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Engineered Masonry Design Course Saturday April 28, 2018
© 2018 Canada Masonry Design Centre 1
Lateral Load Distribution: Walls with Openings, Shearlines and
Buildings8:30 AM – 10:30 AM
Bennett Banting
Lecture Outline1. Wall Deflections and Rigidity
a) Solid Shear Walls (15)b) Shear Walls with Openings and
Shearlines (45)2. Distribution of Lateral Loads
a) Rigid diaphragm structures (40)b) Flexible diaphragm structures
(20)
Engineered Masonry Design Course Saturday April 28, 2018
© 2018 Canada Masonry Design Centre 2
Deflections of Solid Shear Walls(Pages 469-479)
Flexural Deflection
• Shear Wall as Cantilever with Point Load at End
• From Mechanics
∆3
3
VΔ
Engineered Masonry Design Course Saturday April 28, 2018
© 2018 Canada Masonry Design Centre 3
Shear Deflection
• Normally neglected for flexural members
• Significant for squat walls
VΔ
∆ 1.20
0.4
Total Wall Deflection ∆
Vh33EI
1.20Vh0.4EAA tℓwI tℓ12
0.0%
20.0%
40.0%
60.0%
80.0%
100.0%
0 2 4 6 8 10
% C
ontri
butio
n to
Tot
al
Def
lect
ion
Aspect Ratio (Height / Length)
Shear
Flexure
Engineered Masonry Design Course Saturday April 28, 2018
© 2018 Canada Masonry Design Centre 4
Connected Shear Walls in a
“Shearline”
• Earthquake loads and Wind loads act on the entire structure
• Whole building loads are resisted by shear walls
• Forces carried to foundation
• Walls connected • Displace the same
Vshearline
V1 V2 V3 V40.03950.343 11.5%
0.0950.343 27.7%
0.1690.343 49.3%
0.03950.343 11.5%
Masonry Shearline Analysis
• Incorporates walls with openings
• Use with multiple shearlines in a building plan
• Determine whole building elastic force distribution
• Estimate torsional sensitivity
Engineered Masonry Design Course Saturday April 28, 2018
© 2018 Canada Masonry Design Centre 5
Validity of
Analysis
Deflections of Solid Shear Walls with Openings(Pages 469-479)
Engineered Masonry Design Course Saturday April 28, 2018
© 2018 Canada Masonry Design Centre 6
Stiffness Reduction
from Openings
• Define shear wall behavior as a composite of smaller wall segments and piers
• Cantilever/Fixed-Fixed boundary conditions
• Account for changes to stiffness of height• Piers aligned vertically vs. horizontally
Detailing around
Openings
• Presence of movement joints
• Continuity of reinforcement
• Size and Span
Engineered Masonry Design Course Saturday April 28, 2018
© 2018 Canada Masonry Design Centre 7
Fixed-Fixed Boundary Conditions• Some elements are restricted
against rotation at top and bottom
• Typically termed ‘piers’• Often shear critical
0%
20%
40%
60%
80%
100%
0 2 4 6 8 10
% C
ontri
butio
n to
Tot
al D
efle
ctio
n
Aspect Ratio (Height / Length)
Total Wall Deflection
∆ VEthℓw 3
hℓw
Shear
Flexure
Engineered Masonry Design Course Saturday April 28, 2018
© 2018 Canada Masonry Design Centre 8
Discretizing a Wall with Openings
V
Discretizing a Wall with Openings• Top Slab
• Link member• Free Rotations • Lateral Displacements
• Consistent Wall Properties• Block Strength, Grouting,
Unit Size
1
2
34 5
6
Engineered Masonry Design Course Saturday April 28, 2018
© 2018 Canada Masonry Design Centre 9
Solid Wall 1
• Cantilever behaviour• Determine unit rigidity• Factor out E, t, normalized
by Vℓ ℓ
Solid Walls 2-6
• The behavior of the wall with openings would not be the same as if it were a solid wall
• R is Reduced• Δ is Increased
• 3, 4, 5 Assume fixed against rotation
• Engineering judgement
Engineered Masonry Design Course Saturday April 28, 2018
© 2018 Canada Masonry Design Centre 10
Vertical and Horizontal Alignment • Horizontally Aligned
• Wall Segments Act in “Parallel”• Share Equal Displacement
• Vertically Aligned• Wall Segments Act in “Series”• Displacement is algebraic sum of
each
, ,1∆, ,
1∆1 ∆2 ∆3
, ,1
11
12
13
∆1 ∆2 ∆3∆ 11
∆11∆2
1∆3
∆ ∆1 ∆2 ∆3
, , 1 2 3
Segment Addition and Subtraction
Includes Rotation
Neglects Rotation ∆ ∆1 ∆2 ∆3
Engineered Masonry Design Course Saturday April 28, 2018
© 2018 Canada Masonry Design Centre 11
Assumptions• We are considering an idealized analysis of the
wall to estimate lateral load distribution• Elastic conditions Equally at the same time• Aligned openings• Single storey• Point loading• Neglect Axial Load Effects on Stiffness
Assumptions• Once we deviate away from
these parameters this approach will become less valid
• Push-over analysis• Finite element• Strut and tie
Engineered Masonry Design Course Saturday April 28, 2018
© 2018 Canada Masonry Design Centre 12
Solution Strategy
• Determine deflection of wall with opening:
1. Start with deflection an equivalent solid wall• Defined as Solid2,3,4,5,6
2. Subtract the deflection of an equivalent solid base• Defined as Solid3,4,5,6
3. Add back the deflection of the lower segments 3,4,5,6
2
34 5
6
Solid2,3,4,5,6
Solid3,4,5,6 34 5
6
Solution Strategy
• Determine displacement of 4,5,6
• Same process1. Start with Solid4,5,62. Subtract Solid4,53. Add Piers 4 and 5
4 5
6Solid4,5,6
Solid4,5 4 5
Engineered Masonry Design Course Saturday April 28, 2018
© 2018 Canada Masonry Design Centre 13
“Parallel” or “Series”
Cantilever or Fixed-Fixed
Piers 3,4,5,6 Fixed-Fixed
Solid3,4,5,6 Cantilever
Solid2,3,4,5,6 Cantilever
Act as Series
Top Displacements algebraically added/subtracted
Act in Parallel
• Piers 3,4,5,6• Fixed-Fixed
• Pier 3• Fixed-Fixed
• Pier4,5,6• Fixed-Fixed
Engineered Masonry Design Course Saturday April 28, 2018
© 2018 Canada Masonry Design Centre 14
Act in Series
• Act in Series• Solid4,5,6
• Fixed-Fixed• Solid4,5
• Fixed-Fixed• Piers 4, 5
• Fixed-Fixed
• Act in Parallel• Pier 4• Pier 5
∆ , , , , ∆ , , , , ∆ , , , ∆ , , ,
∆ , , , 11∆
1∆ , ,
∆ , , ∆ , , ∆ , ∆ ,
∆ , 11∆
1∆
Review
Engineered Masonry Design Course Saturday April 28, 2018
© 2018 Canada Masonry Design Centre 15
Boundary Conditions
• Strong Beam – Weak Column• Strong Column – Weak Beam
Solution:Start with
Solid Sections
Rf 1hℓw 3
hℓw
Rc 14 hℓw 3
hℓw
Wall Segment
h (m) ℓ (m) R Δ
1c 8 4 0.026 38.5Solid2,3,4,5,6c 8 9 0.18 5.56Solid3,4,5,6c 4 9 0.59 1.693f 4 2 0.071 14.1Solid4,5,6f 4 5 0.343 2.92Solid4,5f 2 5 0.791 1.264f 2 2 0.25 4.05f 2 1 0.071 14.1
Engineered Masonry Design Course Saturday April 28, 2018
© 2018 Canada Masonry Design Centre 16
Distribution of Lateral and Axial Load in a Shearline
V
16.0%V 84.0%V
21.2%V 48.9%V 13.9%V
Vi VRi∑Ri
Review
• Approximate method• Distributes loads based on relative stiffness of
elastic sections• Limitations to analysis
• Distribute shear• To walls within a shear line• Within a wall with openings to individual piers
• Solution• Relate section back to equivalent solid sections
adding and subtracting as required• Boundary conditions by judgement
• Would this member be restrained by rotations?
Engineered Masonry Design Course Saturday April 28, 2018
© 2018 Canada Masonry Design Centre 17
Rigid Diaphragm Behaviour(Pages 469-479)
Distributing Loads to Many Walls
• Walls within a shearline• Share the same top displacement• Move via linked members
• Walls within a building• Rigid diaphragm action• Relative position on diaphragm does
not change
Engineered Masonry Design Course Saturday April 28, 2018
© 2018 Canada Masonry Design Centre 18
Wind Forces vs. Seismic Forces• “Lateral Force Resisting System”
• A general term for the structural system of a building which resists lateral loads
• “Seismic Force Resisting System”• SFRS is a specific term designated by
the national building code to resist seismic loads
• SFRS are defined by the NBCC
Wind Forces vs. Seismic Forces
R1
R2
V
Engineered Masonry Design Course Saturday April 28, 2018
© 2018 Canada Masonry Design Centre 19
Resultant Loads in Buildings• Wind Forces
• Act on exterior face• Half of the Resultant acts at centroid of
windward face• Other half goes to foundation
• Seismic Forces• Generated as an acceleration acting on
building mass• Resultant acts at centroid of building mass
Diaphragm Action
• “Rigid Diaphragm” Behavior• Floor/roof diaphragms are unlikely to
deform in-plane significantly under design loads relative to shear walls
• Shear walls are effectively linked together in their movement
• “Flexible Diaphragm” Behavior• Floor/roof diaphragms are likely to
deform in-plane under design loads relative to shear walls
• Shear walls effectively move independently outside of a shearline
Engineered Masonry Design Course Saturday April 28, 2018
© 2018 Canada Masonry Design Centre 20
Rigid Diaphragm Behaviour• The diaphragm remains stiff under lateral
loading • Most concrete floor systems can be
considered as rigid systems• Rigid Diaphragms = Rigid Motion, i.e.
the diaphragm is assumed not to deform under loading
• Translation + Rotation
Resultant Resistance in Buildings
Net resistance acts at centre of rigidity
VrxVry
VfyVfx
C.M.
C.R.x
C.R.y
Engineered Masonry Design Course Saturday April 28, 2018
© 2018 Canada Masonry Design Centre 21
Centre of Rigidity
• Consider a simple single-storey building• 10 m × 10 m Plan• C.M. = 5.0 m, 5.0 m• hw = 4.0 m• Cantilever boundary conditions, solid walls• Neglect out-of-plane stiffness
• ‘Goes along for ride’
C.M.
Vfx
#1
#3#2
Elastic Wall Rigidities
• R1 = 0.5• R2 = 0.143• R3 = 0.143ℓ ℓ
• V1x = 63.6% Vfx• V2x = 18.2% Vfx• V3x = 18.2% Vfx
∑
“Direct Shear”
Engineered Masonry Design Course Saturday April 28, 2018
© 2018 Canada Masonry Design Centre 22
Centre of Rigidity
C.M.Vfx
#1
#3#2
C.R.x Vrx
ey
#1
#3#2
#4
#6
#5
0,0
Wall # xref yref Rx Ry
1 5.0 9.905 0.5
2 2.0 0.095 0.143
3 8.0 0.095 0.143
4 0.095 5.0 0.314
5 9.905 2.5 0.074
6 9.905 7.5 0.074
Centre of Rigidity• Unique x,y coordinates
• Torsional Shear Based on Relative distance
• Relative Coordinates
xCR ∑ Ryi xrefi∑ Ryi
yCR ∑ Rxi yrefi∑ Rxi
ey yCM yCRWall # xref yref xi yi
1 5.0 9.905 3.572 2.0 0.095 6.253 8.0 0.095 6.254 0.095 5.0 3.155 9.905 2.5 6.676 9.905 7.5 6.67
Engineered Masonry Design Course Saturday April 28, 2018
© 2018 Canada Masonry Design Centre 23
Torsional Shear
VyTi Ryi xi∑ Ryi xi2 Rxi yi2
Vxey
VxTi Rxi yi∑ Ryi xi2 Rxi yi2
Vxey
Distribution of Shear
Wall #
Direct% of V
Torsion% of V
1 63.6% -8.8%2 18.2% 4.4%3 18.2% 4.4%4 - 4.9%5 - 2.4%6 - 2.5%
#1
#3#2
#4
#6
#5
Engineered Masonry Design Course Saturday April 28, 2018
© 2018 Canada Masonry Design Centre 24
Additional Considerations
• Seismic Forces along other axes• Minimum eccentricities (NBCC)• Combined Loads
Review
• Distribute loads to numerous walls• Within a shearline and elsewhere
• Seismic Forces• Generated by seismic weight• Act at centre of mass
• Wind Forces• Generated by windward surface• Reaction loads carried into roof diaphragm and
foundation• Act at centroid of windward surface edge
• Direct and Torsional Shear
Engineered Masonry Design Course Saturday April 28, 2018
© 2018 Canada Masonry Design Centre 25
Flexible Diaphragm Behaviour
Flexible Diaphragms
• The flexural and shear stiffness of the diaphragm
• The span between resisting supports• The rigidity of supports• Special Considerations in NBCC• No Resultant
• Loads distributed by tributary area
Engineered Masonry Design Course Saturday April 28, 2018
© 2018 Canada Masonry Design Centre 26
Tributary Loads in Buildings• Wind Forces
• Act on exterior face• Seismic Forces
• Generated as an acceleration acting on building mass
• Distributed by tributary area• To in-plane members
Seismic Forces
• Direct Shear only• Proportionate to tributary
seismic weight (tributary area)
• V1x = Vfx × A1/A
A1
A2 A3
Vfx
Engineered Masonry Design Course Saturday April 28, 2018
© 2018 Canada Masonry Design Centre 27
Review
• Diaphragm flexible relative to walls• Relative displacement between walls is not
preserved• Depends on materials, spans, stiffness
• No resultant forces• Torsional effects are negated as independent
action of walls is assumed• Real World
• There is some component of rigidity• Walls in the same shearline often consider to be
linked at top still• Flexibility may not exist over entire structure