Homework answers / question archive / 1) (Uniaxial bar: total Lagrangian formulation) Using the total Lagrangian formulation, solve displacement at the tip, stress, and strain of the uniaxial bar in Fig

1) (Uniaxial bar: total Lagrangian formulation) Using the total Lagrangian formulation, solve displacement at the tip, stress, and strain of the uniaxial bar in Fig

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1) (Uniaxial bar: total Lagrangian formulation) Using the total Lagrangian formulation, solve displacement at the tip, stress, and strain of the uniaxial bar in Fig. 3.11 under tip force F=100 N. Use a two-node bar element. Assume St. Venant–Kirchhoff material with E=200 Pa and cross-sectional area A=1.0 m².

2) (Large displacement and rotation) A four-node element undergoes large displacement and rotation in the XY plane, as shown in Fig. 3.2. The element is rotated counterclockwise by 90^o, its length is stretched to 2, and width is reduced to 0.7. Calculate the deformation gradient, Lagrangian strain, Eulerian strain, and engineering strain.

3) (Tension of an elastoplastic bar) An elastoplastic bar in the dimension of 1 cm*1 cm*2 cm is under axial load as shown in Fig. 2.27. Using two eight-node finite elements, solve for displacements and stresses for both elements. Assume material properties of λ=110.7 GPa, μ=80.2 GPa, σ_Y=400 MPa, and H=100 MPa.

4) (Nonlinear springs [modified Newton–Raphson method]) Using the modified Newton–Raphson method, solve the displacements of the two nonlinear springs in Example 2.2. Use the initial estimate, u0={0.3 0.6}^T. Compare the number of iterations with that of Example 2.3. Also, check the convergence rate.

5) A) (Equilibrium of a spring) Consider a spring component, which is fixed at one end and under an applied force, f, at the other end. Calculate the displacement, u, at the load application point. B) (Equilibrium of a bar) For the bar in Example 1.10, calculate the displacement, u(x), using the principle of minimum potential energy. Assume that the virtual displacement is in the same form as the displacement.

6) (Equilibrium of a bar) The bar in Fig. 1.14 has length, L; Young’s modulus, E; and cross-sectional area, A. Assume that displacement is in the form of u(x)=c1x + c₂; calculate the displacement (1) by solving the governing differential equation and (2) by using the stationary condition of the potential energy.

7) A) (Inner product of two tensors) Consider the inner product of two rank-2 tensors: C=A. B. Using the dyadic representation method as in Eq. (1.6), calculate the Cartesian components of C in terms of that of A and B. B) (Symmetric and skew part of displacement gradient) A displacement gradient, ∇u, is a rank-2 tensor. Calculate the symmetric and skew part of the displacement gradient.

8) NHTSA car crash tests. Refer to Exercise 4.26 (p. 221),in which you found the probability distribution for x, thenumber of stars in a randomly selected car’s driver-sidecrash rating. Find m = E1x2 for this distribution and interpretthe result practically.