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

6.2 Bending Deformation and Strain Key Points: Bending moment causes beam to deform. X = longitudinal axis Y = axis of symmetry Neutral surface – does not undergo a change in length Figure: 06-20

Internal bending moment causes beam to deform. Figure: 06-21a Key Points: Internal bending moment causes beam to deform. For this case, top fibers in compression, bottom in tension.

Neutral surface – no change in length. Figure: 06-22a-b Key Points: Neutral surface – no change in length. All cross-sections remain plane and perpendicular to longitudinal axis. Neglect deformation of cros section within its own plane.

Radius of curvature Figure: 06-23a-b Length decreases Length = constant Figure: 06-23a-b

Says normal strain is linear Maximum at outer surface (where y = c)

6.2 Bending Stress – The Flexure Formula What about Stress???? Recall from section 6.1: Figure: 06-24 Therefore, it follows that

Figure: 06-26a,b

This is the moment of inertia, I Sum moments about cut: Figure: 06-26-02c This is the moment of inertia, I

The Flexure Formula: Or in general: Internal bending moment, lb-in Max bending stress, psi Distance from NA to outer fiber, in Moment of inertia, in4 Or in general:

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. Draw shear and moment diagram. Identify location and magnitude of Mmax. 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? What is the largest internal moment the beam can resist if sallow = 2 ksi?

45K

Given: Typical household pliers as shown. 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. Do this for homework. See solution Link Side: what is the shear stress in pin and bending stress in handle? SofM

Given: Crane structure as shown. 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. Do this for homework. See solution Link Side: what is the normal stress in cables (average normal only) and normal stress in boom and post (combined loading)? SofM

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