For the frame shown below, P=15 kN and a=1.5m. Assume I=20000 in^4 and E=30000 ksi for all beam. Use virtual work method to find the horizontal displacement of node H.
The correct answer and explanation is :
To find the horizontal displacement of node H in the given frame using the virtual work method, we must first outline the steps and formulas required to apply this method.
Step 1: Understand the Setup
We have the following values:
- Load ( P = 15 \, \text{kN} )
- Distance ( a = 1.5 \, \text{m} )
- Moment of inertia ( I = 20,000 \, \text{in}^4 )
- Modulus of elasticity ( E = 30,000 \, \text{ksi} )
- Node H is where we want to find the horizontal displacement.
We assume that the structure has only horizontal displacements (not vertical). The virtual work method calculates displacements by relating work done by virtual forces to the strain energy in the structure.
Step 2: Virtual Work Method
The virtual work method involves two key steps:
- Apply a virtual load: This is a hypothetical load, usually small, applied in the direction of the displacement we want to find. For this case, a unit horizontal force is applied at node H (where we want the displacement).
- Calculate the strain energy: The total strain energy is determined from the internal moments and deformations in the beam. We use beam deflection formulas to find the strain energy due to the real load ( P ) and the virtual load.
Step 3: Apply the Formulas
Strain Energy Due to Bending:
The strain energy due to bending in a beam is given by:
[
U = \int_0^L \frac{M^2(x)}{2EI} \, dx
]
Where:
- ( M(x) ) is the moment at a point ( x ) along the length of the beam.
- ( E ) is the Young’s Modulus.
- ( I ) is the moment of inertia of the cross-section.
- ( L ) is the length of the beam.
Virtual Work Equation:
The virtual work equation is:
[
\delta = \frac{\sum U_{virtual}}{\text{Applied Virtual Force}}
]
Where ( \delta ) is the displacement (horizontal displacement at node H) and the sum of the virtual strain energy is calculated based on the real and virtual loadings.
Step 4: Calculate the Displacement
Given the real applied load ( P = 15 \, \text{kN} ), and using the formulas for strain energy, we would integrate over the beam’s length to determine the total strain energy and use the virtual displacement method to find the horizontal displacement.
This type of calculation can be quite involved and usually requires step-by-step analysis of moments and strains, which would depend on how the structure is configured. However, the displacement ( \delta_H ) can be expressed as:
[
\delta_H = \frac{P \cdot \text{Virtual Displacement Force}}{EI} \cdot \text{Factor from the Moment Curvature Relation}
]
Step 5: Conclusion
By applying the virtual work method and the above relationships, you can solve for the displacement at node H. However, to achieve an accurate result, the exact beam geometry, boundary conditions, and applied loads must be considered. If this was a real problem, further detailed calculations with the appropriate moment-curvature diagrams and integration would yield the numerical displacement value.
