ph.d. preliminary examination analysis

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UNIVERSITY OF CALIFORNIA, BERKELEY Dept. of Civil and Environmental Engineering Spring Semester 2020 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. Formulas The deformation v-basic force q relation of a homogeneous, prismatic, 2d frame element is: v = f q + v 0 with f = L EA 0 0 0 L 3EI - L 6EI 0 - L 6EI L 3EI v 0 = ε 0 L + ΔL 0 - κ 0 L 2 κ 0 L 2 + 0 - wL 3 24EI wL 3 24EI where L is the element length, EA the axial stiffness, EI the flexural stiffness, ε 0 an initial axial strain, ΔL 0 a lack-of-fit, κ 0 a uniform curvature field, and w a uniformly distributed transverse load. The inverse relation between the basic forces q and the element deformations v of a homogeneous, prismatic, 2d frame element is: q = k v + q 0 with k = EA L 0 0 0 4EI L 2EI L 0 2EI L 4EI L q 0 = -EA ( ε 0 + ΔL 0 L ) EIκ 0 -EIκ 0 + 0 wL 2 12 - wL 2 12 The symbolic inverse of a 2x2 matrix M is M = M 11 M 12 M 21 M 22 M -1 = 1 det(M ) M 22 -M 12 -M 21 M 11 with det(M )= M 11 M 22 - M 12 M 21 1

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Page 1: Ph.D. Preliminary Examination Analysis

UNIVERSITY OF CALIFORNIA, BERKELEY Dept. of Civil and Environmental EngineeringSpring Semester 2020 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.

Formulas

The deformation v-basic force q relation of a homogeneous, prismatic, 2d frame element is:

v = f q + v0 with f =

L

EA0 0

0L

3EI− L

6EI

0 − L

6EI

L

3EI

v0 =

ε0L+∆L0

−κ0L2κ0L

2

+

0

− wL3

24EI

wL3

24EI

where L is the element length, EA the axial stiffness, EI the flexural stiffness, ε0 an initial axial strain,∆L0 a lack-of-fit, κ0 a uniform curvature field, and w a uniformly distributed transverse load.

The inverse relation between the basic forces q and the element deformations v of a homogeneous,prismatic, 2d frame element is:

q = kv + q0 with k =

EA

L0 0

04EI

L

2EI

L

02EI

L

4EI

L

q0 =

−EA

(ε0 + ∆L0

L

)EIκ0

−EIκ0

+

0

wL2

12

−wL2

12

The symbolic inverse of a 2x2 matrix M is

M =

[M11 M12

M21 M22

]→ M−1 =

1

det(M)

[M22 −M12

−M21 M11

]with det(M) = M11M22−M12M21

1

Page 2: Ph.D. Preliminary Examination Analysis

1. Problem (50% weight)

The tied frame in Fig. 1 consists of an inextensible beam with flexural stiffness EI, two inextensiblecolumns with flexural stiffness EI, and a tie (truss) with axial stiffness EA.

� Under the application of a horizontal force Ph = 10 the axial force in the tie is 4.8544 units.

� Under the application of a vertical force Pv = 20 the axial force in the tie is 1.4563 units.

Ph

Pv

10

5 5

EA

EI

EI

EI

Figure 1: Tied frame

You are asked to answer the following questions:

1. Determine the axial stiffness EA of the tie and the flexural stiffness EI of the beam and columns.

2. Discuss whether there are values of the axial stiffness EA that result in the same bending momentvalue in absolute terms at the left and at the right beam-column joints under the horizontal forcePh and under the vertical force Pv as separate load cases.

2

Page 3: Ph.D. Preliminary Examination Analysis

2. Problem (50% weight)

The structural model of a cable-stayed bridge in Fig. 2 consists of three inextensible frame elementsa, b, and c, and the truss element d.

The frame elements have linear elastic, perfectly-plastic flexural response with flexural stiffness EI of200,000 units and plastic flexural capacity Mp of 200 units. The frame elements have very high plasticaxial capacity Np. It is assumed that the presence of an axial force in a frame element does notaffect its plastic flexural capacity .

The truss element has linear elastic, perfectly-plastic axial response with axial stiffness EA of 40,000units and plastic axial capacity Np of 40 units.

The structure is subjected to a downward nodal force of 40 units at node 2, as shown in Fig. 2.

6 8

8

8

401 2 3

4

5a b

d c

Figure 2: Cable-stayed bridge

You are asked to answer the following questions:

1. Determine the collapse load factor λc.

2. Draw the bending moment distribution M(x) at incipient collapse.

3. What is the change of the collapse load factor λc, if the truss element d is prestressed with a forceof 40 units?

4. What is the required prestressing force so that the incipient collapse state is reached in a singleevent, i.e. that all hinges necessary for the formation of the collapse mechanism form at the sametime?

3

Page 4: Ph.D. Preliminary Examination Analysis

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Page 10: Ph.D. Preliminary Examination Analysis

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Page 17: Ph.D. Preliminary Examination Analysis

UNIVERSITY OF CALIFORNIA, BERKELEY Dept. of Civil and Environmental EngineeringSpring Semester 2018 Structural Engineering, Mechanics, and Materials

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

Ph.D. Preliminary Examination

Analysis

Notes:

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.

1

Page 18: Ph.D. Preliminary Examination Analysis

1. Problem (50% weight)

The one-story braced frame in Fig. 1 consists of 2 frame elements a and b and a truss (brace) elementc. The frame elements are assumed to be inextensible and have flexural section stiffness EI of 50,000.The axial stiffness of EA of the brace element is 50,000. The frame is subjected to a horizontal forceat node 2 of 50 units acting to the left. In addition, the brace element c is prestressed, but the initialprestressing force q0 was not recorded.

Determine the prestressing force q0 so that the brace element c is not deformed under the given loading.

Figure 1: Structural geometry and loading

2

Page 19: Ph.D. Preliminary Examination Analysis

2. Problem (50% weight)

The structure in Fig. 2 carries a reference downward vertical force of 50 units and a reference horizontalforce of 80 units at node 3. The plastic flexural capacity Mp of elements a and c is 180 units and theplastic flexural capacity Mp of elements b, d, e is 200 units. The plastic axial capacity Np of the braceelement f is 30 units. Elements a through e are assumed inextensible and have very large plastic axialcapacity. The flexural stiffness EI of the frame elements a through e is 200,000 units and the axialstiffness EA of the brace element f is 10,000 units. The plastic hinge distribution at incipient collapseis given in Fig. 2.

6

6

8050

a

b

c

d

e

f

1

8 82

34

5

6

Figure 2: Structure geometry, and loading, and plastic hinge locations

You are asked the following question:

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

2. Draw the bending moment distribution indicating clearly which way the frame elements bend.

3

Page 20: Ph.D. Preliminary Examination Analysis

UNIVERSITY OF CALIFORNIA, BERKELEY Dept. of Civil and Environmental EngineeringSpring Semester 2017 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.

1

Page 21: Ph.D. Preliminary Examination Analysis

1. Problem (50% weight)

The continuous beam over two spans in Fig. 1 is subjected to a concentrated force Pv at the middle ofeach span. The left span of length 2L has flexural stiffness EI1 and the second span also of length 2Lhas flexural stiffness EI2.

L L L L

Pv Pv

Figure 1: Continuous beam over two spans

The analysis of the continuous beam over two spans under the given load pattern gives a bendingmoment in the middle of the left span of 0.23992PvL and a bending moment in the middle of the rightspan of 0.35484PvL.

You are asked to answer the following questions:

1. Determine the stiffness ratio EI1/EI2.

2. Determine the vertical deflection under the concentrated force of the left span in terms of Pv, L andEI1.

2

Page 22: Ph.D. Preliminary Examination Analysis

2. Problem (50% weight)

The structural model in Fig. 2 consists of five frame elements a through e. The frame elements havevery large axial capacity and the following flexural capacity values Mp: elements a, b and e have flexuralcapacity Mp of 250 units, element c has flexural capacity Mp of 300 units, and element d has flexuralcapacity Mp of 350 units. Fig. 2 shows the location of the plastic hinges at incipient collapse: theyform at both ends of element a, end j of element b, both ends of element c, and end j of element enoting that end i corresponds to the lower numbered node to which the element connects.

a

b6

6 6

8

8

8

d

50

50

c

e

10

2

1

2

3

4

5

6

Figure 2: Simply supported girder with flexibly supported overhang

You are asked to answer the following questions:

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

2. Draw the bending moment diagram of the structure at incipient collapse in Fig. 3 and supply allbending moment values in the figure.

3. Determine the vertical support reaction at node 6.

3

Page 23: Ph.D. Preliminary Examination Analysis

6

6 6

8

8

810

2

Figure 3: Bending moment diagram

6

6 6

8

8

810

2

Figure 4: Auxiliary figure

4

Page 24: Ph.D. Preliminary Examination Analysis

6

6 6

8

8

810

2

Figure 5: Auxiliary figure

5