Mechanics of Solids I
Transverse Loading
Introduction
0 0
0
0
x x x xz xy
y xy y x
z xz z x
F dA M y z dA
F dA V M z dA
F dA M y M
o Distribution of normal and shearing stresses
satisfies
o Transverse loading applied to a beam
results in normal and shearing stresses in
transverse sections.
Shear V is the result of a transverse shear-
stress distribution that acts over the beam’s
x-section.
Introduction
o When shearing stresses are exerted on
the vertical faces of an element, equal
stresses must be exerted on the horizontal
faces.
o Longitudinal shear stress (yx) must exist
o Shear does not occur in a beam subjected to pure bending
Shear on the Horizontal Face of a Beam Element
o Consider prismatic beam
o For equilibrium of beam element
0x D CA
D C
A
F H dA
M MH y dA
I
A
D C
Q y dA
dMM M x V x
dx
o Note,
VQH x
I
H VQq shear flow
x I
o Substituting,
VQq shear flow
I • Shear flow,
where
1
2
'
first moment of area above
second moment of full cross section
A
A A
Q y dA
y
I y dA
• Same result found for lower area
H H
Shear on the Horizontal Face of a Beam Element
Example 6.1
A beam is made of three planks, nailed
together. Knowing that the spacing between
the nails is 25 mm and that the vertical shear
in the beam is V = 500 N, determine the shear
force in each nail.
Shearing Stress in a Beam
o The average shearing stress on the horizontal
face of the element is obtained by dividing
the shearing force on the element by the
area of the face: Shear formula.
ave
H q x VQ x
A A I t x
ave
VQ
It
Midterm 2/49
A beam with rectangular cross section
subject to a vertical shear Vy and a horizontal
shear Vz as shown. Determine shear stress at
point A, B, C and D on the beam cross
section. Vz = 30 kN
x
y
z
50 mm
80 mm
Vy = 20 kN
10 mm
10 mm
A
C
D
B
xy and yx exerted on a transverse and a horizontal plane through D’ are equal.
o If the width of the beam is comparable or large
relative to its depth, the shearing stresses at D1
and D2 are significantly higher than at D.
Shearing Stress in a Beam
o On the upper and lower surfaces of the beam,
yx= 0. It follows that xy= 0 on the upper and
lower edges of the transverse sections.
Shearing Stresses xy in Common Types of Beams
• For a narrow rectangular beam,
2
2
max
31
2
3
2
xy
VQ V y
Ib A c
V
A
parabola
• By comparison, max is 50% greater than the average
shear stress determined from avg = V/A.
Shearing Stresses xy in Common Types of Beams
max
ave
web
VQ
It
V
A
Wide-flange beam (W-beam) and Standard beam (S-beam)
A wide-flange beam consists of two (wide) “flanges” and a “web”.
Using analysis similar to a rectangular x-section, the shear stress
distribution acting over x-section is shown
o There is a jump in shear stress at the flange-web junction since x-sectional thickness changes at this point o The web carries significantly more shear force than the flanges
Example 6.2
A timber beam is to support the three
concentrated loads shown. Knowing
that for the grade of timber used,
determine the minimum required
depth d of the beam.
all12 MPa, 0.8 MPaall
Longitudinal Shear on an Arbitrary Shape Beam
Earlier, we learn how to calculate shear flow
along horizontal surfaces
How to calculate q along vertical surfaces?
0x D CA
F H dA
H VQq
x I
Shear flow is calculated by using the same
equation.
But by cutting through the vertical surface!
Example 6.3
A square box beam is constructed from four planks as shown. Knowing that the
spacing between nails is 44 mm. and the beam is subjected to a vertical shear of magnitude V = 2.5 kN, determine the shearing force in each nail.
Shearing Stresses in Thin-Walled Members
o Consider a segment of a wide-flange beam
subjected to the vertical shear V.
o The longitudinal shear force on the element
is VQ
H xI
zx xz
H VQ
t x It
o The corresponding shear stress is
o NOTE: 0xy
0xz
in the flanges
in the web
o Previously found a similar expression for the
shearing stress in the web
xy
VQ
It
• The variation of shear flow across the
section depends only on the variation of
the first moment.
VQq t
I
• For a box beam, q grows smoothly from
zero at A to a maximum at C and C’
and then decreases back to zero at E.
• The sense of q in the horizontal portions
of the section may be deduced from
the sense in the vertical portions or the
sense of the shear V.
Shearing Stresses in Thin-Walled Members
o For a wide-flange beam, the shear flow
increases symmetrically from zero at A
and A’, reaches a maximum at C and
then decreases to zero at E and E’.
o The continuity of the variation in q and
the merging of q from section branches
suggests an analogy to fluid flow.
Shearing Stresses in Thin-Walled Members
Directional sense of q is such that
shear appears to “flow” through the x-section
inward at beam’s top flange
“combining” and then “flowing” downward
through the web
then separating and “flowing” outward at the
bottom flange
Shearing Stresses in Thin-Walled Members
Important notes
If a member is made from segments having thin walls, only the
shear flow parallel to the walls of member is important
Shear flow varies linearly along segments that are perpendicular
to direction of shear V
Shear flow varies parabolically along segments that are inclined
or parallel to direction of shear V
On x-section, shear “flows” along segments so that it contributes
to shear V yet satisfies horizontal and vertical force equilibrium
Shearing Stresses in Thin-Walled Members
Knowing that a given vertical shear V causes a maximum shearing stress of 75 Mpa in the hat-shaped extrusion shown, determine the corresponding shearing stress at (a) point a, (b) point b.
Example 6.4