Diagrid Structures: Introduction & Literature Survey

Guided by : Dr. S. A. Bhalchandra,
As. Prof. AMD
Presented by : Uday M.

Introduction
• Firstly introduced by Vladimir
Grigoryevich Shukhov as hyperboloid
towers for water tanks. Then revived
by Froster Nohman
• Consists of structure with an RCC core
and grid of diagonal members around
• Use of perimeter diagonals—hence
the term ‘diagrid’—for structural
effectiveness and esthetics
• In late 19th century early designs of tall buildings recognized the
effectiveness of diagonal bracing members in resisting lateral force
• Almost all the conventional vertical columns are eliminated
• Triangulation provides efficient load carrying system
• Diagonal members in diagrid structural systems can carry gravity
loads as well as lateral forces

• Carry shear by axial action of the diagonal members
and lateral by bending of diagonal member
• Provides both bending and shear rigidity
• Behave like a 3-dimensional box resisting both
compression and tension
• Diagrid structures are less prone to a lock-in
condition
• Very effective in case of buildings up to 70-100 story

Why to be interested in diagrids ?
• 45-75 story buildings are economical hence are in
abundance of the total world buildings
Worlds building
No. of building vs. storey
India’s buildings
Height of buildings vs. storey

Components of typical diagrid structure
• h is the module height of the building indicated in
terms of story. Typical module is 4-6 story high
• H/B ratio of the building is the aspect ratio of that
building

• Tie beams transfers the load
from RC core to diagrid
• Ring beam resists the
unbalance forces

Connections in Diagrids
• Nodes are fabricated and
then raised piece by
piece
• In case of an irregular
structure, each node has
to designed separately
• Welded nodes are preferred but
are time and skill demanding
• Bolded connections are used for
speedier erections
• The sections used – circular or I-
beam or rectangular

• Each node joint is designed for vertical loading
as well as for the lateral loading

Load Transfer in Diagrids
• Load transfer in concentrated,
ULD and Lateral load
• Effect on triangular elements

Literature Review
G. Subramanian & N. Subramonian (1970)
• After obtaining the partial difference equation governing the
deflection of laterally loaded uniform diagrids, closed-form solutions
are presented for the simply supported case.
• Recognizing that the above formulation for diagrids can easily be
extended to study the lateral oscillations by replacing the static loads
at the nodes with inertia loads, assuming masses to be lumped at the
nodes, expressions are obtained for natural frequencies of vibration,
for the simply supported uniform diagrids.
Kyoung-Sun Moon et al (2007)
• Presents A Simple Methodology For Determining Preliminary
Member Sizes
• Examined The Influence Of The Diagonal Angle On The Behavior Of
Diagrid Type Structures
• For 60-story Diagrid Structures Having An Aspect Ratio Of About 7,
The Optimal Range Of Diagrids Angle Is From About 65° To 75°

Charnish (2008)
• Studied various types of diagrid structures of different optimal
configuration and of various aspect ratios with varying module sizes
• Concluded that A 20% savings in the weight of the structural steel is
possible using A diagrid versus A braced tube
Dong-Kyu Lee et al (2010)
• Research provides both a design and analysis tool for diagrid
structural designs and diagrid structural analyses
• Considering both static and dynamic behaviors, appropriate diagrid
is designed
• It is verified that diagrids are redundant and loads follow the
diagonals through the structure
• For 42-story buildings having an aspect ratio of about 5, the range is
lower by around 10° because the importance of bending to the total
lateral displacement is reduced as the building height decreases
• A stiffness-based methodology for determining preliminary design
sizes for the diagonals was introduced

T. M. Boak (2010)
• Diagrids are displaying a language of detailing and design that
corresponds to choices in the size of the base module, building
type and three-dimensional geometry of the project.
• Diagrids are demonstrating a dynamic and adaptable structural
system that is more adapt at structuring contemporary
architectural aspirations
• Considering shape or sizing optimization problems, the number of
movable design variables is relatively very large also some
supports may be linked in order to keep the structural system
symmetric or limit the number of design variables treated, hence
the dynamics-based optimization model is developed to find
natural frequency of the system
• Once the fundamental frequency oscillates with topology
movements, the material interpolation technique, called the
penalty law or SIMP will be used to estimate the optimal topology
within the element

Moon et al (2011)
• Analyzed the twisted structures, tilted towers and
freeform complex structures
• For A typical study, in twisted towers and tilted tower
analysis, deflection at top in diagrid structures were about
8-10% lesser
• More effective at more twist rates and leaning, shows
more stiffness
Raghunath .d. Deshpande et al (2015)
• diagrid performs better across all the criterions of
performance evaluation, such as, efficiency, expressiveness
and sustainability.
• Diagrid structure have comparatively less deflection. Their
structural weight is reduced to greater extent. Due to this
structure has more resistance to lateral forces. Diagrid
structures are cost effective and eco-friendly. Diagrid uses
11247 tonnes of steel which is 28% less compared to the
conventional orthogonal building which uses 15255 tonnes.

Rohit Kumar Singh et al.(2014)
A regular five storey RCC building with plan size 15 m × 15 m
located in seismic zone v is considered for analysis.
• diagrid building shows less lateral displacement and drift in
comparison to conventional building
• although volume of concrete used in both building is approx.
same, but diagrid shows more economical in terms of steel
used. Diagrid building saves about 33% steel without affecting
the structural efficiency
• Better resistance to lateral loads: due to diagonal columns on its
periphery, diagrid shows better resistance to lateral loads and
due to this, inner columns get relaxed and carry only gravity
loads.
• While in conventional building both inner and outer column are
designed for both gravity and lateral loads, In diagrids only
columns are designed to resist both.

Design Methodology
There are three ways to design diagrid structures
A. PDE method governing the deflection using the
slope deflection equations
B. Optimization Technique
C. Stiffness Based Methodology

A. PDE method governing the deflection using the slope
deflection equations
• After obtaining the partial difference equation governing the
deflection of laterally loaded uniform diagrids, closed-form
solutions are presented for the simply supported case.
• Recognizing that the above formulation for diagrids can easily
be extended to study the lateral oscillations by replacing the
static loads at the nodes with inertia loads, assuming masses
to be lumped at the nodes, expressions are obtained for
natural frequencies of vibration, for the simply supported
uniform diagrids.
• Investigation revealed that at least for the case of simple
supports on all edges, the lowest natural frequencies are
closely approximated by the lumped mass approximation.

Rectangular diagrid with 45 degree angle

• Consider an R beam extending from (r, s) to (r+ l, s+ 1) the slope
deflection equations for the (r+ l)th span of an R beam may be
written as
• Also the equations for Shear and Torque are developed
• Applying equations of equilibriums as

Solving and using a shift operator Er and Es
Substituting back in equations of slope deflection
Substituting in equilibrium equations and solving

• The obtained value is then used in an explicit formula for the
determination of frequencies.
• The allowable values for k are l, 2……..m-1 and for l are l ,
2….......n-1; Beyond these, values for frequencies merely
repeat themselves. Thus, fundamental frequency can be
determined

It consists of two methodologies
• To evaluate desired angles of diagonal members consisting
of diagrid with respect to stiffness of optimal topology
• To investigate or understand global and topological diagrid
mechanism by using topology optimization technique
• dynamics-based optimization model is, in general, written as
max: ωi (i=1, ..., m)
subject to ajj ≤ aj ≤ aji (j=1,…,n)
ad = f ( aj )
• aj denotes the design variable, representing the j-th independent
finite element and ad is an arbitrary value which derives a
dynamic governing function f depending on aj into zero. aj and aj
denote the lower (almost 0 for voids) and upper (1 for solids)
bounds of the design variable, respectively
• In general, maximization of the first-order Eigen-frequency is
taken into account as objective, since structure with 1st Eigen-
mode has a tendency to the weakest stiffness
B. Optimization Technique

• To execute the optimization procedure, the FE method is utilized
as an analyzer to calculate the natural frequency and associated
vibration mode
• Once the fundamental frequency oscillates with topology
movements, the material interpolation technique, called the
penalty law or SIMP will be used to estimate the optimal topology
• In order to yield the optimum position model of fixed
support models for distributing material density, sensitivity
derivatives need to be studied
• Finally using minimum energy as constraints, optimal
topology boundaries are determined

• For tall buildings with a large height-to-width aspect ratio, the
stiffness constraint generally governs the design
• Important stiffness design parameters to consider in any tall building
design is its maximum deflection which is usually five hundredth of
the building height
• Diagrid structure is modeled as a vertical cantilever beam on the
ground, and subdivided longitudinally into modules
• In order to more accurately estimate the lateral rigidity provided by
diagrids, all the required lateral stiffness is allocated to the perimeter
diagrids
• The diagonal members are assumed to be pin-ended, and therefore
resist the transverse shear and moment through axial action only –
idealization
• Areas of diagonals on flange and web side are given as-
Ad,w =
VLd
2Nd,wEdh γ cos2θ
Ad,f =
2MLd
Nd,f + δ B2Edβ h sin2θ
3. Stiffness Based Methodology

• Optimal stiffness-based design corresponds to a state of uniform
shear and bending deformation under the design loading
• Since Uniform deformation states are possible only for statically
determinate structures and Tall building structures can be
modeled as vertical cantilever beams on the ground, and uniform
deformation can be achieved for these structures
• Deflection at the top, u(H) is given by
u H = γ∗H +
β∗H2
2
H: Building Height
γ∗
: Desired Uniform Transverse Shear Strain
β∗
: Desired Uniform Curvature
• In order to specify the relative contribution of shear versus
bending deformation, a dimensionless factor ‘s’
s =
β∗H2
2
γ∗H
=
β∗H
2γ∗

• The maximum allowable displacement, one of the most important
stiffness-based design parameters for tall buildings, is usually
expressed as a fraction of the total building height
u H =
H
α
• For Lateral Loading, introduce a dimensionless factor f, which is
defined as the ratio of the strain in a web diagonal due to shearing
action to the strain in a flange diagonal due to bending action
f =
εd,web
εd,flange
Where, εd,web= γsinθcosθ and εd,flange =
B
2
β sin2θ
• For braced frame, f > 1 (3 to 6) and for diagrid f < 1
s =
Hβ
2γ
=
H
B f tan θ

• Relationship between Aspect ratios to that of s and f
• Based on these studies, the following empirical relationship
between the optimal s value and the aspect ratio is proposed
for diagrid structures greater than 40 stories with an aspect
ratio greater than about 5 and a diagrid angle between 60°
and 70°.
s =
H
B
− 3

Optimal angle of diagonal members
• Considering only the maximum shear rigidity, the optimal angle for
diagonal members can be estimated using key assumption that the
members carry only axial forces
• The cross-section shear force is related to the diagonal member
forces by:
V = 2 Fd cosθ
• Assuming linear elastic behavior, the member forces are also
related to the diagonal extensional strain, ɛd, by
Fd = Ad σd = Ad Ed ɛd
• And extensional strain is given by –
• V = DT x axial strain deformation due to shear
: . DT = AD ED sin2θ cosθ

• The plot of sin 2θ cosθ is s that the optimal angle for maximum
shear rigidity of the system is about 35°
• In typical braced frames, the bending moment is carried by the
axial forces in the vertical columns i.e. maximum rigidity at 90°
• optimal angle of the diagonal members of diagrid structures will
fall between these angles

1. Diagrids of Uniform Angle
• For a preliminary design 60 story building with various
uniform angle configuration is considered
Experimental Case Studies:
Case Study - I
- Kyoung S. Moon (2007)

With Corner Columns
Without Corner
Columns

• With Corner Columns
• Without Corner Columns

• Relative stiffness of diagrid and
braced core

• Preliminary sizes of the
diagrid for 60 story and 42
story building

Height (Aspect Ratio)
Near Optimal
Uniform Angle
Near Optimal ‘s’
40 Stories (4.3) 63 Degrees 4
Story Module Diagrid Angle Diagrid Steel
Mass (Ton)
Percentile
Difference
2 stories 52 degrees 5700 +50.0%
4 stories 69 degrees 3800 Near Optimal
• Masses has
calculated and
proves that the
angle is optimal
• Different buildings
with this optimal
angle

2. Diagrids of Varying Angle
• Diagrid structure having
varying angles of 63, 69,
and 73 degrees from the top
to the bottom, uniform and
bottom to top are analyzed
• For more height the value
of ‘s’ is more sensitive
Diagrid Height
(Aspect Ratio)
Diagrid Angles
(Top to Bottom)
Near Optimal ‘s’
60 Stories (6.5) 63, 69, and 73 Degrees 2.2

• For a typical 60 story building
but with various values of s,
different optimal parameters
are obtained.
S = 1 S = 8
S = 4

Comparison of uniform and varying angle diagrids
Case Alt. Angle
Description
Optimal 's' Steel Mass
(tons)
1 2 76, 73, 69,
63,52
0.9
1068
1 69, 63, 52 2.7 1009
Uniform Angle 63 4.1 883
2 1 52, 63, 69 5.1 1906
2 52, 63, 69, 73,
76
2.1
1597
• For a 40 story building
with varying angles
and varying
geometrical positions,
optimum steel is
computed

Case Alt. Angle
Description
Optimal 's' Steel Mass
(tons)
1 2 79, 76, 73, 69,
63
0.9 5791
1 73, 69, 63 2.2 4104
uniform Angle 69 3.9 3820
2 1 63, 69, 73 3.7 4482
2 63, 69, 73,
76, 79
1.4 6549
• For a 60 story building
with varying angles
and varying
geometrical positions,
optimum steel is
computed

- R. D. Deshpande et al.
DESCRIPTION VALUE
Height 180m
Width 24m(Square plan)
Core wall dimension 12mx12m
No of storey 60
Core wall thickness 250mm
Storey height 3m
Floor slab 150mm
• 6-storey Module is
considered for analysis
• Loads are applied as per the
IS 875 code
Case Study - II

Modul
e
Floor Height
(m)
P(Avg.)
kn/m2
V(Avg.)
kn/m
M10 54-60 180 1.03 3.10
M9 48-54 162 1.01 3.05
M8 42-48 144 1.00 3.00
M7 36-42 126 0.98 2.95
M6 30-36 108 0.96 2.90
M5 24-30 90 0.93 2.80
M4 18-24 72 0.90 2.70
M3 12-18 54 0.88 2.65
M2 6-12 36 0.80 2.40
M1 1-6 18 0.76 2.30
• A point atop each module
is considered as tracking
node, deflection, shear
graphs are plotted for
these points representing
the module
Varying Wind
Load

Outcomes from study-
• More opening area
• Less Deflection
• 28% less steel usage

- Rohit Kumar Singh et al.
• Analysis and design of concrete diagrid building and its
comparison With conventional frame building
• A regular floor plan of 15m x 15m is considered in both
buildings
• The design dead load and live load are 4.5 kN/m2 and 4
kN/m2 respectively
• analyzed for seismic zone V, with seismic parameter as
per Indian code IS 1893(Part 1) : 2002
Case Study - III

3D view of
Conventional
building
3D view of
RCC Diagrid
building
Elevation of
RCC Diagrid
building

• Comparison of max. Shear forces (Fy) and bending
moments (Mz) in ground floor beams between
conventional building and diagrid building

• Shear Forces and Bending Moment in Beams

• Shear Forces and Bending Moment in Diagonal
Columns

Storey Drift comparison
• Structural performance: Diagrid building shows less lateral
displacement and drift in comparison to conventional building.
• Material saving property: Although volume of concrete used in
both building is approx. same, but diagrid shows more
economical in terms of steel used. Diagrid building saves
about 33% steel without affecting the structural efficiency.
Study Output:

• Better resistance to lateral loads: Due to diagonal columns
on its periphery, diagrid shows better resistance to lateral
loads and due to this, inner columns get relaxed and carry
only gravity loads.
• Lesser Design Efforts: While in conventional building both
inner and outer column are designed for both gravity and
lateral loads.

- Prashant T G et al.
• Comparison Of Symmetric And Asymmetric Steel
Diagrid Structures By Non-Linear Static Analysis
• 12 storey steel diagrid structure having height 36m and
lateral dimensions as 18mX18m is considered for the
analysis
• Nonlinear Static (Pushover) Analysis has been
conducted
Sl.No Building Details
1 Symmetric structure Model-1
2 Asymmetric structure Model-2
3 Plan dimensions of building 18m X 18m
4 Height of building 36 m
5 No. of stories 12
6 Storey height 3 m
7 Type of structure Steel diagrid structure
8 Type of analysis Nonlinear static analysis
Case Study - IV

Asymmetric
Structure
Symmetric
Structure

Asymmetric structure
• it can be observed that is pushover step 1 that
assigned hinges are in a state of immediate
occupancy.
• In step 2 it can be observed that some hinges shifts
from immediate occupancy state to state of life
safety and from figure that is step 3 it can be seen
that some hinges shifts from life safety to collapse
state.
Symmetric structure
• it can be observed that pushover step 2 that the
assigned hinges are in a state of immediate
occupancy.
• In step 3 and step 4 one can observe that some hinges
shifts from state of immediate occupancy to state of
life safety due to the incremental increase in lateral
load.

Examples of Diagrid Structures
Capital Gate,
Abu Dhabi
O-14, DubaiCOR, Miami
Carpet, Russia

CCTV, China
Coin Building,
Dubai
King’s Cross
Station

• As the lateral loads are resisted by diagonal columns, the top
storey displacement is very much less in diagrid structure as
compared to the simple frame building.
• The storey drift and storey shear is very much less for diagrid
structural system
• Diagrid provide more resistance in the building which makes
system more effective
• Diagrid structure provides more economy in terms of
consumption of steel and concrete as compare to the simple
fame building
• Diagrid structure provides more flexibility in planning interior
space and façade of the building.
• Aesthetics and functionality of a building are improved
simultaneously through diagrid.
Conclusions

• For every diagrid structure, it is recommended that diagrid structure is
to be designed distinctively.
• Any methodology can be applied provided the maximum structural
responses are within the limits
• Analysis of the Concrete diagrid structures is very little available and
further research needs to be done towards it.
• Optimization techniques through different algorithms can be more
advancely applied and research towards it is needed.
• They require skilled workers to work with nodes, fabrication is aslo
needed hence expensive in construction

Future Scopes
• Diagrid Structures even due to their advantages and
economy are handful in India due to demanding skillset
of workers. The node connector can be fabricated in
factories by training the workers and engineers the
required knowledge and skills.
• The Diagrid Structures are to be compared with the
available structural systems such as hexagrid and
pentagrid
• Research in the analysis of Diagrid structure with core
eccentricity is expected for practical applications of this
structure.

• Khushbu Jania, Paresh V. Patel, “Analysis and Design of Diagrid
Structural System for High Rise Steel Buildings”, 3rd Nirma
University International Conference on Engineering (NUICONE-
2012)
• Mir M. Ali and Kyoung Sun Moon, “Structural Developments in
Tall Buildings: Current Trends and Future Prospects”,
Architectural Science Review, Volume 50.3, Pp 205-223
• Terri Meyer Boake, Diagrids, “The New Stablity System:
Combining Architecture with engineering”, Aei (Asce), 2013, P
578-583
• Kyoung-Sun Moon, Jerome J. Connor and John E. Fernandez,
“Diagrid Structural Systems for Tall Buildings: Characteristics
and Methodology for Preliminary Design”, Struct. Design Tall
Spec. Build. 16, 205–230 (2007)
• Kyoung Sun Moon, “Structural Engineering for Complex-
Shaped Tall Buildings”, ASCE, 2011
References

• T. M. Boake,” Diagrid Structures: Innovation and Detailing”,
School Of Architecture, University Of Waterloo, 2012
• Vinubhai Patel and N. Panchal, “Diagrid Structural System:
Strategies to Reduce Lateral Forces on Building”, IJRET, Vol. 5,
2012
• Raghunath .D. Deshpande, Sadanand M. Patil, Subramanya
Ratan, “Analysis and Comparison of Diagrid and Conventional
Structural System”, IRJET, Volume: 02, Issue: 03, June -2015
• Khushbu D. Jani and Paresh V. Patel, “Design of Diagrid
Structural System for High Rise Steel Buildings as Per Indian
Standards”, Structures Congress, 2013
• Dong-Kyu Lee, Uwe Starossek2 and Soo-Mi Shin,” Optimized
Topology Extraction of Steel-Framed Diagrid Structure for Tall
Buildings”, International Journal of Steel Structures June 2010,
Vol 10, No 2, 157-164

• Kyoung Sun Moon,” Optimal Grid Geometry of Diagrid
Structures for Tall Buildings”, Architectural Science Review,
2008, Volume 51.3, Pp 239-251
• Rohit Kumar Singh, Dr. Vivek Garg, Dr. Abhay Sharma,
“Analysis And Design Of Concrete Diagrid Building And Its
Comparison With Conventional Frame Building”,
International Journal Of Science, Engineering And
Technology, Volume 2, Issue 6, August 2014
• Raghunath .D. Deshpande, Sadanand M. Patil,
Subramanya Ratan, “Analysis And Comparison Of Diagrid
And Conventional Structural System”, International Journal
Of Science, Engineering And Technology, Volume 02, Issue
03,June 2015

Diagrid Structures: Introduction & Literature Survey

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