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RC Beam Deflection & Inertia Calculations

This document provides information about calculating deflection in reinforced concrete beams. It includes a problem describing a beam and outlines 7 calculation steps to determine instantaneous and long-term deflection. It also includes two sample problems, one involving calculating effective moment of inertia at supports for a continuous beam, and determining additional deflection after 1 year. The other problem asks to determine the narrowest dimension between form sides and clear bar spacing for a simply supported beam and maximum coarse aggregate size.

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Paul Roque
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3K views9 pages

RC Beam Deflection & Inertia Calculations

This document provides information about calculating deflection in reinforced concrete beams. It includes a problem describing a beam and outlines 7 calculation steps to determine instantaneous and long-term deflection. It also includes two sample problems, one involving calculating effective moment of inertia at supports for a continuous beam, and determining additional deflection after 1 year. The other problem asks to determine the narrowest dimension between form sides and clear bar spacing for a simply supported beam and maximum coarse aggregate size.

Uploaded by

Paul Roque
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF, TXT or read online on Scribd
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#1.

Problem
A 200mm x 400mm RC beam with an effective
depth of 340 mm and with tensile reinforcement of
804 mm2 is simply supported over a 6 meter span.
The beam is to carry a total load of 7.5 kN/m.
f’c = 21 MPa, fs = 140 MPa and modular ratio n = 9.

1. Find the moment of inertia of 200 mm


the gross section, Ig.
2. Calculate the moment of inertia
of cracked section, Icr.
3. Find the actual maximum 400 mm
340 mm
moment, Ma.
4. Determine the cracking 804mm2
moment, Mcr.
DEFLECTION CONTROL (NSCP 2010):
409.6.2.3 Unless stiffness values are obtained
by a more comprehensive analysis, immediate
deflection shall be computed with the modulus of
elasticity Ec for concrete as specified in Sec. 5.8.5.1
and with the effective moment of inertia, Ie , as
follows, but not greater than Ig.

 M cr 
3
  M cr  
3
(409-8)
Ie    I g  1     I cr
 Ma    M a  
f rI g
where M cr  (409-9)
yt
And for normal weight concrete,
fr  0.62 f ' c (409-10)
#1. Problem
A 200mm x 400mm RC beam with an effective
depth of 340 mm and with tensile reinforcement of
804 mm2 is simply supported over a 6 meter span.
The beam is to carry a total load of 7.5 kN/m.
f’c = 21 MPa, fs = 140 MPa and modular ratio n = 9.

1. Find the moment of inertia of the gross section, Ig.


2. Calculate the moment of inertia of cracked section, Icr.
3. Find the actual maximum moment, Ma.
4. Determine the cracking moment, Mcr.
5. Find the effective moment of inertia, Ie.
409.6.2.5 Unless values are obtained by a more
comprehensive analysis, additional long-term deflection
resulting from creep and shrinkage of flexural members
shall be determined by multiplying the immediate
deflection caused by the sustained load considered, by the
factor λΔ.

 
1  50  ' (409-11)
where ’ shall be the value at midspan for simple and
continuous spans, and at support for cantilevers. It is
permitted to assume the time-dependent factor  for
sustained loads to be equal to
5 years or more . . . . . . . . . . . . . . . . . 2.0
12 months . . . . . . . . . . . . . . . . . . . . . 1.4
6 months . . . . . . . . . . . . . . . . . . . . . . 1.2
3 months . . . . . . . . . . . . . . . . . . . . . . 1.0
#1. Problem
A 200mm x 400mm RC beam with an effective
depth of 340 mm and with tensile reinforcement of
804 mm2 is simply supported over a 6 meter span.
The beam is to carry a total load of 7.5 kN/m.
f’c = 21 MPa, fs = 140 MPa and modular ratio n = 9.

1. Find the moment of inertia of the gross section, Ig.


2. Calculate the moment of inertia of cracked section, Icr.
3. Find the actual maximum moment, Ma.
4. Determine the cracking moment, Mcr.
5. Find the effective moment of inertia, Ie.
6. Calculate the instantaneous deflection yi due to the
given load.
7. Determine the deflection due to the same loads after 5
years assuming that 70% of the load is sustained.
#2. CE Board November 2002
A continuous T-beam span supports a dead load including its
own weight of 16 kN/m and a live load of 32 kN/m of which 20%
is assumed to be sustained. f’c = 17.2 MPa. The moment for the
full dead load and live load varies from +145 kN-m at midspan to
-202 kN-m at supports. The T-beam section has a flange width
of 1.9 m, width of web of 0.36m, total depth of 0.62m and an
effective depth of 0.55m. At midspan, the tension steel is 3-
32mm with NA of the gross concrete section 0.195 m from the
top, Ig = 0.0138 m4 and Icr = 0.00573 m4. At the supports, the
tension steel is 5 – 32 mm with Ig = 0.00715 m4 and Icr =
0.00578 m4.

1. Compute the effective moment of inertia at the supports.


2. Compute the effective moment of inertia for the continuous
member.
3. If the instantaneous deflection is 5 mm, what is the addi-
tional deflection of the sustained loadings after one year.
4. Solve question no. 3 if 2 – 32 mm are provided at
compression face of the beam.
#3. CE Board
A simply supported beam is 280
600
mm x 600 mm and with two layers of mm
3 – 28 mm tension reinforcement.
6-28mm
Stirrups are 10 mm with concrete
cover of 40 mm.
280mm
1. What is the narrowest
dimension between 403.4.2 Nominal maximum size of
sides of forms? coarse aggregate shall not be larger
than:
2. What is the clear (a) 1/5 the narrowest dimension
spacing between bars? between sides of forms, or
3. What is the maximum (b) 1/3 the depth of slabs, or
(c) 3/4 the minimum clear spacing
nominal size of coarse between individual reinforcing bars or
aggregate that can be wires, bundles of bars, or prestressing
used? tendons or ducts.
END of LECTURE

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