CVE 412/CVE 511 Matrix Analysis of Framed Structures

Spring Semester, 2011

Final Exam Take Home Name Student Identification # Please show all your work. Output from computer programs are not acceptable, e.g., MathCAD, MatLab, Mathmatica, etc. Answers without justification can not be given partial credit. Answers without the proper units are considered incorrect. Neatness may improve your score. You have until 5:00 PM Friday May 13, 2011 to complete this exam. You may turn the exam in earlier in a sealed envelope to the Department Secretary. Problem #1 (40 points) Assume torsional stiffness as well as axial stiffness are significantly larger than flexural stiffness. Also assume no biaxial bending. Further assume a right handed global coordinate system for the structure where the Y-axis is up on the page, the X-axis is to the right and the Z-axis is out of the page. Under these assumptions for the beam shown below:

(a) Determine the degree of kinematic indeterminacy (5 points) (b) Develop the [SM] matrix for each beam segment. (10 points) (c) Number the joints consecutively (i.e., A , B 2, C 3 and D 4) and number

the potential displacements at each joint in the order YT and ZR (T translation, R rotation) and determine the size of the [SJ] matrix (5 points)

(d) Develop a joint restraint list and a scheme for renumbering the joint restraints (5

points) (e) Develop the initial [SJ] matrix and identify the original restraint numbering system

across the top and along the left of the matrix along with your renumbering scheme along the bottom and down the right hand side of the matrix. (15 points)

CVE 412/CVE 511 Matrix Analysis of Framed Structures - Midterm Exam

Name Problem #2 (40 Points) Assume torsional stiffness as well as axial stiffness are significantly larger than flexural stiffness. Also assume no biaxial bending. The beam is constructed from the same material in all segments of the beam. For the continuous beam shown below:

a. Formulate the stiffness matrix [S] and all pertinent matrices to find the shear and bending

moment (in that order) to the left of joint B and the shear and bending moment reaction (in that order) at joint A. (30 points)

b. Show the changes necessary in the calculations of the shear and bending moment at joint

A if there is a support displacement at joint C. (10 points)

CVE 412/CVE 511 Matrix Analysis of Framed Structures - Midterm Exam

Name Problem #3 (40 Points) Develop the joint stiffness matrix [SJ] for the beam shown below. Then rearrange the stiffness matrix and extract from a partitioned [SJ] the stiffness matrix [S]

CVE 412/CVE 511 Matrix Analysis of Framed Structures - Midterm Exam

Name Problem #4 (40 Points) GRADUATE STUDENT PROBLEM Develop the [SM] matrix for each beam segment shown below. Neglect axial deformations, torsion and biaxial bending. Then develop the joint stiffness matrix [SJ] for the beam shown below. All segments: E = 200,000 MPa = 0.3 Segment AB: I = 100 x 106 mm4 J = 100 x 103 mm4 Segment BC: I = 300 x 106 mm4 J = 500 x 103 mm4 Segment CD: I = 200 x 106 mm4 J = 300 x 103 mm4

CVE 412/CVE 511 Matrix Analysis of Framed Structures - Midterm Exam

Name Problem #5 (40 Points) Explain how to find the displacements underneath the point loads and member end actions for the continuous beam shown below using the stiffness method. Use matrix based equations from the class notes to develop your explanation. This explanation need not be lengthy, but it must be thorough, as in you must explain how to find the displacements and how to find the end actions. This was an in class problem and a homework problem. Based on your assumptions in your explanation develop the [SM] matrix for all beam segments. Neglect axial deformations, torsion and biaxial bending. Then develop the joint stiffness matrix [SJ] for the beam. Once again assume the distributed load is such that

The beam has constant flexural rigidity EI.

PwL