Direct integration of a beam

Direct integration is a structural analysis method for measuring internal shear, internal moment, rotation, and deflection of a beam. Direct integration is a structural analysis method for measuring internal shear, internal moment, rotation, and deflection of a beam. For a beam with an applied weight w ( x ) {displaystyle w(x)} , taking downward to be positive, the internal shear force is given by taking the negative integral of the weight: The internal moment M ( x ) {displaystyle M(x)} is the integral of the internal shear: The angle of rotation from the horizontal, θ {displaystyle heta } , is the integral of the internal moment divided by the product of the Young's modulus and the area moment of inertia: Integrating the angle of rotation obtains the vertical displacement ν {displaystyle u } : Each time an integration is carried out, a constant of integration needs to be obtained. These constants are determined by using either the forces at supports, or at free ends. Take the beam shown at right supported by a fixed pin at the left and a roller at the right. There are no applied moments, the weight is a constant 10 kN, and - due to symmetry - each support applies a 75 kN vertical force to the beam. Taking x as the distance from the pin,