lecture5混凝土结构设计原理英文课件
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Compute moment capacity of the beam.
Mn
As
fy
d
a 2
2.4 in2
60
ksi 15.5
in.
3.53 2
in.
1979 k-in. 164.8 k-ft.
Flexural Stress – Non-Rectangular Example
For a non-rectangular beam For the given beam with concrete rated at fc = 6 ksi and the steel is rated at fs = 60 ksi. d = 12.5 in.
Example of rectangular reinforced concrete beam.
Given a rectangular beam fc = 4000 psi fy = 60 ksi (4 #7 bars) b = 12 in. d = 15.5 in. h= 18 in. Find the neutral axis. Find the moment capacity of the beam.
3.53 in.
0.85 fcb 0.854 ksi12 in
The neutral axis is c a 3.53 in. 4.152 in.
b1 0.85
Flexural Stress – Rectangular Example
Check to see whether or not the steel has yielded.
Flexural Stress – Rectangular Example
Determine the area of steel, #7 bar has 0.6 in2.
As 4 0.6 in2 2.4 in2
The b value is b1 = 0.85 because the concrete has a fc =4000 psi.
b1 0.85 for fc 4000 psi
Flexural Stress – Rectangular Example
From equilibrium (assume the steel has yielded)
C T
0.85 fcba f y As
a
f y As
60 ksi 2.4 in2
b1 0.85 for fc 4000 psi
b1
0.85 0.05*
fc
4000
0.65
1000
Flexural Stress
Requirements for analysis of reinforced concrete beams
[1] Stress-Strain Compatibility – Stress at a point in member must correspond to strain at a point. [2] Equilibrium – Internal forces balances with external forces
Lecture 5 - Flexure
June 11, 2003 CVEN 444
Lecture Goals
Rectangular Beams Safety factors Loading and Resistance Balanced Beams
Flexural Stress
The compressive zone is modeled with a equivalent stress block.
d
a 2
Flexural Stress
Example of rectangular reinforced concrete beam.
(3) Need to confirm es > ey
ey
y
Es
a c
b1
es
d
c
c
e
c
ey
Flexural Stress – Rectangular Example
Flexural Stress – Non-Rectangular Example
T As fs C 0.85 fcab a As fs
0.85 fcb
Flexural Stress
Example of rectangular reinforced concrete beam. (2) Find flexural capacity.
Mn T moment arm
As
fs
(a) Determine the area of the steel for a balanced system for shown area of concrete.
(b) Determine the moment capacity of the beam. Mn
(c) Determine the NA.
Flexural Stress
Example of rectangular reinforced concrete beam. (1) Setup equilibrium.
Fx 0 T C
As fs 0.85 fcab
M 0
T
d
a 2
Mn
Flexural Stress
Example of rectangular reinforced concrete beam. (2) Find flexural capacity.
ey
fy Es
60 ksi 29000 ksi
0.00207
Check the strain in the steel
es
d
c
c
0.003
Steel yielded!
15.5
in. 4.152 4.152 in.
ຫໍສະໝຸດ Baiduin.
0.003
0.0082
0.000207
Flexural Stress – Rectangular Example
Flexural Stress
The equivalent rectangular concrete stress distribution has what is known as a b1 coefficient is proportion of average stress distribution covers.
Mn
As
fy
d
a 2
2.4 in2
60
ksi 15.5
in.
3.53 2
in.
1979 k-in. 164.8 k-ft.
Flexural Stress – Non-Rectangular Example
For a non-rectangular beam For the given beam with concrete rated at fc = 6 ksi and the steel is rated at fs = 60 ksi. d = 12.5 in.
Example of rectangular reinforced concrete beam.
Given a rectangular beam fc = 4000 psi fy = 60 ksi (4 #7 bars) b = 12 in. d = 15.5 in. h= 18 in. Find the neutral axis. Find the moment capacity of the beam.
3.53 in.
0.85 fcb 0.854 ksi12 in
The neutral axis is c a 3.53 in. 4.152 in.
b1 0.85
Flexural Stress – Rectangular Example
Check to see whether or not the steel has yielded.
Flexural Stress – Rectangular Example
Determine the area of steel, #7 bar has 0.6 in2.
As 4 0.6 in2 2.4 in2
The b value is b1 = 0.85 because the concrete has a fc =4000 psi.
b1 0.85 for fc 4000 psi
Flexural Stress – Rectangular Example
From equilibrium (assume the steel has yielded)
C T
0.85 fcba f y As
a
f y As
60 ksi 2.4 in2
b1 0.85 for fc 4000 psi
b1
0.85 0.05*
fc
4000
0.65
1000
Flexural Stress
Requirements for analysis of reinforced concrete beams
[1] Stress-Strain Compatibility – Stress at a point in member must correspond to strain at a point. [2] Equilibrium – Internal forces balances with external forces
Lecture 5 - Flexure
June 11, 2003 CVEN 444
Lecture Goals
Rectangular Beams Safety factors Loading and Resistance Balanced Beams
Flexural Stress
The compressive zone is modeled with a equivalent stress block.
d
a 2
Flexural Stress
Example of rectangular reinforced concrete beam.
(3) Need to confirm es > ey
ey
y
Es
a c
b1
es
d
c
c
e
c
ey
Flexural Stress – Rectangular Example
Flexural Stress – Non-Rectangular Example
T As fs C 0.85 fcab a As fs
0.85 fcb
Flexural Stress
Example of rectangular reinforced concrete beam. (2) Find flexural capacity.
Mn T moment arm
As
fs
(a) Determine the area of the steel for a balanced system for shown area of concrete.
(b) Determine the moment capacity of the beam. Mn
(c) Determine the NA.
Flexural Stress
Example of rectangular reinforced concrete beam. (1) Setup equilibrium.
Fx 0 T C
As fs 0.85 fcab
M 0
T
d
a 2
Mn
Flexural Stress
Example of rectangular reinforced concrete beam. (2) Find flexural capacity.
ey
fy Es
60 ksi 29000 ksi
0.00207
Check the strain in the steel
es
d
c
c
0.003
Steel yielded!
15.5
in. 4.152 4.152 in.
ຫໍສະໝຸດ Baiduin.
0.003
0.0082
0.000207
Flexural Stress – Rectangular Example
Flexural Stress
The equivalent rectangular concrete stress distribution has what is known as a b1 coefficient is proportion of average stress distribution covers.