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The concept of bending stress in beams, focusing on the moment of inertia, neutral axis, and maximum compressive and tensile stress. It includes formulas and examples to help understand these concepts.
Typology: Study notes
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X
L
X Surface area
d
b
Simply supported beam
W x
W x
B.M.D
(+ ve)
(+ ve)
EF = E’F’ = R
Panjang asal, PQ = EF = R
Panjang akhir, P’Q’ = (R+y)
PanjangAsal
PerubahanPanjangPQ Terikan PQ
R
(R y) -R
R
y
R
y Tegasan padaPQ, E
R
E
y
Persamaan ‘Bending stress’ bolehlah dirumuskan dengan
R
E
I y
M
^ M - Momen Lentur maksima (Nm)
4
2
2
x b/
y d/
bd I
3
P.N.
bd I
3
xx
x d/
y d/
r
d I
4 4
P.N.
c r
x x
4r y
4 IP.N 0.11r
r I
4
xx
c
x x
y h/
bh I
3
P.N.
bh I
3
xx
hb I
3
yy
P.N.
y
x
b
d
y
d
y
P.N
y
P.N.
x
Menentukan Pusat Sentroid ( ȳ ) bagi bentuk yang mempunyai
dua paksi simetri
Sentroid bagi bentuk ini adalah garispusat persilangan kedua-dua
paksi. Ianya sangat mudah untuk ditentukan, hanya perlu
membahagi dua.
h
b
Cx = b/
Cy = h/
Oleh itu : Ixx = Ipn + A h 2 di mana h ialah jarak antara pusat sentroid (neutral axis)
bentuk dengan bahagian tengah bentuk yang di ambil dan
A ialah luas bentuk
Moment of inertia (I)
Example 1
y 57.1mm
( 60 x 20 ) ( 100 x 20 ) ( 120 x 20 )
( 60 x 20 )(130) ( 100 x 20 )( 70 ) ( 120 x 20 )(10)
Ay Ay Ay
Ay y
1 2 3
1 1 2 2 3 3
y 3 y 2
y 1
y
x x
P.N (57.1mm).
h 1
h 2
h 3
To find maximum moment using formula
𝑀
=
𝑊𝐿
4 𝑀𝑥 =
𝑊𝐿
8
𝑀
=
𝑊𝐿
2
𝑀
= 𝑊(𝐿)
A 6m beam is subjected with two concentrated load. Each load is 16
kN at a distance of 1 m from both ends of the beam. Determine;
i) The position of neutral axis
ii) The moment of inertia
iii) The radius of the beam at mid span
iv) The maximum tensile stress and maximum compressive stress
Given E= 200 GN / m
Example 2
To find the moment of inertia
PART A ( mm^2 )
y ( mm )
h (mm) I (mm 4 )
20 x 60
= 1200
80 x 15
= 1200
y 1 −ȳ
=90− 65
=
ȳ− y 2
=65−
=
3
3
40000 ( 1200 x 25 ) 640000 (1200x 25 )
I A h I A h
I (I Ah )
2 2
2 pn2 2 2
2 pn1 1 1
2 PN G
6 𝑚𝑚 4 @ 2.18 × 10 − 𝑚 4
I y
To find the radius of the beam
From the loading shown, we find that the
order of loading is symmetrical, so the
reaction: R 1 = R 2 = 16 kN
To find the maximum tensile stress and maximum compressive stress
the maximum tensile stress,
ymax = 65 mm = 0.065 m
477 x 10 N/mm (tensile)
16 x 10 (0.065)
M y σ
6 2
3 maks maks
The maximum compressive stress,
ymaks = 35 mm = 0.035 m
256.8x 10 N/m (compress)
16 x 10 (0.035)
M y σ
6 2
3 maks maks
Example 3
A 10 m cantilever beam is subjected with 20kN/m uniformly distributed
load along the beam. Determine :
i) The position of neutral axis
ii) The moment of inertia
iii) The maximum tensile stress and maximum compressive stress
iv) Sketch the stress distribution in the beam
