HORIZONTAL SHEAR STRESS IN BEAM

HORIZONTAL SHEAR STRESS IN BEAM

Let us consider a differential length dx of the beam shown

In this section,fv will be used for shearing stress instead of the standard symbol .

For the upper shaded portion of the beam, the forces acting are the total normal forces FR and FL due to the bending stresses to the left and to the right of the beam. These forces will be resisted by the shearing force fvb dx acting at the boundary surface between the shaded and the unshaded portions.

For equilibrium of the upper shaded portion

Where

But where represents the shear at the section in , and represents the first moment of an area of the shaded section about N.A. in mm3 which we will denote as , then

PROBLEM 1

A timber beam 80 mm wide by 160 mm high is subjected to a vertical shear V = 40 kN. Determine the shearing stress developed at layers 20 mm apart from the top to bottom of the section.

Solution 1

Layer 1 (20 mm from top and bottom layers)

answer

Layer 2 (40 mm from top and bottom layers)

answer

Layer 3 (60 from top and bottom layers)

answer

Layer 4 (The Neutral Axis, NA)

answer

Checking:For rectangular section

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PROBLEM 2

The T section shown in Fig. P-572 is the cross-section of a beam formed by joining two rectangular pieces of wood together. The beam is subjected to a maximum shearing force of 60 kN. Show that the NA is 34 mm from the top and the INA = 10.57 106 mm4. Using these values, determine the shearing stress (a) at the neutral axis and (b) at the junction between the two pieces of wood.

Solution 2

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By transfer formula for moment of inertia

Thus,

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(a) At the Neutral Axis

answer

(b) At the junction between the two pieces of wood

Flange:

answer

Web:

answer

PROBLEM 3

Determine the maximum and minimum shearing stress in the web of the wide flange section in Fig. P-575 if V = 100 kN. Also, compute the percentage of vertical shear carried only by the web of the beam.

Solution 3

Where

Maximum horizontal shear stress occurs at the neutral axis

answer

Minimum horizontal shear stress in the web occurs atthe junction of flange and web

answer

The horizontal shearing stresses vary parabolically from the top to the bottom of the web. Recall that the average height of parabolic segment is 2/3 of its altitude measured from its base. Thus,

Shear force carried by web aloneForce = Stress Area

Percentage of shear force carried by web alone

answer

PROBLEM 4Rework problem 3 assuming that the web is 200 mm instead of 160 mm.

Solution 4

Where

At the neutral axis

answer

At the junction of flange and web

Average shear stress

Shear force carried by web aloneForce = Stress Area

Percentage of shear force carried by web alone

answer

PROBLEM 5

SOLUTION 5

PROBLEM 6

SOLUTION 6

PROBLEM 7

SOLUTION 7

PROBLEM 8

SOLUTION 8

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