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4. (10 points) Consider the isoparametric plane strain two-dimensional finite element shown. (a) Construct the Jacobian...
Problem 2. Consider a finite element with shape functions N1 (ξ) and N2Ģ) used to interpolate...
Problem 2. Consider a finite element with shape functions N1 (ξ) and N2Ģ) used to interpolate the displacement field within the element shown below. Derive an expression for the strain-displacement matrix B where strain E-Bq, in terms of N1 and N2. (Do not assume any specific form for N and N2.) (Note: q[1 21.) 72--
3.4. Consider a finite element with shape functions N(E) and N2(E) used to interpolate the displacement...
3.4. Consider a finite element with shape functions N(E) and N2(E) used to interpolate the displacement field within the element (Fig. P3.4) 2 1 9-> 92-> +1 6-1 FIGURE P3.4 Derive an expression for the strain-displacement matrix B, where strain e Bq, in terms of N and N. (Do not assume any specific form for N or N.) (Note: qa )
QUESTION 2 21 Finite elements can appear in many forms such as two-dimensional and three- dimensional domains Give two...
QUESTION 2 21 Finite elements can appear in many forms such as two-dimensional and three- dimensional domains Give two examples and a sketch for each domain. (4) 22 Explain the following terms as used in Finite element equations a Plain stress b Plain strain 23 Use the Finite element method to develop the stiffness matrix for element 2 of the steel cantilever beam structure shown in Figure 2 The elastic modulus IS 200 kN/mm2 with a thickness of 1 unit...
)Given a 4-node element in x-y plane as shown here: Node X 3 3 1 8 a) Using the shape functions in u-v plane, determine an expression for mapped points from u-v to x-y, i.e. x- x(u, v) and y -y(u...
)Given a 4-node element in x-y plane as shown here: Node X 3 3 1 8 a) Using the shape functions in u-v plane, determine an expression for mapped points from u-v to x-y, i.e. x- x(u, v) and y -y(u, v), for points within the 4-node element in u-v plane. Then, determine value of x and y for a point with (u, v)-(0.3,0.3). (10 points) b) Determine the value of Jacobian matrix, [J], and its determinant for such mapping...
Figures 1 (a) and (b) show a linear square plane stress element in st coordinates and...
Figures 1 (a) and (b) show a linear square plane stress element in st coordinates and its distorted form mapped into quadrilateral in x-y coordinates, respectively. Edge - 1 u (*3. y) (-1,1) 1 1 (1,1) Edge PNJA 5 -Edge s = 1 (XY) 2 (xz. Y2) 2 Edge 1 = -1 (-1,-1) (1, -1) (b) Figure 1 (a) Specify the interpolation functions for the four-node isoparametric quadrilateral element. (4 marks) (b) Discuss how the isoparametric formulation is applied into...
3- Consider a thin elastic plane of thickness t with a given in-plane displacement of (u(x,...
3- Consider a thin elastic plane of thickness t with a given in-plane displacement of (u(x, y) =şd,67)+ d2m2 |u(x,y) = d763)2 +d263 a) Find an expression for all the strain components on this plane. (10 points) b) Find an expression for all the stress components on this plane. (10 points) c) Write down the integral equation to calculate the force on edge A of this plane. (you do not need to solve the integral). (5 points) d) Write down...
3- Consider a thin elastic plane of thickness t with a given in-plane displacement of ſu(x,...
3- Consider a thin elastic plane of thickness t with a given in-plane displacement of ſu(x, y) = -4,6)3 + d2012 (u(x,y) = _d203)2 ++d793 a) Find an expression for all the strain components on this plane. (10 points) b) Find an expression for all the stress components on this plane. (10 points) c) Write down the integral equation to calculate the force on edge A of this plane. (you do not need to solve the integral). (5 points) d)...
Problem 4. Give an example of a linear operator T on a finite-dimensional vector space such...
Problem 4. Give an example of a linear operator T on a
finite-dimensional vector space such that T is not nilpotent, but
zero is the only eigenvalue of T. Characterize all such
operators.
Problem 5. Let A be an n × n matrix whose characteristic
polynomial splits, γ be a
cycle of generalized eigenvectors corresponding to an
eigenvalue λ, and W be the subspace spanned
by γ. Define γ′ to be the ordered set obtained from γ by
reversing the...
Question 1 (10 marks) Consider the bar element shown in Figure 1. Cross sectional area A...
Question 1 (10 marks) Consider the bar element shown in Figure 1. Cross sectional area A = 200 mm and Young's modulus E = 200 x 107 kPa. If u = 3 mm and u2 = 2 mm, determine: (a) the displacement at point P, (b) the strain, (c) the stress, and (d) the strain energy in the element. U] U2 x = 400 mm xi = 200 mm x2 = 700 mm Figure 1 EAL 2 Note: the strain...
Problem 2. Consider a finite element with shape functions N1 (ξ) and N2Ģ) used to interpolate the displacement field within the element shown below. Derive an expression for the strain-displacement matrix B where strain E-Bq, in terms of N1 and N2. (Do not assume any specific form for N and N2.) (Note: q[1 21.) 72--
3.4. Consider a finite element with shape functions N(E) and N2(E) used to interpolate the displacement field within the element (Fig. P3.4) 2 1 9-> 92-> +1 6-1 FIGURE P3.4 Derive an expression for the strain-displacement matrix B, where strain e Bq, in terms of N and N. (Do not assume any specific form for N or N.) (Note: qa )
QUESTION 2 21 Finite elements can appear in many forms such as two-dimensional and three- dimensional domains Give two examples and a sketch for each domain. (4) 22 Explain the following terms as used in Finite element equations a Plain stress b Plain strain 23 Use the Finite element method to develop the stiffness matrix for element 2 of the steel cantilever beam structure shown in Figure 2 The elastic modulus IS 200 kN/mm2 with a thickness of 1 unit...
)Given a 4-node element in x-y plane as shown here: Node X 3 3 1 8 a) Using the shape functions in u-v plane, determine an expression for mapped points from u-v to x-y, i.e. x- x(u, v) and y -y(u, v), for points within the 4-node element in u-v plane. Then, determine value of x and y for a point with (u, v)-(0.3,0.3). (10 points) b) Determine the value of Jacobian matrix, [J], and its determinant for such mapping...
Figures 1 (a) and (b) show a linear square plane stress element in st coordinates and its distorted form mapped into quadrilateral in x-y coordinates, respectively. Edge - 1 u (*3. y) (-1,1) 1 1 (1,1) Edge PNJA 5 -Edge s = 1 (XY) 2 (xz. Y2) 2 Edge 1 = -1 (-1,-1) (1, -1) (b) Figure 1 (a) Specify the interpolation functions for the four-node isoparametric quadrilateral element. (4 marks) (b) Discuss how the isoparametric formulation is applied into...
3- Consider a thin elastic plane of thickness t with a given in-plane displacement of (u(x, y) =şd,67)+ d2m2 |u(x,y) = d763)2 +d263 a) Find an expression for all the strain components on this plane. (10 points) b) Find an expression for all the stress components on this plane. (10 points) c) Write down the integral equation to calculate the force on edge A of this plane. (you do not need to solve the integral). (5 points) d) Write down...
3- Consider a thin elastic plane of thickness t with a given in-plane displacement of ſu(x, y) = -4,6)3 + d2012 (u(x,y) = _d203)2 ++d793 a) Find an expression for all the strain components on this plane. (10 points) b) Find an expression for all the stress components on this plane. (10 points) c) Write down the integral equation to calculate the force on edge A of this plane. (you do not need to solve the integral). (5 points) d)...
Problem 4. Give an example of a linear operator T on a
finite-dimensional vector space such that T is not nilpotent, but
zero is the only eigenvalue of T. Characterize all such
operators.
Problem 5. Let A be an n × n matrix whose characteristic
polynomial splits, γ be a
cycle of generalized eigenvectors corresponding to an
eigenvalue λ, and W be the subspace spanned
by γ. Define γ′ to be the ordered set obtained from γ by
reversing the...
Question 1 (10 marks) Consider the bar element shown in Figure 1. Cross sectional area A = 200 mm and Young's modulus E = 200 x 107 kPa. If u = 3 mm and u2 = 2 mm, determine: (a) the displacement at point P, (b) the strain, (c) the stress, and (d) the strain energy in the element. U] U2 x = 400 mm xi = 200 mm x2 = 700 mm Figure 1 EAL 2 Note: the strain...
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