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CTC / MTC 222 Strength of MaterialsPowerPoint Presentation

CTC / MTC 222 Strength of Materials

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Chapter Objectives

- List the situations where shear stress in a beam is likely to be critical.
- Compute the shear stress in a beam using the general shear formula.
- Compute the maximum shear stress in a solid rectangular or circular section using the appropriate formulas.
- Compute the approximate maximum shear stress in a hollow thin-walled tube or thin-webbed section using the appropriate formulas.

Shear Stresses

- To determine shear stress at some point in a beam, first must determine shear force.
- Construct V diagram to find distribution and maximum shear.

- Often calculate vertical shear at a section
- Horizontal shear at the section is equal.

- Shear stress is not usually critical in steel or aluminum beams
- Beam is designed or selected to resist bending stress.
- Section chosen is usually more than adequate for shear

- Shear stress may be critical in some cases:
- Wooden beams
- Wood is weaker along the grain, subject to failure from horizontal shear

- Thin-webbed beams
- Short beams or beams with heavy concentrated loads
- Fasteners in built-up or composite beams
- Stressed skin structures

- Wooden beams

The General Shear Formula

- The shear stress, , at any point within a beams cross-section can be calculated from the General Shear Formula:
- = VQ / I t, where
- V = Vertical shear force
- I = Moment of inertia of the entire cross-section about the centroidal axis
- t = thickness of the cross-section at the axis where shear stress is to be calculated
- Q = Statical moment about the neutral axis of the area of the cross-section between the axis where the shear stress is calculated and the top (or bottom) of the beam
- Q is also called the first moment of the area

- Mathematically, Q = AP ̅y̅ , where:
- AP = area of theat part of the cross-section between the axis where the shear stress is calculated and the top (or bottom) of the beam
- ̅y̅ = distance to the centroid of AP from the overall centroidal axis
- Units of Q are length cubed; in3, mm3, m3,

- = VQ / I t, where

Distribution of Shear Stress in Beams

- The maximum shear stress, , at any point in a beam’s cross-section occurs at the centroidal axis, unless, the thickness of the cross-section is less at some other axis.
- Other observations:
- Shear stress at the outside of the section is zero
- Within any area of the cross-section where the thickness is constant, the shear stress varies parabolically, decreasing as the distance from the centroid increases.
- Where an abrupt change in the thickness of the cross-section occurs, there is also an abrupt change in the shear stress
- Stress will be much higher in the thinner portion

Shear Stress in Common Shapes

- The General Shear Formula can be used to develop formulas for the maximum shear stress in common shapes.
- Rectangular Cross-section
- max = 3V / 2A

- Solid Circular Cross-section
- max = 4V / 3A

- Approximate Value for Thin-Walled Tubular Section
- max ≈ 2V / A

- Approximate Value for Thin-Webbed Shape
- max ≈ V / t h
- t = thickness of web, h = depth of beam

- max ≈ V / t h

- Rectangular Cross-section

Design Shear Stress, d

- Design stress, d , varies greatly depending on material
- Wood beams
- Allowable shear stress ranges from 70 - 100 psi
- Allowable bending stress is 600 – 1800 psi
- Allowable tension stress is 400 – 1000 psi
- Failure is often by horizontal shear, parallel to grain

- Steel beams
- d = 0.40 SY
- Allowable stress is set low, because method of calculating stress (max ≈ V / t h ) underestimates the actual stress

Shear Flow

- Shear flow – A measure of the shear force per unit length at a given section of a member
- The shear flow q is calculated by multiplying thr shear force at a given section by the thickness at that section: q = t
- By the General Shear Formula: = VQ / I t
- Then q = t = VQ / I
- Units of q are force per unit length, N / m, kips / inch, etc.

- Shear flow is useful in analyzing built-up sections
- If the allowable shear force on a fastener, Fsd , is known, the maximum allowable spacing of fasteners required to connect a component of a built-up section, smax , can be calculated from: smax= Fsd / q

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