Transcript BEAMS - Majed Dabdoub

BEAMS
SHEAR AND MOMENT
1
Beam Shear
 Shear and Moment
Diagrams
 Vertical shear:
tendency for one part
of a beam to move
vertically with respect
to an adjacent part
2
Beam Shear
 Magnitude (V) = sum of vertical forces on
either side of the section
– can be determined at any section along the
length of the beam
 Upward forces (reactions) = positive
 Downward forces (loads) = negative
 Vertical Shear = reactions – loads
(to the left of the section)
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Beam Shear
 Why?
 necessary to know the maximum value of
the shear
– necessary to locate where the shear changes
from positive to negative
• where the shear passes through zero
 Use of shear diagrams give a graphical
representation of vertical shear throughout
the length of a beam
4
Beam Shear –
 Simple beam
– Span = 20 feet
– 2 concentrated loads
 Construct shear
diagram
5
Beam Shear – Example 1
1)
Determine the reactions
1  Fx  0
2  Fy  0  R1  8000lb.  1200lb.  R2
3  M1  0  (8000lb.  6' )  (1200lb. 16' )  ( R2  20' )
Solving equation (3):
20' R2  48,000lb. ft.  19,200lb. ft.
67,200lb. ft.
R2 
 3,360lb. 
ft.
20
Solving equation (2):
R1  8,000lb.  1,200lb.  3,360lb.
R1  5,840lb. 
Figure 6.7a =>
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Beam Shear – Example 1 (pg. 64)

Determine the shear at various
points along the beam
V( x 1)  5,480 0  5,480lb.
V( x 8)  5,480 8,000  2,160lb.
V( x 18)  5,480 8,000 1,200  3,360lb.
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Beam Shear – Example 1

Conclusions
–
max. vertical shear = 5,840 lb.
•
–
max. vertical shear occurs at greater
reaction and equals the greater reaction
(for simple spans)
shear changes sign under 8,000 lb. load
•
where max. bending occurs
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Beam Shear – Example 2
 Simple beam
– Span = 20 feet
– 1 concentrated load
– 1 uniformly distr. load
 Construct shear diagram,
designate maximum shear,
locate where shear passes
through zero
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Beam Shear – Example 2

Determine the reactions
1  Fx  0
2  Fy  0  R1  (1,000lb ft. 12' )  6,000lb.  R2
3  M1  0  [(1,000lb ft. 12' )(6' )]  (6000lb. 16' )  ( R2  24' )
Solving equation (3):
24' R2  72,000lb. ft.  96,000lb. ft.
168,000lb. ft.
lb.

R2 

7
,
000

ft.
24
Solving equation (2):
R1  12,000lb.  6,000lb.  7,000lb.
R1  11,000lb. 
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Shear and Moment Diagrams
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Beam Shear – Example 2

Determine the shear at various
points along the beam
V( x 1)  11,000 (11,000)  10,000lb.
V( x 2)  11,000 (2 1,000)  9,000lb.
V( x 12 )  11,000 (121,000)  1,000lb.
V( x 16  )  11,000 (121,000)  1,000lb.
V( x 16  )  11,000 121,000  6,000  7,000lb.
V( x 24 )  11,000 121,000  6,000  7,000lb.
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Beam Shear – Example 2

Conclusions
–
max. vertical shear = 11,000 lb.
•
–
at left reaction
shear passes through zero at some
point between the left end and the
end of the distributed load
•
x = exact location from R1
–
at this location, V = 0
V  0  11,000 ( x 1,000)
x  11 feet
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Beam Shear – Example 3
 Simple beam with
overhanging ends
– Span = 32 feet
– 3 concentrated loads
– 1 uniformly distr. load
acting over the entire beam
 Construct shear diagram,
designate maximum shear,
locate where shear passes
through zero
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Beam Shear – Example 3
1  Fx  0
2  Fy  0  2,000lb.  R1  12,000lb.  4,000lb.  (500lb ft.  32' )  R2
3  M 1  0  (2,000lb.  8' )  (12,000lb. 14' )  (4,000lb.  24' )  (500lb. ft.  32' )32 2  8 R2 (20' )
4  M 2  0  (4,000lb.  4' )  (12,000lb.  6' )  (2,000lb.  28' )  (500lb. ft.  32' )32 2  4 R1 (20' )
20' R2  16,000lb. ft.  168,000lb. ft.  96,000lb. ft.  128,000lb. ft.
376,000lb. ft.
R2 
 18,800lb. 
ft.
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20' R1  16,000lb. ft.  72,000lb. ft.  56,000lb. ft.  192,000lb. ft.
304,000lb. ft.
R1 
 15,200lb. 
20'
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Determine the reactions
Solving equation (3):
Solving equation (4):
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Beam Shear – Example 3

Determine the shear at various
points along the beam


V( x 8  )  0  2,000lb.  500lb. ft.  8'  6,000lb.

 500

V( x 8  )  15,200lb.  2,000lb.  500lb. ft.  8'  9,200lb.
V( x  22  )  15,200lb.  2,000lb.
lb. ft .

 22'  2,200lb.

 500

 28'  12,800
 500  28'  6,000
 500  32'  4,000
V( x  22  )  15,200lb.  2,000lb.  12,000lb.  500lb. ft.  22'  9,800lb.
V( x  28  )  15,200lb.  2,000lb.  12,000lb.
lb. ft .
V( x  28  )  15,200lb.  18,800lb.  2,000lb.  12,000lb.
V( x 32  )  15,200lb.  18,800lb.  2,000lb.  12,000lb.
lb.
lb. ft .
lb.
lb. ft .
lb.
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Beam Shear – Example 3

Conclusions
– max. vertical shear = 12,800
lb.
•
disregard +/- notations
– shear passes through zero at
three points
•
R1, R2, and under the 12,000lb.
load
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Bending Moment
 Bending moment: tendency of a beam to bend due
to forces acting on it
 Magnitude (M) = sum of moments of forces on
either side of the section
– can be determined at any section along the length of the
beam
 Bending Moment = moments of reactions –
moments of loads
• (to the left of the section)
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Bending Moment
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Bending Moment – Example 1
 Simple beam
– span = 20 feet
– 2 concentrated loads
– shear diagram from
earlier
 Construct moment
diagram
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Bending Moment – Example 1
1)
Compute moments at
critical locations
–
under 8,000 lb. load &
1,200 lb. load
 M ( x6')  (5,840lb.  6' )  0  35,040lb. ft.
 M ( x16')  (5,840lb. 16' )  (8,000lb. 10' )  13,440lb. ft.
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Bending Moment – Example 2
 Simple beam
– Span = 20 feet
– 1 concentrated load
– 1 uniformly distr. Load
– Shear diagram
 Construct moment
diagram
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Bending Moment – Example 2
1)
Compute moments at
critical locations
–
When x = 11 ft. and under
6,000 lb. load
 M ( x11')  (11,000lb. 11' )  1,000lb ft 11112   60,500lb. ft.
 M ( x16')  (11,000lb. 16' )  1,000lb ft 1212 2  4  56,000lb. ft.
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Negative Bending Moment
 Previously, simple beams subjected to positive
bending moments only
– moment diagrams on one side of the base line
• concave upward (compression on top)
 Overhanging ends create negative moments
• concave downward (compression on bottom)
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Negative Bending Moment
 deflected shape has
inflection point
– bending moment = 0
 See example
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Negative Bending Moment Example
– Simple beam with
overhanging end on right
side
• Span = 20’
• Overhang = 6’
• Uniformly distributed load
acting over entire span
– Construct the shear and
moment diagram
– Figure 6.12
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Negative Bending Moment Example
1) Determine the
reactions
1  Fx  0
2  Fy  0  R1  (600lb ft.  26' )  R2
3  M 1  0  (600lb ft.  26' )26 2  ( R2  20' )
-
Solving equation (3):
20' R2  202,800lb. ft.
202,800lb. ft.
R2 
 10,140lb. 
ft.
20
-
Solving equation (4):
R1  15,600 lb.  10,140 lb.
R1  5,460 lb.  
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Negative Bending Moment Example
2) Determine the shear at
various points along
the beam and draw
the shear diagram
V( x 1)  5,460 (1 600)  4,860lb.
V( x 10 )  5,460 (10 600)  540lb.
V( x  20  )  5,460 (20 600)  6,540lb.
V( x  20  )  5,460 10,140 (20 600)  3,600lb.
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Negative Bending Moment Example
3) Determine where the
shear is at a
maximum and where
it crosses zero
–
max shear occurs at the right
reaction = 6,540 lb.
V  0  5,460 ( x  600)
x  9.1 feet
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Negative Bending Moment Example
4) Determine the
moments that the
critical shear points
found in step 3) and
draw the moment
diagram

 20' )  600
  24,843
2
 20'20   10,800
2

M ( x 9.1)  (5,460lb.  9.1' )  600lb ft  9.1' 9.1
lb. ft .
M ( x  20 )  (5,460lb.
lb. ft .
lb ft
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Negative Bending Moment Example
4) Find the location of the inflection
point (zero moment) and max.
bending moment

 2 
M  0  (5,460 x)  600lb ft  x x
 x2 
M  0  5,460x  600 
 2
M  0  5,460x  300x 2  300x 2  5,460x
 5,460 (5,460) 2  4(300)(0)
x
2(300)
 5460 5,460
x
 0 feet;18.2 feet
 600


since x cannot =0, then we use x=18.2’
Max. bending moment =24,843 lb.-ft.
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Rules of Thumb/Review
 shear is dependent on the loads and reactions
– when a reaction occurs; the shear “jumps” up by the
amount of the reaction
– when a load occurs; the shear “jumps” down by the
amount of the load
 point loads create straight lines on shear diagrams
 uniformly distributed loads create sloping lines of
shear diagrams
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Rules of Thumb/Review
 moment is dependent upon the shear diagram
– the area under the shear diagram = change in the
moment (i.e. Ashear diagram = ΔM)
 straight lines on shear diagrams create sloping
lines on moment diagrams
 sloping lines on shear diagrams create curves on
moment diagrams
 positive shear = increasing slope
 negative shear = decreasing slope
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Typical Loadings
 In beam design, only
need to know:
–
–
–
–
reactions
max. shear
max. bending moment
max. deflection
35