Beams

Beams:

Comparison with trusses, plates

Examples:

1. simply supported beams 2. cantilever beams

L, W, t: L >> W and L >> t

L W

t

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:

left section right section

positive shear:

negative shear:

positive moment

negative moment

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.

Relationship between Loads, Shear Forces, and Bending Diagram

dV dM

q Vdx dx

Beam - Normal Strain

Pure bending problem

no transverse load

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

MM

Normal Strain - Analysis

x

neutral axis (N.A.):

radius of curvature:

Coordinate system:

longitudinal strain:

y

'x

L x x

L xy y

N.A.

Beam - Normal Stress

x x

EyE

Hooke’s Law:

Maximum stresses:

MMM x

y

Neutral axis:0 0 0

0 0

x x

A A

c

A

EyF dA dA

ydA y

Flexure Formula

M M

22 1

x

A A

y E MdM dA y M E dA y dA

EI

2 : second moment of inertial (with respect to the neutral axis)

A

I y dA

M x

y

Moment balance:

x x

EyE

x

M y

I

Axially loaded members Torsional shafts:

Comparison:

Moment of Inertia - IdAyI

A2

Example 2:

Example 3:

w

h

4hww

w

h

Design of Beams for Bending Stresses

Design Criteria:

nuY

allowableallowable

or , 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 Stresses

Procedure:

• Find Mmax

• Calculate the required section modulus

• Pick a beam with the least cross-sectional area or weight

• Check your answer

; : section modulusx

M y M IS

I S y

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 sizeof 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 Beamsshear force: V

Horizontal shear stresses:

V

1

1

, : first momenth

H

y

VQQ ydA

Iw

x

yh1

h2

y1

Shear Stresses inside Beams

H

VQ

Iw

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

Shear stresses - force balance

Iw

VQ

V: shear force at the transverse cross sectionQ: 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 evaluatedI: second moment of inertial of the cross section

xh1

h2

y1

y

1

1

h

y

Q ydA

Shear Stresses inside Beams

2

L4

L

Example 5: Find shear stresses at points A, O and B located at cross sectiona-a. P

a

a 4h

4

L

4h4h

4h

w

A

BO

Shear Stress Formula - Limitations

Iw

VQ - 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 Beams

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

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

1

M

EI

2

2

3 22

1

1

d ydx

dydx

2

2

11

dy d y

dx dx

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

x

y

x

y

Derivation:

2 2

2 2

d y M d yEI M

dx EI dx

2

2

dM d d yV V EI

dx dx dx

2 2

2 2

dV d d yq q EI

dx dx dx

Moment-curvature relationship:

Curvature of the deflection curve:

Small deflection:

(1)

(2)

(3)

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

P

Deflections by Integration of the Moment Differential Equation

Example 8 (approach 1):

Deflections by Integration of the Load Differential Equation

Example 8 (approach 2):

Method of Superposition

Pq

P

Deflection: y

Deflection: y1Deflection: y2

1 2y y y

Method of Superposition

Example 9

Statically Indeterminate Beam

Number of unknown reactions is larger than the number of independentEquilibrium equations.

Propped cantilever beam

Clamped-clamped beam

Continuous beam

Statically Indeterminate Beam

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