Normative europee sul reinforced concrete

Steel Material properties The data obtained from the tension test are generally plotted

as a stress-strain diagram. The initial linear portion of the

curve is the elastic region within which Hooke's law is obeyed.

In this region stress and strain are proportional. The constant

of proportionality is called the Elastic Modulus or Young's

Modulus (E). The relationship between stress and strain in this

region is given by Equation:

=

The Elastic Modulus is also the slope of the curve in this

region, E of aluminium is 1/3 of the steel. Point 2 is the elastic

limit, defined as the greatest stress that the metal can

withstand without experiencing a permanent strain when the

load is removed. The determination of the elastic limit is quite tedious, not at all routine, and dependent on

the sensitivity of the strain-measuring instrument. For these reasons it is often replaced by the proportional

limit, point 1. The proportional limit is the stress at which the stress-strain curve deviates from linearity. For

engineering purposes the limit of usable elastic behaviour is described by the yield strength, point 3. The

yield strength is defined as the stress which will produce a small amount of permanent deformation, generally

equal to a strain of.0.002. Plastic deformation begins when the elastic limit is exceeded. As the plastic

deformation of the specimen increases, the metal becomes stronger (strain hardening) so that the load

required extending the specimen increases with further straining. Eventually the load reaches a maximum

value. The maximum load divided by the original area of the specimen is the ultimate tensile strength, point

4. For a ductile metal the diameter of the specimen begins to

decrease rapidly beyond maximum load, so that the load required

continuing deformation drops off until the specimen fractures, point

5.

Comparison between steel, and carbon and glass FRP. Make notice

that most of the time steel can have a higher elastic modulus of

composite materials. But the strength is usually higher for composite

material due to the fact that high strength textile are associated with

outstanding tensile properties. But what we cannot be reaching with

composite materials is the ductility. So this capacity of elongate

significantly and maintaining at the same time an high value of

strength. So composite material will always fail suddenly in a brittle

way.

Design should be based on the nominal cross-section area of the

reinforcement and the design values derived from the characteristic

values given in 3.2.2. For normal design, either of the following

assumptions may be made:

A) an inclined top branch with a strain limit of and a maximum

stress of / at , where k = . This models will keep

( / )

into account the presence of strain hardening, so the increase of the

gain of strength beyond the elastic limit.

B) a horizontal top branch without the need to check the strain limit. For our calculation that we are

proposing to refer to the elastic perfectly plastic steel model which beyond the design value of the yielding

strength that is just obtained as a ratio between the characteristic value and the partial safety coefficient we

will rely on a constant strength, maintain its theoretical or ideally up to infinite, according to what Eurocode

say, steel will never fail due to tension. but we will limit that to a certain value that I would suggest to be 1%.

So we are not relying on this infinite deformation ability of the steel according to you Eurocode. beyond the

1% of deformation, the compatibility of strains between steel and concrete can no longer be respected. And

this is one of the governing hypothesis we are following. So limiting the elastic of effective plastic behavior

to 1% would be a very safe, yet effective way to design.

Concrete Material properties

For the stress-strain diagram of concrete it is possible to adopt appropriate models representative of the real

behaviour of the material, suggested by the Eurocode and ntc, defined on the basis of the design resistance

and the ultimate design

deformation . (a) parabola-

rectangle; (b) triangle-rectangle

behaviour; (c) rectangle (stress block).

The difference between the three is the

level of approximation. The simplest of

the three is the stress block. In

particular, for resistance classes equal

to or lower than C50/60 the following can be set: and

= 0.2%, = 0.175%,, = 0.35% =

2 3 4

If we are going to use the stress block we are not relying on the elastic behavior of the concrete, but

0.07%.

you are pretending that the concrete behave maintaining a constant value of strength till these value of the

deformation of 0.35% will be attained.

The semi-probabilistic limit state method - Bending

The design of the beam cross section is the same probabilistic, limited state design. The assumptions of such

approach are:

Cross-beam sections remain plane after deflection up to failure

• Perfect bond exists between concrete and steel reinforcement, this is a consequence of hypothesis

• number one, and that is the reason for me enforcing to limit steel strain to 1%. Because beyond that,

cracks in concrete would be so high that the compatibility could no longer be assumed.

Concrete does not react in tension.

•

The objective is the definition of the neutral axis. What is going to state above the neutral axis will be in

compression while below neutral axis will staying in tension, on that side the concrete will not be working,

according to our assumption which means it will not produce any capacity. In an ideal case of a simple

supported beam, tensile stress and crack are developing only on the along the bottom of the beam therefore

longitudinal rebars are placed close to the

bottom side the beam. There is no inversion in

bending moment, so there is no need of steel

bars on the upper side. According to the

hypostasis that the beam is going to bent, the

curvature is proportional to the bending

moment, and the curvature will make any

cross-section of the beam partialized, which

means that the upper side will be compressed,

the bottom side will be in tension. The neutral

axis position would no longer pass through the

centre of mass of the cross-section and this is a matter of the fact that the section is not homogeneously

made of the same material, but it has to material, a composed cross-section made of concrete and steel. By

hypothesis the actual deformation of the cross-section at each depth with respect to the upper edge of the

cross-section would be linear so, We assume the linearity of the cross-section that entails a linear axial strain

profiles for the beam. Besides that, since perfect bonding was assumed between the steel and concrete,

along within this strain profile, I can reading both the deformation of the steel and that of the surrounding

concrete. Accordingly, this linear strain profile allow us to set a geometrical relationship between the

maximum compressive strain of concrete and the tension strain of rebar. According to the hypothesis

introduced it and the properties of the material, the failure of the cross section can occur ideally in two ways:

1. Compression failure of concrete: Large amount of reinforcement is used. Concrete fails by crushing

when strains become so large. Failure is sudden, it occurs with no warning (Brittle Failure), > 3.5%

2. Tension failure of steel: Moderate amount of reinforcement is used. Steel yields suddenly and

stretches a large amount, tension cracks become visible and widen and propagate upward (Ductile

Failure), > 1%

within this two limit failure

mode, there are an infinite

number of intermediate

failure configuration. A strain

profile of the cross-section

that follows the first dashed

line is a non failure

configuration because

indeed neither was

obtained at the top compressive side neither 1% was reached for the bottom tension side. So this strain

configuration that would certainly correspond to certain loading action, is not the failure of configuration.

There are some configuration that are incompatible with the material properties, like second the dashed line

in such case Still is still acceptable within the limits, but on the compressive side, we have a theoretical strain

that goes beyond the concrete strain limits. We cannot design according to, to that configuration of strain to

letds to non capacity in the beam. The red line is the limited configuration, limit because it passes through

which identified the the limit condition where a compressive failure of concrete was obtained.

Failure regions – Pure bending Among all the possible strain

configuration, Some very similar cases

have been collected within regions or

domains. So those domains that were

referred to with the numbers one two,

three, four and five collects together

failure configuration that are

associated to the same physical

meaning. For a generic strain

Region 1,

configuration belonging to

concrete on the upper side is in tension,

in contrast to our assumption. Besides

this the line is passing through point B

That means simultaneously reaching of

the limit strain in the tension steel. So region 1 means that the whole cross-section would be subjected to

region 2

tension. In the upper side failure is not occurring, but for sure all of the shaft of segments passing

through point B belonging to Region two will be associated with the tensile failure at the bottom side of the

Region 3

steel rebars. In the compressed concrete reached the limit strain and the tensile rebars passed the

yielding point. This is a good region for designers because both materials exploited their maximum capacity,

region 4a

we are plateauing at the maximum strength that the steel can provide. The is not optimal because

the failure of concrete was attained, but steel is neither yielded so is still in its elastic range, so we're not

optimizing at all the steel capacity on the tension side. What makes our failure configuration to belong in a

region respect to another to a region Is the properties of the materials and the geometry of cross section,

including the sizing of the cross section and sizes of the bars.

Region 3

A generic failure configuration belonging to that region was highlighted, to show how the materials behavior

can be associated to this strain profile in the cross section. The generic failure configuration passes through

the zero in that location, which means that this is the zero strain, So zero stress location, So location in which

the neutral axis will pass through here, from this point