Chapter 6 Section 3,4 Bending Deformation, Strain and Stress in Beams.

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Chapter 6 Section 3,4 Bending Deformation, Strain and Stress in Beams

Transcript of Chapter 6 Section 3,4 Bending Deformation, Strain and Stress in Beams.

Chapter 6Section 3,4

Bending Deformation, Strain and Stress in Beams

Key Points:

1. Bending moment causes beam to deform.

2. X = longitudinal axis

3. Y = axis of symmetry

4. Neutral surface – does not undergo a change in length

6.2 Bending Deformation and Strain

Key Points:

1. Internal bending moment causes beam to deform.

2. For this case, top fibers in compression, bottom in tension.

Key Points:

1. Neutral surface – no change in length.

2. All cross-sections remain plane and perpendicular to longitudinal axis.

3. Neglect deformation of cros section within its own plane.

yy

s

ssss

)(lim

'lim

00

Radius of curvature

Length = constant

Length decreases

y

Says normal strain is linear

Maximum at outer surface (where y = c)

max

c

y

max

c

y

6.2 Bending Stress – The Flexure Formula

What about Stress????

Recall from section 6.1:

Therefore, it follows that

max

c

y

dAyc

M

dAc

yydAyydFM

A

A A A

2max

max)(

Sum moments about cut:

This is the moment of inertia, I

The Flexure Formula:

I

Mcmax

I

MyOr in general:

Max bending stress, psi

Internal bending moment, lb-in

Distance from NA to outer fiber, in

Moment of inertia, in4

Examples:

• Find maximum moment• Find area properties, I and c• Calculate stress

WHERE IS BENDING STRESS MAXIMUM???

Answer: • Outer surface

(furthest away from Neutral Axis)

• Value of x along length where moment is maximum!!

Example: The T-shape beam is subjected to the loading below.

1. Draw shear and moment diagram. Identify location and magnitude of Mmax.

2. Determine location and magnitude of maximum bending stress and draw stress

profile. Is the beam safe if the material is aluminum w/ sy = 15 ksi?

3. What is the largest internal moment the beam can resist if sallow = 2 ksi?

45K

Statics: Example 1 - Pliers

Given: Typical household pliers as shown.Find: Force applied to wire and force in pin that connects the two parts of the pliers.

Side: what is the shear stress in pin and bending stress in handle? SofM

Do this for homework.

See solution Link

Statics: Example 2 – Crane Structure

Given: Crane structure as shown.Find: Forces and FBD’s for cables A-B and A-E, boom DEF and post AFC.

Side: what is the normal stress in cables (average normal only) and normal stress in boom and post (combined loading)? SofM

Do this for homework.

See solution Link

Example 4: Determine the resultant internal loadings acting on the cross sections located through points D and E of the frame. (1-114)