Let’s work on Group 6 values from the table:

Given Parameters for Group 6:

1. H = 3.5 m (Column height)
2. L = 3.0 m (Beam span)
3. q = 5.0 kN/m (Uniformly distributed load)
4. b/h = 200/400 mm (Beam and column dimensions: width/depth)
5. E = 30 GPa (Modulus of elasticity)

Problem requirements:

• Calculate node displacements and reaction forces.
• Draw force diagrams for bending moment (M), shear force (V), and axial force (N).
• Show how results can be validated using finite element analysis software.

Approach:

1. Structural Analysis Setup:

   • Use the stiffness matrix method (finite element method) for a planar frame.
   • Assume fixed supports at both ends of the columns.
   • Model the frame as two columns and one beam connected at rigid nodes.

2. Properties of Structural Members:

   • Moment of inertia ($I$): $I = \frac{b \cdot h^3}{12}$ Substituting $b = 200 \, \text{mm} = 0.2 \, \text{m}$ and $h = 400 \, \text{mm} = 0.4 \, \text{m}$: $I = \frac{0.2 \cdot (0.4)^3}{12} = 0.001067 \, \text{m}^4$
   • Axial stiffness ($EA/L$): $EA = E \cdot A, \quad A = b \cdot h = 0.2 \cdot 0.4 = 0.08 \, \text{m}^2$ Substituting $E = 30 \, \text{GPa} = 30 \cdot 10^9 \, \text{Pa}$: $EA = 30 \cdot 10^9 \cdot 0.08 = 2.4 \cdot 10^9 \, \text{N}$

3. Beam Analysis under Uniform Load:

   • Maximum bending moment at mid-span: $M_{\text{max}} = \frac{q \cdot L^2}{8}$ Substituting $q = 5.0 \, \text{kN/m} = 5000 \, \text{N/m}$ and $L = 3.0 \, \text{m}$: $M_{\text{max}} = \frac{5000 \cdot (3.0)^2}{8} = 5625 \, \text{N·m}$

   • Shear force at supports: $V = \frac{q \cdot L}{2}$ Substituting $q = 5000 \, \text{N/m}$: $V = \frac{5000 \cdot 3.0}{2} = 7500 \, \text{N}$

4. Frame Displacements and Reactions:

   • Use finite element formulation to calculate nodal displacements and reaction forces considering boundary conditions (fixed supports). This involves assembling the global stiffness matrix and solving for unknowns.

Results Summary (Partial):

• Maximum Bending Moment (Beam): $M_{\text{max}} = 5625 \, \text{N·m}$
• Shear Force at Supports: $V = 7500 \, \text{N}$

Would you like me to continue with detailed stiffness matrix formulations, solve for displacements, or plot diagrams?

Related Questions:

1. What are the reaction forces at the supports?
2. How does increasing $q$ affect the bending moment and shear forces?
3. Can the structure remain stable if $E$ decreases significantly?
4. How would varying the beam depth $h$ change the moment of inertia $I$?
5. What software tools can verify the results accurately?

Tip: In practice, use finite element software (e.g., SAP2000 or ANSYS) for complex structures to save time and ensure accuracy.