chapter 5 analysis and design of beams for bending · pdf file5-10 relationship between load,...
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5-1 Internal Forces in Members
Chapter 5 Analysis and Design of Beams for Bending
INTRODUCTION
Internal Forces in Members
Shear and Bending-Moment Diagrams
Design of Prismatic Beams for Bending
A
B
A
B
C
P
A E
F
DB
C
5'10'
4'
8'
80' 40' 40'
B A
P P
C D
80' 40' 40'
B A
P P
C D
Cross-section
A
5-2 Internal Forces in Members
A
B
7 14 7
7
10
A
B
D
C
E
F
A
B
C
P
A E
F
DB
C
5'10'
4'
8'
C
E
INTERNAL FORCES IN MEMBERS
5-3
A
B
Example Determine the internal forces at point J. P= 5000 lb. Units: Lb, ft.
P
A E
F
DB
C
5 10
4
8
J
1.5
5-4 Shear and Bending Moment Diagrams
Shear and Bending Moment Diagrams
M
V
Sign Convention M V
5-5 Shear and Bending Moment Diagrams
Why do we Need to Draw Shear and Bending Diagrams?
80' 40' 40'
B A
P P
C D
5-6
Support Reactions
Segment A to C
C to D
D to B
Shear Moment
Example Draw the shear and bending-moment diagrams for the beam and loading shown. Label all points of change, maximums and minimums, and the axes. Units: Lb, ft.
8 4 4
B A
P P
C D
x
A
P
x
A
P
x
A
P
5-7
Support Reactions
Segment Shear Moment
Example Draw the shear and bending-moment diagrams for the beam and loading shown. Label all points of change, maximums and minimums, and the axes.
L BA
W
x
A
W
5-8
Caution:
L BA
W
L BA
When drawing FBDs, always use the original loading and not the equivalent.
5-9 Relations Among Load, Shear, and Bending Moment
RELATIONS AMONG LOAD, SHEAR, AND BENDING-MOMENT
L BA
w w
dV wdx
= −
dM Vdx
=
The change in shear is equal to the area under the load curve.
The change in moment is equal to the area under the shear curve.
The slope of the shear diagram is equal to the value of the w load.
The slope of the moment diagram is equal to the value of the shear.
M V xΔ = Δ
V w xΔ = − Δ
5-10 Relationship Between Load, Shear, and Bending Moment
A B
Observations about the Shape of Shear/ Moment Diagrams Shear Diagrams: -Are a plot of forces (note the units). -Discontinuities occur at concentrated forces. Moment Diagrams: -Are a plot of moments (note the units). -Discontinuities occur at concentrated moments. Miscellaneous: -Check your work by noting that you always start and end at zero. -Always use the original FBD and not the equivalent.
15 kN 25 kN 20 kN
30 kN
4 kips/ft
M
5-11
Support Reactions
Example Draw the shear and bending-moment diagrams for the beam and loading shown. Label all points of change, maximums and minimums, and the axes. Units: kN, m
3 BA
8 8
C D
10
10
3 3 3
5-12
Support Reactions
Example Draw the shear and bending-moment diagrams for the beam and loading shown. Label all points of change, maximums and minimums, and the axes. Units: kN, m
3.2 BA
2 kN/m
4
0.8
5-13
Example Draw the shear and bending-moment diagrams for the beam and loading shown. Label all points of change, maximums and minimums, and the axes. Units: Lb, ft.
8 4 4 B A
30 lb/ft 20 lb/ft
5-14
Example Sketch the shear and bending-moment diagrams for the beam and loading shown. Label all points of change, maximums and minimums, and the axes. Units: Lb, ft.
8 4 B A
30 lb/ft
4
5-15
Example Sketch the shear and bending-moment diagrams for the beam and loading shown. Label all points of change, maximums and minimums, and the axes. Units: Lb, ft.
5 4 3 4 B A
30 lb/ft
5-16
Example Sketch the shear and bending-moment diagrams for the beam and loading shown. Label all points of change, maximums and minimums, and the axes. Units: Lb, ft.
5 4 3 4 B A
30 lb/ft
20 lb/ft
5-17
Example Here is an example of how the shape of the girder reflects the shear and bending diagrams.
AB
L
5-18
So why did they put that gap in the bridge?
BA
BA
5-19
Pins
5-20
Example Draw the shear and bending-moment diagrams for the beam and loading shown. Label all points of change, maximums and minimums, and the axes. The addition of the internal pin at the center of the beam allows additional head room because rather than the moment being a maximum in the center it becomes zero. This design is used at Wings Air West in SLO. Total span= 75 ft.
500 lb/ft
A B
500 lb/ft
A B
5-21
Example Draw the shear and bending-moment diagrams for the beam and loading shown. Label all points of change, maximums and minimums, and the axes. This example demonstrates that with the addition of an internal pin we get an additional equation, otherwise we would have too many unknowns.
Support Reactions C
A B
3
3.5 1 2
C
B
3
5-22 Design of Prismatic Beams for Bending
DESIGN OF PRISMATIC BEAMS FOR BENDING
Example Design the cross section’s minimum height of the beam, knowing that the grade of wood used has an allowable bending stress of 12 MPa. Units: N, m.
6
B A
3000
1.5 1.5
3000
h
100
V (N)
M (N•m)
5-23
Cross-section
Example Knowing that the grade of copper used has an allowable bending stress of 15 ksi, determine the minimum wall thickness for the 6" diameter pipe. Units: lb, ft.
3
700 500 lb/ft
V (lb)
M (k•in)
5-24
Example Knowing that the allowable bending stress for steel is 160 MPa, determine the most economical W410-shape to support the load. Units: kN, m.
20 kN/m
6
B A
100
1.5 1.5
100
Cross-section
X X
V (kN)
M (kN•m)
5-25
Cross-section
A B
6.5
500 50 lb/ft
4
Example Knowing that the allowable bending stress for steel is 24 ksi, determine the most economical C7-shape to support the load. Units: lb, ft.
Y Y
V (lb)
M (lb•ft)
5-26
6
BA
30 kN/m
Example Knowing that the allowable bending stress for steel is 160 MPa, determine the most economical W310-shape to support the load. Units: kN, m.
80
3 Cross-section
X X
V (kN)
M (kN•m)
5-27
Example Two rolled-steel C150 channels are welded back to back. Knowing that the allowable bending stress for steel is 160 MPa, determine the most economical channels to support the load. Units: kN, m.
6
BA
8
1.5
5 4
1.5 1.5
Cross-section
X X
V (kN)
M (kN•m)
5-28 Summary
SUMMARY
Internal Forces in Members
Shear and Bending-Moment Diagrams
Design of Prismatic Beams for Bending
A
B
A
B
C
P
A E
F
DB
C
5'10'
4'
8'
80' 40' 40'
B A
P P
C D
80' 40' 40'
B A
P P
C D
Cross-section
A
Sign Convention
V
M
M V xΔ = Δ V w xΔ = − Δ