Masonry Out-of- Plane Walls Ultimate Limit States: Flexure and Shear · 2018. 5. 3. · Flexure and...

Engineered Masonry Design Course Friday April 27, 2018

© 2018 Canada Masonry Design Centre 1

Masonry Out-of-Plane Walls Ultimate Limit States: Flexure and Shear12:30 PM – 2:00 PMBennett Banting

Lecture Outline1. Overview (10)2. Flexure and Axial Load Resistance

a) Fully-Grouted Wall Close-Spaced Reinforcement (35)b) Fully-Grouted Wall Wide-Spaced Reinforcement (15)c) Partially-Grouted Wall Wide-Spaced Reinforcement (15)d) Eccentric Reinforcement (5)

3. Shear Resistancea) Fully- and Partially-Grouted Shear Walls (10)



Flexure

• One-way Bending• Weak axis bending of masonry walls

• Wind• Earthquake• Lateral Earth

• Unit Wall Design• Design for a “per meter” equivalent wall• Effective Compression Zone Width

Axial Load

• Loadbearing vs. Non-loadbearing• Concentric vs. Eccentric vs. Virtual

Eccentric• Euler Buckling• Second Order Slenderness Effects



Flexure and Axial Load Interaction• Interaction Diagrams Useful• Complex Relationship• General Solution Strategy

• Solve Force Equilibrium for Pf = Pr• Solve Mr

• Equivalent Compression Block• In face shell versus in grouted cell

Slenderness Effects• Slenderness Ignored• Slenderness Considered• Special Provisions for Slenderness

• Wall Loads versus Wall Resistance• Wall Resistance Focus Here



Outside Scope

Two-way BendingTwo-way Bending

Walls with OpeningsWalls with Openings

Blast LoadingBlast Loading

Wall Resistance



Flexure & Axial Load Resistance:

-Fully-Grouted -Out-of-Plane Walls-Closely-Spaced Reinforcement

(Pages 376-380)Cl. 10 CSA S304

Masonry Out-of-Plane Loads

• Factored Axial Load• Applied Wind Load on Face

• Determine Moment Resistance• 6.0 m Long• 25 cm Units, tf = 38.6 mm• 30 MPa Block, Type S Mortar, Fully-Grouted• 15M Vertical Reinforcement @ 0.2 m



Determine Moment Capacity of Cross-section

Normal to Bed Joints

Out-of-Plane Wall Reinforcement• Single Bar Equivalent • Single Layer

• Double Layer in Some Instances (e.g. 30 cm units)

• Tolerances • Tension Reinforcement is not Required to Yield

• … except for Slender Walls with kh/t > 30 • Compression Reinforcement not Typical

• Difficult to provide



Design Assumptions

Equivalent Section for Analysis• Need not analyze entire wall

length• Determine equivalent “per

meter” effective cross-section

• Aligns with distributed loadsbeff = 1,000 mm/m

f′m,gr

f′m,gr

As, eff As 1,000mm/m

s

d

d

As = 200 mm2200 mm



No Axial LoadLet Pf = 0

Assumptions1. β1c < tf (solving with f′m,ug)2. εs > εy

εmu

εs

C

T

εc

εd c

f′m,gr or f′m,ug

Strain Compatibility

Force Equilibrium

When Compression Block Lies in Face Shell



Moment Resistance

C

TMr C d

β1c2

Property Wall6.0 m

Equivalent 1,000mm/m(Including Grout)

Equivalent 1,000mm/m(Neglecting Grout)

b 6,000 mm 1,000 mm/mAs 200 mm2 1,000 mm2/ms 200 mm 1,000 mm/m

β1c 49.4 mm 38.1 mm

εs 0.0028 0.00456

C 2,040.6 kN 340 kN/mMr 194.4 kNꞏm 206.0 kNꞏm 32.4 kNꞏm/m 34.3 kNꞏm/mPr 0 kN 0 kN/m 0 kN/m

When Ignoring

Grout Improves

Wall Capacity

f′m,ug

f′m,gr



Moderate Axial LoadLet Pf = 170 kN/m

Assumptions1. β1c > tf2. εs < εy

εmu

C

T

εc

εd cεs

Pf

fsε d c

c Es

Ignoring Grout

• Maximum compressive force β1c = tf• c > cb

• cgr = 79.0 mm• cug = 89.4 mm

C T P

ϕm0.85 f m, ug befftf ϕsAs, effε d c

c Es P



Moment Resistance

C

TMr C d

β1c2Pf

Property Wall6.0 m

Equivalent 1,000mm/m(Including Grout)

Equivalent 1,000mm/m(Neglecting Grout)

b 6,000 mm 1,000 mm/mAs 200 mm2 1,000 mm2/ms 200 mm 1,000 mm/m

β1c 63.2 mm 38.6 mm (c = 89.4 mm)εs 0.00156 0.00103

C 2,610 kN 2,067 kN 435.0 kN/m 344.5 kN/mMr 231 kNꞏm 208 kNꞏm 38.5 kNꞏm/m 34.7 kNꞏm/mPr 1,020 kN 170 kN/m 170 kN/m

High Axial LoadLet Pf = 670 kN/m

Assumptions1. c > d2. Section MUST be

designed as Unreinforced

C

Pf

εmu εmc

εmt



Unreinforced Behaviour in a Reinforced Wall

• Out-of-Plane Walls• Reinforcement at

mid-depth• Once under

compression, wall behaves as unreinforced

• Analysis per Cl. 7.2

When c ≥ d

Mf / Pf > 0.33t Must Remain “Uncracked”

Linear Elastic

Compression Controlled Tension Controlled

Mf / Pf ≤ 0.33t Permitted to be

“Cracked”

Equivalent Stress Block


σ PfAeMfSe

0.5ϕmf m PfAeMfSe ϕmft

PfAe

MfSe

Mr 0.5ϕmf mPfAe Se

Mr ϕmftPfAe Se

Pr = C = ϕmχ(0.85 f′m,gr)bβ1c Mr = C × (d-β1c/2)



Section Permitted to Crack?

• Assume Mf / Pf < 0.33t • Pf = Pr = 670 kN/m = C• Mf = Mr

Mr C d β1c2 47.8kN m/m

MrPr

47.8670

71.3mm 80mm

Moment Resistance

CMr C d

β1c2

Property Wall Equivalent 1,000mm/mb 6,000 mm 1,000 mm/mAs 200 mm2 1,000 mm2/ms 200 mm 1,000 mm/m

β1c 97.3 mmεs N/AC 4,020 kN 670 kN/mMr 286.8 kNꞏm 47.8 kNꞏm/mPr 4,020 kN 670 kN/m

Pf



Maximum Axial LoadSolve for Pf = 1,321.9 kN/m


designed as UnreinforcedC

Pf

εmu

Moment Resistance

CMr C d

β1c2

Property Wall Equivalent 1,000mm/mb 6,000 mm 1,000 mm/mAs 200 mm2 1,000 mm2/ms 200 mm 1,000 mm/m

β1c 192 mmεs N/AC 7,931.4 kN 1,321.9 kN/mMr 190.2 kNꞏm 31.7 kNꞏm/mPr 7,931.4 kN 1,321.9 kN/m

Pf

MrPr

31.71,321.9 24mm 80mm



Review

• Fully-Grouted Wall with Closely-Spaced Reinforcement• Moment Resistance Determined

• Effects of Axial Load and Reinforcement Ratio• Equivalent Section for Analysis

• In terms of a “per meter” equivalency• As,eff, beff, Loads and moments “/m”

• Moment about centre of wall• Cancel out reinforcement and axial load

Moment Capacity

High Axial Loads Reinforcement under

compressionWall behaves as “Unreinforced”

Moderate Axial Loads & Moderate ρCompression zone outside

face shell May count grout and use

f′m,gr, or f’mug; εs > εy or εs < εy

Low Axial Loads & Low ρSmall compression zone

could be in face shellMay be able to neglect grout

and use f′m,ug, εs >> εy

• Effects of changing Pf in Out-of-Plane walls

• Changes Depth of Effective Compression Block

• When to Ignore Grout

• Interaction with Axial Load




-Fully-Grouted -Out-of-Plane Walls-Wide-Spaced Reinforcement

(Pages 376-380)Cl. 10 CSA S304

Equivalent Section for Analysis• Wide-Spaced Reinforcement• Consider a 20M @ 1.2m

• From Before• f’m,gr = 13.5 MPa• f’m,ug = 17.5 MPa• tf = 38.6• d = 120 mm

beff = 1,000 mm/m

f′m,gr

f′m,gr

As, eff As 1,000mm/m

s

d

d

As = 300 mm21,200 mm



Effective Compression Zone Width

Similar to Flanged Beams or Walls

1.2 m

4 x 240 = 960 mm

Equivalent Section for Analysis• When Reinforcement is under

Tension• β1c ≤ tf

• f’m,ug• β1c > tf

• f’m,gr

1,000 mm/m

f′m,gr

f′m,grorf′m,ug

d

d

beff = 800 mm/m

beff 960mm 1,000mm/m1,200mm 800mm/m

As,eff = 250 mm2/m

As = 300 mm21,200 mm



No Axial LoadLet Pf = 0

Assumptions1. β1c < tf2. εs > εy

εmu

εs

C

T

εc

εd c

f′m,gr or f′m,ug

Strain Compatibility

Force Equilibrium

Moment Resistance

C

TMr C d

β1c2

Property Wall Equivalent 1,000mm/mb 6,000 mm 800 mm/mAs 200 mm2 250 mm2/ms 200 mm 1,000 mm/m

β1c 11.9 mmεs 0.0212C 510 kN 85 kN/mMr 51.6 kNꞏm 8.6 kNꞏm/mPr 0 kN 0 kN/m




Assumptions1. c > d 2. No Effective Compression Zone

from Cl. 10.6 3. Section MUST be designed as

Unreinforced

C

Pf

εmu εmc

εmt

beff = 1,000 mm/m

Review

• Differences from Closely-Spaced Reinforcement

• Equivalent Section for Analysis• Effective Compression Zone Width • Similar to Flanges• Only when Reinforcement is Under Tension

• Reinforcement under Compression• Exact same analysis• No Effective Compression Zone



Moment Capacity

High Axial Loads

Reinforcement under compressionNo Effective Compression Zone

Exact same design as closely-spaced wall

Moderate Axial Loads & Moderate ρSame process as closely-spaced reinforcement with new

valuesNew beff, As,eff

Check εs, β1c < tf

Low Axial Loads & Low ρ

Effective Compression Zone Width Include or Neglect Grout


-Partially-Grouted -Out-of-Plane Walls-Wide-Spaced Reinforcement

(Pages 376-380)Cl. 10 CSA S304



Equivalent Section for Analysis• Grouted Cell• May neglect grout for only face

shells1,000 mm/m

f′m,ugd

d

beff = 800 mm/m

f′m,gr

As,eff = 250 mm2/m

As = 300 mm21,200 mm

Considering Grout

beff,ug = 633.3 mm/m

beff,gr = 166.7 mm/m

beff,ug = 800 mm/m

beff, gr 200mm 1,000mm/m1,200mm 166.67mm/m



Moderate Axial LoadLet Pf = 250 kN/m

Assumptions1. β1c > tf2. εs < εy

εmu

T

εc

εd cεs

Pf

fsε d c

c EsCug = ϕmχ (0.85)f′m,ugbeff,ugtfCgr = ϕmχ (0.85) f′m,grbeff,grβ1c

CugCgr

Moment Resistance

Cug

T

Mr Cug dtf2 Cgr d

β1c2

Property Wall Equivalent 1,000mm/mb 6,000 mm 1,000 mm/m

beff,ug 4,800 mm 633.3 mm/mbeff,gr 1,200 mm 166.7 mm/mAs 300 mm2 250 mm2/ms 1,200 mm 1,000 mm/m

β1c 69.7 mmεs 0.00113

Cug 1,309 kN 218.2 kN/mCgr 480 kN 80.0 kN/mMr 172.7 kNꞏm 28.8 kNꞏm/mPr 1,500 kN 250 kN/m

Pf

Cgr



When Ignoring

Grout Improves

Wall Capacity

f′m,ug

f′m,gr

f′m,ugf′m,ug



designed as Unreinforced

Pf

εmu εmc

εmt

As c > d

CugCgr

beff,ug = 833.3 mm/m

beff,gr = 166.7 mm/m



Section Permitted to Crack?

• Assume Mf / Pf < 0.33t • Pf = Pr = 400 kN/m

• Cug = 287.1 kN/m• Cgr = 112.9 kN/m

• Mf = Mr

MrPr

37.0400 92.5mm 80mm

Mr Cug dtf2 Cgr d

β1c2 37.0kN · m/m

Maximum Design Moment

Take the Greater of:

1. Limit Mr for crack analysisa) Mr = Pf × 0.33t =

32.0 kNꞏm/m

2. Check if higher eccentricity possible for “section not permitted to crack”a) Tension controlledb) Compression controlled



When c ≥ d

Mf / Pf > 0.33t Must Remain “Uncracked”

Linear Elastic

Compression Controlled Tension Controlled

Mf / Pf ≤ 0.33t Permitted to be

“Cracked”

Equivalent Stress Block


σ PfAeMfSe

0.5ϕmf m PfAe

MfSe ϕmft

PfAe

MfSe

Mr 0.5ϕmf mPfAe Se

Mr ϕmftPfAe Se

Pr = C = ϕmχ(0.85 f′m,gr)bβ1c Mr = C × (d-β1c/2)

Elastic Wall Properties

• Textbook• Table B.4• Page 753

• Neglect Reinforcement• Equivalent Wall Section

f mbeff, ugAe, ugf m, ug beff, grAe, grf m, grbeff, ugAe, ug beff, grAe, gr

16.0MPa

ftbeff, ugAe, ugft, ug beff, grAe, grft, grbeff, ugAe, ug beff, grAe, gr

0.50MPa

Aebeff, ugAe, ug beff, grAe, grbeff, ug beff, gr

104,300mm2/m

Sebeff, ugSe, ug beff, grSe, grbeff, ug beff, gr

7,100,000mm3/m

Iobeff, ugIo, ug beff, grIo, grbeff, ug beff, gr

852,000,000mm4/m



Tension or Compression Controlled

• Elastic Stress Analysis• Best Practice

• Limiting compressive stress in masonry to 0.5f′m as elastic limit

• Determine which moment governs

Compression Controlled

Mr 0.5ϕmf mPfAe Se

Tension Controlled

Mr ϕmftPfAe Se


-Partially-Grouted -Out-of-Plane Walls-Wide-Spaced Eccentric Reinforcement

(Pages 376-380)Cl. 10 CSA S304



Out-of-Plane Loads

• Positive and Negative wind pressures

• Often equal or close• 2-Layers of vertical

reinforcement • When loads greater in one

direction• 1-Layer of vertical

reinforcement close to tension face

Eccentric Rebar



Shear Resistance:

-Out-of-Plane Shear-Sliding Shear

(Pages 398-399)Cl. 10 CSA S304

Out-of-Plane Shear



Out-of-Plane Shear Resistance

• f′m,eff = 16.0 MPa• Mf/Vfdv = 1• v = 0.64 MPa• d = 120 mm• b = bwebs + beff,gr

• Includes webs• CSA A165 estimate min.

Out-of-Plane Sliding Shear

Masonry Out-of- Plane Walls Ultimate Limit States: Flexure and Shear · 2018. 5. 3. · Flexure and...

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