Calculation of Reactions, Internal Shears and Internal Moments ...

Thus the maximum possible reaction on a simply supported beam due to a uniformlydistributed load of any length can be determined using the influence line of Figure 1.First, the influence line tells us where to place the uniform load for maximum positiveeffect, namely across the entire length of the beam where the influence line is positive.WxNext, the value of the reaction can be computed by multiplying the magnitude of theuniform load w by the area under the influence line over which the uniform load isapplied:Ra = Height of uniform load * Area under the influence line triangleRa = w kips/ft * (L ft * (1.0+0)/2) = wL/2 kipsNow admittedly this result is rather obvious, and most people would not create aninfluence line for such a simple case. However, we will later show that determiningwhere to place these loads for maximum effect can be quite confusing unless influencelines are available.Calculation of maximum positive shear at a point in a beam: Previously we showedthe generation of an influence line for shear at the quarter point in a simply supportedoverhanging beam, with critical placement of a set of concentrated and uniform loads,which will be repeated here. Note that for maximum positive shear, the uniform live loadis placed only in the ranges where the influence line is positive.wL/4w0.250.750.250.50Assuming a uniform load of 2,000 pounds/foot, and L = 10 feet, we can calculate themaximum positive shear at the quarter point by:V ¼ positive =Concentrated Load*height + Concentrated Load*height+ Uniform Load*Area under diagram+ Uniform Load*Area under diagramV ¼ positive = 20k * 0.75 + 10k * (2/3)*0.75+ 2k/ft * 0.25 * (10ft/4)/2 + 2k/ft * 0.75 * (3*10ft/4)/2= 26.25 kipsCalculation of maximum negative shear at a point in a beam: Previously we showedthe generation of an influence line for shear at the quarter point in a simply supported

M ¼ positive = Concentrated Load*height + Concentrated Load*height+ Uniform Load*Area under influence line+ Uniform Load*Area under influence line= 20kip * 0.1875*10ft + 10kip * 0.1875*10ft*2/3+ 2kip/ft * (0.1875*10ft*(10ft/2))= 406.25 kip ftCalculation of maximum negative moment at a point in a beam: Previously weshowed the generation of an influence line for moment at the quarter point in a simplysupported overhanging beam, with critical placement of a set of concentrated anduniform loads, which will be repeated here:wwAssuming a uniform load of 2,000 pounds/foot, and L = 10 feet, we can calculate themaximum negative moment at the quarter point by:M ¼ negative =Concentrated Load*height + Concentrated Load*height+ Uniform Load*Area under influence line+ Uniform Load*Area under influence line= 20kip * -0.375*10ft + 10kip * (-0.375*10ft)/2+ 2kip/ft * (-0.375*10ft * (10ft/2)/2)+ 2kip/ft * (-0.03125*10ft)*(10ft/8)/2= 131.64