To find the maximum deflection of a beam subjected to a uniformly distributed load, we use the formula for deflection of a simply supported beam:

$\delta_{\text{max}} = \frac{5qL^4}{384EI}$

Where:

• $q = 14,000 \, \text{N/m}$ (uniform load),
• $L = 6 \, \text{m} = 6000 \, \text{mm}$ (beam length),
• $E = 200 \, \text{GPa} = 200 \times 10^9 \, \text{Pa}$ (modulus of elasticity),
• $I = 99 \times 10^6 \, \text{mm}^4$ (moment of inertia).

Step-by-step solution:

1. Convert the given values into consistent units:

   • Convert $q$: $14,000 \, \text{N/m} = 14 \, \text{N/mm}$,
   • $L$ is already $6000 \, \text{mm}$,
   • $E$: $200 \times 10^9 \, \text{Pa}$,
   • $I = 99 \times 10^6 \, \text{mm}^4$.

2. Substitute the values into the formula: $\delta_{\text{max}} = \frac{5 \cdot 14 \cdot (6000)^4}{384 \cdot (200 \times 10^9) \cdot (99 \times 10^6)}$

3. Perform the calculations:

   • Calculate $L^4 = (6000)^4 = 1.296 \times 10^{15} \, \text{mm}^4$,
   • Calculate the numerator: $5 \cdot 14 \cdot 1.296 \times 10^{15} = 9.072 \times 10^{16} \, \text{N} \cdot \text{mm}^4$,
   • Calculate the denominator: $384 \cdot 200 \cdot 99 \cdot 10^9 \cdot 10^6 = 7.6032 \times 10^{18} \, \text{N} \cdot \text{mm}^2$,
   • Divide numerator by denominator: $\delta_{\text{max}} = \frac{9.072 \times 10^{16}}{7.6032 \times 10^{18}} \approx 0.01193 \, \text{mm}.$

4. Convert to meters: $\delta_{\text{max}} = 0.01193 \, \text{mm} = 0.00001193 \, \text{m}.$

Final Answer:

The maximum deflection of the beam is approximately: $\delta_{\text{max}} \approx 0.00001193 \, \text{m}.$

Would you like additional details or assistance with related concepts?

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5 Related Questions:

1. How does the length of a beam affect its maximum deflection under a uniform load?
2. What changes in deflection occur if the modulus of elasticity $E$ is halved?
3. How does the moment of inertia $I$ influence the beam's stiffness?
4. Can this formula be used for a cantilever beam? Why or why not?
5. What are the real-world factors that might affect the accuracy of this deflection calculation?

Tip: Always check units carefully when performing structural calculations—convert everything to a consistent system!