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Mechanics of Materials (MindTap Co...

9th Edition
Barry J. Goodno + 1 other
ISBN: 9781337093347

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BuyFindarrow_forward

Mechanics of Materials (MindTap Co...

9th Edition
Barry J. Goodno + 1 other
ISBN: 9781337093347
Textbook Problem

A wood beam ABC with simple supports at A and B and an overhang ÄChas height h = 300 mm (sec figure). The length of the main span of the beam is L = 3.6 m and the length of the overhang is L/3 = 1,2 m. The beam supports a concentrated load 3P = 18 kN at the midpoint of the main span and a moment PLI2 = 10,8 kN - m at the free end of the overhang. The wood has a weight density y = 5,5 kN/m3.

  1. Determine the required width h of the beam based upon an allowable bending stress of 8,2 MPa,
  2. Determine the required width based upon an allowable shear stress of 0.7 MPa.

  

(a)

To determine

Width of the beam.

Explanation

Given Information:

The following figure is given:

  

  L=3.6 m, 3P=18 kN, PL/2=10.8 kNm, γ=5.5 kN/m3 . The permissible bending stress is 8.2 MPa.

Calculation:

Consider the following diagram:

  

The distributed load due to weight density is:

  q=γA=5.5×0.3×b×103 kN/m

Value of P is 6 kN.

Calculate the upward reaction force at B by taking equilibrium of moments about A as,

  MA=0ByL=q(L+L/3)22+3P×L2+PL2By=q(L+L/3)22L+3P2+P2=q(L+L/3)22L+2P

There is no reaction force in horizontal direction.

Calculate the upward reaction force at A by taking equilibrium of forces in y -direction as:

  Fy=0Ay+By=q(L+L/3)+3PAy=By+q(L+L/3)+3PAy=(q(L+L/3)22L+2P)+q(L+L/3)+3PAy=q(L+L/3)22L+q(L+L/3)+P

The bending moment at x is given by:

  M={Ayxq×x22                                  &#

(b)

To determine

Width of the beam

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