The flanges of an I-girder are 30 cm wide by 2.5 cm thick and the web is 60 cm deep by 1.25 cm thick. At a particular section the sagging bending moment is 500 kNm and the shearing force is 500 kN. Consider a point in the section at the top of the web and calculate for this point; (i) the

Question

The flanges of an I-girder are 30 cm wide by 2.5 cm thick and the web is 60 cm deep by 1.25 cm thick. At a particular section the sagging bending moment is 500 kNm and the shearing force is 500 kN. Consider a point in the section at the top of the web and calculate for this point; (i) the longitudinal stress, (ii) the shearing stress, (iii) the principal stresses. (Cambridge)

Accepted Answer

First calculate the second moment of area about the neutral axis; the second moment of area of the web is

[latex]\frac{1}{2}(0.0125)(0.6)^{3} = 0.225 imes 10^{-3} m^{4}[/latex]

The second moment of area of each flange is

[latex](0.3) (0.025) (0.3125)^{2} =0.733 imes 10^{-3} m^{4}[/latex]

The total second moment of area is then

[latex]I_{x}=[0.225 + 2(0.733)]10^{-3} = 1.691 imes 10^{-3} m^{4}[/latex]

Next, for a point at the top of the web,

[latex]A \bar{y} = (0.3 imes 0.025) (0.3125) = 2.34 imes 10^{-3} m^{4}[/latex]

Then, for this point, with M = 500 kNm we have

[latex]\sigma =\frac{My}{I_{x}} =\frac{(500 imes 10^{3}) (0.3)}{1.691 imes 10^{-3}}=88.6 MN/m^{2} (compressive)[/latex]

[latex] au =\frac{FA\bar{y}}{I_{x}t} =\frac{(500 imes 10^{3})(2.342.34 imes 10^{-3})}{(1.691 imes 10^{-3})(0.0125)}=55.3 MN \cdot m^{2}[/latex]

The principal stresses are then

[latex]\frac{1}{2}\sigma \pm \left[\frac{1}{4}\sigma ^{2}+ au ^{2} ight] ^{\frac{1}{2} }=(-44.3 \pm 70.9) MN/m^{2}[/latex]

[latex]= 26.6 and -115.2 MN/m^{2}[/latex]

It should be noticed that the greater principal stress is about [latex]30\%[/latex] greater than the longitudinal stress. At the top of the flange the longitudinal stress is -96 MN/m², so the greatest principal stress at the top of the web is [latex]20\%[/latex] greater than the maximum longitudinal stress.

Question 10.2: The flanges of an I-girder are 30 cm wide by 2.5 cm thick an...

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