June 11, 2017
Computer Methods in Civil Engineering - PROBLEM SET, ver.3 1. Write the virtual work principle for a tensile bar. Explain quantities used.
2. Describe the FEM algorithm using the example of a tensile bar.
3. Write the differential equation of vibrations of a tensile bar with example boundary and initial conditions. Neglect damping, explain variables used.
4. Describe briefly the sequence of models in computer simulation and list errors involved.
5. How can displacement discontinuities be accounted for in FE analysis?
6. Write the virtual work principle for a continuum exhibiting small deformations.
7. Write the general matrix equation of dynamic equilibrium for a linear system discretized with FEM. Explain the symbols used. How is it solved?
8. Write the matrix equation of free vibrations of a discretized continuum model. What is the solution of the equation?
9. Which equations of mechanics are satisfied exactly and which approximately in the displacement-based FEM version?
10. 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?
11. List 2D problem types in structural mechanics. What are their features? Considering linear elasticity, how is 33 computed for plane stress and σ33for plane strain?
12. What are the options of mesh refinement? What is the Zienkiewicz-Zhu error estimator?
13. What are the possible sources of nonlinearities in structural response?
14. One of nonlinear elastic material models is described by Ramberg-Osgood equation:
= σ
E +σ0α E
σ σ0
m
in which E, σ0, α, m are material model parameters. Derive for this model an expression for tangent modulus ET. Show on a sketch the interpretation of tangent stiffness of a material with nonlinear properties.
15. 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
16. 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), σy0 is initial yield strength, σ∞y ≥ σy0 and β are parameters of saturation hardening (¯σ0(κ)|κ→∞= 0), and H is linear hardening modulus.
17. 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.
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June 11, 2017
18. 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 (per- pendicular 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.
19. Linear isotropic hardening is assumed in classical elasto-plasticity. Given Young mod- ulus E=210GPa, initial yield strength σ0y=300MPa and tangent modulus ET=1GPa, compute the equivalent plastic strain value κ for which the yield strength reaches the value σy=500MPa.
20. What is the definition of out-of-balance forces in the Newton-Raphson algorithm? Explain the notation used.
21. What is the purpose of convergence criterion in the Newton-Raphson algorithm and how is it expressed in terms of forces?
22. In which situations does one need to use displacement control or arc-length control to trace a nonlinear equilibrium path?
23. What are the components of the tangent stiffness operator in geometrically nonlinear analysis?
24. Write the general formula for the tangent stiffness matrix in a physically nonlinear prob- lem. State which matrices in the formula represent this type of nonlinearity.
25. Which tensors in plastic flow theory are related in bijective manner (one-to-one) via the Hooke’s tensor? Write a plastic flow rule and explain the meaning of the variables used in it. When is the flow rule called associated/associative?
26. 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?
27. The response of concrete is strongly sensitive to the sign of strain/stress. How is this represented in nonlinear modelling? What is the physical sense of fracture energy in the analysis of concrete cracking?
28. 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.
29. What is main difference between the Kirchhoff-Love plate/shell theory and the one due to Reissner-Mindlin?
30. What is the difference between the linear theory of thin slabs and the von Karman theory?
31. How many degrees of freedom (dofs) does the simplest 4-noded plate bending element have? What is their physical interpretation? What dofs does a node of a shell finite element usually have?
32. Write the assumptions of linear buckling analysis. What is the equation of initial buckling problem? What is the solution of the problem?
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