Beam Deflection and Shearing Force (Civil Engineering) Solution

Problem Overview

We evaluate the shear force distribution $V(x) = \frac{dM}{dx}$ on a structural highway bridge girder given the bending moment equation:

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Step-by-Step Mathematical Solution

Step 1: Pre-Process Bending Moment Expression

Express the radical denominator term using rational and negative exponents:

Step 2: Differentiate to Obtain Shear Force $V(x)$

Differentiate term-by-term using the Power Rule:

Step 3: Post-Process and Evaluate at Specified Coordinates

Convert the negative fractional exponent back into radical form:

1. At Support Coordinate $x = 0\text{ m}$:
   $$V(0) = 48 + \frac{8}{\sqrt{(0 + 4)^3}} - 4(0) = 48 + \frac{8}{\sqrt{64}} = 48 + \frac{8}{8} = 49\text{ kN}$$
2. At Coordinate $x = 5\text{ m}$:
   $$V(5) = 48 + \frac{8}{\sqrt{(5 + 4)^3}} - 4(5) = 48 + \frac{8}{\sqrt{9^3}} - 20 = 28 + \frac{8}{27} \approx 28.296\text{ kN}$$

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Engineering Conclusion

By algebraically converting the root term into a negative fractional exponent prior to differentiation, structural engineers directly calculate shear force diagrams without tedious numerical approximations. The girder experiences maximum shear ($49\text{ kN}$) at the left abutment support ($x = 0$), guiding the required thickness and bolt spacing of the web stiffeners.