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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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beams
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
Beams - loads and internal loads

Loads: concentrated loads, distributed loads, couples (moments)

Internal loads: shear force and bending moments

shear forces bending moments sign conventions
Shear Forces, Bending Moments - Sign Conventions

right section

left section

Shear forces:

positive shear:

negative shear:

Bending moments:

positive moment

negative moment

shear forces bending moments static equilibrium approach
Shear Forces, Bending Moments - Static Equilibrium Approach

Procedure:

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.

beam normal strain
Beam - Normal Strain

M

M

no transverse load

Pure bending problem

no axial load

no torque

Observations of the deformed beam under pure bending

Length of the longitudinal elements

Vertical plane remains plane after deformation

Beam deforms like an arc

normal strain analysis
Normal Strain - Analysis

neutral axis (N.A.):

radius of curvature:

Coordinate system:

q

longitudinal strain:

r

y

N.A.

beam normal stress
Beam - Normal Stress

Hooke’s Law:

y

M

M

M

x

Maximum stresses:

Neutral axis:

flexure formula
Flexure Formula

y

Moment balance:

M

x

Comparison:

Axially loaded members Torsional shafts:

moment of inertia i
Moment of Inertia - I

Example 2:

h

w

Example 3:

h

w

w

4h

w

design of beams for bending stresses
Design of Beams for Bending Stresses

Design Criteria:

1.

2. cost as low as possible

Design Question:

Given the loading and material, how to choose the shape and the size

of the beam so that the two design criteria are satisfied?

design of beams for bending stresses12
Design of Beams for Bending Stresses

Procedure:

  • Find Mmax
  • Calculate the required section modulus
  • Pick a beam with the least cross-sectional area or weight
  • Check your answer
design of beams for bending stresses13
Design of Beams for Bending Stresses

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.

6 ft

shear stresses inside beams
Shear Stresses inside Beams

shear force: V

V

Horizontal shear stresses:

y

h1

y1

x

h2

s2

s1

tH

shear stresses inside beams15
Shear Stresses inside Beams

Relationship between the horizontal shear stresses and the vertical shear stresses:

y

h1

y1

x

h2

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

evaluated

I: second moment of inertial of the cross section

shear stresses inside beams16
Shear Stresses inside Beams

Example 5: Find shear stresses at points A, O and B located at cross section

a-a.

P

a

A

O

a

B

w

shear stress formula limitations
Shear Stress Formula - Limitations

- elementary shear stress theory

Assumptions:

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

shear stresses inside beams18
Shear Stresses inside Beams

Example 6: The transverse shear V is 6000 N. Determine the vertical shear stress

at the web.

beams examples
Beams - Examples

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

deflections of beam
Deflections of Beam

Deflection curve of the beam: deflection of the neutral axis of the beam.

y

P

y

x

x

Derivation:

Moment-curvature relationship:

Curvature of the deflection curve:

(1)

(2)

Small deflection:

(3)

Equations (1), (2) and (3) are totally equivalent.

method of superposition
Method of Superposition

P

q

Deflection: y

P

Deflection: y2

Deflection: y1

statically indeterminate beam
Statically Indeterminate Beam

Number of unknown reactions is larger than the number of independent

Equilibrium equations.

Propped cantilever beam

Clamped-clamped beam

Continuous beam

statically indeterminate beam26
Statically Indeterminate Beam

Example 10. Find the reactions of the propped beam shown below.