FEM Structures

Question 1

Explain at least three different sources of nonlinearity for finite element problems.

Answer 1

Nonlinearities arise from material nonlinearity (e.g., plasticity, hyperelasticity), geometric nonlinearity (e.g., large deformations or rotations), and boundary condition nonlinearity (e.g., contact problems where contact points are not predefined).

Question 2

Define the small strain tensor in terms of the displacement vector.

Answer 2

The small strain tensor is defined as ε_ij = 1/2(∂u_i/∂x_j + ∂u_j/∂x_i), where u_i is the displacement in the i-direction.

Question 3

Considering an FE approximation of displacements, show how to compute Voigt matrix strain.

Answer 3

Using u ≈ N a, the strain is ε = B a, where B is the strain-displacement matrix derived from shape function derivatives.

Question 4

Explain how to define a nonlinear problem in terms of a residual g and how this impacts the solution procedure.

Answer 4

The residual g represents the imbalance in the system: g(u) = f_int - f_ext. Iterative methods like Newton-Raphson minimize g to achieve equilibrium.

Question 5

Define the Jacobian matrix associated with Newton iterations.

Answer 5

The Jacobian (tangent stiffness matrix) is K = ∂g/∂u, where g is the residual vector.

Question 6

Explain what is meant by convergence of Newton iterations.

Answer 6

Convergence occurs when the residual norm ||g|| falls below a tolerance ε. It is detected by monitoring ||g|| or the reduction rate over iterations.

Question 7

Based on the quasistatic balance of linear momentum, derive the internal and external virtual work for small strains.

Answer 7

Internal virtual work: δW_int = ∫_V δε_ij σ_ij dV. External virtual work: δW_ext = ∫_V δu_i f_i dV + ∫_S δu_i t_i dS, where σ_ij is stress, f_i is body force, and t_i are tractions.

Question 8

Introduce a 2D finite element approximation and derive internal and external force vectors.

Answer 8

Using u ≈ N a and δu ≈ N δa, internal force vector: f_int = ∫_V B^T σ dV, and external force vector: f_ext = ∫_V N^T f dV + ∫_S N^T t dS.

Question 9

Define the residual vector for a nonlinear material problem under small strains.

Answer 9

The residual vector is g = f_int - f_ext, representing the imbalance between internal and external forces.

Question 10

Define the Jacobian matrix for Newton iterations and how material tangent behavior enters.

Answer 10

The Jacobian matrix is K = ∫_V B^T C_tangent B dV, where C_tangent is the material tangent stiffness matrix.

Question 11

Show how the internal force vector and tangent stiffness matrix are computed from element contributions.

Answer 11

Internal force: f_e,int = ∫_Ve B^T σ dV. Tangent stiffness: K_e = ∫_Ve B^T C_tangent B dV. Global matrices are assembled by summing element contributions.

Question 12

Explain why linear triangular elements may be less suitable for plasticity problems.

Answer 12

Linear triangular elements assume constant strain, which poorly captures stress gradients in plastic zones, leading to inaccuracies.

Question 13

Compute the Jacobian matrix for a 6-noded isoparametric triangular element.

Answer 13

For a 6-noded element, F_isop = [[∂x/∂ξ ∂x/∂η], [∂y/∂ξ ∂y/∂η]]. Compute derivatives using nodal coordinates.

Question 14

Show that [∂Na/∂Xi] = [F_isop^-T][∂Na/∂ξj].

Answer 14

Using the chain rule, ∂Na/∂Xi = ∂Na/∂ξj * ∂ξj/∂Xi. The inverse transpose of the Jacobian matrix transforms derivatives from local to global coordinates.

Question 15

Show how to compute the B_e-matrix for a 6-noded isoparametric triangular element.

Answer 15

The B_e matrix is derived from shape function derivatives transformed using the inverse Jacobian. B_e relates nodal displacements to strains.

Question 16

Explain how the Jacobian matrix enters FE equations for internal force and stiffness matrix.

Answer 16

The Jacobian maps integrals from global to parent domains: ∫V (…) dV = ∫ξ (…) |det(J)| dξ.

Question 17

Show how numerical integration can evaluate element contributions f_e,int and K_e.

Answer 17

Use Gaussian quadrature: f_e,int = Σ B^T σ w |det(J)|, K_e = Σ B^T C_tangent B w |det(J)|, summing over Gauss points.

Question 18

Explain path-dependent material behavior and its impact on FE implementation.

Answer 18

Path-dependent materials require history variables (e.g., plastic strain) stored and updated at integration points during simulations.

Question 19

Describe von Mises plasticity with isotropic hardening.

Answer 19

Von Mises plasticity involves yielding based on stress invariants: σ_y = σ_0 + H ε_p. Elastic loading occurs when σ_von Mises < σ_y.

Question 20

Compute von Mises stress for a 3D stress state.

Answer 20

Von Mises stress: σ_von Mises = √(3/2 * s_ij * s_ij), where s_ij = σ_ij - 1/3 σ_kk δ_ij is the deviatoric stress.

Question 21

Explain the concept of trial stress and show two ways to compute it.

Answer 21

Trial stress: σ_trial = σ^n + C : (ε^n+1 - ε^n), where σ^n is the previous stress state, and C is the elasticity matrix.

Question 22

Show how to determine elastic or plastic loading based on trial stress.

Answer 22

If f = σ_trial - σ_y ≤ 0, the loading is elastic. If f > 0, plastic loading occurs, requiring stress corrections.

Question 23

Describe continuum motion φ(X, t) for basic motions.

Answer 23

Continuum motion maps a material point X in the reference configuration to x in the current configuration: x = φ(X, t). Examples include pure elongation, simple shear, and pure rotation.

Question 24

Define the deformation gradient F and compute it from φ(X, t) or displacement u(X, t).

Answer 24

The deformation gradient is F = ∂φ/∂X = I + ∂u/∂X, where u = x - X. It transforms line elements in the reference configuration to the current configuration.

Question 25

Compute the Cauchy-Green deformation tensor C and Green-Lagrange strain tensor E for a given F.

Answer 25

Right Cauchy-Green tensor: C = F^T F. Green-Lagrange strain tensor: E = 1/2(C - I), where I is the identity tensor.

Question 26

Provide a spectral representation of symmetric second-order tensors such as C and E.

Answer 26

A symmetric tensor T is represented as T = Σ λ_i n_i ⊗ n_i, where λ_i are eigenvalues and n_i are eigenvectors.

Question 27

Describe the polar decomposition of F into R and U.

Answer 27

Polar decomposition: F = R U, where R is a rotation tensor and U is a stretch tensor. U is related to C by U = √C, and R = F U^(-1).

Question 28

Compute the change of volume and area in terms of F.

Answer 28

Change of volume: dV = det(F) dV_0. Change of area (Nanson's formula): dA n = det(F) F^(-T) dA_0 N.

Question 29

Define traction stress t and scaled traction vector t_0.

Answer 29

Traction stress: t = σ · n, where n is the normal in the current configuration. Scaled traction: t_0 = P · N, where N is the normal in the reference configuration.

Question 30

Derive the relation between 1st Piola-Kirchhoff stress P and Cauchy stress σ.

Answer 30

The relation is P = det(F) σ F^(-T), where F is the deformation gradient.

Question 31

Derive δW_int and δW_ext for large deformations.

Answer 31

Internal work: δW_int = ∫_V0 P : δF dV_0. External work: δW_ext = ∫_V0 δu · f_0 dV_0 + ∫_A0 δu · t_0 dA_0.

Question 32

Show that P and F are work-conjugated.

Answer 32

From δW_int = ∫_V0 P : δF dV_0, the variation of F, δF, is directly conjugated to P.

Question 33

Derive -P · ∇_0 = f_0 as the equilibrium equation.

Answer 33

From the principle of virtual work and divergence theorem, the equilibrium equation in the reference configuration is derived as -P · ∇_0 = f_0.

Question 34

Show that S and E are work-conjugated.

Answer 34

Internal virtual work in terms of 2nd Piola-Kirchhoff stress: δW_int = ∫_V0 S : δE dV_0, showing S and E are work-conjugated.

Question 35

For a given hyperelastic material model, compute S, P, and σ for a given F.

Answer 35

Using the strain energy density W(E) or W(C), compute S = ∂W/∂E, P = F S, and σ = 1/det(F) P F^T.

Question 36

Derive expressions for internal and external forces in FE formulation for large deformations.

Answer 36

Internal force: f_int = ∫_V0 B^T P dV_0. External force: f_ext = ∫_V0 N^T f_0 dV_0 + ∫_A0 N^T t_0 dA_0.

Question 37

Assume plane strain conditions and define quantities in F.

Answer 37

For plane strain: F = [[1 + ∂u_x/∂X, ∂u_x/∂Y, 0], [∂u_y/∂X, 1 + ∂u_y/∂Y, 0], [0, 0, 1]].

Question 38

Define quantities in f_int = ∫_V0 B_0^T P dV_0 under plane strain.

Answer 38

Internal force: f_int = ∫_V0 B_0^T P dV_0, where B_0 contains derivatives of shape functions N_a.

Question 39

Show that ∂f_int/∂a = ∫_V0 B_0^T ∂P/∂F B_0 dV_0.

Answer 39

Tangent stiffness matrix: K = ∂f_int/∂a = ∫_V0 B_0^T C_tangent B_0 dV_0, where C_tangent = ∂P/∂F.

Question 40

For an isoparametric element, show that ∂Na/∂Xi = J^(-T) ∂Na/∂ξj.

Answer 40

Using the Jacobian J for isoparametric mapping, the transformation of derivatives is ∂Na/∂Xi = J^(-T) ∂Na/∂ξj.

Question 41

Illustrate and define the kinematic assumptions for the Mindlin plate.

Answer 41

In Mindlin theory, transverse normals may rotate but not remain perpendicular to the midplane, and transverse shear deformation is included. Displacements: u_α = u_α^0 - zθ_α, w = w(x, y).

Question 42

Show how to compute the out-of-plane shear strains γ_αz.

Answer 42

The out-of-plane shear strains are γ_αz = ∂w/∂x_α - θ_α.

Question 43

Define the in-plane section forces per unit length N_αβ for given in-plane stress components.

Answer 43

In-plane section forces: N_αβ = ∫_(-h/2)^(h/2) σ_αβ dz, where σ_αβ are in-plane stresses.

Question 44

Define the out-of-plane section shear forces per unit length V_αz.

Answer 44

Out-of-plane section shear forces: V_αz = ∫_(-h/2)^(h/2) σ_αz dz.

Question 45

Define the bending moments per unit length M_αβ for given in-plane stress components.

Answer 45

Bending moments: M_αβ = ∫_(-h/2)^(h/2) z σ_αβ dz.

Question 46

Derive the weak form of the Mindlin equilibrium equations.

Answer 46

From the principle of virtual work: δW_int = ∫_V δε_ij σ_ij dV, weak form involves contributions from membrane forces, bending moments, and shear forces.

Question 47

Express the weak form equations in Voigt matrix form.

Answer 47

Using σ = D ε and τ = G γ_z, the weak form becomes K a = f, where K includes bending, membrane, and shear stiffness matrices.

Question 48

Derive sectional forces and moments (N_sec, V_sec, M_sec) in terms of displacements.

Answer 48

Membrane forces: N_sec = D_m ε. Shear forces: V_sec = G γ_z. Bending moments: M_sec = D_b κ.

Question 49

Discuss boundary conditions for the Mindlin plate problem.

Answer 49

Dirichlet BCs: Prescribed displacements (u_0, w, θ). Neumann BCs: Prescribed forces (N_αβ, M_αβ, V_αz).

Question 50

Explain why a shear correction factor is needed for Mindlin theory.

Answer 50

Shear stresses are assumed constant across thickness, which overestimates stiffness. A correction factor accounts for the true parabolic distribution of shear stresses.

Question 51

Derive the FE form for the Mindlin plate problem.

Answer 51

Using approximations u_0 ≈ N_u a, w ≈ N_w a, θ ≈ N_θ a, the FE form includes contributions from bending, shear, and membrane stiffness matrices.

Question 52

Define element degrees of freedom (DOFs) for a 4-noded quadrilateral Mindlin element.

Answer 52

Each node has 5 DOFs: u_0, v_0, w, θ_x, θ_y. Total: 4 × 5 = 20 DOFs.

Question 53

Explain shear locking in quadrilateral Mindlin plate elements.

Answer 53

Bilinear approximations cannot accurately represent shear strain variations, causing excessive stiffness (shear locking).

Question 54

Explain ways to alleviate shear locking in Mindlin plate elements.

Answer 54

Use reduced integration to avoid over-constraining shear strains or mixed formulations to interpolate w and θ separately.

Question 55

Define the kinematic assumptions for the Kirchhoff plate.

Answer 55

In Kirchhoff theory, transverse normals remain perpendicular to the midplane, and transverse shear deformation is neglected. Displacements: u_α = -z(∂w/∂x_α), w = w(x, y).

Question 56

Show that the shear strains γ_αz = 0.

Answer 56

Since transverse normals remain perpendicular to the mid-surface, γ_αz = ∂w/∂x_α - θ_α = 0, where θ_α = ∂w/∂x_α.

Question 57

For linear isotropic elasticity, show M_αβ = -h^3/12 D^* ∇^2 w.

Answer 57

Bending moments are M_αβ = D κ_αβ, where κ_αβ = ∂²w/∂x_α∂x_β. D = Eh^3/12(1 - ν²) is the flexural rigidity.

Question 58

Show δW_int = -∫_A κ^T M dA.

Answer 58

Internal virtual work: δW_int = ∫_V δε_ij σ_ij dV reduces to bending work: δW_int = -∫_A δκ^T M dA after integrating over thickness.

Question 59

Derive δW_ext for the Kirchhoff plate.

Answer 59

External virtual work: δW_ext = ∫_A δw q dA - ∫_L ∂(δw)/∂n M_nn dL + ∫_L δw (V_n + ∂M_nm/∂m) dL, where n is the normal.

Question 60

For clamped, simply supported, and free boundaries, give the boundary conditions.

Answer 60

Clamped: w = ∂w/∂n = 0. Simply supported: w = M_nn = 0. Free: M_nn = 0, V_n + ∂M_nm/∂m = 0.

Question 61

What boundary conditions prevent rigid body motions?

Answer 61

At least three points must be fixed to prevent rigid body motion: two for translation and one for rotation. Alternatively, constrain edges with displacements and rotations.

Question 62

Derive κ = B a for w = N a.

Answer 62

Curvatures κ_αβ = ∂²w/∂x_α∂x_β can be expressed as κ = B a, where B contains second derivatives of shape functions.

Question 63

Express M in terms of h, D, B, and a.

Answer 63

Bending moments: M = D κ = D B a, where D is the flexural rigidity, B is the curvature matrix, and a is the nodal displacement vector.

Question 64

Define DOFs for a quadrilateral Kirchhoff plate element.

Answer 64

Each node has 3 DOFs: w, θ_x, θ_y. Total DOFs for a 4-noded element: 4 × 3 = 12.

Question 65

Outline the procedure for finding shape functions for an isoparametric element.

Answer 65

Define element geometry with nodal coordinates, use parametric coordinates (ξ, η), derive shape functions N using interpolation polynomials, and transform derivatives with the Jacobian matrix.

Question 66

Define the governing equations for a non-linear dynamic problem.

Answer 66

The governing equation is M ü + C ũ + f_int(u) = f_ext(t), where M is the mass matrix, C is the damping matrix, f_int is the internal force, and f_ext is the external force.

Question 67

Describe the principle of virtual work for a dynamic system and derive the weak form.

Answer 67

The principle of virtual work: ∫_V δu^T(M ü + C ũ + f_int - f_ext) dV = 0. The weak form is obtained by integrating over the domain and reducing derivative orders using boundary conditions.

Question 68

Explain the concept of mass matrices and damping matrices in FEM.

Answer 68

The mass matrix relates nodal accelerations to inertial forces, and the damping matrix relates velocities to damping forces. Lumped mass matrices are diagonal, while consistent mass matrices are derived from shape functions.

Question 69

Derive the equation of motion for a dynamic problem with non-linearities.

Answer 69

The equation of motion is M ü + C ũ + f_int(u) = f_ext(t). Non-linearities in f_int(u) arise from material or geometric effects.

Question 70

Discuss time integration schemes for dynamic problems.

Answer 70

Explicit methods (e.g., central difference) require small time steps for stability. Implicit methods (e.g., Newmark-beta) are unconditionally stable but computationally expensive.

Question 71

For the generalized-alpha method, derive expressions for displacement, velocity, and acceleration.

Answer 71

Displacement: u^(n+1) = u^n + Δt ũ^n + Δt²(γ ü^(n+1) + (1-γ) ü^n). Velocity: ũ^(n+1) = ũ^n + Δt(β ü^(n+1) + (1-β) ü^n).

Question 72

Discuss the role of damping in dynamic systems.

Answer 72

Damping dissipates energy and controls oscillations. Rayleigh damping uses coefficients α and β to combine mass and stiffness matrices. Modal damping applies damping to specific modes.

Question 73

Explain how to compute modal properties for a linearized non-linear system.

Answer 73

Linearize the system at an equilibrium point to compute the tangent stiffness matrix K_t. Solve the eigenvalue problem (K_t - ω²M) φ = 0 to obtain natural frequencies ω and mode shapes φ.

Question 74

Define and compute the residual vector and tangent stiffness matrix for a non-linear dynamic system.

Answer 74

Residual vector: g = M ü + C ũ + f_int - f_ext. Tangent stiffness: K = ∂g/∂u, which accounts for non-linearities in f_int.

Question 75

Describe how energy conservation can be assessed in non-linear dynamic simulations.

Answer 75

Compute total energy E = 1/2 ũ^T M ũ + U(u), where U(u) is strain energy. Energy conservation is checked by verifying that dE/dt = 0 or consistent dissipation if damping is present.

Question 76

Discuss challenges in solving non-linear dynamic problems.

Answer 76

Challenges include convergence issues in iterative methods, stability concerns in explicit time integration, and selecting appropriate time steps to balance accuracy and computational cost.

Question 77

Define the kinematic assumptions for thin and thick shell theories.

Answer 77

Thin shells (Kirchhoff): Transverse normals remain normal to the mid-surface, neglecting transverse shear deformation. Thick shells (Mindlin): Includes transverse shear deformation and allows rotation of cross-sections.

Question 78

Show how the displacement field for a shell element can be expressed in terms of mid-surface displacements and rotations.

Answer 78

Displacement: u = u_0 + zR, where u_0 = [u_x^0, u_y^0, w]^T (mid-surface displacements) and R = [-θ_y, θ_x, 0]^T (cross-sectional rotations).

Question 79

Define the Green-Lagrange strain tensor for shell elements in terms of mid-surface kinematics.

Answer 79

The strain tensor is ε_ij = 1/2(∂u_i/∂x_j + ∂u_j/∂x_i + ∂u_k/∂x_i ∂u_k/∂x_j). For shells, it includes membrane strain ε^m, bending strain κ, and shear strain γ_z.

Question 80

Derive the expression for the internal virtual work in shell elements.

Answer 80

Internal virtual work: δW_int = ∫_V δε_ij σ_ij dV. Using stress resultants, it becomes δW_int = ∫_A (δε^m N + δκ M + δγ_z V) dA.

Question 81

Describe the challenges associated with numerical integration in shell elements and methods to address them.

Answer 81

Challenges include shear locking in thick shells and over-constrained bending in thin shells. Solutions include reduced integration for shear terms and selective integration.

Question 82

Define degrees of freedom for a 4-noded quadrilateral shell element.

Answer 82

Each node has 6 DOFs: u_x, u_y, u_z, θ_x, θ_y, θ_z. Total DOFs for a 4-noded element: 4 × 6 = 24.

Question 83

Explain the concept of stress resultant-based formulations for shells.

Answer 83

Stress-resultant formulations simplify shell mechanics by integrating 3D stress components through the thickness to compute N, M, and V. They differ from full 3D stress formulations by reducing complexity.

Question 84

Discuss the importance of geometric stiffness in shell formulations and its role in stability analyses.

Answer 84

Geometric stiffness accounts for pre-stress or large deformations in shells, crucial for stability analyses, predicting buckling, and capturing post-buckling behavior.