# Simple Beam Design - Portal

Urban and Civil

Nov 25, 2013 (4 years and 6 months ago)

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Beam Design

RC BEAM DESIGN

Based on BS 8110

Partial Safety Factor
γ
m

Cl. 2.4.4.1

Given in Table 2.2

m
strength

stic
Characteri

strength
Design

Durability

Cl. 2.2.4 and 3.1.5

Exposure conditions (Integrity of rc concrete

ability to prevent corrosion) : moderate,
severe, very severe, most severe and
abrasive (Table 3.2)

Protection a agaist corrosion of steel (Table
3.3)

Fire resistance requirement (Table 3.4)

Nominal cover

Cl. 3.3.1.1

Bar Size (Cl. 3.3.1.2)

Single Bars: nominal cover
≥ main bar diameter, d
1

Paired bars:
nominal cover

√2

d1

Bundles bars: 2
√(A
equivalent
/
π
)

Example:

Using the data given, determine:

(i) the nominal cover required to the underside of
the beam, and

(ii) the minimum width of beam required.

Data:

Exposure condition mild

Characteristic strength of concrete (
f
cu) 40 N/mm2

Nominal maximum aggregate size (
h
agg) 20 mm

Diameter of main tension steel 25 mm

Diameter of shear links 8 mm

Minimum required fire resistance 1.5 hours

SOLUTION

Clause 3.3.1.2 Nominal cover ≥ (main bar diameter − link
diameter)

≥ (25 − 8) = 17 mm

Clause 3.3.1.3 Nominal cover ≥ nominal maximum
aggregate size > 20 mm

Clause 3.3.3 Exposure condition is mild

Table 3.3 Nominal cover ≥ 20 mm*

Clause 3.3.6 Minimum fire resistance = 1.5 hr

The beam is simply supported

Table 3.4 Nominal cover ≥ 20 mm

The required nominal cover = 20 mm

Flexural Strength of Sections

The flexural strength (i.e. the ultimate
moment of resistance of a cross
-
section)
is determined assuming the following
conditions as given in Clause 3.4.4.1,
BS8110:Part 1

Concrete Capacity

The maximum compressive force which can be
resisted by the concrete corresponds to the
maximum depth permitted for the neutral axis,
as shown in Figure 5.28 (i.e.
x
=
d
/
2
).

Concrete Capacity

Consider the moment of the compressive force about the
line of action of
F
t :

M
ult,concrete = (
F
c

×

z
)

where
F
c = compressive force = (stress
×

area)

= [0.45
f
cu

×

(
b
×

0.9
x
)]

(for the maximum concrete force
x
=
d
/2)

= [0.45
f
cu

×

(
b
×

0.45
d
)] = 0.2
bdf
cu

z
= lever arm

= [
d
− (0.5
×

0.45
d
)] = 0.775
d

{
Note
: In general
z
= [
d
− (0.5
×

0.9
x
)] }

M
ult,concrete = (0.2
bdf
cu )
×
(0.775
d
) =0.156
bd
2
f
cu

This equation can be rewritten as 0.156 =
M/bd
2
f
cu

Steel Capacity

M
ult,steel = (
F
s
×

z
)

where
F
s = tensile force = (stress
×

area)

= (0.95
f
y
×

A
s)

z
= lever arm

M
ult,steel = 0.95
f
y
A
s
z

A
s =
M
/0.95
f
y
z

Consider
z
, the lever arm:

z
= [
d
− (0.5
×

0.9
x
)] ,

0.9
x
= 2(
d

z
)

M
ult,concrete = [0.45
f
cu
×

(
b
×

0.9
x
)]
×

z
(substitute for
0.9
x
)

= 0.9
f
cu
b
(
d

z
)
z

Steel Capacity

M
ult,concrete =
K bd
2

f
cu

K bd
2

f
cu = 0.9
f
cu
b
(
d

z
)
z

Kd
2

= 0.9
f
cu
d
z − 0.9
z
2

z
2

dz
+ Kd2/0.9 = 0

The solution of this quadratic equation (
ax
2 +
bx
+
c
= 0) gives an expression which can be used to
determine the lever arm,
z
.

Summary

*The smaller of these two values being the critical case

When
K

K
′ for a singly
-
reinforced section the maximum
moment permitted, based on the concrete strength, is equal to
0.156
bd
2
f
cu

When
K
>
K
′ a section requires compression reinforcement.

Area of steel reinforcement, A
s

Areas/metre width for various pitches
of bars

Example 1:

A rectangular beam section is shown
in Figure 5.29. Using the data given,
determine the maximum ultimate
moment which can be applied to the
section assuming it to be singly
reinforced.

Data:

Characteristic strength of concrete (
f
cu)
40 N/mm2

Characteristic strength of steel (
f
y)
460 N/mm2

Solution:

Strength based on concrete:

M
ult = 0.156
bd
2
f
cu

= (0.156
×

250
×

420
2

×

40)/10
6

= 275.2 kNm

Strength based on steel:

A
s = 1260 mm
2

z = 0.775 d = 0.775 x 420 = 325.5

M
ult = 0.95
f
y
A
s
z

= (0.95
×

460
×

1260
×

325.5)/10
6

= 179.5 kNm

The maximum design moment which can be applied is:
M
ult = 179.5 kNm

Example 2:

The cross
-
section of a simply
supported rectangular beam
is shown in Figure 5.30. Using
the data given, determine the
maximum ultimate moment
which can be applied to the
section assuming it to be
singly
-
reinforced.

Data:

Characteristic strength of
concrete (
f
cu) 30 N/mm2

Characteristic strength of steel
(
f
y) 460 N/mm2

Nominal maximum aggregate
size (
h
agg) 20 mm

Diameter of main tension steel
32 mm

Diameter of shear links 8 mm

Exposure condition mild

Minimum required fire
resistance 1.0 hour

Solution:

Clause 3.3.1 Nominal cover to all steel

Clause 3.3.1.2 ≥ bar size

cover ≥ (32 − 8) = 24 mm

Clause 3.3.1.3 ≥ Nominal maximum aggregate size

cover ≥ 20 mm

Clause 3.3.3 Exposure condition: mild

Table 3.3 and
f
cu = 30 N/mm2

cover ≥ 25 mm

Clause 3.3.6 Min. fire resistance: 1 hr

beam is simply supported

Table 3.4 cover ≥ 20 mm

The required nominal cover to all steel = 25 mm

Solution

Figure 3.2 Minimum width
b
for 1 hour fire resistance = 200
mm

Effective depth

d
= (
h
− cover − link diameter − bar diameter/2)

= (475 − 25 − 8 − 16) = 426 mm

Strength based on concrete:

M
ult = 0.156
bd
2
f
cu = (0.156
×

250
×

426
2

×

30)/10
6

= 212.3 kNm

Strength based on steel:

A
s = 2410 mm
2

z = 0.775 d

M
ult = 0.95
f
y
A
s
z
= (0.95
×

460
×

2410
×

0.775
×

426)/10
6

= 349.9 kNm

The maximum design moment which can be applied is:
M
ult =
212.3 kNm

Example 3:

The cross
-
section of a simply supported rectangular
beam is shown in Figure 5.31. Using the data given,
and assuming the section to be singly
-
reinforced,
determine the area of tension reinforcement
required to resist an applied ultimate bending
moment of 150 kNm.

Data:

Characteristic strength of concrete (
f
cu) 40 N/mm2

Characteristic strength of steel (
f
y) 460 N/mm2

Nominal maximum aggregate size (
h
agg) 20 mm

Diameter of main tension steel Assume 25 mm

Diameter of shear links 10 mm

Exposure condition moderate

Minimum required fire resistance 2.0 hours

Solution:

Clause 3.3.1 Nominal cover to all steel

Clause 3.3.1.2 ≥ bar size

cover = (25 − 10) = 15 mm

Clause 3.3.1.3 ≥ Nominal maximum

aggregate size

cover = 20 mm

Clause 3.3.3 Exposure condition: moderate

Table 3.3
f
cu = 40 N/mm2

cover ≥ 30 mm

Clause 3.3.6 Min. fire resistance: 2.0 hr

beam is simply supported

Table 3.4 cover ≥ 40 mm

The required nominal cover to the main steel = 40 mm

Figure 3.2 Minimum width
b
for 2 hours, fire resistance =
200 mm

Figure 5.31

Solution:

Effective depth
d
= (
h
− cover − link diameter − bar
diameter/2)

= (450 − 40 − 10 − 13) = 387 mm

Clause 3.4.4.4

Check that the section is singly
-
reinforced:

K
= M/bd
2
fcu = 0.125 < 0.156

Since
K
<
K
′ the section is singly
-
reinforced

A
s =
M
/0.95
f
y
z
= [(150
×

10
6
) / (0.95
×

460
×

0.83
×

387)]

= 1068 mm
2

Adopt 4/ 20 mm diameter HYS bars providing 1260
mm2
.

d
d
K
d
z
95
.
0
83
.
0
9
.
0
25
.
0
5
.
0

The equation in BS 8110 (Cl 3.4.5.2)

Where

v
= design shear stress

V

= design shear force due to ultimate loads

b
v

d
=effective depth

v

0.8
√(
f
cu)

or

5 N/mm2

Shear Strength of Sections

d
b
V
v
v

Shear Strength of Sections

This is the equation given in Table 3.7 of the
code for the cross
-
sectional area of
designed
when the design shear stress
v
at a cross
-
section is greater than (
v
c + 0.4). It is recognised
that the truss analogy produces conservative
results and the code specifies that:
designed
are required when:

(
v
c + 0.4) <
v
< 0.8 (
fcu)
1/2

or 5 N/mm2

Shear Strength of Sections

It is important to provide minimum areas of steel in
concrete to minimize thermal and shrinkage cracking,
etc.

Minimum links are specified in the code which provide a
shear resistance of 0.4N/mm2 in addition to the design
concrete shear stress
v
c, and consequently designed
v
> (
v
c + 0.4) as indicated.

In Table 3.7:
are required when
0.5
v
c <
v
< (
v
c + 0.4)

The cross
-
sectional area of the links required is given by:

Example 3:

A concrete beam is simply supported over a 5.0
m span as shown in Figure 5.48. Using the data
given, determine suitable shear reinforcement.

Data:

Characteristic strength of concrete (
f
cu) 40 N/mm2

Characteristic strength of mild steel (
f
yv) 250 N/mm2

Maximum shear force at the support 40 kN

Solution:

Solution:

Consider the end of the beam at the support
where the shear force is a maximum = 40 kN

Clause 3.4.5.2

2
2
2
3
/
0
.
5
&
8
.
0
/
06
.
5
40
8
.
0
8
.
0
/
89
.
0
225
x
200
10
x
40
mm
N
f
v
mm
N
f
mm
N
d
b
V
v
cu
cu
v

Solution:

Table 3.7 To determine the required
reinforcement, evaluate
v

A
s = area of 2/16 mm diameter bars = 402 mm
2

mm
d
d
b
A
v
s
225
;
893
.
0
225
x
200
402
x
100
100

From Table 3.8

2
2
3
3
2
2
/
22
.
1
)
4
.
0
82
.
0
(
)
4
.
0
(
/
82
.
0
)
25
/
40
(
7
.
0
)
25
/
(
7
.
0
/
25

Since
/
7
.
0
]}
25
.
0
/
)
75
.
0
893
.
0
(
x
)
66
.
0
73
.
0
[(
66
.
0
{
mm
N
v
mm
N
f
v
mm
N
f
mm
N
v
c
cu
c
cu
c

Solution:

Since 0.5
v
c <
v
< (
v
c + 0.4), minimum links are required for the
whole length of the beam.

The cross
-
sectional area of the links required is given by

Option 1:

Assume 2
-
legged / 6 mm diameter mild steel links.

A
sv = 56.6 mm
2

Table 3.7 The spacing required ,

Clause 3.4.5.5

The maximum spacing of the links

0.75
d

= (0.75 x 225) = 169 mm

throughout the length of the beam.

mm
b
A
f
s
v
sv
yv
v
169
x200
0.4
6
.
56
x
250
x
95
.
0
4
.
0
95
.
0

Solution:

Option 2:

Clause 3.4.5.5

The maximum spacing of the links ≤ 0.75
d

= (0.75
×

225) = 169 mm

Assume
s
v = 150 mm.

*Adopt 6 mm diameter mild steel links (56.6 mm2) @ 150 mm
centres throughout the length of the beam as in option 1.

Example 4:

A concrete beam is simply supported over a 7.0
m span as shown in Figure 5.49. Using the data
given, determine suitable shear reinforcement.

Data:

Characteristic strength of concrete (
f
cu
) 40 N/mm2

Characteristic strength of mild steel (
f
yv
) 250 N/mm2

g
k) 5.0 kN/m

q
k) 30.0 kN/m

Solution:

×

5.0) + (1.6
×

30.0)]
= 55.0 kN/m

Ultimate design shear force at the support
V =
(55.0
×

3.5) = 192.5 kN

Solution:

Table 3.7 : To determine the required
reinforcement evaluate
v
c and to determine the
requirement for either minimum links or

Solution:

Table 3.7 Since (
v
c + 0.4) <
v
< 0.8
√(
fcu)

< 5 N/mm2

The
maximum shear force
which can be resisted by the
provision of
is given by:

V =
(
v
c + 0.4)
b
v
d
= (1.06
×

240
×

570)/10
3

= 145 kN

Solution:

Provide designed links for the first metre from each end

The cross
-
sectional area of the designed links required
is given by

Assuming 8 mm diameter mild steel links,
A
sv = 101
mm
2

and
f
yv

= 250 N/mm
2
.

Solution:

Clause 3.4.5.5

The maximum spacing of the links ≤ 0.75
d

= (0.75
×

570) = 428 mm

for 1.0 m from each end of the beam.

The cross
-
sectional area of the minimum links required is
given by Asv =

throughout the central 5.0 m of the beam.

Deflections (Cl. 3.4.6)

Allowances for these factors to reflect the actual
behaviour are made in the BS 8110 in terms of:

using a basic ‘span/
effective
depth’ ratio (span/
d
)
rather than a ‘span/
h
’ ratio (Table 3.9),

applying a modifying factor which is dependent on the
steel strength (steel percentage and steel strength
are directly related through the factor
K
=
M
/
bd
2
) and
the steel stress (
f
s) under service conditions
(Table 3.10),

applying a modifying factor which is dependent on the
percentage of compression steel provided
(Table 3.11).

Deflections

The basic ratios given in Table 3.9 relate
to three support conditions:

cantilevers,

simply supported beams,

continuous beams.

*
Note:
Continuous beams are considered to
be any beam in which at least one end of
the beam is continuous, i.e. this includes
propped cantilevers at the end of a series
of continuous beams.

Example 5:

A rectangular concrete beam 250 mm wide
×

475 mm overall depth is simply supported over a
7.0 m span. Using the data given, check the
suitability of the beam with respect to deflection.

Data:

Characteristic strength of concrete (
f
cu) 40 N/mm2

Characteristic strength of main steel (
f
y) 460 N/mm2

Design ultimate bending moment at mid
-
span (
M
)
150.0 kNm

Assume the distance to the centre of the main steel
from the tension face is 50 mm

Solution:

Effective depth
d
= (475 − 50) = 425 mm

Check that section is singly
-
reinforced:

Adopt 3/20 mm diameter HYS bars providing 943 mm
2

Solution:

Clause 3.4.6 There is no redistribution of
moments

β b = 1.0

Table 3.9 The beam is simply supported:

L/d = 20

Clause 3.4.6.5

M
/
bd
2

= (
K
×

f
cu )

= (0.082
×

40) = 3.32 N/mm2

2
/
7
.
291
1
x
3
2
mm
N
As
fyAs
f
b
provided
required
s

Interpolate from Table 3.10

0.92

Solution:

L/d x value from Table 3.10 = L/d x 0.92

Allowable deflection = 20 x 0.92 = 18.4

Actual deflection = L/d = 6000/425 = 14.12

So, the deflection is acceptable.

Minimum Areas of Steel (Cl. 3.12.5)

Minimum areas of steel are required in structural
elements to ensure that any unnecessary
cracking due to thermal/shrinkage effects or
minimized.

The required minimum steel areas for different
situations, i.e. tension or compression
reinforcement, rectangular beams/slabs, flanged
beams and columns/walls, are given in
Table 3.25

Example 5: Minimum Areas of Steel

Determine the minimum areas of steel required
for the cross
-
sections shown in Figure 5.57; in
all cases
f
y = 460 N/mm
2
.

(a)
Rectangular beam

The required area of steel is given by:

100
A
s /
A
c = 0.13

where:

A
s is the minimum recommended area of
reinforcement,

A
c is the total area of concrete.

A
c = (550
×

320) = 176
×

10
3

mm
2

A
s = (0.13
×
176
×
10
3
)/100= 229 mm
2

(b)
Flanged beam
(see Cl
3.4.1.5)

where:

b
is the breadth of the section
b
effective breadth of the rib; for a box, T or I section,
b
w is

taken as the average breadth of the concrete below the
flange,

h
is the overall depth of the cross
-
section of a reinforced
member.

b
w

/
b
= (300 / 1200) = 0.25 < 0.4

the required area of steel is given by:

100
A
s /
b
w
h
= 0.18

bw h
= (300
×

600) = 180
×

10
3

mm
2

A
s = 0.18
×
180
×
10
3
/100 = 324 mm
2

Maximum Areas of Steel (Cl. 3.12.6
)

The maximum percentages of steel given
in the code are based on the physical
requirements for placing the concrete and
fixing the steel. They are:

In beams (Clause 3.12.6.1):

A
s ≤ 4%
A
c

As
′ ≤ 4%
A
c

Minimum Spacing of Bars (Cl.
3.12.11.1)

Guidance is given in the code for minimum bar
spacing to ensure that members can be
compaction of the concrete to enable the
reinforcement to perform as designed.

It is important that reinforcing bars are
surrounded by concrete for two main reasons:

(i) to develop sufficient bond between the concrete
and the bars such that the required forces are
transferred between the steel and the concrete, and

(ii) to provide protection to the steel against corrosion,
fire, etc.

Maximum Spacing of Bars (Clause
3.12.11.2)

The requirement to limit the maximum spacing of
reinforcement is to minimize surface cracking.

Using Table 3.28.

Since very small bars when mixed with larger
bars would invalidate the assumptions on which
the Table 3.28 values are based, the code
specifies that any bar in a section with a
diameter < (0.45
×

the largest bar in the section)
should be ignored except when considering
those in the side faces of deep beams.

Maximum Spacing of Bars

The minimum size of bar to be used in the
side faces is given in Clause 3.12.5.4 as:

bar diameter ≥

where:
s
b

is the bar spacing (e.g. 250 mm)

b
is the breadth of the section ≤ 500 mm