A beam of length /= 1m, which is clamped at both ends, is modelled using four flexural elements. The first modeshape of the beam is described by: O e O 0.544 3.045 1.000 0.000 0.544 -3.045 Note that Y₁ = ₁=Y₁ == 0 due to the boundary conditions. Recalling that displacement of a flexural element is given by: 1 1 5 5 y=S₂(x) y₂ +S₂(x) + S₂(x) y₂ + ₂(x) ₂ O 1 2 where the shape functions are given by: $₁(x) = 1-3 ( ² ) ² + 2 (²) ² S₂(x)=x-2-² +²2 L L2 $₂(x) = 3 ( ² ) ² - ² (+) ² x² x3 + S₂(x)=-I L² Find the displacement of the first modeshape at position x=//8. Express your answer to three decimal places and do not include units in your answer.

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There is a beam AD subjected to two concentrated forces, as shown in the figure below, the Young's coefficient of the full beam material is E:(a) Try to find the slope of point A after deformation.(b) Try to find the (vertical) displacement of point B after deformation.

A steel rod (E = 200 GPa) is fixed at A and subjected to the loading shown in the figure. The cross-sectional area of the rod is 50 mm². Neglect the size of the couplings at B, C and D. Calculate the displacement of the rod at D when F₁ = 2.8 kN, F₂ = 4.3 kN and F3 = 7.3 kN. Input your answer as a value only in mm to two decimal places without +/-sign. 1m 1.5 m 1.25 m C F2 D F3 A F₁ B

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The collar C is fixed to AB and supports a weight W=15 N, acting in the negative z direction. What is the vector component of the weight that is parallel to rod?

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Rigid bar ABC is supported by bronze rod (1) and stainless steel rod (2) as shown in the figure. A concentrated load of P = 16 kips is applied to the free end of bronze rod (3). Determine the magnitude of the deflection of rod end D after the temperature of all rods has increased by 125°F. Use the following dimensions and properties: a = 2.8 ft, b = 6.3 ft. Rod (1): d₁ = 0.90 in., L₁ = 11 ft, E₁ = 15,200 ksi, α₁ = 12.20 × 10-6/°F. Rod (2): d₂ = 0.80 in., L₂ = 9 ft, E2 = 28,000 ksi, a₂ = 9.60 × 10-6/F. Rod (3): d3 = 1.25 in., L3 = 5 ft, E3 = 15,200 ksi, a3 = 12.20 x 10-6/°F. VD = i L in. (1) A a L3 B (3) D b Rigid bar L2

PROBLEM2: The displacement amplitude u, of an SDF system due to harmonic force is known for two excitation frequencies. At w=wn, up = 15 cm; at w=5w, u, = 0.06 cm. Estimate the damping ratio of the system. @= PROBLEM3: Find the response of the rigid bar shown in Figure 2 when the end P of the spring PQ is subjected to the displacement, x(t) = xoSin10t. Data: k 500, 1 = 1 m, m = 10 kg, xo = 1 cm. N = m 00000 Uniform bar, mass m - Li+ Figure 2-Rigid bar undergoing rotational motion

Use the stiffnes method to determine all the displacement components at node 2 and all the reactions including the internal reactions at node 2. Show all calculations. Draw the Bending Moment Diagram of the frame. Show all calculations. Node 1 and 3 are fixed. L3 = 6.5m  L4 = 13m  w = 6.0 kNm P = 20 kN I = 300 x 10^6 mm^4 A = 10 x 10^3 mm^2 E = 200 GPa