Skew Loads and Non-Symmetric Cross Sections (Notes 3.10)

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Skew Loads and Non-Symmetric Cross Sections
(Notes 3.10)

• MAE 316 Strength of Mechanical Components
• NC State University Department of Mechanical and
  Aerospace Engineering

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Introduction

• Will perform advanced stress and deflection
  analysis of beams with skew loads and
  non-symmetric cross sections.
• Challenge Need to calculatemoments of inertia
  Iyy, Izz,and Iyz and principal moments of
  inertia.

Skew load
Non-Symmetric
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Moments of Inertia

• For any cross-section shape

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Moments of Inertia

• The moments of inertia can be transformed to
  y1-z1 coordinates by
• Does this look familiar??

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Moments of Inertia

• Similar to transformation of stress, principal
  angle (angle to the principal axes of inertia)
  can be found from
• Where ?P is the angle at which Iyz is zero.

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Example

• Find Iyy and Izz for a rectangle

C
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Example

• Find Iyy and Izz for a Z-section (non-symmetric
  about y-z) Let b 7 in, t 1 in, and h 16 in.

C
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Example

• Find Iyy and Izz for an L-section (non-symmetric
  about y-z) Let b 4 in, t 0.5 in, and h 6 in.

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Skew Loads (3.10)

• Skew loads for doubly symmetric cross sections
• Beam will bend in two directions
• Py P cos a
• Pz P sin a

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Skew Loads (3.10)

• Find bending moments
• Side view
• From statics Mz Py(L-x) P cos a (L-x)
• Why is Mz positive?
• beam is curving in direction of positive y

Side
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Skew Loads (3.10)

• Find bending moments
• Top view
• From statics My Pz(L-x) P sin a (L-x)
• Why is My positive or negative?

Top
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Skew Loads (3.10)

• Bending stress
• From side view
• From top view
• Combine to get ?

Top
Side
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Skew Loads

• What about the neutral axis?
• When there is only vertical bending, sxx0
  because y0 at the neutral axis.

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Skew Loads

• But with a skew load
• It turns out deflection will be perpendicular to
  this line.

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Skew Loads

• Curvature due to moment
• From side view
• From top view
• Where vy and vz are deflections in the positive y
  and z directions, respectively.

Top
Side
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Skew Loads

• Find deflection at free end.
• Apply B.C.s vy(0)0 vy(0)0
• The tip deflection in the y-direction is

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Skew Loads

• Continued
• Apply B.C.s vz(0)0 vz(0)0
• The tip deflection in the z-direction is

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Skew Loads

• The resultant tip deflection is

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Example

• Consider a cantilever beam with the cross-section
  and load shown below. Find the stress at A and
  the tip deflection when a 0o and a1o. Let L
  12 ft, P 10 kips, E 30x106 psi and assume an
  S24x80 rolled steel beam is used.

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Non-Symmetric Cross-Sections

• Bending of non-symmetriccross-sections
• Iyz ? 0
• Iyy Izz are not principal axes
• Use generalized flexure formula

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Non-Symmetric Cross-Sections

• Generalized moment-curvature formulas

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Non-Symmetric Cross-Sections

• A special case which we discussed previously
  is whenIyz 0 and y z are the principal axes.

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Example

• Analysis choices
• Work in principal coordinates simple formulas
• Work in arbitrary coordinates more complex
  formulas
• Calculate the stress at A and the tip deflection
  for the beam shown below.

Mz 10,000 in-lbs(pure bending) Cross-section
dimensions6 x 4 x 0.5 in Iyy 6.27 in4 Izz
17.4 in4 Iyz 6.07 in4 E 30 x 106 psi