WORLD FOR CIVIL

STATICS & STRENGTH OF MATERIALS - Example

A loaded, simply supported W 8 x 28 beam is shown below. For this beam:
A. Determine the maximum bending stress 10 feet from the left end of the beam.
B. Determine the horizontal shear stress at a point 6 inches above the bottom of the beam cross section and 10 feet from the left end of the beam.
Unless otherwise indicated, all joints and support points are assumed to be pinned or hinged joints.

Solution:
Part A:
STEP 1: Determine the external support reactions:
1.) FBD of structure (See Diagram)
2.) Resolve all forces into x/y components
3.) Apply equilibrium conditions:

Sum F_x = 0 none
Sum F_y = B_y + D_y - 1,000 lbs/ft (8 ft) - 4,000 lbs - 6,000 lbs = 0
Sum T_B = 1,000 lbs/ft (8 ft) (4 ft) + 4,000 lbs (8 ft) - 6,000 lbs (4 ft) + D_y (8 ft) = 0
Solving: B_y = 23,000 lbs; D_y = -5,000 lbs

STEP 2: Determine the shear force and bending moment at x=10 ft.
1.) Cut beam at 10 ft. Draw the FBD of left end of beam, showing and labeling all external forces.
2.) Resolve all forces into x/y directions.
3.) Apply equilibrium conditions:

Sum F_x = 0 none
Sum F_y = -4,000 lbs - 8,000 lbs + 23,000 lbs - V₁₀ = 0
Sum T_A = -8,000 lbs (4 ft) + 23,000 lbs (8 ft) -V₁₀(10 ft) + M₁₀ = 0
Solving: V₁₀ = 11,000 lbs; M₁₀ = -42,000 ft-lbs

STEP 3: Apply the Flexure Formula to determine the Maximum Bending Stress (MBS) at 10'.

MBS = M_10'/S (Where M_10' is the bending moment at 10 ft, and S is the section modulus for the beam. The section modulus is available from the Beam Tables. The W 8 x 28 beam has a section modulus for the beam from the beam tables is 24.3 in³.)

MBS = 42,000 ft-lbs(12 in/ft)/24.3 in³ = 20,740 lbs/in²

Part B:
STEP 4: To determine the Horizontal Shear Stress (HSS) at 10 ft from the end of the beam and 6 inches above the bottom of the beam, apply the horizontal shear stress formula.
The form we will use is: HSS = Vay'/Ib
Where:
V = Shear force 10 ft from the end of the beam
a = cross sectional area from 6 in above the bottom of the beam to bottom of beam
y' = distance from neutral axis to the centroid of area a
I = moment of inertia of the beam (97.8 in⁴ for W 8 x 28 beam)
b = width of beam a 6 in above the bottom of the beam

HSS = [(11,000 lbs)(4.62 in²)(2.76 in)]/[(97.8 in⁴)(.29 in)] = 4,950 psi