Question 7.6: The beam shown is subjected to a distributed load. For the c......

First we need to consult our FBD and find the reactions at the supports.

We are interested in the cross section at x = 0.6 m. We know that the average shear stress depends on the internal shear force in the beam at the point of interest, so we need to calculate the shear V(x = 0.6 m). To do this, we will make an imaginary cut at x = 0.6 m:

In this 0.6-m-long span, the distributed load has a maximum intensity of \left(130 \frac{\mathrm{kN}}{\mathrm{m}}\right) \frac{0.6 \mathrm{~m}}{1.4 \mathrm{~m}}=55.7 \frac{\mathrm{kN}}{\mathrm{m}} , so the equivalent concentrated load acting on the 0.6-m-long segment is the area under this load: \frac{1}{2}\left(55.7 \frac{\mathrm{kN}}{\mathrm{m}}\right)(0.6 \mathrm{~m})=16.7 \mathrm{~kN} .
Equilibrium of our 0.6 m segment:

The second moment of area for the cross section is

I=\frac{1}{12} b h^3=\frac{1}{12}(0.04)(0.06)^3=7.2 \times 10^{-7} \mathrm{~m}^4 .

At the height of the centroid or neutral axis, Q^{\prime} is the first moment of area of the area above (or below) the centroid:

Q^{\prime}=z^{\prime} A^{\prime}=0.015(0.03 \times 0.04)=1.8 \times 10^{-5} \mathrm{~m}^3 .

Then the shear stress at the centroid is

\sigma_{x z}=\frac{V Q^{\prime}}{I b}=\frac{(31.8 \mathrm{~kN})\left(1.8 \times 10^{-5} \mathrm{~m}^3\right)}{\left(7.2 \times 10^{-7} \mathrm{~m}^4\right)(0.04 \mathrm{~m})}=19.9 \mathrm{~MPa} .

At z = 0.02 m above the neutral axis

Q^{\prime}=z^{\prime} A^{\prime}=0.02(0.02 \times 0.04)=1.0 \times 10^{-5} \mathrm{~m}^3 .

This is the only change from the calculation at z = 0, and the shear stress at this point is 11.0 MPa.