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Combined Flexure and Axial Load
Interaction Diagram
Solidly grouted bearing wall
Partially grouted bearing wall
Bearing Walls: Slender Wall Design Procedure Stren th
Serviceability Deflections
Example Pilaster
Prestressed Masonry
Combined Flexural and Axial Loads 1
Interaction Diagram
Assume strain/stress distribution
Compute forces in masonry and steel
Sum forces to get axial force
Sum moment about centerline to get bending moment
Ke oints
Pure axial load
Pure bending
99180.08.02
hhAfAAfP stystnmn
=0.9 rr
9970r80.080.02
hAfAAfP stystnmn
Ast is area of
laterall tied steel
Combined Flexural and Axial Loads 2
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Example 8 in. CMU Bearing Wall
Given: 12 ft high CMU bearing wall, Type S masonry cement mortar;
Grade 60 steel in center of wall; #4 @ 48 in.; solid grout
Required: Interaction diagram in terms of capacity per foot
Pure Moment: ys
ysnbfAdfAM
'8.021
ftftkftftkMn /834.0/927.00.9
'8.0
bf
Aa
m
ys
Combined Flexural and Axial Loads 3
8.0
c
Example 8 in. CMU Bearing WallPure Axial: tr
12
1 4.65
201.2
144
in
in
r
h
ftkPn /7.680.9 99180.08.02
hhAfAAfP stystnmn
ftk/8.61
Combined Flexural and Axial Loads 5
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Example 8 in. CMU Bearing Wall
Maximum Moment at Pn = 61.8k/ft
a cs:
Combined Flexural and Axial Loads 7
Example 8 in. CMU Bearing WallChoose strain distribution (alternatively c)
Balanced conditionsT
Strain
Stress
Cm
mC
T
-TCP mn n
nM
Combined Flexural and Axial Loads 9
nM
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Example - Interaction Diagram
Point c (in) Cm (kip/ft) T (kip/ft) Pn (kip/ft) Mn(kip-ft/ft)
. . . .
a = d 4.76 54.8 0 49.4 7.85
c = d 3.8125 43.9 0 39.5 7.53
2.95 34.0 1.1 29.6 6.71
Balanced 2.09 24.0 3.0 18.9 5.37
1.8 20.7 3.0 16.0 4.81
1.5 17.3 3.0 12.8 4.16
1.2 13.8 3.0 9.7 3.46
. . . . .
0.6 6.9 3.0 3.5 1.85
Pure Moment 0.26 3.0 3.0 0 0.83
Combined Flexural and Axial Loads 11
Interaction Diagram
Combined Flexural and Axial Loads 12
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Interaction Diagram Below Balanced
Below the balanced point, the interaction diagram is a straight line:
22
/sp
ys
sp
ysun
tdfA
atfAPM
bf
PfAa
m
uys
80.0
/
These are equations 3-28 and 3-29 in the code except:
modified to account for non-centered steel (ignores any tension
in a possible second layer of steel near the compression face)
corrected Pu to Pu/
For centered bars: / adfAPM sun
Combined Flexural and Axial Loads 13
Partially Grouted Bearing Wall Small _______ forces
Partially grouted walls act as ______ walls
Compression area is in _____________
Strength design
Hi her axial loads act as _________
Very high axial loads act as ________
Need to calculate rbased on grouted cross-section.
Combined Flexural and Axial Loads 14
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Interaction Diagram: Solid vs. Partial Grout
Combined Flexural and Axial Loads 15
Walls: Slenderness EffectsSecond-order procedure (3.3.5): Assumes simple support conditions.
uP 05.0 No height limit mu
m fP
f 20.005.0 h/t 30
nA g
Complementary moment: Design moment
2
p us momen n uce y wa e ec ons
uuu
ufu
u PPM 28
Puf= Factored floor load
Puw = Factored wall load
ufuwu
Assumes maximum moment is at midheight
Combined Flexural and Axial Loads 16
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Walls: Deflections
Deflection Calculation
Mh52
hMMhM 55 22cr
nmIE
48 crcrm
cr
nm
cr MMIEIE
48
48
Deflection Limit hs 007.0 Calculated under service loads
cr.
3
2 bccd
tPAnI
spuscr
PfA
cuys
'
y m.
For centered bars:32 bccdPAnI u
Combined Flexural and Axial Loads 17
3fy
Walls: DesignDesign Procedure
1. Solve for Mu. Compare to Mn.. . , .
Solving for M222
cr
crnm
crf
crm
MMIIE
PIE
M
482848
1
22
crf
nm
MMPIE
M
2848
1
.
Combined Flexural and Axial Loads 18
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Walls - Design Procedure
1. Determine a, depth of compressive stress block
Preliminary estimate of steel; assume steel yields
bf
MtdPdda
m
uu
8.0
2/22
ums
PbafA
/8.0
. s
y
This neglects increase in moment due to second order effects. Can
,
amount of reinforcement.
Combined Flexural and Axial Loads 19
Example - Slender WallGiven: 18 ft high CMU bearing wall, with 2.5 ft parapet (total height is 20.5
ft); Type S masonry cement mortar; Grade 60 steel in center of wall; Dead
load from roof of 500 lb/ft; Roof live load of 400 lb/ft; Lateral wind load of
ps , ps on parape ; n up o ; oo orces ac on n.
wide bearing plate at edge of wall.
Required: Reinforcing steel.
Determine eccentricit
Solution:
Cross-section
of top of wall
e = 7.625in/2 1.0 in.
= 2.81 in.
Combined Flexural and Axial Loads 20
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Example - Slender Wall: Estimate Steel
Use 1.2D+1.0W+0.5Lr without second-order effects, parapet, and wall weight
.
b
MtdPdda uu
8.0
2/22
m
um Pbaf /8.0
y
sf
Combined Flexural and Axial Loads 21
Try #__@__ in., As = ______in2/ft
Example - Slender WallSummary of Strength Design Load Combination Axial Forces
(wall weight is 38 psf for 48 in. grout spacing)
Load CombinationPuf
(kip/ft)
Puw
(kip/ft)
Pu
(kip/ft)
. + . . . .
1.2D+1.6Lr+0.5W 1.240 0.524 1.764
1.2D+1.0W+0.5L 0.440 0.524 0.964
Puf= Factored floor load (just eccentrically applied load)
Puw = Factored wall load (includes wall and parapet weight; found at
midheight of wall between supports (9 ft from bottom)
Combined Flexural and Axial Loads 23
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Example - Slender Wall: McrFind modulus of rupture; use linear interpolation between no grout and full grout
Ungrouted (Type S masonry cement): 38 psi
Full routed T e S masonr cement : 153 si
psicells
groutedcell
psipsipsifr 2.576
1
)38153(38
Find Mcr, cracking moment:
Commentary allows one to include axial load
,
/ IfAPM nrnu
2/tcr
Combined Flexural and Axial Loads 24
Example - Slender WallCheck strength; 1.2D+1.0W+0.5Lr
ufe = m - e g momen rom momen a op o wa .
There will be a moment at the top of the wall from eccentric load and
a era orces on parape
Method assumes wind is providing suction on wall
Moments from lateral wind and eccentric load add together Lateral wind load on parapet will cause moment at top to decrease
Decrease in moment from parapet wind is ignored in calculations
(that is, wind load is considered to be zero on parapet)
Earthquake lateral forces on parapet would be included; first mode
has motion in opposite directions.
Combined Flexural and Axial Loads 26
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Example - Slender Wall: Moment of Inertia
Find In, net moment of inertia
ininftininininin
inftinIn
25.1*2625.7/22/25.181.325.112
25.1/122
32
3
ftinIn /1.331
4
5.2129000 ksiEn s
n cr, crac e moment o nert a.
b
PfAc
uys
'64.0
m
m
3
2 bccd
PAnI uscr
y
Combined Flexural and Axial Loads 27
Example - Slender WallCheck strength; 1.2D+1.0W+0.5Lr
Solvin for M rucruu MM
hPMeP
hwhPM
1155
1222
crgmcrm
uhP
M5
12
crmIE48
115 22 hPMehw
4828 crgm
ucruf
u
IIEP
Combined Flexural and Axial Loads 29
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Example - Slender Wall
Check strength; 1.2D+1.0W+0.5Lr
Compare to capacity:
PfAa uys /
/ adfAPM ysun
m.
Combined Flexural and Axial Loads 31
Example - Slender WallSummary of Strength Design Load Combinations
Load Combination M M M / M Second Order M /
(kip-in/ft) (kip-in/ft) First Order M
0.9D+1.0W 16.6 16.9 0.98 1.06
. + . r+ . . . . .
1.2D+1.0W+0.5Lr 18.2 18.6 0.98 1.12
Combined Flexural and Axial Loads 33
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Example - Slender Wall
Check deflections:
cr M(in4/ft) (k-in/ft)
D+0.6W 0.466 21.9 10.28 0.767+ +. . . r . . . .
0.6D+0.6W 0.434 20.7 9.71 0.708
22
Deflection Limit ininhs 51.1)216(007.0007.0
cr
crm
cr
nm
cr MMIEIE
48
48
Combined Flexural and Axial Loads 34
Example - Pilaster DesignGiven: Nominal 16 in. wide x 16 in. deep CMU pilaster; fm=1500 psi;
Grade 60 bar in each corner, center of cell; Effective height = 24 ft; Dead
load of 9.6 kips and snow load of 9.6 kips act at an eccentricity of 5.8 in. (2
n. ns e o ace ; n oa o ps pressure an suc on an up o
8.1 kips (e=5.8 in.); Pilasters spaced at 16 ft on center; Wall is assumed to
span horizontally between pilasters; No ties.
Solution:
e=5.8 in 2.0 in
ning
Lateral Load
w = 26psf(16ft)de
rticalSpaLoad
Insi
Combined Flexural and Axial Loads 35
Vd=11.8 in d = 15.625 7.625/2 = 11.8 in
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Example - Pilaster Design
Weight of pilaster:
Weight of fully grouted 8 in wall (lightweight units) is 75 psf. Pilaster is like a
double thick wall. Weight is 2(75psf)(16in)(1ft/12in) = 200 lb/ft
1.2D + 1.6SCritical location is to of ilaster. P = 26.9 ki s M = 156.0 ki -in
bf
MhdPdda
m
uu
8.0
2/2
2
. . .
y
ums
f
aA
.
Combined Flexural and Axial Loads 36
Example - Pilaster DesignWhy the negative area of steel?
Sufficient area from just masonry to resist applied forces.
Determine a from ust com ression.
in
inksi
kip
bf
Pa
m
u 44.16.155.18.0
9.26
8.0
Find the moment
ininat 44.16.15inipipPM u
190
229.26
22
Sufficient ca acit from ust masonr . No steel needed.
u -
. .
Combined Flexural and Axial Loads 38
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Example - Pilaster Design
0.9D + 1.0W Check wind suction
At top of pilaster. Pu = 0.9(9.6) 1.0(8.1) = 0.54 kips
Mu = 0.54kips(5.8in) = 3.1 kip-in
2
22
max 282 wL
MwLMM
wL
MLx
2
If xL, Mmax=M
. .
Combined Flexural and Axial Loads 39
Example - Pilaster Design0.9D + 1.0W At top: Pu=0.5 k Mu=3 k-in
a = 2.04 in A = 0.59 in2
x=144 in Pu=2.7 k Mu=361 k-in
1.2D + 1.0W + 0.5S At top: Pu=8.2k Mu=48k-in
a = 2.39 in A = 0.54 in2
x = 139in Pu=11.0k Mu=384k-in
1.2D + 1.6S + 0.5W At top: Pu=22.8k Mu=132k-inx=117in P
u=25.2k M
u=252k-in
. .
a = 1.92 in As = 0.14 in2
Re uired steel = 0.59 in2
Use 2-#5 each face, As = 0.62in2
Total bars, 4-#5, one in each cell
Combined Flexural and Axial Loads 41
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Example - Pilaster Design
Combined Flexural and Axial Loads 42
Columns Isolated vertical member; width 3 x thickness; height 4 x thickness
Minimum side dimension is 8 in. (1.14.1.1)
Effective height / least dimension 25 (2.1.6.1)
Distance between lateral supports 30 width (3.3.4.4.2)
Minimum eccentricity is 0.1t (2.1.6.2)
Minimum reinforcement is 0.0025An (1.14.1.2)
Maximum reinforcement is 0.04An (1.14.1.2) Additional maximum reinforcement requirements in strength design
Minimum of 4 bars, one in each corner(1.14.1.2)
. . . .
Allowable compressive stress is smaller of 0.4fy or 24 ksi (2.3.2.2.2)
Solid grouted (3.3.4.4.1)
: . . .
1/4 in. diameter; located in mortar joint or grout
spacing 16 longitudinal bar diameter, 48 tie diameter, or least cross-
Combined Flexural and Axial Loads 43
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Slenderness Effects
Long column reduction factor
Reduce ordinate (axial force) of interaction diagram
Piers, columns, and in-plane walls
3.3.4.1.1 Nominal axial and flexural strength The nominalaxial strength, Pn , and the nominal flexural strength, Mn , of
a cross section shall be determined in accordance with the
design assumptions of Section 3.3.2. Using the
slenderness-dependent modification factors of Eq. (3-17) [1-
r an q. - r , as appropr a e, e
nominal axial strength shall be modified for the effects of
slenderness.
Combined Flexural and Axial Loads 44
Interaction Diagram: Slenderness Effects
80
.,
h = 0 ft
60
70
h = 12 ft
40
50
P(kip-ft/ft)
20
30h = 20 ft
0
10
0 1 2 3 4 5 6 7 8
Combined Flexural and Axial Loads 45
M (kip-ft/ft)
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Bearing Walls
Location of Reaction:Wall section
Bearing area (1.9.5):
2A
w/3
1
1
1A
o A
Members that rotate will cause
reaction to shift towards edge
A2
A2 ends at edge
o mem er or
head joint in
stack bond
w/2
Members that experience
A1A2 Strength Design:
= 0.6 (3.1.4.5)
Bn = 0.6fmAbr(3.1.7)
Combined Flexural and Axial Loads 46
little rotation (deep truss) Plan view
Bearing WallsDistribution of Concentrated Loads Along Wall: (1.9.7)
Load is dispersed along a 2 vertical: 1 horizontal line.
Load
Load is dispersed
at 2:1 slope
Check bearing
on hollow wall
(a) Distribution of conenctrated load through bond beam
Combined Flexural and Axial Loads 47
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Bearing1
Load
1
Load Load
Walls 2
h
h/2
Effective Length Effective
Length
Effective
Length
1
2
Load
1
2
Load
Effective
Length
Effective
Length
Combined Flexural and Axial Loads 48
(b) Distribution of conenctrated load in wall
Prestressed Masonry
Combined Flexural and Axial Loads 49
www.masoncontractors.org/newsandevents/masonryheadlines/892004950.php
www.durowal.com/prod/pdf/catalog/07-sure-stress_post_tensioning.pdf
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Prestressed Masonry
Load indicating washer (LIW)
Combined Flexural and Axial Loads 50