Given:

• Weight of rail carrier (W) = 200 kN = 200,000 N

• Velocity (v) = 5 km/h = $\frac{5\times 1000}{3600}=1.389\text{m/s}$

• Number of springs (n) = 4

• Spring index (C) = $\frac{D}{d}=5$

• Deflection per spring (δ) = 220 mm = 0.22 m

• Maximum allowable shear stress (τ) = 400 MPa = $400\times {10}^6\text{Pa}$

• Modulus of rigidity (G) = 80,000 MPa = $80\times {10}^9\text{Pa}$

• Ends of the spring are squared and ground.

Step 1: Calculate the Kinetic Energy Absorbed by the Springs

The kinetic energy of the rail carrier is absorbed by the 4 springs:

$$\mathrm{<span\; style="background-color:\; white;">K</span>}\mathrm{<span\; style="background-color:\; white;">E</span>}<span\; style="background-color:\; white;">=</span>\frac{\mathrm{<span\; style="background-color:\; white;">1</span>}}{\mathrm{<span\; style="background-color:\; white;">2</span>}}\mathrm{<span\; style="background-color:\; white;">m</span>}{\mathrm{<span\; style="background-color:\; white;">v</span>}}^{\mathrm{<span\; style="background-color:\; white;">2</span>}}$$

First, find the mass ($m$):

$$\mathrm{<span\; style="background-color:\; white;">m</span>}<span\; style="background-color:\; white;">=</span>\frac{\mathrm{<span\; style="background-color:\; white;">W</span>}}{\mathrm{<span\; style="background-color:\; white;">g</span>}}<span\; style="background-color:\; white;">=</span>\frac{\mathrm{<span\; style="background-color:\; white;">200</span>}<span\; style="background-color:\; white;">,</span>\mathrm{<span\; style="background-color:\; white;">000</span>}}{\mathrm{<span\; style="background-color:\; white;">9.81</span>}}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">20</span>}<span\; style="background-color:\; white;">,</span>\mathrm{<span\; style="background-color:\; white;">387.36</span>}\text{<span style="background-color: white;">\hspace{0.17em}</span>}\text{<span style="background-color: white;">kg</span>}$$

Now, calculate $KE$:

$$\mathrm{<span\; style="background-color:\; white;">K</span>}\mathrm{<span\; style="background-color:\; white;">E</span>}<span\; style="background-color:\; white;">=</span>\frac{\mathrm{<span\; style="background-color:\; white;">1</span>}}{\mathrm{<span\; style="background-color:\; white;">2</span>}}<span\; style="background-color:\; white;">\times </span>\mathrm{<span\; style="background-color:\; white;">20</span>}<span\; style="background-color:\; white;">,</span>\mathrm{<span\; style="background-color:\; white;">387.36</span>}<span\; style="background-color:\; white;">\times </span><span\; style="background-color:\; white;">(</span>\mathrm{<span\; style="background-color:\; white;">1.389</span>}{<span\; style="background-color:\; white;">)</span>}^{\mathrm{<span\; style="background-color:\; white;">2</span>}}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">19</span>}<span\; style="background-color:\; white;">,</span>\mathrm{<span\; style="background-color:\; white;">684.5</span>}\text{<span style="background-color: white;">\hspace{0.17em}</span>}\text{<span style="background-color: white;">J</span>}$$

This energy is absorbed by 4 springs:

$$\mathrm{<span\; style="background-color:\; white;">K</span>}{\mathrm{<span\; style="background-color:\; white;">E</span>}}_{\text{<span style="background-color: white;">per spring</span>}}<span\; style="background-color:\; white;">=</span>\frac{\mathrm{<span\; style="background-color:\; white;">19</span>}<span\; style="background-color:\; white;">,</span>\mathrm{<span\; style="background-color:\; white;">684.5</span>}}{\mathrm{<span\; style="background-color:\; white;">4</span>}}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">4</span>}<span\; style="background-color:\; white;">,</span>\mathrm{<span\; style="background-color:\; white;">921.13</span>}\text{<span style="background-color: white;">\hspace{0.17em}</span>}\text{<span style="background-color: white;">J</span>}$$

Step 2: Relate Energy to Spring Parameters

The energy stored in a spring is:

$$\mathrm{<span\; style="background-color:\; white;">E</span>}<span\; style="background-color:\; white;">=</span>\frac{\mathrm{<span\; style="background-color:\; white;">1</span>}}{\mathrm{<span\; style="background-color:\; white;">2</span>}}\mathrm{<span\; style="background-color:\; white;">P</span>}\mathrm{<span\; style="background-color:\; white;">\delta </span>}$$

Where $P$ is the load on the spring and $\delta$ is the deflection.

$$\mathrm{<span\; style="background-color:\; white;">4</span>}<span\; style="background-color:\; white;">,</span>\mathrm{<span\; style="background-color:\; white;">921.13</span>}<span\; style="background-color:\; white;">=</span>\frac{\mathrm{<span\; style="background-color:\; white;">1</span>}}{\mathrm{<span\; style="background-color:\; white;">2</span>}}<span\; style="background-color:\; white;">\times </span>\mathrm{<span\; style="background-color:\; white;">P</span>}<span\; style="background-color:\; white;">\times </span>\mathrm{<span\; style="background-color:\; white;">0.22</span>}$$$$\mathrm{<span\; style="background-color:\; white;">P</span>}<span\; style="background-color:\; white;">=</span>\frac{\mathrm{<span\; style="background-color:\; white;">4</span>}<span\; style="background-color:\; white;">,</span>\mathrm{<span\; style="background-color:\; white;">921.13</span>}<span\; style="background-color:\; white;">\times </span>\mathrm{<span\; style="background-color:\; white;">2</span>}}{\mathrm{<span\; style="background-color:\; white;">0.22</span>}}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">44</span>}<span\; style="background-color:\; white;">,</span>\mathrm{<span\; style="background-color:\; white;">737.5</span>}\text{<span style="background-color: white;">\hspace{0.17em}</span>}\text{<span style="background-color: white;">N</span>}$$

Maximum load on each spring ($P$) = 44.74 kN

Step 3: Find Wire Diameter ($d$)

The shear stress in the spring is given by:

$$\mathrm{<span\; style="background-color:\; white;">\tau </span>}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">K</span>}\frac{\mathrm{<span\; style="background-color:\; white;">8</span>}\mathrm{<span\; style="background-color:\; white;">P</span>}\mathrm{<span\; style="background-color:\; white;">C</span>}}{\mathrm{<span\; style="background-color:\; white;">\pi </span>}{\mathrm{<span\; style="background-color:\; white;">d</span>}}^{\mathrm{<span\; style="background-color:\; white;">2</span>}}}$$

Where $K$ is Wahl's correction factor:

$$\mathrm{<span\; style="background-color:\; white;">K</span>}<span\; style="background-color:\; white;">=</span>\frac{\mathrm{<span\; style="background-color:\; white;">4</span>}\mathrm{<span\; style="background-color:\; white;">C</span>}<span\; style="background-color:\; white;">-</span>\mathrm{<span\; style="background-color:\; white;">1</span>}}{\mathrm{<span\; style="background-color:\; white;">4</span>}\mathrm{<span\; style="background-color:\; white;">C</span>}<span\; style="background-color:\; white;">-</span>\mathrm{<span\; style="background-color:\; white;">4</span>}}<span\; style="background-color:\; white;">+</span>\frac{\mathrm{<span\; style="background-color:\; white;">0.615</span>}}{\mathrm{<span\; style="background-color:\; white;">C</span>}}<span\; style="background-color:\; white;">=</span>\frac{\mathrm{<span\; style="background-color:\; white;">4</span>}<span\; style="background-color:\; white;">\times </span>\mathrm{<span\; style="background-color:\; white;">5</span>}<span\; style="background-color:\; white;">-</span>\mathrm{<span\; style="background-color:\; white;">1</span>}}{\mathrm{<span\; style="background-color:\; white;">4</span>}<span\; style="background-color:\; white;">\times </span>\mathrm{<span\; style="background-color:\; white;">5</span>}<span\; style="background-color:\; white;">-</span>\mathrm{<span\; style="background-color:\; white;">4</span>}}<span\; style="background-color:\; white;">+</span>\frac{\mathrm{<span\; style="background-color:\; white;">0.615</span>}}{\mathrm{<span\; style="background-color:\; white;">5</span>}}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">1.3105</span>}$$

Now, substitute $\tau =400\times {10}^6$, $P=44,737.5$, $C=5$:

$$\mathrm{<span\; style="background-color:\; white;">400</span>}<span\; style="background-color:\; white;">\times </span>{\mathrm{<span\; style="background-color:\; white;">10</span>}}^{\mathrm{<span\; style="background-color:\; white;">6</span>}}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">1.3105</span>}<span\; style="background-color:\; white;">\times </span>\frac{\mathrm{<span\; style="background-color:\; white;">8</span>}<span\; style="background-color:\; white;">\times </span>\mathrm{<span\; style="background-color:\; white;">44</span>}<span\; style="background-color:\; white;">,</span>\mathrm{<span\; style="background-color:\; white;">737.5</span>}<span\; style="background-color:\; white;">\times </span>\mathrm{<span\; style="background-color:\; white;">5</span>}}{\mathrm{<span\; style="background-color:\; white;">\pi </span>}{\mathrm{<span\; style="background-color:\; white;">d</span>}}^{\mathrm{<span\; style="background-color:\; white;">2</span>}}}$$$${\mathrm{<span\; style="background-color:\; white;">d</span>}}^{\mathrm{<span\; style="background-color:\; white;">2</span>}}<span\; style="background-color:\; white;">=</span>\frac{\mathrm{<span\; style="background-color:\; white;">1.3105</span>}<span\; style="background-color:\; white;">\times </span>\mathrm{<span\; style="background-color:\; white;">8</span>}<span\; style="background-color:\; white;">\times </span>\mathrm{<span\; style="background-color:\; white;">44</span>}<span\; style="background-color:\; white;">,</span>\mathrm{<span\; style="background-color:\; white;">737.5</span>}<span\; style="background-color:\; white;">\times </span>\mathrm{<span\; style="background-color:\; white;">5</span>}}{\mathrm{<span\; style="background-color:\; white;">\pi </span>}<span\; style="background-color:\; white;">\times </span>\mathrm{<span\; style="background-color:\; white;">400</span>}<span\; style="background-color:\; white;">\times </span>{\mathrm{<span\; style="background-color:\; white;">10</span>}}^{\mathrm{<span\; style="background-color:\; white;">6</span>}}}$$$${\mathrm{<span\; style="background-color:\; white;">d</span>}}^{\mathrm{<span\; style="background-color:\; white;">2</span>}}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">0.001865</span>}$$$$\mathrm{<span\; style="background-color:\; white;">d</span>}<span\; style="background-color:\; white;">=</span>\sqrt{\mathrm{<span\; style="background-color:\; white;">0.001865</span>}}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">0.0432</span>}\text{<span style="background-color: white;">\hspace{0.17em}</span>}\text{<span style="background-color: white;">m</span>}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">43.2</span>}\text{<span style="background-color: white;">\hspace{0.17em}</span>}\text{<span style="background-color: white;">mm</span>}$$

Wire diameter ($d$) = 43.2 mm

Step 4: Find Coil Diameter ($D$)

Given $C=\frac{D}{d}=5$:

$$\mathrm{<span\; style="background-color:\; white;">D</span>}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">5</span>}\mathrm{<span\; style="background-color:\; white;">d</span>}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">5</span>}<span\; style="background-color:\; white;">\times </span>\mathrm{<span\; style="background-color:\; white;">43.2</span>}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">216</span>}\text{<span style="background-color: white;">\hspace{0.17em}</span>}\text{<span style="background-color: white;">mm</span>}$$

Coil diameter ($D$) = 216 mm

Step 5: Find Number of Turns ($N$)

The deflection formula for a spring is:

$$\mathrm{<span\; style="background-color:\; white;">\delta </span>}<span\; style="background-color:\; white;">=</span>\frac{\mathrm{<span\; style="background-color:\; white;">8</span>}\mathrm{<span\; style="background-color:\; white;">P</span>}{\mathrm{<span\; style="background-color:\; white;">D</span>}}^{\mathrm{<span\; style="background-color:\; white;">3</span>}}\mathrm{<span\; style="background-color:\; white;">N</span>}}{\mathrm{<span\; style="background-color:\; white;">G</span>}{\mathrm{<span\; style="background-color:\; white;">d</span>}}^{\mathrm{<span\; style="background-color:\; white;">4</span>}}}$$

Given $\delta =0.22\text{m}$, $P=44,737.5\text{N}$, $D=0.216\text{m}$, $G=80\times {10}^9\text{Pa}$, $d=0.0432\text{m}$:

$$\mathrm{<span\; style="background-color:\; white;">0.22</span>}<span\; style="background-color:\; white;">=</span>\frac{\mathrm{<span\; style="background-color:\; white;">8</span>}<span\; style="background-color:\; white;">\times </span>\mathrm{<span\; style="background-color:\; white;">44</span>}<span\; style="background-color:\; white;">,</span>\mathrm{<span\; style="background-color:\; white;">737.5</span>}<span\; style="background-color:\; white;">\times </span><span\; style="background-color:\; white;">(</span>\mathrm{<span\; style="background-color:\; white;">0.216</span>}{<span\; style="background-color:\; white;">)</span>}^{\mathrm{<span\; style="background-color:\; white;">3</span>}}\mathrm{<span\; style="background-color:\; white;">N</span>}}{\mathrm{<span\; style="background-color:\; white;">80</span>}<span\; style="background-color:\; white;">\times </span>{\mathrm{<span\; style="background-color:\; white;">10</span>}}^{\mathrm{<span\; style="background-color:\; white;">9</span>}}<span\; style="background-color:\; white;">\times </span><span\; style="background-color:\; white;">(</span>\mathrm{<span\; style="background-color:\; white;">0.0432</span>}{<span\; style="background-color:\; white;">)</span>}^{\mathrm{<span\; style="background-color:\; white;">4</span>}}}$$$$\mathrm{<span\; style="background-color:\; white;">0.22</span>}<span\; style="background-color:\; white;">=</span>\frac{\mathrm{<span\; style="background-color:\; white;">8</span>}<span\; style="background-color:\; white;">\times </span>\mathrm{<span\; style="background-color:\; white;">44</span>}<span\; style="background-color:\; white;">,</span>\mathrm{<span\; style="background-color:\; white;">737.5</span>}<span\; style="background-color:\; white;">\times </span>\mathrm{<span\; style="background-color:\; white;">0.01008</span>}\mathrm{<span\; style="background-color:\; white;">N</span>}}{\mathrm{<span\; style="background-color:\; white;">80</span>}<span\; style="background-color:\; white;">\times </span>{\mathrm{<span\; style="background-color:\; white;">10</span>}}^{\mathrm{<span\; style="background-color:\; white;">9</span>}}<span\; style="background-color:\; white;">\times </span>\mathrm{<span\; style="background-color:\; white;">3.49</span>}<span\; style="background-color:\; white;">\times </span>{\mathrm{<span\; style="background-color:\; white;">10</span>}}^{<span\; style="background-color:\; white;">-</span>\mathrm{<span\; style="background-color:\; white;">6</span>}}}$$$$\mathrm{<span\; style="background-color:\; white;">0.22</span>}<span\; style="background-color:\; white;">=</span>\frac{\mathrm{<span\; style="background-color:\; white;">3</span>}<span\; style="background-color:\; white;">,</span>\mathrm{<span\; style="background-color:\; white;">612.6</span>}\mathrm{<span\; style="background-color:\; white;">N</span>}}{\mathrm{<span\; style="background-color:\; white;">279.2</span>}<span\; style="background-color:\; white;">\times </span>{\mathrm{<span\; style="background-color:\; white;">10</span>}}^{\mathrm{<span\; style="background-color:\; white;">3</span>}}}$$$$\mathrm{<span\; style="background-color:\; white;">N</span>}<span\; style="background-color:\; white;">=</span>\frac{\mathrm{<span\; style="background-color:\; white;">0.22</span>}<span\; style="background-color:\; white;">\times </span>\mathrm{<span\; style="background-color:\; white;">279.2</span>}<span\; style="background-color:\; white;">\times </span>{\mathrm{<span\; style="background-color:\; white;">10</span>}}^{\mathrm{<span\; style="background-color:\; white;">3</span>}}}{\mathrm{<span\; style="background-color:\; white;">3</span>}<span\; style="background-color:\; white;">,</span>\mathrm{<span\; style="background-color:\; white;">612.6</span>}}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">17</span>}\text{<span style="background-color: white;">\hspace{0.17em}</span>}\text{<span style="background-color: white;">turns</span>}$$

Number of active turns ($N$) = 17

Step 6: Find Free Length ($L_f$)

For squared and ground ends, the free length is:

$${\mathrm{<span\; style="background-color:\; white;">L</span>}}_{\mathrm{<span\; style="background-color:\; white;">f</span>}}<span\; style="background-color:\; white;">=</span><span\; style="background-color:\; white;">(</span>\mathrm{<span\; style="background-color:\; white;">N</span>}<span\; style="background-color:\; white;">+</span>\mathrm{<span\; style="background-color:\; white;">2</span>}<span\; style="background-color:\; white;">)</span>\mathrm{<span\; style="background-color:\; white;">d</span>}<span\; style="background-color:\; white;">+</span>\mathrm{<span\; style="background-color:\; white;">\delta </span>}<span\; style="background-color:\; white;">+</span>\mathrm{<span\; style="background-color:\; white;">0.15</span>}\mathrm{<span\; style="background-color:\; white;">\delta </span>}$$$${\mathrm{<span\; style="background-color:\; white;">L</span>}}_{\mathrm{<span\; style="background-color:\; white;">f</span>}}<span\; style="background-color:\; white;">=</span><span\; style="background-color:\; white;">(</span>\mathrm{<span\; style="background-color:\; white;">17</span>}<span\; style="background-color:\; white;">+</span>\mathrm{<span\; style="background-color:\; white;">2</span>}<span\; style="background-color:\; white;">)</span><span\; style="background-color:\; white;">\times </span>\mathrm{<span\; style="background-color:\; white;">43.2</span>}<span\; style="background-color:\; white;">+</span>\mathrm{<span\; style="background-color:\; white;">220</span>}<span\; style="background-color:\; white;">+</span>\mathrm{<span\; style="background-color:\; white;">0.15</span>}<span\; style="background-color:\; white;">\times </span>\mathrm{<span\; style="background-color:\; white;">220</span>}$$$${\mathrm{<span\; style="background-color:\; white;">L</span>}}_{\mathrm{<span\; style="background-color:\; white;">f</span>}}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">19</span>}<span\; style="background-color:\; white;">\times </span>\mathrm{<span\; style="background-color:\; white;">43.2</span>}<span\; style="background-color:\; white;">+</span>\mathrm{<span\; style="background-color:\; white;">220</span>}<span\; style="background-color:\; white;">+</span>\mathrm{<span\; style="background-color:\; white;">33</span>}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">820.8</span>}<span\; style="background-color:\; white;">+</span>\mathrm{<span\; style="background-color:\; white;">220</span>}<span\; style="background-color:\; white;">+</span>\mathrm{<span\; style="background-color:\; white;">33</span>}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">1</span>}<span\; style="background-color:\; white;">,</span>\mathrm{<span\; style="background-color:\; white;">073.8</span>}\text{<span style="background-color: white;">\hspace{0.17em}</span>}\text{<span style="background-color: white;">mm</span>}$$

Free length ($L_f$) = 1,074 mm

Final Answers:

• Maximum load on each spring ($P$) = 44.74 kN

• Wire diameter ($d$) = 43.2 mm

• Coil diameter ($D$) = 216 mm

• Number of active turns ($N$) = 17

• Free length of the spring ($L_f$) = 1,074 mm