Beams. Beams:. t. L, W, t: L >> W and L >> t. W. L. Comparison with trusses, plates. Examples:. 2. cantilever beams. 1. simply supported beams. Beams - loads and internal loads. Loads: concentrated loads, distributed loads, couples (moments).
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L, W, t: L >> W and L >> t
Comparison with trusses, plates
2. cantilever beams
1. simply supported beams
Loads: concentrated loads, distributed loads, couples (moments)
Internal loads: shear force and bending moments
1. find reactions;
2. cut the beam at a certain cross section, draw F.B.D. of one piece of the beam;
3. set up equations;
4. solve for shear force and bending moment at that cross section;
5. draw shear and bending moment diagrams.
Example 1: Find the shear force and bending diagram at any cross section of
the beam shown below.
no transverse load
Pure bending problem
no axial load
Observations of the deformed beam under pure bending
Length of the longitudinal elements
Vertical plane remains plane after deformation
Beam deforms like an arc
neutral axis (N.A.):
radius of curvature:
Axially loaded members Torsional shafts:
2. cost as low as possible
Given the loading and material, how to choose the shape and the size
of the beam so that the two design criteria are satisfied?
Example 4: A beam needs to support a uniform loading with density of
200 lb /ft. The allowable stress is 16,000 psi. Select the shape and the size
of the beam if the height of the beam has to be 2 in and only rectangular and
circular shapes are allowed.
shear force: V
Horizontal shear stresses:
Relationship between the horizontal shear stresses and the vertical shear stresses:
Shear stresses - force balance
V: shear force at the transverse cross section
Q: first moment of the cross sectional area above the level at which
the shear stress is being evaluated
w: width of the beam at the point at which the shear stress is being
I: second moment of inertial of the cross section
Example 5: Find shear stresses at points A, O and B located at cross section
- elementary shear stress theory
1. Linearly elastic material, small deformation
2. The edge of the cross section must be parallel to y axis, not applicable for
triangular or semi-circular shape
3. Shear stress must be uniform across the width
4. For rectangular shape, w should not be too large
Example 6: The transverse shear V is 6000 N. Determine the vertical shear stress
at the web.
Example 7: For the beam and loading shown, determine
(1) the largest normal stress
(2) the largest shearing stress
(3) the shearing stress at point a
Deflection curve of the beam: deflection of the neutral axis of the beam.
Curvature of the deflection curve:
Equations (1), (2) and (3) are totally equivalent.
Example 8 (approach 1):
Example 8 (approach 2):
Number of unknown reactions is larger than the number of independent
Propped cantilever beam
Example 10. Find the reactions of the propped beam shown below.