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1、Lesson 4 Deflections of beams A bar that is subjected to forces acting transverse to its axis is called a beam.The beam in Fig.1,with a pin support at one end and a roller support at the other,is called a simply supported beam,or a simple beam.The essential feature of a simple beam is that both ends
2、 of the beam may rotate freely during bending,but they cannot translate in the lateral direction.Also,one end of the beam can move freely in the axial direction.The beam,which is built-in or fixed at one end and free at the other end,is called a cantilever beam.At the fixed support the beam can neit
3、her rotate nor translate,while at the free end it may do both.Loads on a beam may be classified into concentrated force,such as P in Fig.1,or distributed loads which is expressed in units of force per unit distance along the axis of the beam.The axial force N acting normal to the cross section and p
4、assing through the centroid of the cross section,shear force V acting parallel to the cross section,and bending moment M acting tin the plane of the beam are known as stress resultants.The relationship between the shear force V,bending moment M,and the loads on a beam is given by.This equation shows
5、 that t the rate of change of the bending moment is equal to the algebraic value of the shear force,provided that a distributed load acts on the beam.If the beam is acted upon by a concentrated force,however,there will be an abrupt change,or discontinuity,in the shear force at the point of applicati
6、on of the concentrated force.Lateral loads acting on a beam will cause the beam to deflect.As shown in Fig.1,before the load P is applied,the longitudinal axis of the beam is straight.After bending,the axis of the beam becomes a curve,as represented by the line ACB.let us assume that the xy plane is
7、 a plane of symmetry of the beam and that all loads act in this plane.Then the curve ACB,called the deflection cure of the beam,will lie in this plane also.From the geometry of the figure we see that K.where K is the curvature,equal to the reciprocal of the radius of curvature.Thus,the curvature K i
8、s equal to the rate of change of the angle with respect to the distance S measured along the deflection curve.The basic differential equation for the deflection curve of a beam is given as follows,where V is the deflection of the beam from its initial position.It must be integrated in each particula
9、r case to find the deflection V.The procedure consists of successive integration of the equations,with the resulting constants of integration being evaluated from the boundary conditions of the beam.It should be realized that Eq(3)is valid only when Hooke s law applies for the material and when the
10、slopes of the deflection curve are very small.Another method for finding deflections of beams is the moment-area method.The name of this method comes from the fact that it utilizes the area of the bending moment diagram.This method is especially suitable when it is desired to find the deflection or
11、slope at only one point of the beam rather than finding the complete equation of the deflection curve.The normal and shear stresses acting at any point in the cross section of a beam can be obtained by using the equations in which I is the second moment of the cross=sectional area with respect to th
12、e neutral axis,and Q is the first moment of the plane area of a beam.It can be seen that the normal stress is a maximum at the outer edges of the beam and is zero at neutral;the shear stress is zero at the outer edges and usually reaches a maximum at the neutral axis.The shear force V and bending mo
13、ment M in a beam will usually vary with the distance x defining the location of the cross section at which they occur.when designing a beam,it is desirable to know the values of V and M at all cross sections of the beam,and a convenient way to provide this information is by a graph showing how they
14、vary along the axis of the beam.To plot the graph,we take the abscissa as the position of the cross section,and we take the ordinate as the corresponding value of either the shear force or the bending moment.Such graphs are called shear force and bending moment diagrams.The simple beam shown in Fig.
15、1 is one of the statically determinate beams.The feature of this type of beams is that all their reactions can be determined from equations of static equilibrium.The beams that have a large number of reactions than the number of equations of static equilibrium are said to be statically indeterminate
16、.For statically determinate beams we could immediately obtain the reactions of the beam by solving equations of static equilibrium.however,when the beam is statically indeterminate,we cannot solve for the forces on the basis of statics alone.Instead,we must take into account the deflections of the beam and obtain equations of compatibility to supplement the equations of statics.