UNIVERSITY OF CALIFORNIA, BERKELEY
Spring Semester 2020
Dept. of Civil and Environmental Engineering
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:



 

L
0
ε0 L + ∆L0
0
0
 EA


  wL3 

 − κ0 L

 −
L
L 
 0


+
−
v = f q + v0 with f = 
v0 = 

2
24EI




3EI
6EI 





3


κ
L
0
wL
L
L
0
−
2
24EI
6EI
3EI
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:




EA

0

0
0
∆L0
−EA ε0 + L
 L

2 



  wL 
4EI 2EI 
 0





q0 = 
q = kv + q0 with k = 
EIκ0

 +  12 
L
L 





wL2
−EIκ0
2EI 4EI
−
0
12
L
L
The symbolic inverse of a 2x2 matrix M is
1
M11
M12
M22 −M12
M=
→ M−1 =
with det(M ) = M11 M22 − M12 M21
M21
M22
M11
det(M ) −M21
1
1. Problem (50% weight)
The tied frame in Fig. 1 consists of an inextensible beam with flexural stiffness EI, two inextensible
columns with flexural stiffness EI, and a tie (truss) with axial stiffness EA.
ˆ Under the application of a horizontal force Ph = 10 the axial force in the tie is 4.8544 units.
ˆ Under the application of a vertical force Pv = 20 the axial force in the tie is 1.4563 units.
Pv
Ph
EI
10
EI
EI
EA
5
5
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 moment
value in absolute terms at the left and at the right beam-column joints under the horizontal force
Ph and under the vertical force Pv as separate load cases.
2
2. Problem (50% weight)
The structural model of a cable-stayed bridge in Fig. 2 consists of three inextensible frame elements
a, b, and c, and the truss element d.
The frame elements have linear elastic, perfectly-plastic flexural response with flexural stiffness EI of
200,000 units and plastic flexural capacity Mp of 200 units. The frame elements have very high plastic
axial capacity Np . It is assumed that the presence of an axial force in a frame element does not
affect its plastic flexural capacity .
The truss element has linear elastic, perfectly-plastic axial response with axial stiffness EA of 40,000
units 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.
4
8
d
c
40
5
1
6
a
2
8
b
3
8
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 force
of 40 units?
4. What is the required prestressing force so that the incipient collapse state is reached in a single
event, i.e. that all hinges necessary for the formation of the collapse mechanism form at the same
time?
3
01234562789 29512 454 48 44
6!"#$%
6&'&4(')'!(!
1*
+,-.-+/0123245/6785324592:4
;4516<2<
=>?@*
$!!! '!!&!A!
" B&!C!'
D! E9 !F!)&A&A! )C)C!&A
' !' G7)H'!!)C)C&A)I)C)
))&A
7)JKA!''L)&!!'"!*
QT
Z
] `PTOaTPc ] # c
#
#
RRUV T T [
gd
^
^
d
[
fT
b
T
P
^
^
d
dd
R
[
X
X
^
^
d
O
JMNLOJP C) NMRR # DUW XYUW[
J
M
P
"
"
G
UW
^
^
d
de
[
_
_
e
g
S
\
b
T
P
fT
T T
# XYUW
"
"
GUW
DUW
C)T!))UV)h!i!!UW)jh&!i!!`Ph!
aTP'kKKBbP&'&&Bf&l!A&!!
7)!AC)A!''!L)!J)&!
!'"!*
QUV
Z
] #c
c
]
#
#
XUV`Pd ^
RR T
[
[
fTn d
^
d
^
G
UW
"
UW
R
[
^
d
d
^
UW
b
O
LMmJOLP C) mMRR # T T [
M
L
P
P
$
"
^
d
e
_
[
_
e
n
S "UW GUW\
fT
XUW
b
P
X
# T T
$"
$
012345678 9 7
!" #$%#&'()*#+,#-$ ./$#
$0$1# 2/$#$.'#-$$3+$1
4"4#564#" 78 9 (//$#3!#1#$$1
4"4#564#" 789(:/$#';$11"&6#4#"'
<13= 6#$564#" $$$+$&'
%#&'()+*&#14##$&#$#$#4#$4>"#$? $3+$
$+$ $'
@
K
A
B
G
E
I
@F
C
A
H
D
AF
J
@
A
A
@
A
A
A
)*L"$#$&
)+*M##$&#$#4#$4
%#&(>!N"%
OP$! !#$&Q#$>
('R#$4 STU/ &#1$$'
:'R!4$##$%#&':)*'
0'V$W&#1$4##$&#$'
.'R!+$#$&$##+#$#$#4#$4#$%#&':)+*'X$
$1$#$W&#"#$&"#&$$1$#$ 4##1Y$&#1
$'
:
0
0
0
1
1
0
2345677389
0
39
0
0
24 6 938 677389
012345678 90 7
!"#$%%&%$&'()*+)(,'-.)/&%0&&1%%
"#/& 2&/%0&341/%&5%%67$/&85&0
%$&&679067:67;$&&% &1%%/&%4/%&5%%6<
%%1=&&$1$/8%&1&/$>/&
G
?
E
@
C
B
A
@
F
D
?
?
@
@
@
"#H$&8/$
"#2I&JK%
1!LM
N$1%O&$%P&$//$PQ1%&$%L
R 2&8&I&%$$"JK%# !"#
S T&1I%8$///8&%&5%%&4UV&&/$W&$XY$&1
&W/$
Z//&&&Q1/1Q1&$%$&%I/&&$$/8%%&O&$
XV[UV\V]XY
PXV &II/$/$%\V &I&JK%I/&%
^
012345678 9 7
!"#$%#&'("$#")*+,-.,+/*01,)!!$2$$
34,/.4,//,5!$'67 #8$9:")!$2$#;<2<<< $#
$7##8$9=")#>2<<< $#'#?" $#)"!
## "@A;<#$!$'
$#")34,/.4,//,5 $@&#B$7#)"");;'C> $##$
$#67 )"DEAFG<HII)"!$J$#$&!"!$K7LL"M'
Q
T
U
P
WXRY
N
O
T
S
U
U
U
%#&(Z[ $ $#)"!!$"#$!$
V
R
%#&>Z\$#$&!"!$## #"$
%#&IZ])"!L
^
012345367589136 5495 11 256916
43 95581598694291 1495694292452 854 !395434"#
"1246 1$591 14%169$5&539$51596'562459145148 !9558159
$3256383"1954455$39159$3256319559 1 355596395462%%"32#
39161495485954391
(5954595$325 14959436391 39 1856(38)
*43 36324395"36%1665 23939$5361 9585145863%519569429245
2 854 +
,
UNIVERSITY OF CALIFORNIA, BERKELEY
Spring Semester 2018
Dept. of Civil and Environmental Engineering
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 manipulate
expressions symbolically as far as possible and substitute numbers only at or near the end.
1
1. Problem (50% weight)
The one-story braced frame in Fig. 1 consists of 2 frame elements a and b and a truss (brace) element
c. 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 force
at node 2 of 50 units acting to the left. In addition, the brace element c is prestressed, but the initial
prestressing 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
2. Problem (50% weight)
The structure in Fig. 2 carries a reference downward vertical force of 50 units and a reference horizontal
force of 80 units at node 3. The plastic flexural capacity Mp of elements a and c is 180 units and the
plastic flexural capacity Mp of elements b, d, e is 200 units. The plastic axial capacity Np of the brace
element f is 30 units. Elements a through e are assumed inextensible and have very large plastic axial
capacity. The flexural stiffness EI of the frame elements a through e is 200,000 units and the axial
stiffness EA of the brace element f is 10,000 units. The plastic hinge distribution at incipient collapse
is given in Fig. 2.
6
e
50
80
c
4
3
b
1
6
f
d
6
5
a
2
8
8
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
UNIVERSITY OF CALIFORNIA, BERKELEY
Spring Semester 2017
Dept. of Civil and Environmental Engineering
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 manipulate
expressions symbolically as far as possible and substitute numbers only at or near the end.
1
1. Problem (50% weight)
The continuous beam over two spans in Fig. 1 is subjected to a concentrated force Pv at the middle of
each span. The left span of length 2L has flexural stiffness EI1 and the second span also of length 2L
has flexural stiffness EI2 .
Pv
Pv
L
L
L
L
Figure 1: Continuous beam over two spans
The analysis of the continuous beam over two spans under the given load pattern gives a bending
moment in the middle of the left span of 0.23992Pv L and a bending moment in the middle of the right
span of 0.35484Pv L.
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 and
EI1 .
2
2. Problem (50% weight)
The structural model in Fig. 2 consists of five frame elements a through e. The frame elements have
very large axial capacity and the following flexural capacity values Mp : elements a, b and e have flexural
capacity Mp of 250 units, element c has flexural capacity Mp of 300 units, and element d has flexural
capacity Mp of 350 units. Fig. 2 shows the location of the plastic hinges at incipient collapse: they
form at both ends of element a, end j of element b, both ends of element c, and end j of element e
noting that end i corresponds to the lower numbered node to which the element connects.
3
50
5
8
b
d
50
2
10
c
6
e
4
6
2
a
8
1
6
8
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 all
bending moment values in the figure.
3. Determine the vertical support reaction at node 6.
3
8
6
2
10
8
6
8
6
Figure 3: Bending moment diagram
8
6
2
10
8
6
8
Figure 4: Auxiliary figure
4
6
8
6
2
10
8
6
8
Figure 5: Auxiliary figure
5
6