Joint Analysis during Vault Lowering of the Officer Sean Collier Memorial
by
William 0. Cord
B.S., Civil Engineering
Iowa State University, 2014
Submitted to the Department of Civil and Environmental Engineering in Partial Fulfillment of
the Requirements for the Degree of
Master of Engineering
ARCHNES
at the
MASSACHU;ETTS 1\NSTITU.JTE
OF IECHNOLOLGY
Massachusetts Institute of Technology
JUL 02 2015
June 2015
LIBRARIES
C2015 William Cord. All rights reserved.
The author hereby grants to MIT permission to reproduce and to distribute publicly paper and
electronic copies of this thesis document in whole or in part in any medium now known or
hereafter created.
Signature redacted
Signature of
A uth o r...............................................................................................................................................
Department of Civil and Environmental Engineering
May 21, 20
Certified
b y .......................................................................................
Sicinature redacted
t1"
John A. Ochsendorf
Class of 1942 Professor of Architecture
Professor of Civil and Environmental Engineering
Thesis Supervisor
Accepted
Signature red acted
by.................................... . . .. . . .. . . .. . ..5.........I.......
i
f
Donald and Martha Harleman Professor of Civil and Environmental Engine4eng
Chair, Graduate Program Committee
Joint Analysis during Vault Lowering of the Officer Sean Collier Memorial
by
William 0. Cord
Submitted to the Department of Civil and Environmental Engineering on May 21, 2015 in Partial
Fulfillment of the Requirements for the Degree of Master of Engineering in Civil and
Environmental Engineering
ABSTRACT
The unique geometry and indeterminacy of the Collier Memorial make it a highly complex
structure to analyze. During the critical vault-lowering phase of the construction of the memorial,
joints were monitored to understand where and how the memorial deformed. Joint displacements
are aggregated into figures to allow for a visual representation of the joint displacements working
in conjunction with one another. With the visualization of the displacement data, it was
determined that a hinging mechanism formed in one of the legs. The majority of the self-weight,
about 96%, was transferred to the supporting walls in compression during the lowering of the
vault.
Joint displacements were also used to assess the actual thrust states of two legs to be compared to
analytical and numerical models generated during the design phase. The mechanics of the
multipolymer shim used as the intermediary between the granite blocks were studied. The joint
displacements in combination with the shim mechanics produced thrust values within the
minimum and maximum thrust range established by a thrust line analysis.
Thesis Supervisor: John A. Ochsendorf
Title: Class of 1942 Professor of Civil and Environmental Engineering and Architecture
ACKNOWLEDGEMENTS
It was a privilege to work on the MIT Memorial to Officer Sean Collier. By the age of 27, Sean had
entrenched himself as a positive figure and role model for the students at MIT. During the eight months
working on the memorial, I received glimpses of Sean's personality through family and friends. The
Collier Memorial truly exemplifies the void left by the loss of Sean.
Professor John Ochsendorf gave the opportunity to be involved with the construction of the Collier
Memorial to the Master of Engineering students in the early fall. John's openness to allow more hands
touch an already cluttered project appropriately displays his ranks among the best educators at MIT. In
addition to his expertise, John gave his heart to the design and construction of the Collier Memorial. The
significance of this project and learning experience is not lost on me. John, thank you.
An incredibly personable and talented cast of MIT students and staff were instrumental in my progress
with this project. Dr. Corentin Fivet and William Plunkett were key to bringing me up to speed and
coordinating efforts with the construction team, and their photographic documentation of the project was
an incredible resource. Stephen Rudolph was always available with equipment, space and advice during
the experiment stages. Andrew Smith provided a much needed additional set of eyes and inquiries
throughout my involvement with the project. Thanks to Christopher Porst, Michael Laracy and Natalia
Zawisny for answering the call to measure the joint openings during day of the vault lowering. A special
thanks goes to my colleague on this project, Nathaniel Lyon, for his strong work ethic and stellar attitude.
It was a joy to spend the majority of my weeks collaborating with you.
The design and construction teams involved with the Collier Memorial were especially accommodating
considering the tight deadlines. I am grateful to Professor Meejin Yoon and her team at H6weler + Yoon
Architecture LLP for allowing me to instrument the Collier Memorial with the joint monitors during the
construction process. Rob Rogers of Suffolk Construction and Phoenix Bay State Construction Company
were responsive and helpful during my times on site.
Thanks go to Professor Connor, Dr. Ghisbain and my classmates. They've helped with my development
into a structural engineer.
Mom and Dad, your love, involvement and genuine interest have allowed me to be confident, purposeful
and happy.
TABLE OF CONTENTS
1
IN TROD U CTION ................................................................................................................................
1.1
1.2
1.3
1.4
2
M OTIVATION FOR THE SEAN COLLIER M EMORIAL..................................................................
JOINT D ISPLACEM ENT M EASUREMENTS..................................................................................
JOINT D ISPLACEMENT A NALYSIS .............................................................................................
SHIM M ECHANICS TO ESTIMATE LOAD TRANSFER .................................................................
BACK GROU ND ................................................................................................................................
STRUCTURAL ANALYSIS AND D ESIGN ....................................................................................
FABRICATION AND CONSTRUCTION...........................................................................................13
V AULT LOW ERING .....................................................................................................................
2.1
2.2
2.3
3
M ETH OD O LO G Y .............................................................................................................................
14
17
17
19
25
V AULT LOW ERING ACCOUNT .................................................................................................
JOINT D ISPLACEMENTS ..............................................................................................................
25
26
AN ALY SIS OF JOINT D ISPLACEMEN TS .................................................................................
33
M ECHANISM FORMATION ......................................................................................................
THRUST STATE OF LEGS.............................................................................................................34
ANCILLARY FINDINGS................................................................................................................34
ESTIMATION OF THRUST BASED ON DEFORMATIONS ........................................................
BACKGROUND ............................................................................................................................
FIRST ITERATION OF THRUST ESTIMATION USING SHIM MECHANICS ..................................
SECOND ITERATION OF THRUST ESTIMATION USING SHIM MECHANICS ...............................
FORCE R ESULTANTS ..................................................................................................................
SHIM M ECHANICS D ISCUSSION..............................................................................................
6.1
6.2
6.3
6.4
6.5
7
10
MEASUREMENTS OF JOINT DISPLACEMENTS ...................................................................
5.1
5.2
5.3
6
10
22
22
4.1
4.2
5
8
9
9
9
JOINT M ONITORING SYSTEM ..................................................................................................
M EASUREM ENTS ........................................................................................................................
D ETERM INING H INGE FORMATION .......................................................................................
SHIM M ECHANICS ......................................................................................................................
3.1
3.2
3.3
3.4
4
8
CON CLU SIONS ................................................................................................................................
7.1
7.2
7.3
7.4
D ISPLACEM ENT M EASUREM ENTS..............................................................................................43
M ECHANISM FORMATION ......................................................................................................
LOAD TRANSFER ........................................................................................................................
LASTING IM PACT........................................................................................................................44
APPENDIX A: MATERIALS PROPERTIES OF THE COLLIER MEMORIAL ................
33
36
36
36
39
40
41
43
43
43
45
APPENDIX B: BLOCK NAMES AND SETTING SEQUENCE.............................................................46
APPEND IX C: RA W M ONITOR DATA .................................................................................................
47
JOINT M EASUREM ENT M ETHOD .............................................................................................................
47
4
LIST OF TABLES
Table 2.1: Minimum and Maximum Horizontal Thrusts Estimated by Thrust Line Analysis to Achieve
3D Equilibrium, kN (kips) (ODB 2014) .......................................................................
Table
Table
Table
Table
Table
Table
Table
Table
Table
Table
10
36
6.1: Calculation of Total Normal Force across Joint A2A3..........................................................
37
6.2: Calculation of Total Shear Force across Joint A2A3 ............................................................
37
6.3: Calculation of Total Normal Force across Joint E2E3..........................................................
37
6.4: Calculation of Total Shear Force across Joint E2E3............................................................
6.5: Horizontal Thrust Comparison of ODB Thrust Network Analysis (TNA) and First Iteration of
Shim M echanics, kN (kips)...............................................................................................38
6.6: Calculation of Total Normal Force across Joint A2A3 - Second Iteration...........................39
6.7: Calculation of Total Shear Force across Joint A2A3 - Second Iteration............................. 39
6.8: Calculation of Total Normal Force across Joint E2E3 - Second Iteration........................... 40
6.9: Calculation of Total Shear Force across Joint E2E3 - Second Iteration............................... 40
6.10: Horizontal Thrust Comparison of ODB Thrust Line Analysis and Second Iteration of Shim
40
M echanics, kN (kips)..................................................................................................
Table A. 1: Granite M aterial Properties ................................................................................................
Table A.2: Shim M aterial Properties..........................................................................................................45
Table A.3: Grout M aterial Properties.....................................................................................................
45
45
Datum Dimensions ...................................................................................................................
Stage One .................................................................................................................................
Stage Two.................................................................................................................................50
Stage Three...............................................................................................................................51
Stage Four.................................................................................................................................52
48
49
Table C.6: Stage Five .................................................................................................................................
Table C.7: Stage Six...................................................................................................................................54
53
Table
Table
Table
Table
Table
C. 1:
C.2:
C.3:
C.4:
C.5:
5
LIST OF FIGURES
8
Figure 1.1: Completed Collier Memorial (Photo by Iwan Baan)............................................................
2014)
red.
(ODB
thrust
is
is
blue.
Maximum
thrust
Minimum
each
leg.
Figure 2.1: Thrust line analysis of
....... ...... ....... ...... ....... ...... ...... ....... ...... ....... ...... ....... ...... ...... ....... ...... ....... ...... ..... 1
Figure 2.2: Hinging mechanism leading to leg collapse (ODB 2014) ...................................................
12
Figure 2.3: Sliding mechanism leading to sliding rotation (ODB 2014)...............................................12
Figure 2.4: Dowel locations across block joints. Dowels linking the blocks to the grade beam are not
13
sh own . ...............................................................................................................................
Figure 2.5: Vault constructed on scaffolding (Photo by Corentin Fivet)...............................................14
15
Figure 2.6: Plan view of inner and outer scaffolding supporting the vault ............................................
16
Fivet)...................................
Figure 2.7: Screw jacks used for lowering the vault (Photo by Corentin
17
Figure 3.1: Joint Monitor (Photo by Corentin Fivet) ..............................................................................
Figure 3.3: The author measuring the horizontal displacement change of the joint monitor (Photo by
20
C hristopher P orst) .............................................................................................................
Figure 3.4: Change of Cartesian coordinates to calculate normal and parallel displacement of the joint..20
21
Figure 3.5: Variable visualization for Equation 3.3 through Equation 3.5 ............................................
by
(Photo
testing
compression
uniaxial
(right)
from
shim
Figure 3.6: Undeformed shim (left) and crushed
W illiam P lunkett)..............................................................................................................23
Figure 3.7: Korolath stress-strain curve from compression testing........................................................23
25
Figure 4.1: Displacement of joint E2E3 after stage six of lowering. ....................................................
Figure 4.2: Percentage of load removed from scaffolding and joint displacements after stage one of vault
27
low erin g .............................................................................................................................
of
vault
stage
two
after
joint
displacements
and
scaffolding
from
removed
of
load
Percentage
4.3:
Figure
28
low erin g .............................................................................................................................
Figure 4.4: Percentage of load removed from scaffolding and joint displacements after stage three of
29
vau lt low ering ....................................................................................................................
Figure 4.5: Percentage of load removed from scaffolding and joint displacements after stage four of vault
30
low ering .............................................................................................................................
Figure 4.6: Percentage of load removed from scaffolding and joint displacements after stage four of vault
31
low erin g .............................................................................................................................
six
of
vault
stage
Figure 4.7: Percentage of load removed from scaffolding and joint displacements after
32
lo werin g .............................................................................................................................
Figure 5.1: Hinging mechanism occurring in Leg E after stage six of lowering. Deformations are highly
34
ex agg erated . ......................................................................................................................
35
Figure 5.2: Joint slip of the C1C2 joint during stage one of scaffold lowering .....................................
Figure 6.1: Shim A2B2bI falling out of the joint before vault lowering...................................................38
Figure 6.2: The force resultant across A2A3 and E2E3 after stage six of vault lowering was calculated
using the second iteration of shim mechanics...............................................................41
Figure C. 1: Avongard crack monitor used as a datum for measuring joint displacements....................47
6
LIST OF VARIABLES
a
F-
joint orientation from horizontal
normal strain
CTE
elastic stress
plastic stress
ap
shear stress
shim area-assumed 10,300 mm2 (16 in 2
Fs
G
Shear force resultant across a joint
shear modulus
L
Lb
length of joint
distance from lower monitor to lower joint extremity
Lt
N
distance from upper monitor to upper joint extremity
normal joint displacement at extremity (joint opening is positive)
Nb
Nt
normal joint displacement at lower joint extremity (joint opening is positive)
normal joint displacement at upper joint extremity (joint opening is positive)
n
nb
normal joint displacement at joint monitor (joint opening is positive)
normal joint displacement at lower joint monitor (joint opening is positive)
nt
p
normal joint displacement at upper joint monitor (joint opening is positive)
parallel joint displacement (vault side of joint displacing up relative to leg side is positive)
p ave
R
average parallel joint displacement
radius of normal joint change due to differential normal displacements
w
x
shim width
joint monitor width change
y
joint monitor height change
)
A
T
7
I
1.1
INTRODUCTION
Motivation for the Sean Collier Memorial
Officer Sean Collier was only 27 years old when he was shot and killed in the line of duty on April 18,
2013. It was later revealed that the gunmen were the perpetrators of the Boston Marathon bombings that
occurred three days prior. Many impromptu memorials were erected for the popular MIT police officer
across Boston and Cambridge. A committee composed of MIT students, faculty and police officers was
established to aggregate and cultivate design ideas for a permanent memorial for Sean Collier.
On the one year anniversary of Sean Collier's death, MIT revealed the design by Meejin Yoon for the
Collier Memorial. The Collier Memorial-pictured in Figure 1.1-is composed of 32 unique granite
blocks that form a shallow vault near where Officer Collier was killed. Meejin Yoon describes the
memorial as an open hand reaching over a void left by the loss of Officer Collier. With the unveiling of
the finished memorial to take place near the two year anniversary of Collier's death, a large team of
engineers, contractors and fabricators was assembled to make the tight deadline.
Figure 1.1: Completed Collier Memorial (Photo by Iwan Baan)
8
1.2
Joint Displacement Measurements
The Collier Memorial is designed to carry the vault load through only arching action by having the
granite blocks lean into one another. This historic process of building with unreinforced blocks is
unconventional in today's construction industry. Uncertainty of the structural system's performance and
construction was an area of contention among the design and construction teams.
In order to insure safety, confirm the compression-only behavior of the structural system and monitor the
tight architectural tolerances to achieve a flush finish along granite blocks, 24 joint monitors were placed
on the memorial to record displacements. Joint displacements were recorded during the lowering of
scaffolding supporting the vault. At this point in construction, the load of the vault is being transferred
from the scaffolding to the leg-abutments through arching action. These joint measurements show where
the memorial deforms and how much, which is critical to the understanding of the mechanisms and load
distribution in the initial, compression-only state of the memorial.
1.3
Joint Displacement Analysis
With five geometrically unique legs contributing to carry the vault load, the Collier Memorial is an
&
indeterminate and complex structure. In a structural analysis report prepared by Ochsendorf DeJong
Block, LLC (ODB), a range of thrust states for each leg was established for the memorial to be in
equilibrium. Maximum joint openings and potential mechanisms that could cause the collapse of a leg
using rigid block analysis were hypothesized in the ODB report (ODB 2014).
Interpretation of the joint displacements of the Collier Memorial allows for the visualization of the actual
thrust state of each leg. Mechanism formations in the legs can also be found by analyzing the joint
displacements. Ultimately, the theory of thrust line analysis performed by ODB will be compared with the
actual state of the Collier Memorial.
1.4
Shim Mechanics to Estimate Load Transfer
Multipolymer shims were placed between all stones to protect the granite from damage due to direct
contact. Assuming that displacement of the joints also occur in the shims, the force resultant through the
joint could be estimated with knowledge of the shim mechanics. The two legs with the largest anticipated
load transfer, according to the ODB report, are analyzed using this methodology. The results are
compared to the minimum and maximum thrust values estimated by the ODB report.
9
2
BACKGROUND
2.1
Structural Analysis and Design
The Collier Memorial consists of 32 unique granite blocks that form five half-arches (referred to as legs)
that lean into one another to form a shallow vault. The eight blocks composing the vault have a total mass
of 32,900 kg. The joints of these blocks are orientated perpendicular to the predicted thrust lines to carry
the load entirely in compression. The base of the legs are tied together with a robust reinforced concrete
grade beam in order for the arch to stand alone in equilibrium. The reinforced concrete tie eliminates large
horizontal thrusts from entering the foundation abutments and subsequent outward displacements of the
legs.
With five geometrically unique legs contributing to carrying the vault load, the Collier Memorial is an
indeterminate and complex structure. Using thrust line analysis-the idea that a structure is stable if a
thrust can travel through the geometric boundaries of the material from one support to another supportthe 0DB report calculated the minimum and maximum horizontal thrusts for each leg under dead load.
This follows the classical limit analysis assumptions of Heyman (1997). Figure 2.1 provides a
visualization produced by 0DB of the minimum (blue) and maximum (red) thrust lines for each leg. In
order to find minimum and maximum thrusts in each leg with the constraint of 3D equilibrium, a thrust
line analysis was additionally performed. Table 2.1 provides the minimum and maximum horizontal
thrusts calculated by the 0DB thrust line analysis. As the defining characteristic of indeterminate
structures, the exact horizontal thrust in each leg of the Collier Memorial cannot be calculated without
information about the deformations of the structure. This thesis will attempt to use the measured
deformations to validate the assumed values of internal force.
Table 2.1: Minimum and Maximum Horizontal Thrusts Estimated by Thrust Line Analysis to Achieve 3D
Equilibrium, kN (kips) (ODB 2014)
Leg E
Leg D
Leg C
Leg B
Leg A
Min Thrust
244
(55)
113
(25)
104
(23)
154
(35)
182
(41)
Max Thrust
458
(103)
280
(63)
210
(47)
265
(60)
429
(96)
10
A2~
A
I
DI
xI
-L
- -
________ DS x
Figure 2.1: Thrust line analysis of each leg. Minimum thrust is blue. Maximum thrust is red. (ODB 2014)
11
The ODB report also predicted two mechanisms of joint opening that could lead to a rigid-body
deformation of the structure. The first mechanism explored was a hinging mechanism where the
keystone-leg interface hinges and blocks supporting the keystone pivot at a joint causing the keystone to
lower as shown in Figure 2.2. ODB reports that this hinging mechanism is most likely to happen in the
longer legs-Leg A and Leg E-
because they will tend toward a minimum thrust state for the memorial
to be in equilibrium. The second mechanism is a sliding mechanism where the leg slides downward off
the keystone as shown in Figure 2.3. In order for the sliding mechanism to occur, the leg is not supporting
the keystone, so no thrust can develop. Legs B and Leg C are potential candidates for the slip mechanism
due to their short length and proximity to each other (ODB 2014).
leg A
Figure 2.2: Hinging mechanism leading to leg collapse (ODB 2014)
L
-
- - -
-
8
B
Figure 2.3: Sliding mechanism leading to sliding rotation (ODB 2014)
In order to safeguard from a block slipping or falling, stainless steel dowels are placed across select block
joints as shown in Figure 2.4. The dowels-designed to carry tension and/or shear-are structurally
activated after the vault is lowered and the joints are grouted. The tension dowels are 19 mm (0.75 in) or
32 mm (1.25in) in diameter and are threaded to allow for tensile forces to transfer from the jointing grout
to the steel dowel and across a joint. The shear dowels are 19 mm (0.75 in) diameter and have a mirrored
finish, so a large tensile force cannot be transferred from the jointing grout to the dowel.
12
Leg B
4
Leg C
Shear Dowel
- Tension Dowel
Leg D
LLeg
A
Figure 2.4: Dowel locations across block joints. Dowels linking the blocks to the grade beam are not shown.
2.2
Fabrication and Construction
Granite blocks were quarried in Rapidan, Virginia. The ODB report used a modulus of elasticity equaling
48,000 MPa (7,000 ksi) for the granite stone (ODB 2014). The quarry owner reported the granite stones
having an ultimate compressive strength of 290 MPa (42 ksi) (ODB 2014). More information about the
material properties of the granite blocks can be found in Appendix A. The blocks were then cut by a
robotic block saw in order to form the unique shapes. The robotic milling process produced blocks within
0.5 mm tolerance of the design.
The first blocks that arrived on site consisted of the vault and were erected on scaffolding 6.4 mm (0.25
in) above design elevation as shown in Figure 2.5. The initial camber allowed for dimensional
imperfections of the blocks to be shed to the legs. The legs were then slid into place under the vault. The
individual block names and setting sequence are located in Appendix B.
Multipolymer plastic shims were used as the sacrificial material in the joints in order to protect the granite
blocks from damage due to direct contact. The product literature listing the minimum compressive
strength of 35 kN (8 kip) (Korolath of New England, Inc. 2015). A uniaxial compressive test performed
on the shims generated an elastic modulus of 1,100 MPa (160 ksi). More information about the material
properties of the shims can be found in Appendix A. The shims used on the project are 100 x 100 x 6.4
13
mm (4 x 4 x 0.25 in) and were placed 25 mm (1 in) offset from the edges at the corners of the granite
block bearing faces.
The leg joints not touching the vault were grouted before vault lowering. The Portland cement based
grout is highly pumpable to fill the holes around the dowels and the 6.4 mm (0.25 in) joints. The grout has
a 28-day compressive strength of 60 MPa (8.7 ksi) (KPM Industries Ltd. 2015). More information about
the material properties of the grout can be found in Appendix A.
Figure 2.5: Vault constructed on scaffolding (Photo by Corentin Fivet)
2.3
Vault Lowering
On March 31, 2015, screw jacks on the scaffolding supporting the vault were incrementally lowered with
the goal of reaching a state of equilibrium where the entire load of the vault is transferred into the legs.
The vault was cambered 6.4 mm (0.25 mm) above the design elevation and the legs were in contact with
the vault from the start of lowering. Due to architectural concerns, vault lowering would cease if the
keystone descended 16 mm or a joint opened more than 2 mm from initial values. Once the vault
lowering process ended, the scaffolding would be raised until it contacted the vault, and the grouting of
the vault joints would commence.
14
The incremental lowering process was initiated by dividing the vault scaffolding into an inner and outer
support ring as shown in Figure 2.6. The screw jacks-pictured in Figure 2.7 -have
a lead of 6.4 mm
(0.25 mm). The inner ring was first lowered by rotating the jacks 1/8 of one turn. The outer ring was
subsequently lowered 1/ 16 of one turn. This completed one stage of lowering. After each stage of
lowering, the elevation of the keystone and the joint openings of the vault were checked to see if the
architectural limits were exceeded. Detailed records of the monitoring and measurements of the vault
lowering process are located in Chapter 3 and 4.
Leg D
Leg C
Leg B
.
Leg E
0
o Inner Scaffolding Ring
e Oute r Scaffolding Ring
Leg A
Figure 2.6: Plan view of inner and outer scaffolding supporting the vault
15
Figure 2.7: Screw jacks used for lowering the vault (Photo by Corentin Fivet)
16
3
3.1
METHODOLOGY
Joint Monitoring System
In order to check if the architectural joint opening constraint of 2 mm is exceeded after each stage of vault
lowering, the Collier Memorial was instrumented with joint monitors as shown in Figure 3.1. Each side of
the monitor was adhered to the granite stone with industrial strength, weatherproof adhesive. The monitor
bridges the joint, and each side of the monitor moves in conjunction with the attached granite block. The
joint monitor displays the relative planar displacements of the granite blocks. The granite is assumed
infinitely stiff, so the displacement change of the monitor occurs entirely in the joint. The monitor
locations were selected in large part from the "Mechanism of Joint Opening" section of the ODB report.
This section of the report highlights joints that would have the maximum opening due to the expected
thrust line locations being near the intrados or extrados of the vault (ODB 2014). Error! Reference
source not found. displays the names and locations of the 25 joint monitors. Monitor AIBIb could not
be placed because a temporary lateral support covered the location.
Figure 3.1: Joint Monitor (Photo by Corentin Fivet)
17
MONITOR LOCATIONS
9
LEG C
10
8
LEG B
13
221
24
LEG D
2
1
4
3
17
16
17
LEG A
18
LEG E
19
20
Figure 3.2: Names and locations of joint monitors. Monitor A1B1b could not be installed.
No.
Name
1
A2A3t1
2
A2A3t2
3
A2A3bl
4
A2A3b2
5
AlBit
7
BB2t1
8
9
B1B2t2
BICHt
10
BICIb
11
C1C2t1
12
13
C1C2t2
14
D1D22
D1D2td
15
E2E3t1
16
E2E3t2
17
E2E3b1
18
E2E3b2
19
AlElt
20
AlElb
21
22
A1Kb
B1Kb
23
C1Kb
24
D1Kb
25
ElKb
The amount of joint monitors and exact locations were highly influenced by the site and time constraints
on the vault lowering day. In order to gain a more complete understanding of the how the loads of the
vault are distributed to the legs, 40 locations could have been instrumented with joint monitors. A total of
40 joint monitors were deemed impractical to gather displacement information during the short period
between lowering stages. Instrumenting the five joints around the top surface of the keystone would have
been beneficial in understanding the thrust state of each leg, but those joints were thought to be
inaccessible to the students measuring the joints. All monitors except those located under the keystone are
on the side faces of the granite blocks to allow for the normal and relative parallel joint displacements to
be measured. Locating the monitors on the side faces of the granite blocks had the additional advantage of
being much more accessible to measure.. The bottom face of the blocks tended to be difficult to access due
to the scaffolding supports. The torsional joint displacement measurement is sacrificed by locating the
monitors on the side faces, but the torsional joint displacement does not provide relevant information that
could be compared to the thrust line analysis. Joints A2A3 and E2E3 were selected to have the top and
bottom of the joint measured on both sides because the thrust line analysis by ODB and Lyon concluded
that these two legs would likely have the largest thrust resultant (ODB 2014) (Lyon 2015). Having the
joint displacements at all four locations of the joint along with knowledge of the shim mechanics allow
for an estimation of magnitude and location of the force resultant across the joint. Joints AIB1, BlC1 and
Al El were instrumented to confirm if a compression ring forms around the keystone, in which case, the
legs are leaning on one another in addition to the keystone.
3.2
Measurements
A team of students recorded joint displacements at each of the monitors after each stage of vault lowering
as shown in Figure 3.2. The joint monitors were used as a datum for digital calipers with an accuracy of
+/-0.03 mm (+/-0.00 1 in) that measured the cumulative change in width and height of the monitor. The
joint monitors were adhered to the block faces horizontally level, so a Cartesian coordinate change had to
be calculated to represent the joint displacements as normal and parallel displacements. Refer to Equation
3.1, Equation 3.2 and Figure 3.3 for the change of coordinates calculation.
Equation 3.1: n = x cos(a - 7/2) + y sin(a - 7/2)
Equation 3.2: p = y cos(a -
/2) - x sin(a - 7/2)
19
Figure 3.2: The author measuring the horizontal displacement change of the joint monitor (Photo by
Christopher Porst)
pyt
Monitor
x
yb
Mnto r
Figure 3.3: Change of Cartesian coordinates to calculate normal and parallel displacement of the joint
20
The normal changes of the joint extremities were calculated using similar triangles as demonstrated in
Equation 3.3 through Equation 3.5 and Figure 3.4. The joints instrumented with monitors on the upper
and lower portions-A2A3, B 1 C1, E2E3 and AlEl-allowed for an accurate gradient of normal joint
displacements to be calculated for the entire length of the joint. Joints monitored only at the upper section
of the joint were assumed to have zero change ofjoint displacement at the bottom extremity of the joint
throughout the entire vault lowering process.
Equation 3.3: R = nt((L - Lt - Lb)/(nt - nb)) + Lt
Equation 3.4: Nt = R(nt/(R - Lt))
Equation 3.5: Nb = R(Nt/(R - L))
Figure 3.4: Variable visualization for Equation 3.3 through Equation 3.5
During the vault lowering, the values ofjoint displacement were input into a spreadsheet to actively track
if the joint displacement constraints were exceeded. Some joint displacement measurements were used in
combination with the load being shed from the scaffolding to the legs in order to track the stiffness of
each leg. Results of the stiffness of each leg during vault lowering were recorded by Nathaniel Lyon and
can be found in his thesis (Lyon 2015).
21
Measurements of the joint displacement during vault lowering are analyzed in Chapter 4. The joint
displacement measurements are critical to understanding the mechanism formation across the memorial
and the load transfer into the legs.
3.3
Determining Hinge Formation
In a journal article titled "Limit Analysis of Structures Formed from Rigid Blocks," Livesley describes
that if a joint opens between two rigid blocks, then a hinge must form on the opposite side of the joint
opening (Livesley 1978). The plastic shims located in the joints of the Collier Memorial make this
assertion by Livesley ambiguous. The criterion used for determining a hinge formation is if the
cumulative normal joint change is positive at one extremity, a hinge forms on the opposite extremity of
the joint.
3.4
Shim Mechanics
All joints monitored during vault lowering were ungrouted. Multipolymer plastic shims were the only
material in the joints. The shims used on the project are 100 x 100 mm (4 x 4 in) and were placed 25 mm
(1 in) offset from the edges at the corners of the granite block bearing faces. With the gradient of normal
joint displacements calculated from the measurement data, the displacement and therefore strain of the
shims could be hypothesized. Testing on the shims was performed to understand the mechanical
properties with the goal of estimating the force resultant across the monitored joints.
A uniaxial compression test was performed using a universal testing machine on a 50 x 50 x 6.4 mm (2 x
2 x 0.25 in) shim. Figure 3.5 displays the initial and crushed state of the shim. The stress-strain curve
generated from the compression testing is shown in Figure 3.6. A bilinear approximation of the stressstrain curve was generated by the method of least squares. The bilinear line equations were set to fit the
experimental data set closely around the strain range of 0.04 to 0.09, which included the large strains of
the shims during vault lowering. The yield stress is 84 MPa (12 ksi), which occurred at a strain of 0.076.
The elastic modulus is 1,100 MPa (160 ksi), and the linear approximation of the plastic region has a slope
of 275 MPa (40 ksi). The compression test ran until failure, so an unload-reload curve was not generated
with the test data. Strain occurring at stresses greater than the yield stress is assumed inelastic. The
unloading and reloading stress-strain relationship is assumed to have a slope equal to the elastic modulus.
The stresses of the shims from vault lowering are calculated using the bilinear approximation and the
assumed unload-reload curve.
22
Figure 3.5: Undeformed shim (left) and crushed shim (right) from uniaxial compression testing (Photo by
William Plunkett)
Shim Stress vs Strain
120
1
I
80 6= (275 MPa)c + 62.5 MPq.
60
'
140
(YE(10
----
0.01
Elastic Approximati on
-------- Plastic Approximatic n
-
-0.01
Actual
-
-
0.03
0.05
0.07
0.09
Strain (mm/mm)
-
Unloading/Reloadi ng
Assumption
0.11
0.13
0.15
Figure 3.6: Korolath stress-strain curve from compression testing
No experiments were performed to determine the shear modulus of the shims. The shear modulus for the
shims is 1.2 MPa (0.17 ksi), which was gathered from the product literature (Korolath of New England,
23
Inc. 2015). The force resultant along the joint is calculated using the shear modulus and the parallel joint
displacement measurements. Using the normal displacement gradient, each shim is checked to be in
contact with both granite block faces by comparing the normal joint opening to the shim width. If a shim
is not wide enough to fill the joint, it is assumed to not carry shear force. Equation 3.6 and Equation 3.7
calculate the shear force resultant across a joint.
Equation 3.6: r = G(p-ave/w)
Equation 3.7: Fs = Er
In order to encourage consistent results for the force resultants across the monitored joints, a list of
assumptions are presented about the joints and shims.
*
No load is transferred across any joint/shim before vault lowering
*
The initial width of the shim is the minimum normal joint width in the area occupied by the shim.
In other words, the shims barely contact both granite block faces of the joint before vault
lowering.
*
The initial width of the shim is homogeneous.
*
No tension force can develop across a joint because the shims are not adhered to either granite
block.
*
A shim cannot develop any form of force across a joint if it is not wide enough to contact both
sides of the joint.
24
4
4.1
MEASUREMENTS OF JOINT DISPLACEMENTS
Vault Lowering Account
Joint displacement measurements were successfully acquired and checked to the deformation constraints
after each stage of vault lowering. Neither the constraint of the keystone descending 16 mm nor a joint
opening 2 mm were surpassed during the vault lowering process. The vault lowering was halted after the
sixth stage due to the scaffolding under Leg E increasing in load. On closer inspection, joint E2E3 had
displaced downward by about 1mm as shown in Figure 4.1. This occurred late in the day after 96% of the
self-weight had been transferred from the scaffolding into arching action, and the construction schedule
did not allow time for continued lowering the following day. Thus it was decided to not continue
lowering the scaffolding for the remaining 4% of self-weight and the scaffolding was left in place for
grouting to occur the next day. Explaining the deformation of the E2E3 joint became a supplementary
motivation for this thesis.
Figure 4.1: Displacement of joint E2E3 after stage six of lowering.
25
4.2
Joint Displacements
Figures 4.2 through 4.7 showcase the joint displacement data after each stage of vault lowering. The
known displacements gathered from measuring the joint monitors are labeled in the figures. The joint
parallel displacement plot in Figure 4.7 has the vertical displacements of the underside corners of the
keystone. These cumulative vertical displacements of the keystone were shot by on site surveyors.
For the joint normal displacement plots, the joint displacements on each side of the leg are averaged. For
example, the normal joint displacement at the extremity near monitor B lB2t l is averaged with the normal
joint displacement at the extremity near monitor B 1B2t2. Likewise, joint parallel displacements on the
same side of the leg are averaged. The averaging of joint displacement is to mitigate measurement errors
and to simplify the plots. For raw data measurement data, turn to Appendix C.
As stated in Chapter 3, joints monitored only at the upper section of the joint were assumed to have zero
change of normal joint displacement at the bottom extremity of the joint throughout the entire vault
lowering process. Although this assumption likely does not hold true, the correlation between joint
closing at one extremity and joint opening at the other is too erratic to approximate across the joints. Take
the joint normal plot in Figure 4.7 as an example. The top extremity of joint A2A3 opens 0.34 mm and
the bottom extremity closes 0.37 mm. For joint E2E3, the top opens 0.04 mm and the bottom closes 0.43
mm. For joint AlE 1, the top closes 0.43 mm and the bottom opens 0.95. All three of these joints provide
different relationships between joint opening and joint closing for the same lowering stage.
Hinge locations are designated on the joint normal displacement plots with a solid black line by using the
criterion that a hinge forms at one joint extremity if the cumulative normal joint change is positive at the
opposite extremity.
The slider bars in the figures represent the estimated load removed from the scaffolding. This data was
provided by Nathaniel Lyon as part of his thesis to track the load shedding and stiffness of the legs during
lowering of the vault. Scales were placed under the scaffolding of each leg. If the load reading in the
scales reduced as the vault was being lowered, it could be surmised that the load was being transferred to
arching action within the leg (Lyon 2015). As shown in Figure 4.7, Lyon estimates a total of 96% of the
initial load on the scaffolding was shed into the legs at the time the vault lowering was halted.
26
-I
-I
Joint Normal Displacement
Total
'H
Leg C
OP-i(+)
Leg B
I
1.719
1001A
IOW
1.547
1.375
1.203
g]3%
0.359
0-516
0.172
020
~-'
121
.25
0
'-
-
\45
K
7~X~
A1372
-0.34
7~i~I
-0.516
I,
j
-04"3
Log E
4-
I
>
1%0
IV
K
Hinge
Joint Parallel Displacement
Total
Leg C
100%
UPuu.I(+)
(mm)
10i
Leg B
0.733
0.367
0
-0.367
-0.733
-1
AS
-1.100
-1A67
K
-1.833
-2.200
6%
W
\
T---
-2.567
-2.933
-3300
-3.667
-4.033
4A00
~I
4
-4.767
-5.133
--
------ V
Leg E
100%
100%
W
Figure 4.2: Percentage of load removed from scaffolding and joint displacements after stage one of vault
lowering
27
Joint Normal Displacement
Total
Leg C
Opins(+)
[
L eg B
100%
1.719
1.547
1375
IftI
- im
iOM
K
N
- O.aS6
Isis
0.172
~19%
_
A1601
iN
> Leg E
IV0%
Hinge
-
N2%
-
-4
Joint Parallel Displacement
UPw
Total
100%
Leg C
(+)
100%
(->)
Leg B
- 0367
100%
-0
-
.367
-
0.733
4
4
-1.100
2200
-
-2.567
-r
-
-
-1.33
--
-2933
-
-4
-33
-3.667
2%
-4A00
-4767
-5.133
I-,
~Leg E
2A
Ino
-
N
T
09%
Figure 4.3: Percentage of load removed from scaffolding and joint displacements after stage two of vault
lowering
28
-/
Joint Normal Displacement
1.891
1.719
ME
Total
Leg C
O~I.a<+ ()
100lA
139%
eg B
139%
1.547
j4211
1.375
1.203
- 0.MS.
-
-016
0.172
0
11
NN
.- , 7'~N
LegE
106%
H
H
-F
K
-I-
-II
i00%
Joint Parallel Displacement
Total
Leg C
UWr+d()
-0.733
0.367
.- 0367
B
Leg 10OW
100%
IM-
3V%
-0.733
-
.7
-1.100
-1.467
-
-1.33
-
-22
Le
-2.933
-3.667
-4.033
I
71
-3300
NI
\
-5.133
-HF'
F'
<110
F-V
Figure 4.4: Percentage of load removed from scaffolding and joint displacements after stage three of vault
lowering
29
Joint Normal Displacement
Opmn--(+)
-
Leg B
1891
Total
T"W
Leg C
IOM
-1.719
61%
-1.547
-
1.375
62%
1203
a7
-
-1.719
\
-o.on
0-516
659
0.344
1
iIN~
0.172
/e
0
-0,172
-03"
39
-.
---..85
>1
1>
>4
-I
Leg E
K.
j66
51%
Hinge
Joint Parallel Displacement
Total
Leg C
ipwil(+)
(ff0.733
Leg B
O.367
IOW
0
162%
N>
.0.367
>1>
-. 733
61%
-1.100
"vii
-1A67
-2.
-I
-1.833
-2.200
-2.567
-2.933
-3.300
-3.667
4.033
-4AOO
-4.767
-5.133
-
I
Leg 1
y1
>4<
/7
IOA
\/
\
A
Leg E
N
K~t>IU
N>
jEO
10%
Figure 4.5: Percentage of load removed from scaffolding and joint displacements after stage four of vault
lowering
30
Joint Normal Displacement
Total
Leg C
-I
OP-ft(+)
Leg B
(->
1.891
1.719
79%
1.547
1101
1.375
120
1.031
I
a.Msp
X>
On55
-
0.516
0.344
1%
0.172
-0.172
-0514
-oil'
'7,-
74.
> LegE
-1728%
Y
-k
Hinge
Joint Parallel Displacement
Total
Leg C
UPWup-(+)
1WA
168%
(-)
Leg B
0.733
100%
-0.367
-0
-0.367
-
-0.733
-r
-1.100
-
-1A467
-1.933
-
-2200
-
-2567
-
-2333
--
3.300
-
---
T/
-04
|--3.667
-400
-.767
7--
----- 7
>---
Leg E
-' 77
100%
1W8%
1-
'7
172%
Figure 4.6: Percentage of load removed from scaffolding and joint displacements after stage five of vault
lowering
31
Joint Normal Displacement
Total
Leg C
Opcng (+)
100%
100%
L
1.891
II
100,'
1.719
1.547
r
1.375
1.203
I-
1.031
0.859
0.683
rOM
~ CI
0.516
0.344
~>
03
0.172
2
0
0172
--0344
-0.516
AM68
K
{
IcgA
>jr
-9
> Leg E
100%
1-4
L~
Hinge
Joint Parallel Displacement
Total
Leg C
UpwwZI(+)
100%
943%
(in)
0.733
0.367
Leg B
10%
0
-0.367
-0.733
-I
-1.467
-2.200
"4
-2.567
-2933
-
-1.933
-3.3M
4033
-4.400
-4.767
-5.133
N
-:-'5-11'I
\-361
-
-3.667
- 100%
V..
\
ii
, '9
LegE
100%
Figure 4.7: Percentage of load removed from scaffolding and joint displacements after stage six of vault
lowering
32
5
5.1
ANALYSIS OF JOINT DISPLACEMENTS
Mechanism Formation
During vault lowering, all joint openings were found to be less than 1 mm, which was half of the
architectural constraint of 2 mm. By the end of stage five, many screw jacks were no longer in contact
with the memorial, and Lyon estimated 83% of the original load in the scaffolding was supported through
arching action. Then during stage six lowering, the scale readings under Leg E scaffolding started to
increase. The E2E3 joint was inspected, and a joint parallel displacement of 1.4 mm had occurred during
that lowering stage.
Inspection of the joint normal displacement plot in Figure 4.7 reveals that a hinging mechanism described
a
by ODB developed during stage six of vault lowering. Leg E had entered a minimum thrust state with
hinge developing along the upper extremity of the KEI joint. The bottom of the KB1 joint closed in order
to support the rotating keystone. An exaggerated illustration of the hinging mechanism is illustrated in
Figure 5.1.
The cause and sequence of formation of the hinging mechanism is difficult to deduce. It is likely that the
deformation of the E2E3 joint was a small slippage as the thrust engaged across the joint. In order for the
1.4 mm of deformation to be caused by shear deformation of the shims, a shear force of 9 kN (2.0 kips)
would need to develop during just the stage six lowering. However, the arching force most likely acted
perpendicular to the joint, so it is difficult to imagine how a shear force could develop at this shim.
The leads of the screw jacks were inspected to see if differential lowering of the vault potentially caused
the displacement. A screw jack under block El and near the E1E2 joint had a lead of 8 mm, which is an
outlier for all of the other 6.4 mm lead jacks. However, this screw jack was located under a beam working
in coordination with another screw jack and discrepancy in the screw jacks should have been evident in
the parallel displacements for the previous stages of lowering.
V
E4
E2
El
i
K
~
B
E3
E5
E6
33
. ...........
.
. .
........
Figure 5.1: Hinging mechanism occurring in Leg E after stage six of lowering. Deformations are highly
exaggerated.
One hypothesis is that either Leg B or Leg E displaced outward and/or downward. With the outward
displacement, the thrust across joint E2E3 drops to zero, and the joint slips until contact is established
again across the joint. For an outward and downward foundation displacement of 6.4 mm, the ODB report
calculates that the maximum joint opening for the bottom of the KAl joint to be 3.8 mm, assuming the
granite blocks are rigid bodies (ODB 2014). The bottom of the KEl joint only opened 0.87 mm, so a
relatively small foundation displacement may have played a role in forming the Leg E hinging
mechanism.
5.2
Thrust State of Legs
By analyzing the hinge formation around the keystone, the inclination of the thrust state of a leg toward
minimum or maximum thrusts can be qualified. If a hinge forms on top of the keystone joint, the thrust
state of the leg is tending toward a minimum thrust state and vice versa for a hinge forming at the bottom
of the keystone joint. Only the bottom of the keystone joints were instrumented with crack monitors, so
only hinges forming on the top of the keystone joint could be confirmed. After stage six of vault
lowering, hinges formed on top of the keystone joint for Legs A, C, D and E as shown in Figure 4.7. This
shows that these legs are tending toward a minimum thrust state. The bottom of the Leg B keystone joint
is closing, but whether a hinge forms for that particular joint cannot be determined.
5.3
Ancillary Findings
Joint CIC2 had a large initial joint parallel displacement of -1.56 mm as seen in Figure 4.2. This
displacement is nearly equal to the 1.6 mm the screw jack under the joint was lowered. This displacement
does not correspond to the hinge or slip mechanisms described in the ODB report. The parallel
displacement is very likely a joint slip where no thrust force had developed in Leg C during the initial
scaffolding lowering during stage one. A lack of initial contact between the CI and C2 blocks through the
shims due to block Cl being raised too high on the scaffolding is the most probable explanation for the
slip. The C1 block translated downward until contact with the C2 block was made in order to transfer load
as illustrated in Figure 5.2. This would explain why the intrados of the CI and C2 blocks were flush with
each other at the end of vault lowering. It is fascinating to note how the vault seems to move rigidly due
to the slip at CIC2. The sides of A2 and E2 blocks opposite of Leg C displace up, while the remainder of
the parallel displacements of the vault is down. This initial slip at C IC2 ultimately causes the maximum
vertical displacement of the keystone at the corner of BI and Cl as seen in Figure 4.7.
34
-- - --
- -
-A-
-
I-MR-
-
-
-
-
-
. - - - - :-:- --- -11-11- - -- &- -
C1
C2
Figure 5.2: Joint slip of the C1C2 joint during stage one of scaffold lowering
Instrumenting the top of the B 1B2 and D1D2 joints was beneficial for monitoring the joint openings, but
these measurements gave virtually no insight on the thrust state of their respective leg. From the ODB
thrust line analysis, the minimum and maximum thrust locations both pass near the intrados while
crossing the B 1B2 and D1D2 joints as shown in Figure 2.1. No matter the thrust state in these legs, the
top of the B1B2 and D1D2 joints are opening as shown in Figure 4.2 through 4.7.
35
6
ESTIMATION OF THRUST BASED ON DEFORMATIONS
6.1
Background
With the assumption that the initial width of the shim is the minimum normal joint width in the area
occupied by the shim, the strains in the shims can be estimated by using the joint deformation
measurements. Laboratory testing was performed on the shims to understand the mechanical properties
under compression, so the force traversing a joint could be estimated. The bilinear approximationdisplayed in Figure 3.7-of the stress-strain curve of the shims is used to calculate the force resultant
across a joint. Joints A2A3 and E2E3 are only two joints analyzed because a gradient of the joint
displacements can be generated due to monitors located near the top and bottom of the joints.
The horizontal thrust is calculated by using the known joint orientation. The horizontal thrust is then
compared to the ODB thrust line analysis in Table 2.1.
Equation 6.1: Horizontal Thrust = Normal Force * sin a + Shear Force * cos a
6.2
First Iteration of Thrust Estimation Using Shim Mechanics
The first iteration of thrust estimation based on shim mechanics takes into account all assumptions
established in Section 3.4.
Table 6.1: Calculation of Total Normal Force across Joint A2A3
Stage
1
2
3
4
5
6
Shim Strain
Total Force
A2A3
Total Force
A2A3tI
mm/mm
A2A3bI
mm/mm
A2A3t2
mm/mm
A2A3b2
mm/mm
0.000
0.034
0.042
0.001
0.000
0.000
0.009
0.015
400
562
617
90
126
0.035
0.045
0.008
0.010
0.041
0.000
0.000
0.015
0.024
0.024
847
745
870
191
168
0.000
0.000
0.000
0.000
0.000
0.052
36
kN
A2A3
kips
139
196
Table 6.2: Calculation of Total Shear Force across Joint A2A3
Stage
Change in Shear Strain
Total Force
Total Force
A2A3tl
A2A3bl
A2A3t2
A2A3b2
A2A3
A2A3
mm/mm
mm/mm
mm/mm
mm/mm
kN
kips
1
2
0.000
0.000
0.133
-0.148
0.133
0.000
0.000
-0.148
1
0
0
0
3
4
5
6
0.000
0.000
0.000
0.000
0.003
-0.003
-0.030
-0.019
0.003
-0.003
0.000
0.000
0.003
-0.003
-0.030
-0.019
0
0
0
0
0
0
0
0
Table 6.3: Calculation of Total Normal Force across Joint E2E3
Stage
Shim Strain
Total Force
Total Force
E2E3t1
E2E3bI
E2E3t2
E2E3b2
E2E3
E2E3
mm/mm
mm/mm
mm/mm
mm/mm
kN
kips
858
951
991
661
806
193
214
223
149
181
1259
283
1
2
0.002
0.000
0.068
0.058
0.000
0.000
0.006
0.026
3
4
0.000
0.000
0.065
0.050
5
6
0.000
0.000
0.060
0.082
0.000
0.000
0.000
0.006
0.023
0.008
0.011
0.028
Table 6.4: Calculation of Total Shear Force across Joint E2E3
Stage
Change in Shear Strain
Total Force
Total Force
E2E3tl
E2E3bl
E2E3t2
E2E3b2
E2E3
E2E3
mm/mm
mm/mm
mm/mm
mm/mm
kN
kips
1
2
3
0.158
0.000
0.000
0.158
-0.211
-0.057
0.000
0.000
0.000
0.158
-0.211
-0.057
1
0
0
0
0
0
4
0.000
5
6
0.000
0.000
0.012
-0.098
-1.453
0.000
0.000
-1.453
0.012
-0.098
-1.453
0
-1
-9
0
0
-2
37
Table 6.5: Horizontal Thrust Comparison of ODB Thrust Network Analysis (TNA) and First Iteration of
Shim Mechanics, kN (kips)
TNA Min Thrust
TNA Max Thrust
Shim Mechanics Estimation
Leg A
244
(55)
458
(103)
835
(188)
Leg E
182
(41)
429
(96)
1228
(276)
The first iteration of the shim mechanics estimation overestimates the minimum and maximum thrust
ranges established by the thrust line analysis. This overestimation is due to shims not making contact with
both sides of the joint. Shims A2A3b 1 and E2E3b 1 have exceedingly large initial strains, which suggest
that the initial joint width was larger than the shim. In fact, Figure 6.1 displays shim A2A3b I falling out
of the joint before vault lowering started. The second iteration of thrust estimation based on shim
mechanics proposes a method of accounting for the initial gap.
Figure 6.1: Shim A2B2bl falling out of the joint before vault lowering.
38
6.3
Second Iteration of Thrust Estimation Using Shim Mechanics
In order to account for the initial gap between shim and granite, the shims for A2A3b 1 and E2E3bl are no
longer assumed to be in contact with both sides of the joint. The width of the A2A3bl is shrunk by 0.25
mm and the width of E2E3bI is reduced by 0.5 mm to negate the normal joint displacement during stage
one of vault lowering. The widths of the rest of the shims are reduced by 0.05 mm to account for the joint
face of the granite not being flat and smooth. All other assumptions from Section 3.4 continue to be
enforced.
Table 6.6: Calculation of Total Normal Force across Joint A2A3 - Second Iteration
Stage
1
2
3
4
5
6
Shim Strain
Total Force
Total Force
A2A3t1
A2A3bl
A2A3t2
A2A3b2
A2A3
A2A3
mm/mm
mm/mm
mm/mm
mm/mm
kN
kips
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.004
0.000
0.004
0.003
0.011
0.000
0.000
0.004
0.009
0.000
0.000
0.000
0.004
0.006
0.008
0.016
0.016
0
85
120
243
214
316
0
19
27
55
48
71
Table 6.7: Calculation of Total Shear Force across Joint A2A3 - Second Iteration
Change in Shear Strain
A2A3tl
A2A3bI
mm/mm
mm/mm
1
0.000
0.133
2
3
4
0.000
0.000
0.000
-0.148
0.003
-0.003
5
6
0.000
0.000
-0.030
-0.019
Total Force
A2A3
[otal Force
kN
kips
A2A3t2
mm/mm
A2A3b2
mm/mm
0.000
0.000
0.003
0.000
-0.148
0.003
0
-0.003
0.000
-0.003
-0.030
-0.019
0
0
-1
0.000
39
0
0
A2A3
)
Stage
Table 6.8: Calculation of Total Normal Force across Joint E2E3 - Second Iteration
Stage
Shim Strain
Total Force
Total Force
E2E3tI
E2E3bI
E2E3t2
E2E3b2
E2E3
E2E3
mm/mm
mm/mm
mm/mm
mm/mm
kN
kips
2
43
38
1
2
0.000
0.000
0.000
0.000
0.000
0.000
0.001
0.017
10
193
3
4
0.000
0.000
0.000
0.000
5
6
0.000
0.000
0.000
0.011
0.000
0.000
0.000
0.002
0.015
0.004
0.006
0.019
170
43
70
357
10
16
80
Table 6.9: Calculation of Total Shear Force across Joint E2E3 - Second Iteration
Stage
Change in Shear Strain
Total Force
Total Force
E2E3t1
E2E3bI
E2E3t2
E2E3b2
E2E3
E2E3
mm/mm
mm/mm
mm/mm
mm/mm
kN
kips
0
0
0
0
0
0
0
-9
0
0
0
-2
1
2
0.000
0.000
0.000
0.000
0.000
0.000
0.158
-0.211
3
4
0.000
0.000
0.000
0.000
0.000
0.000
-0.057
0.012
5
6
0.000
0.000
0.000
-1.453
0.000
-1.453
-0.098
-1.453
Table 6.10: Horizontal Thrust Comparison of
ODB Thrust Line Analysis and Second Iteration of Shim
Mechanics, kN (kips)
TNA Min Thrust
TNA Max Thrust
Shim Mechanics Estimation
Leg A
244
(55)
458
(103)
302
(68)
Leg E
182
(41)
429
(96)
351
(79)
6.4
Force Resultants
The location of the force resultant across a monitored joint is calculated through a weighted average of
how much force is going through each shim. Figure 6.2 displays the force resultant and location across
joints A2A3 and E2E3. These force resultants fall between the minimum and maximum thrust lines
estimated by ODB.
40
Al2
A3
A4
4.
i
A6
A7
A8
El
E2
E4
E3
E5
E6
Figure 6.2: The force resultant across A2A3 and E2E3 after stage six of vault lowering was calculated using
the second iteration of shim mechanics.
6.5
Shim Mechanics Discussion
The shim mechanics analysis showcases how well orientated the joints of the Collier Memorial are.
Nearly all of the force traversing the A2A3 and E2E3 joints is normal to the joint.
With the adjustments made to the shim widths, the second iteration of the shim mechanics estimation falls
within the minimum and maximum thrusts obtained by the thrust line analysis. However, the reliability of
the shim mechanics estimation is questionable. From the mechanism formation analysis performed in
Chapter 5, Leg E is expected to be near the minimum thrust state, but in the shim mechanics analysis, Leg
E has a higher thrust than Leg A. Some problems with the shim mechanics analysis can be attributed to
being run on a construction site rather than in a laboratory. The shims may have been already loaded
before the vault lowering. The joints between the granite are rough, which may cause the stress-strain
relationship to deviate from the lab results. Measurement errors are magnified because the shims are very
thin. Any measured normal deformation of the joint gets blown up as a large strain on the shim.
Nevertheless, the shim mechanics analysis provides a snapshot of an exact magnitude and location of
41
forces across the joint. Table 6.8 shows the estimated normal load across joint E2E3 after each stage of
vault lowering. The large displacement of the E2E3 joint occurred during stage six of vault lowering. The
force resultants calculated at the end of stage four and stage five show that Leg E was struggling to attract
load, which may have led to the slip in the joint.
42
7
7.1
CONCLUSIONS
Displacement Measurements
The 24 locations monitored for joint displacements were generally placed in critical joint opening areas in
order to monitor the architectural constraint of no joint opening beyond 2 mm. The maximum joint
opening during vault lowering was 0.79 mm at joint D1D2. This comes as no surprise as the thrust line
analysis of Leg D by ODB shows that both the minimum and maximum thrust states pass near the
intrados of the D1D2 joint.
Aggregating the 24 joint displacement readings into a single figure allowed for a visual representation of
the joint displacements working in conjunction with one another. Mechanism formations along with leg
thrust states are quickly surmised by glancing at the displacement plots.
7.2
Mechanism Formation
With the visualization of the displacement data, it was determined that a similar hinging mechanism as
described by the ODB report developed in Leg E during stage six of vault lowering. The hinging
mechanism coincided with the joint slip at E2E3 that ultimately ended the vault lowering process. The
hinging mechanism and joint slip were likely caused by a displacement in the abutting exteriors of the
legs or foundation.
At the end of stage six, four out of five legs were tending toward a minimum thrust state by creating a
hinge at the top of their respective keystone joints. Leg B appeared to be near a maximum thrust state in
order to support the pivoting keystone caused by the hinging mechanism in Leg E.
7.3
Load Transfer
Using the joint displacements and shim mechanics was a serviceable approach for estimating thrusts
across a joint. The caveat is that initial displacements of the joint need to be critically analyzed to ensure
that the shim and joint boundaries are in contact. After inferring the actual shim widths in the joint, the
resultant force across the joint fall within the minimum and maximum thrust bounds established by a
thrust line analysis by ODB.
43
7.4
Lasting Impact
The Collier Memorial is constructed to last for centuries. Design drawings get lost and as-builts may
never be made. This thesis provides a brief, yet thorough, catalogue of materials, design procedures,
construction methods and actual construction results for posterity. The Collier Memorial is a unique and
daring structure for our time.
44
APPENDIX
A:
MATERIALS PROPERTIES OF THE COLLIER MEMORIAL
Table A.1: Granite Material Properties
Virginia Mist Granite
(0DB 2014)
International System of Units
US Customary Units
Density
3100 kg/m 3
193 lb/ft3
Modulus of Rupture
21 MPa
3 ksi
Ultimate Compressive Strength
290 MPa
42 ksi
Modulus of Elasticity
48,000 MPa
7000 ksi
Table A.2: Shim Material Properties
Korolath Shim Strips
International System of Units
(Korolath of New England, Inc. 2015)
US Customary Units
55 MPa
8 ksi
Modulus of Elasticity
1,100 MPa
7000 ksi
Shear Modulus
1.2 MPa
0.17 ksi
Minimum Ultimate Compressive
Strength
Table A. 3: Grout Material Properties
MS Cable Grout
(KPM Industries Ltd. 2015)
International System of Units
US Customary Units
Density
1875 kg/M 3
117 lb/ft3
28 Day Ultimate Compressive Strer gth at
60 MPa
8.7 ksi
21 0C
45
APPENDIX B: BLOCK NAMEs AND SETTING SEQUENCE
Setting
Sequence
Granite
Block
1
2
K
BI
3
4
Al
El
5
6
DI
Cl
7
8
C5
C2
9
10
C4
C3
11
12
B5
B2
13
14
B4
B3
15
16
D5
D2
17
18
D4
D3
19
20
A2
A6
21
22
A3
AS
23
24
A4
E2
25
26
E6
E3
27
28
E5
E4
29
30
B6
A7
31
32
A8
C6
A4
K
Al
A2,
A3
A5
A6
A7
AS
BI
B3
B2
B4
7-
K
B5
B6
K
C3
C2
C4
I
C5
C6
D2
D3
D4I
D4
D5
El
E2
E4
E3
ES
E6
46
K
APPENDIX
C:
RAw MONITOR DATA
Joint Measurement Method
The Avongard* crack monitor was used as a datum for joint displacements as shown below in Figure C. 1.
Horizontal joint displacements are measured with digital calipers from the monitor to the lip of the
fastener legs. Vertical displacements are measured at the middle of the monitor with digital calipers. The
displacement of the granite block closest to the keystone controls the sign of the displacement. If the
block closest to keystone shifts up relative to the outer block, the measurement is positive.
ID
Figure C.A: Avongard crack monitor used as a datum for measuring joint displacements.
47
Table C.1: Datum Dimensions
Joint Width
Monitor Width
Height
Monitor
Monitor Height
Monitor to Joint
(deg)
(mm)
(mm)
(mm)
(mm)
A2A3tI
0.4
5.49
24.96
40.66
9.92
A2A3t2
-0.3
5.48
25.05
40.49
12.91
A2A3bI
0
6.92
24.84
40.39
90.27
A2A3b2
1.1
6.67
24.76
40.42
83.24
A1Bt
-1.6
10.12
24.67
40.48
10.89
B1B2tl
-4.35
7.29
25.08
40.52
10.36
BIB2t2
-1.35
6.34
25.06
40.3
11.75
BICIt
0
3.82
24.91
40.31
14.5
BICIb
0
5.1
24.87
40.33
10.5
CIC2tI
0
9.96
24.77
41.95
9.96
C IC2t2
0
3.85
25.07
40.05
14.41
DID2tI
0
5.19
24.92
40.27
13.14
D ID2t2
0
6.16
24.56
41.1
8
E2E3tI
0
7.35
24.94
41.27
4.69
E2E3t2
0
7.7
24.77
41.49
10.17
E2E3bI
0
7.12
24.86
41.2
90.42
E2E3b2
0
5.48
24.68
41.5
9.3
AIElt
0
8.22
24.78
40.47
7.98
AlEib
0
8.03
24.26
40.46
8
A1Kb
0
8.22
8.22
BI Kb
0
8.54
8.54
ClKb
0
8.45
8.45
DlKb
0
7.85
El Kb
0
7.85
5.63
Name
Monitor
Monitor
Orientation
5.63
48
Monitor to Joint
Extremity
Table C.2: Stage One
Name
Monitor Width
Monitor Height
A2A3tI
A2A3t2
(mm)
24.97
25.08
-40.29
-40.64
A2A3bI
(mm)
40.46
A2A3b2
24.46
24.86
A1B1t
BIB2tI
24.77
25.28
40.68
-40.50
BI B2t2
BICIt
25.19
24.95
40.27
BICIb
ClC2tl
24.7
25.18
40.33
CIC2t2
DlD2tl
25.31
DI D2t2
E2E3tI
25.04
25.13
40.80
-41.97
E2E3t2
E2E3bI
24.71
24.34
-40.45
-41.07
E2E3b2
AlElt
24.6
24.73
-41.43
AIEib
Al Kb
24.26
40.46
BI Kb
ClKb
8.59
8.3
DlKb
ElKb
7.8
6
24.98
40.39
40.31
-41.26
-41.50
-40.35
-40.47
8.1
49
Table C.3: Stage Two
Name
Monitor Width
Monitor Height
(mm)
A2A3tI
A2A3t2
25.09
25.1
(mm)
-40.63
-40.65
A2A3bI
A2A3b2
24.48
24.67
-40.34
40.42
AIBIt
24.77
40.39
BIB2tl
25.57
40.41
BI B2t2
BICIt
25.33
24.96
-40.42
40.31
BICIb
CIC2tI
24.81
25.23
40.26
-41.44
CIC2t2
DID2t1
25.56
25.2
-41.32
-40.35
DID2t2
25.26
E2E3tI
25.14
40.71
-41.33
E2E3t2
24.82
E2E3bI
24.51
-41.57
-41.46
E2E3b2
AlE~t
24.42
24.7
-41.40
-40.41
AIEb
24.7
-40.43
AIKb
8.3
BIKb
CIKb
8.86
8.84
DIKb
EIKb
7.92
6.1
50
Table C.4: Stage Three
Name
Monitor Width
Monitor Height
(mm)
(mm)
-40.64
-40.68
A2A3t2
25.1
25.01
A2A3bI
24.52
A2A3b2
24.64
-40.33
40.40
AIBIt
BIB2tI
24.65
25.7
40.46
40.79
BIB2t2
25.55
BICIt
25
40.34
40.30
BIC b
CIC2tI
24.7
25.53
40.33
-41.47
CIC2t2
25.83
DID2t1
25.4
-41.66
-40.36
DID2t2
E2E3tI
25.44
25.18
E2E3t2
24.87
E2E3bI
24.41
-41.57
-41.25
E2E3b2
AIElt
24.45
24.58
-41.45
-40.50
AlEib
AIKb
25.01
40.45
B1Kb
8.79
ClKb
9.00
DlKb
ElKb
8.23
A2A3tI
40.63
-41.72
7.49
6.25
51
Table C.5: Stage Four
Name
Monitor Width
Monitor Height
A2A3tI
A2A3t2
(mm)
25.29
24.92
(mm)
-40.64
-40.55
A2A3bI
24.49
A2A3b2
24.62
-40.43
40.43
A1B1t
BIB2tI
24.65
25.84
40.49
40.89
BI B2t2
BICIt
25.64
24.93
40.59
40.25
BICIb
C1C2tI
24.70
25.61
40.25
-41.71
CI C2t2
25.91
25.41
-42.09
25.54
25.27
40.43
-41.56
E2E3b1
24.90
24.54
-41.60
-41.35
E2E3b2
24.55
AlElt
24.53
-41.35
40.65
AIEIb
25.02
40.76
A1Kb
8.23
B1Kb
ClKb
8.92
DlKb
ElKb
7.85
6.26
Dl D2tl
DI D2t2
E2E3tI
E2E3t2
-40.34
8.93
52
Table C.6: Stage Five
Name
Monitor Width
Monitor Height
A2A3tI
(mm)
25.31
A2A3t2
25.32
(mm)
-40.59
-40.63
A2A3bI
24.51
A2A3b2
24.56
-40.42
40.42
AIBIt
B1B2t]
24.63
25.8
-40.47
40.89
BIB2t2
BICIt
25.63
24.92
40.72
BICIb
CIC2t1
24.68
25.70
40.32
-41.75
CIC2t2
D1D2tI
25.87
25.50
-42.2
40.32
DID2t2
25.71
E2E3tI
25.41
40.33
-41.75
E2E3t2
E2E3bI
25.05
24.51
-41.63
-41.53
E2E3b2
AIElt
24.51
24.42
40.71
AIEIb
AIKb
25.15
8.18
BIKb
CIKb
8.70
8.60
DIKb
EIKb
8.07
8.36
40.29
-41.3
41.06
53
Table C.7: Stage Six
Name
Monitor Width
Monitor Height
A2A3tI
A2A3t2
(mm)
25.39
25.34
-40.53
A2A3b1
A2A3b2
24.44
24.57
-40.42
40.46
AIBIt
24.69
25.80
40.46
40.90
BICIt
25.63
24.91
40.73
40.23
BiCib
CIC2tI
24.64
25.56
40.27
-41.91
CIC2t2
DlD2tl
25.84
25.63
-42.30
40.32
DID2t2
E2E3tI
25.84
25.52
-40.60
-43.14
E2E3t2
E2E3bI
25.08
24.72
43.11
-43.10
E2E3b2
AlElt
24.72
24.35
-42.70
-40.71
AIEIb
25.19
8.25
41.06
BIB2tI
B1 B2t2
A1Kb
B1Kb
ClKb
8.38
DlKb
ElKb
8.17
6.50
(mm)
-40.72
8.50
54
REFERENCES
Heyman, J. The Stone Skeleton: StructuralEngineeringof Masonry Architecture, Cambridge
University Press, 1997.
Korolath of New England, Inc. 2015. "Korolath R.O.F. Reinforced Bearing Pads." Accessed May 16.
http://www.fisterinc.com/pdf/RO_F_Pads.pdf.
KPM Industries Ltd. 2015. "MS Cable Grout." Accessed May 16.
http://www.kpmindustries.com/KingConstructionProducts/wpcontent/uploads/sites/1 3/2014/02/MS-Cable-Grout-TDS-ENG-REV.0002.pdf.
Livesley, R. K. 1978. "Limit Analysis of Structures Formed from Rigid Blocks." InternationalJournal
for Numerical Methods in Engineering 12 (12): 1853.
Lyon, Nathaniel A. 2015. "The Collier Memorial: A Study of Theory and Construction." M.Eng.,
Cambridge, MA: Massachusetts Institute of Technology.
Ochsendorf DeJong & Block, LLC. 2014. "Structural Analysis Report Officer Sean A. Collier
Memorial."
55