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Problem 1 Consider the bar shown below with a cross-sectional area A, 1.2 m2, and Young's...
5) Consider a bar shown below. Cross-sectional area Ae = 1.2 in., and Young's modulus E...
5) Consider a bar shown below. Cross-sectional area Ae = 1.2 in., and Young's modulus E = 35 x 106 psi. If ui = 0.02 in, and u2 = 0.025 in., considering linear interpolation, determine the following: (a) the displacement at point P; (b) the strain & and stress o; and (c) the element stiffness matrix. u2 2 x = 20 in. x = 15 in. X = 23 in.
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--
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...
Q1 An elastic cantilever beam of varying cross section, as shown in Figure Q1(a), is subjected...
Q1 An elastic cantilever beam of varying cross section, as shown in Figure Q1(a), is subjected to an increase in temperature of 60°C in an unnatural environment. The equation governing the displacement of the elastic column and the finite element stiffness matrix are respectively given as -O and ΑΕ) - where A is the cross sectional area of the beam, E is the Young's modulus of the beam material, u is the displacement and / is the finite element length....
The members of the truss shown below have a cross sectional area of 0.0002m^2 and Young's modulus of E=69GPa. De...
The members of the truss shown below have a cross
sectional area of 0.0002m^2 and Young's modulus of E=69GPa.
Determine the deflection at each joint using Finite Element
Method.
500 N 500 N Fuc (compression) Faa (tension) 2 m 500 N 45° 45" ac (compression) Fas (tension)
500 N 500 N Fuc (compression) Faa (tension) 2 m 500 N 45° 45" ac (compression) Fas (tension)
3. Consider a two-d.o.f bar element, as shown below, but let the cross-sectional area vary linearly...
3. Consider a two-d.o.f bar element, as shown below, but let the cross-sectional area vary linearly with x from Ag atx-0 to 2Ao at x - L Use the direct method to generate the element stiffness matrix. Suggestion: first compute the elongation produced by the axial force. a. b Use the formal procedure to generate the stiffness matrix. Suggestion use Eqn. 2.2.6 c The stiffness matrices of parts a and b do not agree. Why? ก็เ«F21 A E Fiz al...
3. 2090] Consider a uniform bar of Young's modulus E, cross-sectional area A, moment of inertia d...
3. 2090] Consider a uniform bar of Young's modulus E, cross-sectional area A, moment of inertia density p, length L, with an attached end mass, m, connected to a rigid wall via a linear spring of spring constant, k, see Figure. Let the longitudinal vibration of the bar be Wa.f). (a) [4] Write down the boundary conditions. m E, p Boundary condition at x 0 Boundary condition at x L (b) [81 Derive the equation for the natural frequency (c)...
X=0 x = 1/2 x= L u U2 Uz (a) Trial solution for a 1-D quadratic...
X=0 x = 1/2 x= L u U2 Uz (a) Trial solution for a 1-D quadratic elastic bar element can be written as follows: ū(x) = [N]{u} where, [N] = [N1 N2 N3] and {u} u2 13 1 and Ni L2 L2 [N] and {u} are known as interpolation function matrix and nodal displacement, respectively. (272 – 3L + L´), N= = (22- La), Ns = 12 (2=– LE) Derive the expression for element stiffness matrix, (Kelem) and element force...
Figure < 1 of 1 Consider, for instance, a bar of initial length L and cross-sectional...
Figure < 1 of 1 Consider, for instance, a bar of initial length L and cross-sectional area A stressed by a force of magnitude F. As a result, the bar stretches by AL (Figure 1) Let us define two new terms: • Tensile stress is the ratio of the stretching force to the cross-sectional area: stress = 5 • Tensile strain is the ratio of the elongation of the rod to the initial length of the bar strain= 41 It...
A plane truss element is shown in Figure 4, All elements have cross-sectional area of A = 8 in, a...
A plane truss element is shown in Figure 4, All elements have cross-sectional area of A = 8 in, and elastic modulus of E-2 x 10° psi. Use long-hand solution 6. 6.(a). Solve for the unknown displacements. 6.(b). Solve for strains and stresses in al 3 elements. Show your work and follow the finite element method matrix formulation we have covered in lectures. 4 5 kip 10 240 ft 30 ft30 ft Figure 4.
A plane truss element is shown...
5) Consider a bar shown below. Cross-sectional area Ae = 1.2 in., and Young's modulus E = 35 x 106 psi. If ui = 0.02 in, and u2 = 0.025 in., considering linear interpolation, determine the following: (a) the displacement at point P; (b) the strain & and stress o; and (c) the element stiffness matrix. u2 2 x = 20 in. x = 15 in. X = 23 in.
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--
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...
Q1 An elastic cantilever beam of varying cross section, as shown in Figure Q1(a), is subjected to an increase in temperature of 60°C in an unnatural environment. The equation governing the displacement of the elastic column and the finite element stiffness matrix are respectively given as -O and ΑΕ) - where A is the cross sectional area of the beam, E is the Young's modulus of the beam material, u is the displacement and / is the finite element length....
The members of the truss shown below have a cross
sectional area of 0.0002m^2 and Young's modulus of E=69GPa.
Determine the deflection at each joint using Finite Element
Method.
500 N 500 N Fuc (compression) Faa (tension) 2 m 500 N 45° 45" ac (compression) Fas (tension)
500 N 500 N Fuc (compression) Faa (tension) 2 m 500 N 45° 45" ac (compression) Fas (tension)
3. Consider a two-d.o.f bar element, as shown below, but let the cross-sectional area vary linearly with x from Ag atx-0 to 2Ao at x - L Use the direct method to generate the element stiffness matrix. Suggestion: first compute the elongation produced by the axial force. a. b Use the formal procedure to generate the stiffness matrix. Suggestion use Eqn. 2.2.6 c The stiffness matrices of parts a and b do not agree. Why? ก็เ«F21 A E Fiz al...
3. 2090] Consider a uniform bar of Young's modulus E, cross-sectional area A, moment of inertia density p, length L, with an attached end mass, m, connected to a rigid wall via a linear spring of spring constant, k, see Figure. Let the longitudinal vibration of the bar be Wa.f). (a) [4] Write down the boundary conditions. m E, p Boundary condition at x 0 Boundary condition at x L (b) [81 Derive the equation for the natural frequency (c)...
X=0 x = 1/2 x= L u U2 Uz (a) Trial solution for a 1-D quadratic elastic bar element can be written as follows: ū(x) = [N]{u} where, [N] = [N1 N2 N3] and {u} u2 13 1 and Ni L2 L2 [N] and {u} are known as interpolation function matrix and nodal displacement, respectively. (272 – 3L + L´), N= = (22- La), Ns = 12 (2=– LE) Derive the expression for element stiffness matrix, (Kelem) and element force...
Figure < 1 of 1 Consider, for instance, a bar of initial length L and cross-sectional area A stressed by a force of magnitude F. As a result, the bar stretches by AL (Figure 1) Let us define two new terms: • Tensile stress is the ratio of the stretching force to the cross-sectional area: stress = 5 • Tensile strain is the ratio of the elongation of the rod to the initial length of the bar strain= 41 It...
A plane truss element is shown in Figure 4, All elements have cross-sectional area of A = 8 in, and elastic modulus of E-2 x 10° psi. Use long-hand solution 6. 6.(a). Solve for the unknown displacements. 6.(b). Solve for strains and stresses in al 3 elements. Show your work and follow the finite element method matrix formulation we have covered in lectures. 4 5 kip 10 240 ft 30 ft30 ft Figure 4.
A plane truss element is shown...
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