UNIVERSITY OF CALIFORNIA, BERKELEY Spring Semester 2014 Dept. of Civil and Environmental Engineering Structural Engineering, Mechanics and Materials Name:......................................... M.S Comprehensive Examination Analysis Note: 1. Dimensions, properties and loading are given in consistent units in all problems. 2. All figures are drawn to scale. 3. Calculations should be shown in detail with all intermediate steps; it is recommended to manipulate expressions symbolically as far as possible and substitute numbers only at or near the end. 4. Results involving multiplication or division with a matrix larger than 2 x 2 will not receive credit. 1
Problem Fig. 1(a) and (b) show two structural models. Model A consists of two inextensible frame elements a and b and a frame element c, which is in addition inflexible. Model B consists of two inextensible frame elements a and b. All flexible frame elements have the same flexural stiffness EI. All elements have the same length L. Both structural models are subjected to a horizontal force P h. You are asked to answer the following questions for both structural models: 1. Determine the horizontal translation at the point of application of P h in terms of P h, L and EI. 2. Draw the bending moment distribution under P h. P h 2 b 3 P h 2 b 3 a EI c L a L 1 4 1 L L (a) Structural model A (b) Structural model B Figure 1: Two structural models 2
UNIVERSITY OF CALIFORNIA, BERKELEY SPRING SEMESTER 2012 Department of Civil and Environmental Engineering Structural Engineering, Mechanics and Materials Name: M.S. COMPREHENSIVE EXAMINATION ANALYSIS NOTE: 1. Dimensions, properties and loading are given in consistent units in all problems. 2. The following symbol indicates a roller, i.e. free horizontal translation and free rotation. 3. Note that all figures for the structural models are to scale. 4. Calculations should be shown in full detail with numerical results for all intermediate steps. Results involving multiplication or division with a matrix larger than 2 x 2 will not receive credit. 1
1. Problem (50% weight) Determine the collapse load factor λ c of the propped cantilever in the figure for the given concentrated force P in terms of the half span length L and the plastic flexural capacity M. p Determine the location of the last hinge to form and the vertical translation at the point of load application at incipient collapse. P L L 2
2. Problem (50% weight) The horizontal translation of the braced frame in the figure under a horizontal force of 10 units at node 3 is 1.5315 10 2 units. Frame elements a and b can be considered inextensible. Determine the flexural stiffness EI of elements a and b and the axial stiffness EA of the brace element c, if EI/EA=2. 2 b EI 3 1.5315 10 2 10 8 a EI c EA 1 6 3
UNIVERSITY OF CALIFORNIA, BERKELEY SPRING SEMESTER 2011 Department of Civil and Environmental Engineering Structural Engineering, Mechanics and Materials Name: M.S. COMPREHENSIVE EXAMINATION ANALYSIS NOTE: 1. Dimensions, properties and loading are given in consistent units in all problems. 2. The following symbol indicates a roller, i.e. free horizontal translation and free rotation. 3. Note that all figures for the structural models are to scale. 4. Calculations should be shown in full detail with numerical results for all intermediate steps. Results involving multiplication or division with a matrix larger than 2 x 2 will not receive credit. 1
1. Problem (50% weight) For the beam in the figure determine the vertical deflection under the applied force P at incipient collapse. The flexural plastic capacity of the beam is M and the flexural stiffness is EI. p P L L 2
2. Problem (50% weight) The beam in the figure is subjected to a concentrated force of 10 units. The beam has uniform stiffness EI. 1. Determine the bending moment at the fixed end and over the middle support. 2. Draw the bending moment diagram. 3. Sketch the deformed shape of the beam under the given loading. 10 EI EI EI 10 4 6 3
UNIVERSITY OF CALIFORNIA, BERKELEY SPRING SEMSTER 2009 Department of Civil and Environmental Engineering Structural engineering, Mechanics and Materials Name: COMPREHENSSIVE EXAMINATION ANALYSIS NOTE: Dimensions, properties and loading are given in consistent units in all problems.
Problem 1 What is the degree of static indeterminacy of the three structural models shown below? To receive full credit provide the number of basic element forces (force unknowns) and the corresponding number of necessary equilibrium equations. For structure A only, what is the number of independent free global degrees of freedom (dof's), if the elements labeled a, b, and c are assumed to be inextensible? Structure A Structure B Structure C
Problem 2 The structure shown below is subjected to thermal extension strain of α ΔT = 6 10-4 in member d. Members a and b have flexural stiffness EI a =3 10 5 and EI b =2 10 5 and negligible axial deformations. Members c, d and e have axial stiffness EA c =1 10 4, EA d =1.5 10 4, and EA e =5 10 3. a) Set up the equilibrium equation for the smallest number of free degrees of freedom possible. b) Solve for the internal forces in all members. c) Determine the vertical displacement at node 4. 10 20 20
UNIVERSITY OF CALIFORNIA, BERKELEY SPRING SEMESTER 2008 Department of Civil and Environmental Engineering Structural Engineering, Mechanics and Materials Name: M.S. COMPREHENSIVE EXAMINATION ANALYSIS NOTE: Dimensions, properties and loading are given in consistent units in the problem. Problem A computer analysis of the structural model in the figure under the given loading gives the following values for the basic forces of primary interest, i.e. the end moments of elements a through c and the axial force in element d: T ( 87.947 54.824 54.824 22.844 22.844 15.497) Q =. Note that the end moments are numbered in sequence starting from the fixed end of element a. The axial force is listed last. 1. Determine the axial forces in elements a through c. 2. Draw the bending moment diagram. 3. Determine the support reactions and check global equilibrium. 4. Use the principle of virtual displacements to confirm the value of the horizontal support reaction at nodes 1 and 4 without the knowledge of the axial forces in elements a through c. 5. Draw the deformed shape of the structural model under the given loading. Assume that elements a through c are inextensible and prismatic with flexural stiffness EI. 3 30 c 4 b d 6 1 a 2 8 8 1