horizontal shear stress

Example 02: Notched beam with concentrated load

Problem
A 150 mm by 300 mm wooden beam having a simple span of 6 meters carries a concentrated load P at its midspan. It is notched at the supports as shown in the figure. For this problem, all calculations are based on shear alone using the 2010 NSCP specification given below. Allowable shear stress of wood, F_v = 1.0 MPa.

If P = 30 kN, calculate the maximum allowable depth (millimeters) of notches at the supports.
1. 88
2. 62
3. 238
4. 212
If the depth of notches is 100 mm, what is the safe value of P (kiloNewton) the beam can carry.
1. 26.67
2. 17.78
3. 8.89
4. 13.33
If P = 25 kN and the depth of notches is 150 millimeters, what is the shear stress (MegaPascal) near the supports.
1. 0.83
2. 6.67
3. 1.67
4. 3.33

NSCP 2010 Section 616.4: Horizontal Shear in Notched Beams
When rectangular-shaped girder, beams or joists are notched at points of support on the tension side, they shall meet the design requirements of that section in bending and in shear. The horizontal shear stress at such point shall be calculated by:

$f_v = \dfrac{3V}{2bd'}\left( \dfrac{d}{d'} \right)^2$

Where:

$d$ = total depth of beam.
$d'$ = actual depth of beam at notch.

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Example 01: Safe Uniform Load for a Beam that was Notched at the Tension Fibers at Supports

Problem
A 75 mm × 150 mm beam carries a uniform load w_o over the entire span of 1.2 m. Square notches 25 mm deep are provided at the bottom of the beam at the supports. Calculate the safe value of w_o based on shear alone.

Allowable shear parallel to grain = 1.40 MPa
Allowable shear normal to grain = 1.85 MPa

Example 01: Maximum bending stress, shear stress, and deflection

Problem
A timber beam 4 m long is simply supported at both ends. It carries a uniform load of 10 kN/m including its own weight. The wooden section has a width of 200 mm and a depth of 260 mm and is made up of 80% grade Apitong. Use dressed dimension by reducing its dimensions by 10 mm.

Properties of Apitong
Bending and tension parallel to grain = 16.5 MPa
Shear parallel to grain = 1.73 MPa
Modulus of elasticity in bending = 7.31 GPa

What is the maximum flexural stress of the beam?
What is the maximum shearing stress of the beam?
What is the maximum deflection of the beam?

Notching on Beams

NSCP 2001
When rectangular shaped girders, beams or joists are notched at points of supports on the tension side, the horizontal shear stress at such point shall not exceed:

$F_v = \dfrac{3V}{2bd'}\left( \dfrac{d}{d'} \right)$

Where

$d$ = total depth of beam
$d'$ = actual depth of beam at notch

When girder, beams or joists with circular cross section are notched at points of support on the tension side, the actual shear stress at such point shall not exceed:

Bending Stress and Shearing Stress in Timber Beam

Bending Stress $f_b = \dfrac{M}{S} = \dfrac{Mc}{I}$ Horizontal Shear Stress $f_v = \dfrac{VQ}{Ib}$
For Rectangular Sections $f_b = \dfrac{6M}{bd^2}$ $f_v = \dfrac{3V}{2bd}$

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Solution to Problem 599 | Spacing of Rivets or Bolts in Built-Up Beams

Problem 599
A beam is formed by bolting together two W200 × 100 sections as shown in Fig. P-599. It is used to support a uniformly distributed load of 30 kN/m (including the weight of the beam) on a simply supported span of 10 m. Compute the maximum flexural stress and the pitch between bolts that have a shearing strength of 30 kN.

Wide Flange on top of the other and bolted together

Solution to Problem 598 | Spacing of Rivets or Bolts in Built-Up Beams

Problem 598
As shown in Fig. P-598, two C380 × 60 channels are riveted together by pairs of 19-mm rivets spaced 200 mm apart along the length of the beam. What maximum vertical shear V can be applied to the section without exceeding the stresses given in Illustrative Problem 591?

Solution to Problem 597 | Spacing of Rivets or Bolts in Built-Up Beams

Problem 597
A plate and angle girder similar to that shown in Fig. 5-32 is fabricated by riveting the short legs of four 125 × 75 × 13 mm angles to a web plate 1000 mm by 10 mm to form a section 1020 mm deep. Cover plates, each 300 mm × 10 mm, are then riveted to the flange angles making the overall height 1040 mm. The moment of inertia of the entire section about the NA is I = 4770 × 10⁶ mm⁴. Using the allowable stresses specified in Illustrative Problem 591, determine the rivet pitch for 22-mm rivets, attaching the angles to the web plate at a section where V = 450 kN.

Solution to Problem 596 | Spacing of Rivets or Bolts in Built-Up Beams

Problem 596
Three planks 4 in by 6 in., arranged as shown in Fig. P-596 and secured by bolts spaced 1 ft apart, are used to support a concentrated load P at the center of a simply supported span 12 ft long. If P causes a maximum flexural stress of 1200 psi, determine the bolt diameters, assuming that the shear between the planks is transmitted by friction only. The bolts are tightened to a tension of 20 ksi and the coefficient of friction between the planks is 0.40.

Solution to Problem 595 | Spacing of Rivets or Bolts in Built-Up Beams

Problem 595
A concentrated load P is carried at midspan of a simply supported 12-ft span. The beam is made of 2-in. by 6-in. pieces screwed together, as shown in Fig. P-595. If the maximum flexural stress developed is 1400 psi, find the maximum shearing stress and the pitch of the screws if each screw can resist 200 lb.