Answer Happy

Consider the following four structures shown below: A, B, C, and D. Each has extent L = 1 m along the x, direction. Each is loaded with force F in the -x, direction, as shown. All structures are made of aluminum (linear elastic isotropic with Young's modulus E = 70 GPa, Poisson's ratio v=0.3, volumetric density p= 2700 kg/m³). All structures have deflection & in the-x3 direction at the point of application of the force. The structures have following properties: A. Beam with uniform circular cross-section with radius R₁ = 10 mm. B. Shaft with uniform circular cross-section with radius Rg, with a perfectly rigid massless rod of length = 50 mm attached perfectly to the center of the tip of the shaft, with the force F applied at the end of this rod. C. Beam with circular cross-section with radius Rc at the base, and radius 0.2R at the tip. Betweeen these, the radius variues linearly with x₂- D. Pin-jointed truss with both rods having cross-sectional area Ap. Assume this structure is constrained to have no deflection in the x₂ direction, and that it is initially stress-free. Structures A, B, and C are clamped to the foundation at the base as shown, and the nodes of the pin- jointed truss (structure D) are fixed as shown. Assume small deflections and small slopes. D B R₁ P 0.28₂

Determine the stiffness F/8 for structure A. ii. Determine R such that stiffness of structure B (i.e., the ratio F/8) equals that of structure A. Model structure B as a superposition of a beam in bending and a shaft in torsion, with the bending and torsion behaviors uncoupled. That is, the bending of structure B is as though a tip force is applied at the centroid of the cross-section, and the torison of structure B is as though a tip torque is applied. The deflection & may be determined by considering the sum of the deflections from these two effects. iv. Determine Re such that the stiffness F/8 of structure C equals that of structure A. Determine the reduced stiffness matrix for structure D that satisfies the following: [K₁1 K12]| where we use the following bar and node numbering scheme. For this numbering scheme, the equlibrium matrix A is as follows. For our particular loading condition, f11 = 0, and f13 = -F. d33-f33 das fas 1 1/√2 0 -1/√2 -1 0 0 0 0 -1/√2 11 0 1/√2 [₁1] f13 d13-13 f21 A[₁] = f23 dun I f31 f33 1 Ľ You may find it helpful to remember that the full stiffness matrix K for this structure would satisfy the following relation: [d₁1] d₂3 f13 K d21 f21 d23 = f23 d31 f31 Id33] However, you know that d21, d23, d31. d32 are zero, and we are NOT concerned with the reaction forces at the fixed nodes, so we only need the reduced stiffness matrix that is the the sub-matrix (the first two rows x the first two columns) of K that satisfies (+) above. Determine A, such that the stiffness F/8 of structure D equals that of structure A. V. 3 d23-123 A=