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# CTC / MTC 222 Strength of Materials - PowerPoint PPT Presentation

CTC / MTC 222 Strength of Materials. Chapter 9 Shear Stress in Beams. 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.

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### CTC / MTC 222 Strength of Materials

Chapter 9

Shear Stress in Beams

• 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.

• 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

• 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,

• 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

• 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

• 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 – 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