We have already discussed how to design hydraulic cylinder piston.

In this sizing calculation tutorial we will see how to design hydraulic cylinder. Industrial hydraulic cylinder is an example of a typical thick cylinder.

**Thick cylinder vs. thin cylinder**

Cylinders in mechanical design are broadly classified into two categories:

- Thin wall cylinder
**(R**_{i}**: t <10)** - Thick wall cylinder
**(R**_{i : }**t >10**)

Where,

**R**_{i} = inner radius of the cylinder

**t **= wall thickness of the cylinder

**Hydraulic cylinder designing / sizing steps**

**Step 1: Find out internal radius of cylinder**

Internal radius of cylinder**, R**_{i} = outer radius of the piston (which we already calculated here)

**Step 2: Radial stress from Lame’s Theory **

Lame’s Equation for calculating radial stress in a hydraulic cylinder at any given radius is:

**Ϭ**_{r}** ****= ( b/ R**^{2}**) – a ****…………….eq.2.1**

Where,

**Ϭ**_{r}** ***– Radial stress at radius R*

**a, b –*** Constants*

**Step 3: Tangential stress**

Lame’s equation for calculating tangential / circumferential stress at specified radius is:

**Ϭ**_{c}** = ****(b/ R**^{2}**) + a**** …………….eq.2.2**

Where,

**Ϭ**_{c}** ***– Tangential or circumferential stress at radius ***R**

**a, b –*** Constants*

**Step 4: Applying Boundary condition **

Apply the given boundary conditions to the **eq.2.1 & eq.2.2 **to find out the constants **a** & **b.**

Some of the possible boundary conditions are:

- For
**R= Inner radius,****Ϭ**_{r}**= Internal pressure of the cylinder** - For
**R= Inner radius,****Ϭ**_{c}**=Max Permissible stress for the cylinder material**

**Step 5: Solve for the unknown **

As the constants for both the equations are known now, you can easily find out the unknown quantity. We will see in the example, how?

**Worked out example**

Let’s take the same example as discussed in the piston design calculation tutorial with only additional input as external pressure= Atmospheric pressure

Find out the internal and external radius of the hydraulic cylinder.

Also find out the maximum stress in the cylinder.

**Solution:**

**Input given:**

Internal pressure, P_{1} = 2 MPa (Gauge)

External pressure, P_{2} = 0 MPa (Gauge)

Max. Permissible stress for cylinder material, **Ϭ**_{max} **= 407.7 Mpa**

**Step 1:**

We have already calculated, Inner radius, Ri = 12.5 mm (approx.)

**Step 3:**

We will apply following boundary conditions in the **eq.2.1 **& **2.2**:

- For
**R= 12.5 mm,****Ϭ**_{r}**=2 MPa** - For
**R= 12.5 mm,****Ϭ**_{c}**=407.7 MPa**

Solving we get,

**a = 153.85****b = 39664.0625**

By putting the values of the constants the equations become,

**Ϭ**_{r}** ****= (39664.06/ R**^{2}**) – 153.85 ……………………………..eq.2.3**

**Ϭ**_{c}** = (39664.06/ R**^{2}**) + 153.85 ……………………………..eq.2.4**

**Step 4:**

For finding out the thickness (or the outer radius) of the cylinder, we will use the following input conditions:

For **R= required outer radius, ****Ϭ**_{r}** = Given external pressure = 0**

Putting the above conditions in the **eq.2.3 & 2.4** we get:

Required outer radius of the cylinder, **R**_{o}** = 16.05 mm**

In short, Lame’s equations of thick cylinder along with the given boundary conditions are used for calculating the design and sizing parameters of a hydraulic cylinder.

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