11 - Lateral Load Distribution - Walls Shearlines Buildings · 2020. 10. 13. · Lateral Load Distribution: Walls with Openings, Shearlines and Buildings 8:30 AM –10:30 AM Bennett

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)



Deflections of Solid Shear Walls(Pages 469-479)

Flexural Deflection

• Shear Wall as Cantilever with Point Load at End

• From Mechanics

∆3

3

VΔ



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



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



Validity of

Analysis

Deflections of Solid Shear Walls with Openings(Pages 469-479)



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



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



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



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



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



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



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



“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



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



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



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?



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



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



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



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



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”



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



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



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



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



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



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

Documents

11 - Lateral Load Distribution - Walls Shearlines Buildings · 2020. 10. 13. · Lateral Load Distribution: Walls with Openings, Shearlines and Buildings 8:30 AM –10:30 AM Bennett