# A Simply Supported Beam

Calculate by hand the deformation of *joint* 2 of the following simple beam using the *Virtual Work Method*.

**Hint for solution**

According to the method; the integral should be considered separately for the bars between 1-2 and 2-3 nodes.

**Detailed solution**

**E=** Modulus of elasticity **A=** Cross-sectional area **I=** Cross-section moment of inertia

**M=** Moment due to external loading **M'=** Moment due to unit loading **Δ=** Deformation

__Step 1 Finding Moment function by Unit Loading__

**Q:** unit load D_{1} , **D _{3} :**

*Support responses*

**M'(2):**

*2 DN moment value*

**L:**Element length

**M' _{1-2} (x):** 1-2

*Moment function*

**M**2- 3

_{2-3}(x):*Moment function*

**ΣM**Total moment relative to 1 DN.

_{1,dn}:**ΣD:**Total balance

**Moment Function 1-2 :**

**Moment Function 2-3 :**

__Step 2: Finding the Moment Function Based on the Given Load__

**q: **Distributed load D_{1} , **D _{3} : **Support responses

**M(2):**2 DN moment value

**L:**Element length

**M(x):** Moment function** ΣM _{1,dn} : **Total moment with respect to 1 DN.

**ΣD:**Total balance

**Moment Function :**

__Step 3: Multiplying the M and M' Moment Functions in the Previous Steps and Getting the Integral__

__Integral for 1-2 Bar__

__Integral for 2-3 Bar__

**Deformation of node 2 = 0.107m**

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