Solution to Problem 561 | Built-up Beams

Problem 561
A T section has the dimensions given in Fig. P-561. Show that the neutral axis is 3 inches below the top and that INA = 166.7 in4. If the tensile stress at the bottom of the flange is 1000 psi, calculate (a) the total tensile force in the flange and (b) the total compressive force in the cross section. Also compute (c) the moment of the compressive force about the NA, and (d) the moment of the total tensile force about the NA. (e) How does the sum of (c) and (d) compare with the total applied bending moment as computed from the flexure formula?
 

Shear Flow

Shear Flow
If the shearing stress fv is multiplied by the width b, we obtain a quantity q, known as the shear flow, which represents the longitudinal force per unit length transmitted across a section at a level y1 from the neutral axis.
 

Solution to Problem 559 | Built-up Beams

Problem 559
A beam is composed of 6 planks, each 100 mm wide and 20 mm thick, piled loosely on each other to an overall dimension of 100 mm wide by 120 mm high. (a) Compare the strength of such a beam with that of a solid beam of equal overall dimensions. (b) What would be the ratio if the built-up beam consisted of a 12 planks each 100 mm wide by 10 mm thick?
 

Horizontal Shearing Stress

Let us consider a differential length dx of the beam shown
 

Analysis of Horizontal Shearing Stress

 

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

 

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.