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F ormula Sheet: Bending Stresses in Beams
Introduction to Bending Stresses
• Definition : Bending stresses are normal stresses induced in a beam due to
external loads causing bending moments, typically tensile on one side and
compressive on the other .
• Assumptions : Linear elastic material, small deformations, plane sections
remain plane (E uler-Bernoulli beam theory).
K ey F ormulas
• Bending Str ess (Flexure F ormula) :
s =
M ·y
I
wheres = bending stress,M = bending moment,y = distance from neutr al
axis,I = moment of inertia of cross-section.
• Maximum Bend ing Stress :
s
max
=
M ·c
I
wherec = maximum distance from neutr al axis to outermost fiber .
• Bending Mom ent :
M =F ·d (for point load)
or
M =
?
w(x)·xdx (for distributed load)
whereF = force,d = perpendicular distance,w(x) = distributed load inten-
sity .
• Shear F orce :
V =
?
F
y
(sum of vertical forces)
or
V =
?
w(x)dx (for distributed load)
Moment of Inertia (I)
• Rectangular Section :
I =
bh
3
12
whereb = width,h = height.
1
Page 2


F ormula Sheet: Bending Stresses in Beams
Introduction to Bending Stresses
• Definition : Bending stresses are normal stresses induced in a beam due to
external loads causing bending moments, typically tensile on one side and
compressive on the other .
• Assumptions : Linear elastic material, small deformations, plane sections
remain plane (E uler-Bernoulli beam theory).
K ey F ormulas
• Bending Str ess (Flexure F ormula) :
s =
M ·y
I
wheres = bending stress,M = bending moment,y = distance from neutr al
axis,I = moment of inertia of cross-section.
• Maximum Bend ing Stress :
s
max
=
M ·c
I
wherec = maximum distance from neutr al axis to outermost fiber .
• Bending Mom ent :
M =F ·d (for point load)
or
M =
?
w(x)·xdx (for distributed load)
whereF = force,d = perpendicular distance,w(x) = distributed load inten-
sity .
• Shear F orce :
V =
?
F
y
(sum of vertical forces)
or
V =
?
w(x)dx (for distributed load)
Moment of Inertia (I)
• Rectangular Section :
I =
bh
3
12
whereb = width,h = height.
1
• Circular Se ction :
I =
pr
4
4
wherer = r adius.
• Par allel Axis The orem :
I =I
cm
+Ad
2
whereI
cm
= moment of inertia about centroidal axis,A = area,d = distance
between axes.
Shear Stress in Beams
• Shear Str ess F ormula :
t =
VQ
Ib
wheret = shear stress,V = shear force,Q = first moment of area above/below
the point,b = width at the point.
• First Mom ent of Area :
Q =
?
ydA
wherey = distance from neutr al axis to differential area dA .
• Maximum Shea r Stress (Rectangular Section) :
t
max
=
3V
2A
whereA = cross-sectional area.
Deflection of Beams
• Beam Defl ection (Double Integr ation Method) :
EI
d
2
y
dx
2
=M(x)
where E = Y oung’ s modulus, I = moment of inertia, y = deflection, M(x) =
bending moment fu nction.
• Slope of Be am :
? =
dy
dx
=
?
M(x)
EI
dx
• Deflection f or Common Cases :
– Cantilever Beam (Point Load at Free End):
y
max
=
FL
3
3EI
, ?
max
=
FL
2
2EI
– Simply Supported Beam (Point Load at Midspan):
y
max
=
FL
3
48EI
, ?
max
=
FL
2
16EI
2
Page 3


F ormula Sheet: Bending Stresses in Beams
Introduction to Bending Stresses
• Definition : Bending stresses are normal stresses induced in a beam due to
external loads causing bending moments, typically tensile on one side and
compressive on the other .
• Assumptions : Linear elastic material, small deformations, plane sections
remain plane (E uler-Bernoulli beam theory).
K ey F ormulas
• Bending Str ess (Flexure F ormula) :
s =
M ·y
I
wheres = bending stress,M = bending moment,y = distance from neutr al
axis,I = moment of inertia of cross-section.
• Maximum Bend ing Stress :
s
max
=
M ·c
I
wherec = maximum distance from neutr al axis to outermost fiber .
• Bending Mom ent :
M =F ·d (for point load)
or
M =
?
w(x)·xdx (for distributed load)
whereF = force,d = perpendicular distance,w(x) = distributed load inten-
sity .
• Shear F orce :
V =
?
F
y
(sum of vertical forces)
or
V =
?
w(x)dx (for distributed load)
Moment of Inertia (I)
• Rectangular Section :
I =
bh
3
12
whereb = width,h = height.
1
• Circular Se ction :
I =
pr
4
4
wherer = r adius.
• Par allel Axis The orem :
I =I
cm
+Ad
2
whereI
cm
= moment of inertia about centroidal axis,A = area,d = distance
between axes.
Shear Stress in Beams
• Shear Str ess F ormula :
t =
VQ
Ib
wheret = shear stress,V = shear force,Q = first moment of area above/below
the point,b = width at the point.
• First Mom ent of Area :
Q =
?
ydA
wherey = distance from neutr al axis to differential area dA .
• Maximum Shea r Stress (Rectangular Section) :
t
max
=
3V
2A
whereA = cross-sectional area.
Deflection of Beams
• Beam Defl ection (Double Integr ation Method) :
EI
d
2
y
dx
2
=M(x)
where E = Y oung’ s modulus, I = moment of inertia, y = deflection, M(x) =
bending moment fu nction.
• Slope of Be am :
? =
dy
dx
=
?
M(x)
EI
dx
• Deflection f or Common Cases :
– Cantilever Beam (Point Load at Free End):
y
max
=
FL
3
3EI
, ?
max
=
FL
2
2EI
– Simply Supported Beam (Point Load at Midspan):
y
max
=
FL
3
48EI
, ?
max
=
FL
2
16EI
2
Combined Stresses
• T otal Norm al Stress : Sum of bending and axial stresses:
s
total
=
P
A
±
M ·y
I
whereP = axial load.
• Principal St resses (2D) :
s
1,2
=
s
x
+s
y
2
±
v
(
s
x
-s
y
2
)
2
+t
2
xy
• Maximum Shea r Stress :
t
max
=
v
(
s
x
-s
y
2
)
2
+t
2
xy
3
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