Relation Between BM and Slope Lecture No-1 J P Supale Mechanical Engineering Department SKN SITS LONAVALA Strength of Materials

Deflection And Slope Deflection– Deflection at any point is vertical distance between the axis of beam before and after loading at that point. Denoted by ‘y’ or ‘ ‘ Always zero at supports. Slope– Slope at any point is defined as the angle in radian between the tangent to the elastic curve at that point and the original axis of beam. Denoted by ‘dy/dx’ or ‘θ’ Strength of Materials

Deflection And Slope y θ Strength of Materials

Boundary Conditions Simply Supported Beam At supports, the deflection is always zero. At supports, the slope is always maximum. Maximum defection occurs at point where slope is zero or vice versa. θmax y ymax y =0 y =0 θ = 0 Strength of Materials

Boundary Conditions Cantilever Beam At fixed support, the deflection is always zero. At fixed support, the slope is always zero. At free end, the deflection is maximum. At free end, the slope is maximum.

Flexural Rigidity Product of modulus of Elasticity , E and Moment of Inertia, I Its Unit is N.m2 Strength of Materials

Relation between Slope and BM Let L = Span of Beam M = Bending Moment R = Radius of curvature of beam after bending Y = Deflection of beam at centre θ = Slope of beam Strength of Materials

Relation between Slope and BM y D Strength of Materials

Relation between Slope and BM From the fig. AC x BC = CE x CD In practice, deflection of beam y is very small and square of y is negligible. 1 Strength of Materials

Relation between Slope and BM By using Bending Formula, Put value of R in equation 1,we get Strength of Materials

Relation between Slope and BM From the geometry of fig. Since Sin ΘA is very small quantity Sin ΘA = ΘA Measured in radian Strength of Materials

Differential equations of the deflection curve and y y y y y From the Engineering Beam Theory:

Differential equations of the deflection curve Bending Moment Flexural Stiffness Curvature Curvature Since, Slope Deflection Where C1 and C2 are found using the boundary conditions.

Equations of Elastic Curves Strength of Materials