April 19, 2015
Computer Methods in Civil Engineering - PROBLEM SET 2, ver.2
1. One of nonlinear elastic material models is described by Ramberg-Osgood equation for 1D problem:
= σ
E + σ0α E
σ σ0
m
in which E, σ0, α, m are material model parameters. Derive for this model an expres- sion for tangent modulus ET. Show on a sketch the interpretation of tangent stiffness of a material with nonlinear properties.
2. Using the 1D equations of plastic flow theory derive the relationship between Young modulus E, tangent modulus ET and isotropic linear hardening modulus H:
1 E + 1
H = 1 ET
3. In the plastic flow theory the following isotropic hardening expression is assumed:
¯
σ(κ) = σy∞+ (σy0− σy∞) exp(−βκ) + Hκ
in which κ is hardening parameter (plastic strain measure), σ0y is initial yield strength, σy∞ ≥ σ0y and β are parameters of saturation hardening (¯σ0(κ)|κ→∞ = 0), and H is linear hardening modulus. Draw the diagram of function ¯σ(κ) in range κ ∈ (0; 0.01) for model parameters: σy0 = 300, σ∞y = 600, β = 1, H = 3000. For which parameter values the hardening is linear and for which it has a saturation character (approaches horizontal asymptote for growing κ)?
4. What does the yield function/surface describe in plasticity theory? In Voigt’s notation the Huber-Mises-Hencky yield function can be expressed as:
f (σ, κ) = 3 2σTPσ
1/2
− ¯σ(κ)
where P is a constant coefficient matrix. Derive the gradient of the function in the stress space.
5. The Burzynski-Drucker-Prager yield surface can be expressed as:
f (σ, κ) = q + α p − βcp(κ) = 0
where q = p3J2σ, p = 13I1σ. Sketch the section of the surface with the octaedric plane (perpendicular to the hydrostatic axis) which contains the origin of principal stress coordinate system (σ1, σ2, σ3). Sketch the plasticity criterion on (p, q) plane, used in geomechanics. Indicate cohesion and friction angle.
6. Linear isotropic hardening is assumed in classical elasto-plasticity. Given Young mod- ulus E=210GPa, initial yield strength σy0=300MPa and tangent modulus ET=1GPa, compute the equivalent plastic strain value κ for which the yield strength reaches the value σy=500MPa.
7. What are the possible sources of nonlinearities in structural response?
8. How can displacement discontinuities be accounted for in FE analysis?
9. Write the virtual work principle for a continuum.
10. Which equations of mechanics are satisfied exactly and which approximately in the displacement-based FEM version?
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April 19, 2015
11. For a linear elastic problem the FEM nodal displacement vector has been computed.
What is the formula to calculate the stress tensor at any point of the domain?
12. List 2D problem types in structural mechanics. What are they characteristics? Con- sidering linear elasticity, how is 33 computed in plane stress situation and σ33in plane strain?
13. What is the definition of out-of-balance forces in the Newton-Raphson algorithm? Ex- plain the notation used.
14. What is the purpose of convergence criterion in the Newton-Raphson algorithm and how is it expressed in terms of forces?
15. In which situations does one need to use displacement control or arc-length control to trace a nonlinear equilibrium path?
16. What are the components of the tangent stiffness operator in geometrically nonlinear analysis?
17. Write the general formula for the tangent stiffness matrix in a physically nonlinear problem. Indicate which matrices in the formula are responsible for this type of non- linearity.
18. Which tensors in plastic flow theory are related in bijective manner (one-to-one) via the Hooke’s tensor?
19. Write a plastic flow rule and explain the meaning of the variables used in it. When is the flow rule called associated/associative?
20. What are the shapes of the HMH and Mohr-Coulomb yield surfaces in the principal stress space? For which materials is each of them used?
21. The response of concrete is strongly sensitive to the sign of strain/stress. How is this represented in nonlinear modelling?
22. What is the physical sense of fracture energy in the analysis of concrete cracking?
23. Assume that the deflection of a slab with dimensions 2a × 2b is given by:
w(x, y) = (1 − ξ2)2(1 − η2)2, ξ = x/a, η = y/b, ξ, η ∈ [−1, 1]
Compute the functions of curvatures κxx, κyy and twist κxy.
24. What is main differences between the Kirchhoff-Love plate/shell theory and the one due to Reissner-Mindlin?
25. What is the difference between the linear theory of thin slabs and the von Karman theory?
26. Write the constitutive relations between generalized strains and stresses in thin elastic slabs.
27. How many degrees of freedom (dofs) does the simplest 4-noded plate bending element have? What is their physical interpretation?
28. What dofs does a node of a shell finite element usually have?
29. Write the assumptions of linear buckling analysis.
30. What is the equation of initial buckling problem? What is the solution of the problem?
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