the bending strain is then 4.4志.p-n-p地。 ” ,==-E,片 Flexure Elemen Degrees of Freedom of Plane Beam Element △☒ 22 the length after bending at any position y is expressed as: the bending strain is then From basic calculus, the radius of curvature of a planar curve is given by where v = v(x) represents the deflection curve of the neutral surface. In keeping with small deflection theory, slopes are also small, so Equation above is approximated by such that the normal strain in the direction of the longitudinal axis as a result of bending is and the corresponding normal stress is So at a given cross section, the normal stress varies linearly with distance from the neutral surface. As no net axial force is acting on the beam cross section, the resultant force of the stress distribution given by Equation above must be zero. Therefore, at any axial position x along the length, we have Noting that at an arbitrary cross section the curvature is constant, so Equation above implies which is satisfied if the xz plane (y = 0) passes through the centroid of the area.Thus, we obtain the well-known result that the neutral surface is perpendicular to the plane of bending and passes through the centroid of the cross-sectional area. Similarly, the internal bending moment at a cross section must be equivalent to the resultant moment of the normal stress distribution, so The integral term in Equation represents the moment of inertia of the cross sectional area about the z axis, so the bending moment expression becomes Combining Equations , we obtain the normal stress equation for beam bending: Flexure Element Beam elements with identical end deflections but quite different deflection characteristics. Figures below show physically unacceptable discontinuity at the connecting node, so transverse deflection of a beam is such that the variation of deflection along the length is not adequately described by displacement of the end points only. Therefore, the flexure element formulation must take into account the slope (rotation) of the beam as well as end-point displacement