Shear Flow
If the shearing stress f_v 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.
 

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?
 

Problem 558

In Prob. 557, find the values of x and w_o so that w_o is a maximum.
 

Problem 557
A cast-iron beam 10 m long and supported as shown in Fig. P-557 carries a uniformly distributed load of intensity w_o (including its own weight). The allowable stresses are f_bt ≤ 20 MPa and f_bc ≤ 80 MPa. Determine the maximum safe value of w_o if x = 1.0 m.
 

Problem 556
A T beam supports the three concentrated loads shown in Fig. P-556. Prove that the NA is 3.5 in. above the bottom and that I_NA = 97.0 in⁴. Then use these values to determine the maximum value of P so that f_bt ≤ 4 ksi and f_bc ≤ 10 ksi.
 

Problem 555
A beam carries a concentrated load W and a total uniformly distributed load of 4W as shown in Fig. P-555. What safe value of W can be applied if f_bc ≤ 100 MPa and f_bt ≤ 60 MPa? Can a greater load be applied if the section is inverted? Explain.
 

Problem 554
Determine the maximum tensile and compressive stresses developed in the overhanging beam shown in Fig. P-554. The cross-section is an inverted T with the given properties.
 

Let us consider a differential length dx of the beam shown
 

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

If a beam is composed of two or more thin layers placed on each other without any attachment, the separate layers would slide past each other and the total strength of the beam would be the sum of the strengths of the various layers. The strength of this beam is lesser than a solid beam having the same cross-sectional area.
 

Problem 553
Determine the maximum tensile and compressive bending stresses developed in the beam as shown in Fig. P-553.