Formula Sheet: Shear Stresses in Beams | Solid Mechanics - Mechanical Engineering PDF Download

 Page 1


F ormula Sheet: Shear Stresses in Beams
Introduction to Shear Stresses
• Definition : Shear stresses are internal stresses in a beam caused b y shear
forces, acting par allel to the cross-sectional plane, typically varying across
the section.
• Assumptions : Linear elastic material, small deformations, plane sections
remain plane (E uler-Bernoulli beam theory).
K ey F ormulas
• Shear Str ess F ormula :
t =
VQ
Ib
wheret = shear stress at a point,V = shear force,Q = first moment of area
above/below the point,I = moment of inertia of the cross-section,b = width
of the s ection at the point.
• Shear F orce :
V =
?
F
y
(sum of vertical forces)
or
V =
?
w(x)dx (for distributed load)
wherew(x) = distributed load intensity .
• First Mom ent of Area (Q ) :
Q =
?
ydA
wherey = distance from neutr al axis to differential area dA , integr ated over
the area abo ve or below the point of interest.
Shear Stress Distribution
• Rectangular Section :
t =
V
Ib
(
h
2
4
-y
2
)
whereh = total height of the section,y = distance from neutr al axis.
• Maximum Shea r Stress (Rectangular Section) :
t
max
=
3V
2A
whereA = cross-sectional area (A = b·h ).
1
Page 2


F ormula Sheet: Shear Stresses in Beams
Introduction to Shear Stresses
• Definition : Shear stresses are internal stresses in a beam caused b y shear
forces, acting par allel to the cross-sectional plane, typically varying across
the section.
• Assumptions : Linear elastic material, small deformations, plane sections
remain plane (E uler-Bernoulli beam theory).
K ey F ormulas
• Shear Str ess F ormula :
t =
VQ
Ib
wheret = shear stress at a point,V = shear force,Q = first moment of area
above/below the point,I = moment of inertia of the cross-section,b = width
of the s ection at the point.
• Shear F orce :
V =
?
F
y
(sum of vertical forces)
or
V =
?
w(x)dx (for distributed load)
wherew(x) = distributed load intensity .
• First Mom ent of Area (Q ) :
Q =
?
ydA
wherey = distance from neutr al axis to differential area dA , integr ated over
the area abo ve or below the point of interest.
Shear Stress Distribution
• Rectangular Section :
t =
V
Ib
(
h
2
4
-y
2
)
whereh = total height of the section,y = distance from neutr al axis.
• Maximum Shea r Stress (Rectangular Section) :
t
max
=
3V
2A
whereA = cross-sectional area (A = b·h ).
1
• Circular Se ction :
t =
V
I
·
4
3
·
r
2
-y
2
r
wherer = r adius,y = distance from neutr al axis.
• Maximum Shea r Stress (Circular Section) :
t
max
=
4V
3A
whereA = pr
2
.
• I-Section : Maximum shear stress occurs at the neutr al axis, concentr ated
in the web:
t
max
˜
V
A
web
whereA
web
= area of the web.
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.
Shear Flow
• Shear Flow (q ) :
q =
VQ
I
where q = shear force per unit length, used in thin-walled sections or for
fastener anal ysis.
• Application : Used to determine shear stress in welds, bolts, or thin-walled
structures.
2
Page 3


F ormula Sheet: Shear Stresses in Beams
Introduction to Shear Stresses
• Definition : Shear stresses are internal stresses in a beam caused b y shear
forces, acting par allel to the cross-sectional plane, typically varying across
the section.
• Assumptions : Linear elastic material, small deformations, plane sections
remain plane (E uler-Bernoulli beam theory).
K ey F ormulas
• Shear Str ess F ormula :
t =
VQ
Ib
wheret = shear stress at a point,V = shear force,Q = first moment of area
above/below the point,I = moment of inertia of the cross-section,b = width
of the s ection at the point.
• Shear F orce :
V =
?
F
y
(sum of vertical forces)
or
V =
?
w(x)dx (for distributed load)
wherew(x) = distributed load intensity .
• First Mom ent of Area (Q ) :
Q =
?
ydA
wherey = distance from neutr al axis to differential area dA , integr ated over
the area abo ve or below the point of interest.
Shear Stress Distribution
• Rectangular Section :
t =
V
Ib
(
h
2
4
-y
2
)
whereh = total height of the section,y = distance from neutr al axis.
• Maximum Shea r Stress (Rectangular Section) :
t
max
=
3V
2A
whereA = cross-sectional area (A = b·h ).
1
• Circular Se ction :
t =
V
I
·
4
3
·
r
2
-y
2
r
wherer = r adius,y = distance from neutr al axis.
• Maximum Shea r Stress (Circular Section) :
t
max
=
4V
3A
whereA = pr
2
.
• I-Section : Maximum shear stress occurs at the neutr al axis, concentr ated
in the web:
t
max
˜
V
A
web
whereA
web
= area of the web.
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.
Shear Flow
• Shear Flow (q ) :
q =
VQ
I
where q = shear force per unit length, used in thin-walled sections or for
fastener anal ysis.
• Application : Used to determine shear stress in welds, bolts, or thin-walled
structures.
2
Combined Stresses
• T otal Stress : Shear stress ma y combine with bending stress. Use Mohr ’ s
circle or pr incipal stress formulas:
s
1,2
=
s
x
+s
y
2
±
v
(
s
x
-s
y
2
)
2
+t
2
xy
wheres
x
= bending stress ,t
xy
= shear stress.
• Maximum Shea r Stress :
t
max
=
v
(
s
x
-s
y
2
)
2
+t
2
xy
3