ARE 563 SS II
Class 8 – Strength Analysis
Dr. Kyle Larson
Today’s Class
• ACI Strength Design
• Quiz at End
1
Design Stress Block
C
a
C=T
d-a/2
Beam Flexural Resistance
• Design flexural strength is given by fMn
• For singly reinforced rectangular beams:
𝑀 ≤ 𝜙𝑀
0.85𝑓 𝑎𝑏 = 𝐴 𝑓
𝑎=
𝑀 =𝑇 𝑑−
𝐴 𝑓
0.85𝑓 𝑏
𝑎
𝑎
=𝐴 𝑓 𝑑−
2
2
𝜌=
𝐴
𝑏𝑑
𝜌 = 𝑟𝑒𝑖𝑛𝑓𝑜𝑟𝑐𝑒𝑚𝑒𝑛𝑡 𝑟𝑎𝑡𝑖𝑜
2
Strains
0.003
0.85𝑓 𝑎𝑏 = 𝐴 𝑓
𝑎=
ecu
𝐴 𝑓
=𝛽𝑐
0.85𝑓 𝑏
𝑐=
𝑎
𝛽
𝜀
𝑑−𝑐
=
𝜀
𝑐
𝜀 =
𝑑−𝑐
0.003
𝑐
ACI Moment Capacity
d= 21”
b=16”
h= 24”
• Determine ACI Moment Capacity
𝜙𝑀
• With f’c=5 ksi and fy=60 ksi
• Max live load if L=30 feet for simplysupported
As = 6 #8’s
3
ACI Moment Capacity
h= 24”
d= 21”
b=16”
As = 6 #8’s
ACI Moment Capacity
h= 24”
d= 21”
b=16”
As = 6 #8’s
4
ACI Moment Capacity
D= 21”
B=16”
H= 24”
• Determine ACI Moment Capacity
𝜙𝑀
• With f’c=5 ksi and fy=60ksi
• Same problem as last class
period with different concrete
strength
• If does not work what can be
done if same steel is used and
architect would like to keep
depth about the same
As = 5 #9’s
H= 24”
ACI Moment Capacity
D= 21”
B=16”
As = 5 #9’s
5
H= 24”
ACI Moment Capacity
D= 21”
B=16”
As = 5 #9’s
H= 24”
ACI Moment Capacity
D= 21”
B=16”
As = 5 #9’s
6
ACI Moment Capacity
D= 21”
B=16”
H= 24”
• Determine ACI Moment Capacity
𝜙𝑀
• With f’c=4 ksi and fy=60ksi
• Same problem with different
concrete strength
• If does not work what can be
done if same steel is used and
architect would like to keep
depth about the same
As = 5 #9’s
H= 24”
ACI Moment Capacity
D= 21”
B=16”
As = 5 #9’s
7
H= 24”
ACI Moment Capacity
D= 21”
B=16”
As = 5 #9’s
H= 24”
ACI Moment Capacity
D= 21”
B=16”
As = 5 #9’s
8
ACI Moment Capacity
• 𝜌
•
∗
∗
=
=
∗
= 0.018
=
∗
h= 24”
• 𝜌=
d= 21”
b=16”
• Determine ACI Moment Capacity 𝜙𝑀
• With f’c=3 ksi and fy=60 ksi
• If does not work what can be done if same steel
is used and architect would like to keep depth
about the same
= 0.0027
= 0.0033
• 𝜌
= 0.0136 … 𝑡ℎ𝑖𝑠 𝑤𝑖𝑙𝑙 𝑏𝑒 𝑎𝑛 𝑖𝑠𝑠𝑢𝑒
As = 6 #9’s
ACI Moment Capacity
•𝑎=
=
.
∗ ( )
. ( )( )
b=16”
= 8.82“
• From ACI with f’c<4 ksi b1 = 0.85
from equation
•𝜀 =
.
= 10.38“
0.003 =
.
.
.
d= 21”
=
h= 24”
•𝑐=
.003 = .003 > 0.005
• We knew this
As = 6 #9’s
9
ACI Moment Capacity
h= 24”
Increase concrete strength
With 4 ksi rmax=0.0181
Increase b or d
Increasing d might upset architect
Let’s see what new b would have
to be
• 0.0136=6/(b*21)
• b=21 inches
• Let’s run with that
•
•
•
•
•
d= 21”
b=16”
• Options
As = 6 #9’s
ACI Moment Capacity
h= 24”
d= 21”
b=21”
• Determine ACI Moment Capacity 𝜙𝑀
• With f’c=3 ksi and fy=60 ksi
As = 6 #9’s
10
ACI Moment Capacity
h= 24”
d= 21”
b=21”
As = 6 #9’s
ACI Moment Capacity
h= 24”
d= 21”
b=21”
As = 6 #9’s
11
Questions?
Please email me
kyle@ksu.edu
12