Lecture 5-6 Beam Mechanics of Materials Laboratory Sec. 3-4 Nathan Sniadecki University of Washington Mechanics of Materials Lab

$100 Answer: What is the moment of inertia (Second Moment of Area) with respect to the x axis

$100 Answer: What is the moment of inertia (second moment of area) with respect to the y axis

$200 Answer: What is the moment of inertia for a rectangle

$200 Answer: What is the moment of inertia for a circle

$300 The straight line that defines a surface where  x and  x are zero Answer: What is the neutral axis

$300 Answer: What is the bending stress in the x-direction at a distance y from the origin of the coordinate system due to the loading of a couple vector M x acting in x-direction

$400 The point on the stress-strain curve where the material no longer deforms elastically, but also plastically. Answer: What is the proportional limit

$400 The theorem that expresses that the moment of inertia I x of an area with respect to an arbitrary x axis is equal to the moment of inertia I xc with respect to the centroidal x axis, plus the product Ad 2 of the area A and the square of the distance d between the two axis? Answer: What is the Parallel Axis Theorem I x = I xc + Ad 2

$500 The principle that states that the effect of a given combined loading on a structure can be obtained by determining the effects of each load separately and then combining the results obtained together as long as 1) each effect is linearly related to the load that produces it and 2) the deformation resulting from any given load is small and does not affect the conditions of application of the other loads Answer: What is the Principle of Superposition

$500 The principle that states that except in the immediate vicinity of the point of loading, the stress distribution may be assumed to be independent of the actual mode of loading, i.e. for axial loading, at a distance equal to or greater than the width of a member, the distribution of stress across a given section is the same. Answer: What is Saint-Venant’s Principle

FINAL JEOPARDY

Answer: What is the angle of the neutral axis for an asymmetrically loaded beam

Inclined Load Notice the sign convention: positive Mz compress upper part, negative stress; positive My extend front part, positive stress!

Inclined Load Stress Neutral axis 

Example

Stress Distribution

The centroid of the area A is defined as the point C with coordinates ( y c, z c ) which satisfies Asymmetrical Beam If the origin of y and z axes is placed at centroid C (orientation is arbitrary.)

Consider of beam segment AB of length L After deformation, length of neutral surface DE remains L, but JK becomes Pure Bending

Asymmetric Beam If z is a principal axis (symmetry), the product of inertia I yz is zero  M y = 0, bending in x-y plane, analogous to a symmetric beam When z axis is the neutral axis;

Asymmetric Beam If y is a principal axis, the product of inertia I yz is zero  M z = 0, bending in x-z plane, analogous to a symmetric beam When y axis is the neutral axis;

Asymmetric Beam When an asymmetric beam is in pure bending, the plane in which the bending moments acts is perpendicular to the neutral surface if and only if (iff) the y and z axes are principle centroidal axes and the bending moments act in one of the two principle planes. In such a case, the principle plane in which bending moment acts becomes the plane of bending and the usual bending theory is valid

Analysis of Asymmetric Beam Locating the centroid, and constructing a set of principal axes Resolving bending moment into M y and M z Superposition

Principle Axes

Analysis of Asymmetric Beam A channel section C 10  15.3 M = 15 kips-in I y =2.28 in 4, I z =67.4 in 4 Location of Point C c=0.634 in Location of Point A y A =5.00 in z A = = in Calculate bending stress Locate neutral axis

Analysis of Asymmetric Beam

Normal Stress in Beam

Curved Beams What if the beam is already ‘bent’? Where will the beam likely fail?

Bending Stress for Curved Beam #1: Neutral surface remains constant: #2: Deformation at JK:

Bending Stress for Curved Beam #3: Strain: #4: Stress:

Bending Stress for Curved Beam #5: Neutral Axis: #6 Centroid: #7: N.A. Location: Since  > 0 for M > 0 Aside: R = r n

Location of N.A. in Curved Beam Cross-sectional dimensions define neutral axis location for a curve beam about C

Curved Beams Positive M decreases curvature Neutral axis is no longer the centroidal axis

Curved Beam

Curved Beams Curvature is small, e is small, r n is close to r c Recover to straight beam

Curved Beam Pay attention to the sign of s

Curved Beam Pay attention to the sign of s

Read Mechanics of Materials Lab Sec (e), 4.72 posted online Assignment