Calculation of the service-load steel stress in a rectangular RC section

 

Below is calculation formula regarding calculation of steel stress in a rectangular section.(cracked section)

 

Compression            =  Tension

Concrete + As' force =  As force

\[\frac{1}{2}xbf_{c}+f_{s}'A_{s}'-f_{c}'A_{s}'=f_{s}A_{s}\] \[f_{c}=\varepsilon _{c}E_{c}\] \[f_{s}'=\varepsilon _{s}'E_{s}\] \[f_{s}=\varepsilon_{s}E_{s}\] \[\varepsilon _{s}=\varepsilon _{c} \frac{d-x}{x}\]\[\varepsilon _{s}'=\varepsilon _{c}\frac{x-d'}{x}\] \[\frac{1}{2}xb\varepsilon_{c}E_{c}+\varepsilon_{s}'E_{s}A_{s}'-\varepsilon_{c}'E_{c}A_{s}'=\varepsilon_{s}E_{s}A_{s}\] \[\frac{1}{2}xb\varepsilon_{c}E_{c}+\varepsilon_{c}\frac{x-d'}{x}E_{s}A_{s}'-\varepsilon_{c}\frac{x-d'}{x}E_{c}A_{s}'=\varepsilon_{c}\frac{d-x}{x}E_{s}A_{s}\] \[\frac{1}{2}xbE_{c}+\frac{x-d'}{x}E_{s}A_{s}'-\frac{x-d'}{x}E_{c}A_{s}'=\frac{d-x}{x}E_{s}A_{s}\] \[n=\frac{E_{c}}{E_{c}}\]

 

 

  Solve next equation (using EXCEL goal seek..)

\[\frac{1}{2}x^{2}b+(n-1)(x-d')A_{s}=n(d-x)A_{s}\]

 

 

  Concrete stress & Steel stress \[f_{c}=\frac{M_{s}}{\frac{1}{2}xb(d-\frac{x}{3})+(n-1)A_{s}'(\frac{x-d'}{x})(d-d')}\] \[f_{s}=nf_{c}\frac{d-x}{x}\]

 

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