Formula Sheet: Deflection of Beams | Strength of Materials (SOM) - Mechanical Engineering PDF Download

 Page 1


F ormula Sheet: Deflection of Beams
Introduction to Beam Deflection
• Definition : Deflection is the vertical displacement of a beam under applied
loads, whil e slope is the angle of rotation of the beam’ s cross-section.
• Assumptions : Linear elastic material, small deflections, Euler-Bernoulli
beam theory (plan e sections remain plane).
Governing Equations
• Bending Mom ent and Deflection (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 :
y =
? ?
M(x)
EI
dx
2
• Shear F orce and Load Relationship :
V =
dM
dx
, w(x) =
dV
dx
whereV = shear force,w(x) = distributed load inten sity .
Moment of Inertia (I )
• Rectangular Section :
I =
bh
3
12
whereb = width,h = height.
• 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.
1
Page 2


F ormula Sheet: Deflection of Beams
Introduction to Beam Deflection
• Definition : Deflection is the vertical displacement of a beam under applied
loads, whil e slope is the angle of rotation of the beam’ s cross-section.
• Assumptions : Linear elastic material, small deflections, Euler-Bernoulli
beam theory (plan e sections remain plane).
Governing Equations
• Bending Mom ent and Deflection (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 :
y =
? ?
M(x)
EI
dx
2
• Shear F orce and Load Relationship :
V =
dM
dx
, w(x) =
dV
dx
whereV = shear force,w(x) = distributed load inten sity .
Moment of Inertia (I )
• Rectangular Section :
I =
bh
3
12
whereb = width,h = height.
• 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.
1
Deflection for Common Loading Cases
• Cantilever Beam :
– Point Load (F ) at Free End:
y
max
=
FL
3
3EI
, ?
max
=
FL
2
2EI
– Uniformly Distributed Load (w ):
y
max
=
wL
4
8EI
, ?
max
=
wL
3
6EI
• Simply Suppo rted Beam :
– Point Load (F ) at Midspan:
y
max
=
FL
3
48EI
, ?
max
=
FL
2
16EI
– Uniformly Distributed Load (w ):
y
max
=
5wL
4
384EI
, ?
max
=
wL
3
24EI
Moment-Area Method
• First Mom ent-Area Theorem (Slope) :
?
B
-?
A
=
1
EI
?
B
A
M(x)dx
where?
A
,?
B
= slopes at points A and B, area underM/EI diagr am between
A and B.
• Second Mom ent-Area Theorem (Deflection) :
y
B
-y
A
=
1
EI
?
B
A
M(x)(x-x
A
)dx
where y
A
,y
B
= deflections at A and B, x- x
A
= distance from point A to
moment centroid .
Conjugate Beam Method
• Concept : Conjugate beam is loaded withM/EI diagr am, with support con-
ditions modifi ed to match real beam’ s boundary conditions.
• Shear in Con jugate Beam = Slope in real beam.
• Moment in C onjugate Beam = Deflection in real beam.
2
Page 3


F ormula Sheet: Deflection of Beams
Introduction to Beam Deflection
• Definition : Deflection is the vertical displacement of a beam under applied
loads, whil e slope is the angle of rotation of the beam’ s cross-section.
• Assumptions : Linear elastic material, small deflections, Euler-Bernoulli
beam theory (plan e sections remain plane).
Governing Equations
• Bending Mom ent and Deflection (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 :
y =
? ?
M(x)
EI
dx
2
• Shear F orce and Load Relationship :
V =
dM
dx
, w(x) =
dV
dx
whereV = shear force,w(x) = distributed load inten sity .
Moment of Inertia (I )
• Rectangular Section :
I =
bh
3
12
whereb = width,h = height.
• 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.
1
Deflection for Common Loading Cases
• Cantilever Beam :
– Point Load (F ) at Free End:
y
max
=
FL
3
3EI
, ?
max
=
FL
2
2EI
– Uniformly Distributed Load (w ):
y
max
=
wL
4
8EI
, ?
max
=
wL
3
6EI
• Simply Suppo rted Beam :
– Point Load (F ) at Midspan:
y
max
=
FL
3
48EI
, ?
max
=
FL
2
16EI
– Uniformly Distributed Load (w ):
y
max
=
5wL
4
384EI
, ?
max
=
wL
3
24EI
Moment-Area Method
• First Mom ent-Area Theorem (Slope) :
?
B
-?
A
=
1
EI
?
B
A
M(x)dx
where?
A
,?
B
= slopes at points A and B, area underM/EI diagr am between
A and B.
• Second Mom ent-Area Theorem (Deflection) :
y
B
-y
A
=
1
EI
?
B
A
M(x)(x-x
A
)dx
where y
A
,y
B
= deflections at A and B, x- x
A
= distance from point A to
moment centroid .
Conjugate Beam Method
• Concept : Conjugate beam is loaded withM/EI diagr am, with support con-
ditions modifi ed to match real beam’ s boundary conditions.
• Shear in Con jugate Beam = Slope in real beam.
• Moment in C onjugate Beam = Deflection in real beam.
2
Superposition Principle
• T otal deflection or slope is the sum of deflections or slopes from individual
loads:
y
total
=
?
y
i
, ?
total
=
?
?
i
Boundary Conditions
• Fixed Suppo rt : y = 0 ,? = 0 .
• Pinned Suppo rt : y = 0 ,M = 0 .
• Roller Supp ort : y = 0 ,M = 0 .
• Free End : M = 0 ,V = 0 .
3