Slide 1 This is a simply-supported beam. X is the beam axis before deformation. This curve is its axis after deformation under force P. O, C, C1 are points on the beam axis, and C1 is the position of point C after force P is exerted. Point O is the origin of the coordinate system. The horizontal coordinate of Point C is x. W(x) is the deflection at point C, which equals to the length of line CC1. If the beam at point C moves downward under force P, the deflection at C w(x) is positive. Otherwise, negative. θ(x) is the rotation angle of beam cross section at point C under force P. If the beam cross section at point C rotates clockwise under force P , rotation angle of beam cross section at point C θ(x) is positive, otherwise negative. Before force P exerts, the cross section at point C is vertical. Under force P, it’s slanted. The section turns in this direction. That’s clockwise. So the rotation angle of beam cross section at point C θ(x) is positive. Slide 2 T...