Lec.5 strength design method rectangular sections 1

Analysis & Design of
Reinforced Concrete Structures (1) Lecture.5 Strength Design Method
42
Dr. Muthanna Adil Najm
Historical Background:
From the early 1900's until the early 1960's, nearly all reinforced concrete design
in the united states was performed by the working stress design method (WSD). In
1956, as an appendix, the ACI code for the first time included ultimate strength
design method, although the concrete codes for several other countries had been
based on such considerations for several decades. In 1963, the code gave the
ultimate strength design method equal status as the WSD method. The 1977 code
made the ultimate strength design method the predominant method and only
briefly mentioned the WSD method. From 1977 until 2002 each issue of the code
permitted designers to use the WSD method and set out certain provisions for its
application. Beginning with 2002, however, permission is not included for the use
of WSD method.
Advantages of Strength Design:
1- Better estimation of load carrying ability resulted from taking the nonlinearity
of stress-strain diagram into account.
2- With strength design method, a more consistent theory is used throughout the
designs of reinforced concrete structures. While with WSD the transformed
area method is used for beams and a strength design procedure was used for
columns.
3- More realistic factor of safety is used in strength design method by using
different factors of safety for the different types of loads.
4- More economical designs by taking into consideration higher strength of steel
and concrete than used in WSD.
5- The strength method permits more flexible design, where designers can use
larger sections with smaller percentages of steel or smaller sections with higher
percentages of steel .
Structural safety:
Structural safety is assured by:
A- Using factored loads by multiplying service loads by magnification factors, the
ACI code depends on the load factors requirements of the International Building
Code (IBC). In section 9.2 the code presents the following load combinations:
 FDU  4.1 (ACI equation 9-1)
     orSorRLHLTFDU r5.06.12.1  (ACI equation 9-2)
   WLororSorRLDU r 8.00.16.12.1  (ACI equation 9-3)
 orSorRLLWDU r5.00.16.12.1  (ACI equation 9-4)
SLEDU 2.00.10.12.1  (ACI equation 9-5)
Strength Design Method

43
HWDU 6.16.19.0  (ACI equation 9-6)
HEDU 6.10.19.0  (ACI equation 9-7)
The basic load factor equation includes only dead load and live load:
Where:
U = Ultimate design load the structure needs to be able to resist.
D = Dead Load
L = Live Load
F = Fluid pressure load
T = Total effect of temperature, creep, shrinkage, differential settlement and
shrinkage compensating concrete.
H = Load due to lateral earth pressure of soil, bulk materials or ground water
Lr = Roof live load
S = Snow load
R = Rain load
W = wind load
E = seismic load effect.
B- Using reduction factors for concrete strengths, ACI code section 9.3 prescribes
the following values:
9.0 for bending of beams and slabs.
75.0 for shear and torsion.
65.0 for tied reinforced columns and
7.0 for spirally reinforced columns.
65.0 for bearing on concrete.
The reduction factor of tension and compression members depends on the value of
the net tensile strain in the extreme tension steel ( εt ) as shown in Figure R9.3.2 in
the ACI commentary to the 2005 ACI code.
LDU 6.12.1 
= 0.002tε = 0.005tε
0.002) 250/3-tØ = 0.65 +(ε
0/300.002) 2-t+(ε7Ø = 0.
Ø = 0.65
Ø = 0.7
Ø = 0.9
Spiral
Others
Compression Controlled Tension ControlledTransition Zone
Figure R9.3.2
= 0.375tc/d= 0.6tc/d

44
Net tensile strain (εt ):
The nominal flexural strength of a member is reached when the strain in the
extreme compression fiber reached the assumed strain limit 0.003. The net tensile
strain ( εt ) is the tensile strain in the extreme tension steel at nominal strength. The
net tensile strain in the extreme tension steel is determined from a linear strain
distribution at nominal strength, as shown in Figure R.10.3.3 in the 2005 ACI
commentary.
Derivation of beam expression:
Whitney replaced the curved stress block with an equivalent rectangular block of
intensity cf 85.0 and depth ca 1 . The area of this rectangular block should
equal that of the curved stress block.
ACI code section (10.2.7.3) requires that:
85.01  for MPafMPa c 3017 
 30
7
05.0
85.01  cf for MPafc 30
But 1 should not be less than 0.65 ( 65.01  )
The flexural strength of member nM must at least be equal to the factored
moment uM .
b
d
sA yfsAT =
cf 
yield≥εsε
cε
yfsAT =
cf 85.0
c1β=a
Ultimate stress
distribution
Equivalent stress
Block
td
c
0.003 Compression
tε
Reinforcement closet to the
tension face of concrete
Figure R10.3.3

45
un MM 
  0xf  TC 
ysc fAbaf 85.0
bf
fA
a
c
ys


85.0
since
bd
As
 
c
y
f
df
a


85.0

…. (1)
The nominal moment equals:













22
a
dfA
a
dTM ysn
And the flexural strength is:







2
a
dfAM ysn …. (2)
Substitute equation (1) in equation (2)








c
y
yn
f
f
bdfM


7.1
1
12
…. (3)








c
y
yn
f
f
bdfM

 59.012
The above equation used for the design of section dimensions (b & d)
If 2
bd
M
R u
n














c
n
y
c
f
R
f
f
85.0
2
11
85.0

If
c
y
f
f


85.0
 then:









y
n
f
R 


2
11
1
The above equation used for the determination of steel reinforcement.
Minimum steel ratio:
To prevent excessive deflections and failure of concrete in tension due to crack
propagation, the ACI code section (10.5.1) specifies a certain minimum amount of
tensile steel reinforcement that must be used at every section for flexural members.
db
f
f
A w
y
c
s


25.0
min, (Used for MPafc 36.31 ) (ACI equation 10-3)
But not less than db
f
A w
y
s
4.1
min,  (Used for MPafc 36.31 )
yfsT = A
cf 85.0
a
2
a
abc0.85f'=C
2
a
d 

46
Where: bw = width of beam section
yy
c
ff
f 4.125.0
min 


Balanced steel Ratio:
At ultimate load for such a beam, the concrete will theoretically reach a strain of
0.003, and the steel will simultaneously yield.
The steel strain = yield strain =
s
y
E
f
=
200000
yf
From triangular strain diagram, neutral axis can be located as follows:








200000
003.0
003.0
yfd
c
d
f
c
y

600
600
…. (4)
c
y
f
df
a


85.0

 and
1
a
c 
c
y
f
df
c


185.0 

…. (5)
Equating ' c ' from equations 4 and 5;
d
ff
df
yc
y


 600
600
85.0 1
















 

yy
c
b
ff
f
600
60085.0 1

Maximum steel ratio:
To prevent compression failure of concrete and to assure steel yielding before
brittle compression failure of concrete, the (ACI 10-3-4) specifies a maximum
steel percentage. The net tensile strain limit of 0.005 for tension controlled section
was chosen to be a single value that applies to all types of steel. For non pre-
stressed members the strain limit should not be less than 0.004 to assure ductile
failure and to prevent reaching to the compression controlled zone.
005.0t Tension controlled section.
b
d
sA
s
y
E
f
=y=εsε
= 0.003cε
c
d-c
d

47
004.0t Compression controlled section













 

005.0003.0
003.085.0 1
max
y
c
f
f
  Reduction factor 9.0













 

004.0003.0
003.085.0 1
max
y
c
f
f
  Reduction factor 9.0
Under-reinforced and over-reinforced beams:
If the provided steel ratio exceeds the balanced steel ratio, the concrete will reach
its limiting strain ( 0.003 ) before yielding of steel, in this case the section is
considered as compression controlled flexural member and is over-reinforced.
Such section will fail suddenly by crushing of compression zone of concrete
without any previous warning (deflection and tension cracks), such section should
be avoided.
In under-reinforced section, the steel yields before concrete reaches its limiting
value (0.003) and continuous warning (deflection and cracks) are obviously
noticed before failure. In this section, the steel amount shall be less than the
balanced steel ratio of the ACI code.
Under-reinforced b  yt  
003.0c
ys ff 
cc ff 
Ductile failure
Balanced b  yt  
003.0c
ys ff 
cc ff  85.0
Brittle failure
Over-reinforced b  yt  
003.0c
ys ff 
cc ff 
Brittle failure
Minimum thickness ; Deflection requirements:
The ACI code section 9.5 provides minimum permissible beam and one way slab
depths to prevent large deflections. If the ACI Table 9.5.a is used, no need to
compute beam deflection due to applied loads.

48
Minimum thickness, h
Simply
supported
One end
continuous
Both ends
continuous
Cantilever
Member
Members not supporting or attached to partitions or other
construction likely to be damaged by large deflections
Solid one-way
slabs 20
l
24
l
28
l
10
l
Beams or
ribbed one-
way slabs 16
l
5.18
l
21
l
8
l
For light weight concrete and for steel with yield strength other than 420 MPa, the
table values shall be modified as follows:
1- For structural light weight concrete having unit density ( wc ) in the range of
1440-1920 kg/m3
, the values shall modified by ( 1.65-0.003wc ) but not less than
1.09
2- for fy other than 420 MPa, the values shall be multiplied by ( 0.4 + fy/700 ).

Lec.5 strength design method rectangular sections 1

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