We know concrete is very weak in tension, this is only a fraction of compressive strength of it. Near modulus of rupture, concrete is stressed only by 500 psi and if modular ratio n=8, the stress in embedded steel is 500X8=4000psi. The usual yield strength of reinforcing steel (40~75) ksi; using heat treatment and changing constituents, its strength can be increased even more. So 4 ksi is only a fraction of its yield strength.
Thus reinforcement is not effective before concrete is cracked. Where modular ratio, n =Es/Ec. In common practice with usual materials n=8.
In properly designed flexural member like beam, cracks due to flexure are fine and often defined as hairline cracks which usually not allow corrosion to reinforcement, if allow, it will be very little.
When loads are increased gradually exceeding cracking load (exceeds modulus of rupture), number of cracks is also increased; the width of cracks is also increased which reached around 0.016 in (typical width) at stress corresponding service load. A further increase in load accompanied by further increase in crack width, though number remains more or less equal.
As discussed in previous post formation and progression of crack is highly variable as controlled by many factors. Considering this difficulties, cracks widths are now determined based on observations confirmed by test. The equations developed to predict crack width.
Now we will discuss about two equations for determining crack width. It should keep in mind that the equations were produced based on analyzing experimental data. Analysis means application of statistical approach which offers us 90% probability of matching actual crack width.
Let’s explain more easily; about 90% cracks formed in p; i have width below the value calculated by these equations. However, individual cracks may have width exceeding twice the width calculated.
ACI code considered two expressions as prominent to develop its crack control provisions; these are expression developed by
• Lutz and Gergely in 1968
• Frosch in 1999
Two expressions determine maximum width of cracks at tension face of beam.
The respective equations are:
w =maximum crack width, thousandth inches,
fs;= steel stress corresponding to load for which crack width have to be measured, in ksi
Es= modulus of elasticity of steel, in ksi.
The parameters used in equation dc, β, A and S are shown in following figure and are explained as follows:
dc =thickness of concrete measured from tension face to center of rebar. In determining center of rebar, outermost layer of rebar is considered; not considered center of all rebar.
h1 =Distance between neutral axis and centroid of steel
h2 =Distance between neutral axis and tension face
A= Area of concrete surrounding each bar. This is measures by dividing total effective concrete tension area surrounding reinforcement having identical centroid by total number of bars (in2)
S= maximum spacing of bar (in)
Both the equations (1) and (2) only applicable for beams that are reinforced with deformed bars. The β factor is added to consider the effect of distance from the tension face to neutral axis; crack width is increased with increase in this distance.
All the factors that have significant influence on crack width are considered in these equations namely
• Steel stress
• Concrete cover
• Distribution of reinforcing bar in tension zone of concrete