150 MPa and beyond, are still the subject of research involving theconstruction of some trial structures.

Concrete is not an elastic material. As the stress/strain curve does not have a straight portion, one cannot de ne a unique Young’s modulus forconcrete and different codes of practice use different de nitions. The most common de?nitions are the tangent modulus, which is the slope of the curve at one point and the secantmodulus which is the slope of the line that connects the origin to a point on the curve,Figure 3.2 (a) [4]. If the concrete is loaded by a pulse, it will exhibit a dynamic modulusthat is similar to the initial tangent modulus [5]. The Young’s modulus of concreteincreases with its strength, but is typically, as de ned by the British code, of the order of 34,000 MPa for oads applied for a short time on bridge quality concrete.

When concrete fails in compression it exhibits a degree of ductility. The strain of plain concrete at failure in compression is conventionally assumed to be of the order of 2-4 × 1in the UK code of practice), Figure 3.2 (b) [6]. This ductility in compression may be greatlyincreased by the provision of reinforcement speci cally designed for that purpose.When concrete is loaded in compression and the load is maintained, the instantaneous deformation isfollowed by a deferred deformation known as creep. This deferred deformation is frequently upto twice the magnitude of the instantaneous value, and in a large structure such as a bridgedeck, may take several years to complete.

The corollary of this behaviour is that if a xed deformation is imposed on a concrete member,the force required to maintain that deformation would reduce with time. This is knownas relaxation or load decay. Creep and relaxation of concrete are discussed in more detail in 3.9.The effect of creep on the bending moments, de ections and stresses in bridge decks isdiscussed in 6.21 and 6.22.Concrete has a tensile strength that is of the order of 10 per centof its compressive strength. This tensile capability is essential to its strength in shear and to createbond with reinforcement. However, as concrete fails in tension in a brittle manner, andmay be cracked before loading due to internal stresses, this tensile strength is usuallyignored in calculating the bending strength of beams and slabs.

The steel generally used as reinforcement has a yield strength which lies between 400 and 500 MPa and a Young’s modulus that lies between 190,000 and 210,000 MPa. The typicalstress/strain curve is shown in Figure 3.3, and demonstrates a high degree of ductility;usually the strain at rupture is in excess of 12 per cent. The reinforcing bars aredeformed by rolling ribs onto the bar during manufacture. These deformations improve thebond of the bars with the concrete. Higher strength steel is available, but cannot in general beused effectively as it causes excessive cracking of the concrete. Undeformed mild steel, witha yield stress of 250 MPa is used occasionally, generally as starter bars where its greater ductilityallows it to be bent and re-bent.

The ratio of the Young’s moduli of steel to concrete is called the modular ratio. This ratio varies between about 6 for short-term loads, to 15 or over for sustained loads, and is important, as itdetermines how the steel and the concrete in a structural member share the applied loadsbetween them.

For instance, consider a concrete column reinforced with steel bars that represent 3 per cent ofthe total cross-section area. When subjected to an axial compressive load, in the short termthe stress in the steel will be six times higher than the stress in the concrete, and the steel wouldcarry some 16 per cent of the total load. If the load is maintained, the modular ratioincreases as the concrete creeps, and load is shed from the concrete onto the steel, whicheventually will carry nearly one-third of the load.

A reinforced concrete member has to satisfy both the Serviceability Limit State and the Ultimate Limit State. The SLS criteria include considerations of de ection, vibration and of control of thewidth of cracks (3.5), while the ULS is principally the consideration of the collapse of themember in bending, by yield of the steel or crushing of the concrete, or in shear. Althoughbeams are usually designed at the ULS, the beam actually spends its life at workingloads, and its performance at the SLS is just as important.

Most correctly designed beams are ‘under-reinforced’ in bending. This means that the steelreinforcement yields before the concrete crushes, and the mode of failure is ductile. Thesimple methods described below are aimed at demystifying the preliminary sizing of underreinforced concrete members at the ULS. Clearly the detailed design and veri cation of abeam involve much more extensive and careful calculations, at both the SLS and the ULS.

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