Direct integration 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 , taking downward to be positive, the internal shear force is given by taking the negative integral of the weight: The internal moment is the integral of the internal shear: = The angle of rotation from the horizontal, , 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 :

Direct integration 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 , taking downward to be positive, the internal shear force is given by taking the negative integral of the weight: The internal moment is the integral of the internal shear: = The angle of rotation from the horizontal, , 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 :