· the plastic exural capacity m p of element a is 180 units. the plastic axial capacity n p of...

UNIVERSITY OF CALIFORNIA, BERKELEY Dept. of Civil and Environmental EngineeringSpring Semester 2014 Structural Engineering, Mechanics and Materials

Name: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Ph.D. Preliminary 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 manipulateexpressions 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

1. Problem (50% weight)

The structural model in Fig. 1 consists of 4 inextensible frame elements a, b, c, and d. The frameelement b is in addition inflexible. The flexural stiffness EI of the other frame elements is equal to10,000 units.

The structure is subjected to a uniform element loading w of 10 units in element a, as shown in Fig. 1.

You are asked to answer the following questions:

1. Determine the bending moment distribution M(x) under the distributed load w and sketch thebending moment diagram.

2. Determine the vertical translation at midspan of element a?

wa

c

d

b

1 2

8 3

3

4

3

6

3 4

5

EI

Figure 1: Frame with inflexible element b under uniform load w

2


The structural model in Fig. 2 consists of 3 frame elements a through c, and a brace element d. Allframe elements are inextensible and have flexural stiffness EI of 200,000 units. The plastic flexuralcapacity Mp of elements b and c is equal to 160 units. The plastic flexural capacity Mp of element ais 180 units. The plastic axial capacity Np of the brace element d is 20 units and its axial stiffness EAis equal to 40,000 units. All frame elements have very high plastic axial capacity. The presence of anaxial force in the frame elements does not affect the plastic flexural capacity.

The structure is subjected to a horizontal force of 30 units at node 2 and a vertical force of 30 units atnode 3, as shown in Fig. 2.

Under the assumption of linear elastic, perfectly-plastic element response you are asked to answer thefollowing questions:

1. Determine the collapse load factor λc under the given loading.

2. Determine the horizontal translation at node 2 at incipient collapse?

a

c

d

b

1

2 34

3030

4 4

6

Figure 2: Collapse load determination of braced frame

3

UNIVERSITY OF CALIFORNIA, BERKELEY Dept. of Civil and Environmental EngineeringFall Semester 2013 Structural Engineering, Mechanics and Materials

Name: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Ph.D. Preliminary Examination

Analysis

Note:


2. All figures are drawn to scale.

3. Calculations should be shown in detail with all intermediate steps; it is recommended to manipulateexpressions 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


The structural model in Fig. 1(a) consists of 3 frame elements a, b and c, and a brace (truss) elementd. The frame elements can be assumed as inextensible. Furthermore, element b can be assumed asinflexible as well. The frame elements a and c have flexural stiffness EI of 100,000 units, while thetruss element d has axial stiffness EA of 20,000 units.

The structure is subjected to a uniform element loading w of 10 units in element a, as shown in Fig. 1(a).

You are asked to answer the following questions:

1. Determine the bending moment distribution M(x) under the distributed load w and sketch thebending moment diagram in Fig. 1(b).

2. How should the brace (truss) element be prestressed so as to eliminate the horizontal translation atthe roller support (node 1)?

21

3

4

8 8

6

6

a

b

c

d

w=10

(a) Model geometry and loading (b) Bending moment diagram M(x)

Figure 1: Braced frame under uniform element loading w

2


The structural model in Fig. 2 consists of 3 frame elements a through c. All frame elements areinextensible and have flexural stiffness EI of 200,000 units. The plastic flexural capacity Mp of elementsa and c is equal to 240 units. The plastic flexural capacity Mp of element b is 300 units. All frameelements have very high plastic axial capacity. The presence of an axial force in the frame elementsdoes not affect the plastic flexural capacity.

The structure is subjected to a vertical force of 30 units at node 2 and a horizontal force of 50 units atnode 3, as shown in Fig. 2.

Under the assumption of linear elastic, perfectly-plastic element response you are asked to answer thefollowing questions:

1. Determine the collapse load factor λc under the given loading.

2. Determine the horizontal translation at node 3 at incipient collapse?

8 8

6

6

a b

c

1

2

3

4

30

50

Figure 2: Collapse load determination of gable frame

3

1

UNIVERSITY OF CALIFORNIA, BERKELEY Department of Civil and Environmental Engineering SPRING SEMESTER 2012 Structural Engineering, Mechanics and Materials

Name: ________________________________

Ph.D. PRELIMINARY EXAMINATION

ANALYSIS

NOTE:


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.

2


The continuous beam over two spans in the figure is subjected to a concentrated force of 30 units at the middle of the left span, as shown. The middle support consists of a spring with an axial stiffness sk of 2000 units. The flexural stiffness of the beam is 50,000 units. Under this loading the bending moment at the middle support is 32.591 units, as shown.

1. Draw the bending moment diagram of the continuous beam.

2. Determine the vertical translation at the middle of both spans under the given loading.

3. Draw the deformed shape of the continuous beam under the given loading.

10 10 15

3032.591

sk

3


Determine the collapse load factor cλ of the structural model in the figure under the given concentrated force of 50 units at node 2. The plastic moment capacity of elements a and b is 150 units, while their plastic axial capacity is very large. The plastic axial capacity of the truss element c is 30 units.

The flexural stiffness EI of elements a and b is 100,000 units, while their axial stiffness is very large. The axial stiffness EA of the truss element c is 10,000 units. Under these stiffness values, the last hinge forms in the truss element c. Determine the vertical translation at the point of load application at incipient collapse and the plastic hinge rotations in elements a and b.

8 8

6a b

c

50

1

2

3

1


Name: ________________________________


ANALYSIS

NOTE:




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.

2


The structural model in the figure consists of three frame elements, a through c, and one truss element d. It can be assumed that elements a through c are inextensible and that element b is furthermore inflexible. Elements a through c have flexural stiffness EI=20,000. The truss element d has axial stiffness EA=10,000. There is a basic force release at the end of element a next to node #1, as shown. All support dofs at node 4 are restrained. The structure carries a uniform load of 10 units normal to the axis of element a. Furthermore, the truss element is prestressed with an initial force of -50 units (compression).

1. Confirm that the horizontal translation at node 3 under the given loading is 35.2778 10−− ⋅ units (i.e. node 3 displaces to the left). Be sure to provide all intermediate terms of the calculations for full credit.

2. Determine the horizontal translation at node 3 under the separate load case of a unit horizontal translation of the support at node 1.

a

b

c

d

w =10

6 4

3

3

1 2

3

4

3


The structural model in the figure is subjected to a horizontal force of 40 units at node 3 and a vertical force of 60 units at node 5, as shown. All elements can be assumed as inextensible with very large axial plastic capacity. The flexural plastic capacity of the elements is noted next to each element (encircled) in the figure on the next page. The plastic hinge locations at incipient collapse are shown with gray filled circles in the figure on the next page.

1. Determine the collapse load factor λ of the structural model under the given loading.

2. Sketch the collapse mechanism.

3. Determine the basic flexural forces in all elements and draw the bending moment diagram at incipient collapse.

4 4

6

4

a b

c

d

e

60

401

2 3

4

56

4

4 4

6

4

120

120

100

100

120

5


The structural model in the figure is subjected to a vertical force of 40 units, as shown. It consists of two inextensible frame elements a and b and a truss element c. The truss element c is prestressed with an initial tensile force of 20 units.

No element properties are supplied. Instead, the following information is given (see figure on next page): the horizontal translation of the structure without the truss element under a horizontal force of 10 units is 24.5001 10−⋅ units, whereas the horizontal translation of the structure with the truss element under the same horizontal force of 10 units is 21.1803 10−⋅ units.

1. Determine the axial force in the truss element under the vertical force of 40 units and the initial prestressing force of 20 units.

2. Draw the bending moment diagram for elements a and b and supply the necessary value(s).

8 8

6a b

c

40

6

8 8

6

10

8 8

6

a b

a b

c10

24.5001 10−⋅

21.1803 10−⋅

1


Name: ________________________________


ANALYSIS

NOTE:




4. Calculations should be shown in full detail whenever necessary. Results involving multiplication or division with a matrix larger than 2 x 2 will not receive credit.

2


The frame in the figure is subjected to a vertical force of 50 units at node 3 and a horizontal force of 50 units at node 6. The plastic flexural capacity of elements a, c and e is 150 units, while the plastic flexural capacity of elements b and d is 200 units. All elements have very large plastic axial capacity and can be assumed as inextensible. Their flexural stiffness is EI=200,000. The plastic hinge locations at the instant of incipient collapse are shown with gray filled circles in the figure on the next page. The last hinge to form is singled out in the figure.

1. Determine the collapse load factor λ of the model in the figure under the given loading.

2. Determine the basic forces in all elements and draw the bending moment diagram.

For the following questions it suffices to provide symbolic answers by expressing the result in terms of the relevant element basic forces Q. Numerical answers are, of course, welcome, if more convenient for you.

3. Determine the horizontal translation of node 3 at incipient collapse.

4. Determine the plastic deformation at the base of element d at incipient collapse (at node 5).

2 6 6

a

b

c

d

e

6

50

50

12

3 4

5

6

3

2 6 6

6

last hinge to form

4


The braced frame in the figure is subjected to a horizontal force of 50 units, as shown. Elements a through d have flexural stiffness EI and can be assumed as inextensible. The brace elements e and f have axial stiffness EA. Without the braces e and f the horizontal translation under the force of 50 units is 21.7308 10−⋅ units. With the braces the horizontal translation reduces to

21.0397 10−⋅ units.

Determine the value of the axial stiffness EA and of the flexural stiffness EI.

50

8 8

6a

b c

de f

1

2 3 4

5

1

UNIVERSITY OF CALIFORNIA, BERKELEY Department of Civil and Environmental Engineering FALL SEMESTER 2009 Structural Engineering, Mechanics and Materials

Name: ________________________________


ANALYSIS

NOTE:




4. Calculations should be shown in full detail whenever necessary. Results involving multiplication or division with a matrix larger than 2 x 2 will not receive credit.

2


The plastic flexural capacity of the elements a and c is 200 units, the plastic flexural capacity of the elements b and e is 160 units, the plastic flexural capacity of element d is 180 units, and the axial capacity of the brace element f is 20 units (plastic capacities are enclosed in circles in the figure). Elements a through e have very large axial capacity and can be assumed as inextensible. Their flexural stiffness is EI=200,000. The brace element f has axial stiffness EA=30,000. The plastic hinge locations at the instant of incipient collapse are shown with gray filled circles in the second figure. The last hinge to form is singled out in the figure.

1. Determine the collapse load factor λ of the model in the figure under the given loading.

For the following questions it suffices to provide symbolic answers by expressing the result in terms of the relevant element basic forces Q. Numerical answers are, of course, welcome, if more convenient for you.

2. Determine the horizontal and the vertical translation at the points of load application.

3. Determine the plastic deformation at incipient collapse in brace element f.

a

b c

d

e

f

8 6

2

6

5050

160

200

200

160

180 20

3

a

b c

d

e

f

8 6

2

6

5050

1

2 34

56

last hinge to form

5


The structural model in the following figure has three independent free global dofs under the assumption that all elements are inextensible and that element b is inflexible.

Determine the stiffness matrix coefficient K11 and K12 making sure to identify the individual element contributions (subscripts a, c and d for element stiffness EI) and to leave fractions as they appear.

Determine the terms of the initial force vector for the uniformly distributed load in element c.

6 4 5

a

b

c

d

dof 2

dof 1

dof 3

6

3

6 4 5

a

b

c

d

6

3

w=5

6

6 4 5

a

b

c

d

dof 2

dof 1

dof 3

6

3

6 4 5

a

b

c

d

dof 2

dof 1

dof 3

6

3

8


1. Determine the relation between the applied force of 20 units and the force in truss element c in terms of the ratio of the flexural stiffness EI of elements a and b, and the axial stiffness EA of element c. Assume that elements a and b are inextensible.

2. What is the necessary prestressing force in element c to eliminate the vertical translation under the applied force of 20 units?

a bEI EI

c EA

20

8 8

6

UNIVERSITY OF CALIFORNIA, BERKELEY Department of Civil and Environmental Engineering SPRING SEMSTER 2009 Structural engineering, Mechanics and Materials Name: ____________________________________

Ph. D. PRELIMINARY 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 Determine the axial force in the braces of the structure below for the given loading. The flexibility of the portal frame in the figure F with respect to the three given dofs is given below (in terms of 1/EI) under the assumption that axial deformations are negligible. The ratio of flexural to axial stiffness EI/EA=8.

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

231.10385.10439.310385.10308.17

F

Problem 3 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 EIa=3×105 and EIb=2×105 and negligible axial deformations. Members c, d and e have axial stiffness EAc=1×104, EAd=1.5×104, and EAe=5×103.

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

2020

1

UNIVERSITY OF CALIFORNIA, BERKELEY Department of Civil and Environmental Engineering FALL SEMESTER 2008 Structural Engineering, Mechanics and Materials

Name: ________________________________


ANALYSIS


2


The fixed-end moment of a propped cantilever of span 2L under a concentrated force P at midspan is 0.375PL, as shown in the following figure.

P0.375 PL

EI EI

L L

Use the above information to determine the stiffness matrix (or its inverse) for the two dynamic degree of freedom system in the following figure.

L

EI

L 0.4L

m m

EI EI

3


The elements of the structural model in the figure can be assumed inextensible. In addition, element c is inflexible. Elements a, b and d have the same flexural stiffness EI. A linear analysis under the loading in the figure of known direction but unknown magnitude gives the basic forces in the figure.

Determine the horizontal reaction at the left support (point L in the figure).

?

a

b c

d

53.416

10.712

57.743

10.712

?

20.104

41.484

4 8 8

10

L

6

4


The plastic hinges at incipient collapse of the structural model in the figure under the given loading are depicted with gray filled circles. The plastic flexural capacity of element a is 200 units and that of elements b and c is 160 units. The axial capacity of truss element d is 20 units, while the axial capacity of elements a through c is very large.

1. Determine the collapse load factor λ under the given loading.

The flexural section stiffness EI of elements a through c is 50,000 units, while their axial deformations can be neglected. The axial stiffness of element d is 10,000 units.

2. Determine the horizontal translation of node 4 at incipient collapse.

3. Determine the vertical translation at girder midspan (node 3) at incipient collapse.

4. Determine the plastic deformation of the truss element d at incipient collapse and at a horizontal translation at node 4 of 0.1 units.

2 3 3 3

a

b cd

4

50

50

last hinge to form

32 4

1

5

1


Name: ________________________________


ANALYSIS



The structural model in the figure is subjected to a uniform load w of 5 units normal to the axis of element a. Furthermore, the brace element d is delivered 0.016 units longer and then prestressed in place.

It can be assumed that elements a through c are inextensible and that element b is inflexible. The flexural stiffness of elements a and c is EI = 20,000 units and the axial stiffness of element d is EA = 20,000.

Determine the vertical translation at node 2.

8 8

610

a b

cw=5

1

2

3

4

d

2


The continuous beam in the figure is subjected to two concentrated forces of equal magnitude

acting at midspan, as shown. The beam has uniform material properties with flexural stiffness EI

and plastic moment capacity pM . The spring has axial stiffness k and plastic axial capacity pN .

For the case that 3/ 1/ 6EI kL = and 3.5 /p pN M L= answer the following questions.

1. What is the collapse load factor cλ of the continuous beam for the given forces?

2. What is the vertical translation at the point of load application at incipient collapse?

kEI EI

L L L L

P P

· the plastic exural capacity m p of element a is 180 units. the plastic axial capacity n p of...

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