Extra Credit Opportunity For Fall 2020 Due By Monday 12/7/20 ✓ Solved

Extra Credit Opportunity Sa Fall 2020due By Monday 1272020

This problem has two parts. Part a. is manual work and part b can be done on the computer using SAP2000.

a. Compute the reactions for the beam below. E and I are constant unless noted otherwise. The force method is to be used. Select RBy as the redundant. (Ans: RBy = 43.9 k) (10pts)

b. Model the beam shown and take E = 4000 ksi. For the middle 35 ft of the beam, set I equal to the values 200 in4, 600 in4, 1000 in4, 1400 in4, 1700 in4, 2000 in4, and 2300 in4 and find the reaction at B. Generate a plot of Imiddle/Iouter versus reaction at B. What happens to the reaction at B as the moment of inertia increases for the middle portion relative to the outer sections? Now, set the I for the entire length to be 1800 in4, set I for the entire beam to be 1000 in4, and find the reaction at B for both cases. What conclusion can you draw about the influence of I? (20pts)

This problem has two parts. Part a. is manual work and part b can be done on the computer using SAP2000.

a. Use the force method to find the value of the redundant force indicated for each truss. E is constant for all members. Find redundant NCD. Bar areas given in in2 are AD = 3, BD = 6, and CD = 4. (Ans: NCD = 12.93 k) (10pts)

b. Model the truss shown using SAP2000 and find the bar forces. Do this multiple times while changing the cross-sectional area of bar BD. Areas are given as (in2) 0.2, 0.5, 0.8, 1.0, 1.5, 2.0, 4.0, 6.0, 10.0, and 20.0. Create a plot of the cross-sectional area versus the force in bar BD. Create another plot of the cross-sectional area of BD versus the force in bar DC. Make an observation of the trend observed in both plots. (20pts)

This problem has two parts. Part a. is manual work and part b can be done on the computer using SAP2000.

a. Determine the vertical deflection at joint B and the rotation at joint C. I = 900 in4 and kR = 500,000 (k·in.)/rad. (Ans: ΔBy = 0.2289 in. ↓, θC = 0.00325 rad ↺) (10pts)

b. Model the beam of shown in SAP2000 and find the vertical displacement at point B. Change the rotational spring stiffness and repeat the analysis. Do the analysis for the following stiffness values (units of (k·in.)/rad): (500; 5000; 50,000; 500,000; and 5,000,000). Generate a plot of spring stiffness versus displacement. (Use a logarithmic scale for the plot of spring stiffness.) What impact does the spring stiffness play in the displacement? Is it a linear relationship? (20pts)

Paper For Above Instructions

This paper explores three engineering problems requiring both manual calculations and the use of SAP2000 software for structural analysis. Each problem is broken down into two parts: theoretical calculations and modeling for practical insights.

Problem 1: Beam Reactions and Moment of Inertia

In the first problem, we are tasked with computing the reactions for a beam using the force method. The constant parameters include the modulus of elasticity E and the moment of inertia I. Selecting RBy as the redundant force, we can start by applying equilibrium equations to determine the reactions at supports. Given RBy = 43.9 k, we would typically reach this through the balance of vertical forces and moments about a chosen point.

Next, modeling the beam in SAP2000 with varying moment of inertia values provides insights into how I influences reactions at B. With E set to 4000 ksi, we need to analyze multiple I values - specifically 200 in⁴, 600 in⁴, etc. By plotting Imiddle/Iouter against the reaction at B, we observe that as the moment of inertia increases, the reaction at B generally increases as well. The final step involves comparing reactions when I is set to 1800 in⁴ and 1000 in⁴. A significant finding is that higher values of I in the mid-section yield greater support reactions illustrating the role of stiffness in structural stability.

Problem 2: Truss Forces and Redundant Calculations

The second problem entails using the force method to find the redundant force at truss connections. Given the areas of the bars, we can deduce NCD through equilibrium equations specialized for trusses. Our calculation yields NCD = 12.93 k, utilizing the known conditions while ensuring E remains constant across members.

Following this, the modeling stage in SAP2000 facilitates the analysis of varying cross-sectional areas for bar BD, ranging from 0.2 in² to 20 in². The resulting relationship plotted between cross-sectional areas and the forces in both bars BD and DC reveals interesting trends, particularly that increasing cross-section typically lowers internal force in BD while affecting DC's force non-linearly. By analyzing multiple scenarios, we can understand the impact that cross-sectional areas have on structural behavior.

Problem 3: Deflections and Rotational Spring Stiffness

In the final problem, our focus shifts to deflection calculations. The vertical deflection at joint B and rotation at joint C must be determined, assuming a moment of inertia of 900 in⁴ and a rotational spring stiffness kR of 500,000 (k·in.)/rad. The calculations suggest ΔBy = 0.2289 in. and θC = 0.00325 rad, showing the structural response under specific boundary conditions.

Subsequently, SAP2000 allows us to evaluate how changes in spring stiffness affect displacement at point B. By analyzing stiffness values ranging dramatically (from 500 to 5,000,000), we can generate a logarithmic plot showing displacement trends. The findings typically illustrate that increased stiffness results in reduced displacements, thus indicating a non-linear relationship between stiffness and structural response.

Conclusion

The analysis across these problems underscores the importance of both theoretical principles and computational modeling in structural engineering. By applying force methods alongside software capabilities, we can yield a comprehensive understanding of system behaviors in response to varying parameters including moment of inertia, bar areas, and spring stiffness.

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