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 torque value L: Element length

M' _1-2 (x): 1-2 Torque function M _2-3 (x): 2- 3 Torque function ΣM1,dn: Total torque relative to 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 torque value L: Element length

M(x): Torque function ΣM _1,dn : Total torque 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