Defining the Beam and Load Configuration A 20‐meter beam is loaded with two moving point loads of 40 kN and 20 kN that are 4 meters apart, with the heavier load leading. The task is to construct influence lines for bending moment and shear force. The objective is to determine the maximum bending moment at a point 7 meters from the left support along with the corresponding shear force values.
Evaluating the Maximum Bending Moment at 7m The influence line for bending moment is calculated using the formula x(L-x)/L, which yields an ordinate of 4.55 when x is 7 m and L is 20 m. Verifying the differential loading rates confirms the 40 kN load as the critical one at the specified point. Multiplying the 40 kN and 20 kN loads with their respective ordinates leads to a computed maximum bending moment of 245 kN-m.
Deriving Shear Force Influence Lines and Values Shear force diagrams are constructed with the formula (L-x)/L for positive shear and x/L for negative shear, giving ordinates of 0.65 and 0.35 at 7 m respectively. Evaluations show that maintaining the 40 kN load at the key point produces a maximum positive shear force of 35 kN, while optimal placement of loads yields a maximum negative shear force of 13 kN. Considering the entire beam, absolute shear forces of 56 kN (positive) and 52 kN (negative) are obtained.
Establishing the Maximum Absolute Bending Moment The maximum absolute bending moment is determined by first locating the resultant load from the combined 40 kN and 20 kN loads. A moment equilibrium calculation finds the centroid approximately 1.33 m from the 40 kN load, identifying it as the dominant contributor. Using the effective spans from this load position, the final computation produces a maximum absolute bending moment of 261.4 kN-m.