Stability of Structures. Takehome exam

Stability of Structures

Problem S1

For the beam depicted below, the student must evaluate the critical load and the associated critical mode by adopting two cubic finite elements; discuss the results as a function of the parameter β.

Problem S2

Determine the critical load and critical mode of the following 2 d.o.f. system as a function of a ratio between stiffness of extensional (k) and rotational springs (K); discuss the results.

Problem S3

The column with section in the figure has flexural-torsional clamped conditions at each edge. Evaluate its critical load as a function of the length for the flange dimensions indicated below. Compare the elastic results with the maximum compressive stress σ_o.

Data: B = 400, H = 200, t_f = t_w = 20, E = 2.0E+5, G = 8.0E+4, σ_o = 400 (dim N, mm)

Problem 5.1

Problem has 2 d.o.f.

When we use cubic polynomial, we could express elastic and geometric stiffness matrices:

1. elastic stiffness matrix

_ET/_L³

[12 6_L -12 6_L6_L 4_L² -6_L 2_L²-12 -6_L 12 -6_L6_L 2_L² -6_L 4_L²]

2. geometric stiffness matrix

1/_30L

[36 3_L -36 3_L42_L² -3_L 2_L²36 -3_L 4_L²]

d.o.f. of a single element: (we're neglecting axial deform.)

to assemble we could use a connectivity table; and so, our global St.matrix is the following:

K=[_{K33 + K43 K34 + K12}]

[_{K43 + K21 K44 + K22}]

to compute it we need to define st.matrices for each element in our problem.

element 1:

FEM 1

K_E = β_ET/_8L³

[12 12_L -12 12_L16_L -12_L 8_L²12 -12_L{6_L²}]

K_G=1/_60L

[36 6_L -36 6_L{6_L²} -6_L 4_L²36 -6_L {6_L²}]

element 2:

FEM 2

K_E = _ET/_L³

[12 6_L -12 6_L4_L² -6_L 2_L²12 -6_L 4_L²]

K_G=1/_30L

[36 3_L -36 3_L42_L² -3_L 2_L²36 -3_L 4_L²]

Problem S2

Deformed shape:

Relative rotations:

• Lθ₂ = Lθ₁ + 1/2θ₃
• θ₂₃ = θ₂ + θ₃ = 3θ₂, 2θ₁ = θ₂₃
• θ₃ = 2(θ₂ - θ₁)
• θ₃₁ = 1 - θ₁ - θ₃ = -θ₁ - 2θ₂ = -3θ₁ - 2θ₁ = θ₃ = 1

cos θ ≈ 1 - θ²/2!

Δ = L[θ₃²/2 + θ₂²/2 + θ₁²/2]

II⁰ order TPE:

V₂(θ) = 1/2 K [θ₂₃ + 1/2 K (Lθ₂)² - P/2 (Lθ₁² + Lθ₂² + L²θ₁²/2)]

→ 1/2 K [9θ₂² - 12θ₂θ₁ + 4θ₁²)] + 1/2 K L² θ₂² - PL/2 [3θ₁² + 3θ₂² - 4θ₂θ₁]

Define:

• B = κL²/K
• ρ = PL/k

V₂ = 1/2 K [B•θ₂ + (9θ₂² - 12θ₂θ₁ + 4θ₁²) - ρ(3θ₁², 3θ₂² - 4θ₂θ₁)]

→ V₂ = K/2 [θ₂(8θ₄ - 3ρ) + θ₂² (9 - 3ρ) - θ₂ θ₄(12 - 4ρ)]

V₂(θ) = 1/2 K [θ₄ + θ₂] ⋅ [B+4 - 3ρ -6+2ρ, -6+2ρ, 9 - 3ρ] [θ₄, θ₂]

to find critical load we enforce condition det(K) = 0

Critical load as a function of the length

Px (kN) Py (kN) Pδ (kN) P_∗ (kN)

L (m) Px (kN) Py (kN) Pδ (kN) P_∗ (kN) Pc (kN) σ (MPa) Condition 0.100 6.14E+07 3.42E+07 3.95E+06 3.30E+05 3.30E+06 1.65E+05 Unstable 0.500 2.70E+05 1.80E+05 1.45E+05 1.32E+05 1.32E+06 6.66E+04 Unstable 1.000 8.25E+04 5.25E+04 3.73E+04 2.65E+04 5.37E+05 2.69E+04 Unstable 1.500 2.99E+04 1.52E+04 1.07E+04 5.19E+03 2.08E+05 1.04E+04 Unstable 2.000 1.68E+04 9.42E+03 5.32E+03 5.42E+03 1.18E+05 5.90E+03 Unstable 2.100 1.53E+05 7.76E+04 9.61E+03 8.71E+03 8.71E+03 4.35E+02 Unstable 2.200 1.39E+05 7.01E+04 8.89E+03 8.04E+03 8.04E+03 4.07E+02 Unstable 2.205 1.39E+05 7.04E+04 8.85E+03 8.01E+03 8.01E+03 4.00E+02 Unstable 2.210 1.38E+05 7.01E+04 8.82E+03 8.00E+03 8.01E+03 3.99E+02 Stable 2.250 1.33E+05 5.76E+04 8.55E+03 7.14E+03 7.14E+03 3.87E+02 Stable 3.000 1.08E+05 3.86E+04 5.46E+03 4.87E+03 2.92E+03 4.46E+02 Stable 4.000 5.94E+04 1.87E+04 2.34E+03 2.14E+03 1.13E+03 2.15E+02 Stable 5.000 2.70E+04 1.37E+04 2.97E+03 2.47E+03 7.82E+02 1.62E+02 Stable 6.000 1.87E+04 1.13E+04 1.94E+03 1.59E+03 4.84E+02 1.21E+02 Stable 7.000 1.45E+04 9.58E+03 2.28E+03 2.07E+03 2.72E+02 1.01E+02 Stable 8.000 1.05E+04 4.33E+03 2.26E+03 1.98E+03 2.32E+02 1.06E+02 Stable 9.000 8.32E+03 4.36E+03 1.69E+03 1.51E+03 1.58E+02 8.73E+01 Stable 10.000 6.74E+03 3.67E+03 1.43E+03 1.31E+03 1.31E+02 7.38E+01 Stable 11.000 5.57E+03 3.28E+03 1.23E+03 1.13E+03 1.13E+02 6.60E+01 Stable 12.000 4.63E+03 2.83E+03 1.10E+03 1.06E+03 1.00E+02 6.01E+01 Stable 13.000 3.94E+03 2.57E+03 9.60E+02 8.60E+02 8.60E+01 5.60E+01 Stable 14.000 2.98E+03 2.18E+03 9.28E+02 6.95E+02 6.95E+01 4.91E+01 Stable 15.000 2.94E+03 1.89E+03 7.29E+02 6.33E+02 6.33E+01 4.44E+01 Stable 16.000 2.63E+03 1.43E+03 7.66E+02 6.02E+02 6.02E+01 3.82E+01 Stable 17.000 2.29E+03 1.33E+03 6.92E+02 4.69E+02 4.69E+01 3.68E+01 Stable 18.000 2.08E+03 1.06E+03 5.84E+02 4.82E+02 4.82E+01 3.34E+01 Stable 19.000 1.82E+03 9.60E+02 2.84E+02 3.98E+02 3.98E+01 3.21E+01 Stable 20.000 1.68E+03 8.56E+02 1.55E+03 5.76E+02 2.62E+02 3.03E+02 Stable 21.000 1.47E+03 7.00E+02 5.84E+02 4.57E+02 9.82E+01 2.94E+01 Stable 22.000 1.39E+03 5.35E+02 2.46E+03 4.46E+02 1.00E+02 2.57E+01 Stable 23.000 1.27E+03 5.06E+02 1.55E+03 4.79E+02 2.50E+02 3.45E+01 Stable 24.000 1.17E+03 5.94E+02 1.54E+03 4.45E+02 2.48E+02 2.22E+02 Stable 25.000 1.08E+03 5.47E+02 1.53E+03 4.18E+02 1.99E+02 2.09E+01 Stable