Question

The beam has the rectangular cross section shown. If w 1 kN/m, determine the maximum bending stress in the beam. Sketch the stress distribution acting over the cross section. 

Answer 1

Procedure for Analysis
The following procedure provides a method for constructing the shear and moment diagrams for a beam based on the relations among distributed load, shear, and moment.

Support Reactions.

• Determine the support reactions and resolve the forces acting on the beam into components that are perpendicular and parallel to the beam’s axis.

Shear Diagram.

• Establish the V and x axes and plot the known values of the shear at the two ends of the beam.

• Notice how the values of the distributed load vary along the beam, and realize that each of these values indicates the way the shear diagram will slope ( dV>dx = w ). Here w is positive when it acts upward.

• If a numerical value of the shear is to be determined at a point, one can find this value either by using

(1)method of sections

(2)equation of force equilibrium,

or by using !V = 1w(x) dx, which states that the change in the shear between any two points is equal to the area under the load diagram between the two points.

Moment Diagram.

• Establish the M and x axes and plot the known values of the moment at the ends of the beam.

• Notice how the values of the shear diagram vary along the beam, and realize that each of these values indicates the way the moment diagram will slope ( dM>dx = V ).

• At the point where the shear is zero, dM>dx = 0, and therefore this would be a point of maximum or minimum moment.

• If a numerical value of the moment is to be determined at the point, one can find this value either by using

the method of sections and

the equation of moment equilibrium,

or by using !M = 1V(x) dx, which states that the change in moment between any two points is equal to the area under the shear diagram between the two points.

• Since w(x) must be integrated to obtain !V, and V(x) is integrated to obtain M(x), then if w(x) is a curve of degree n, V(x) will be a curve of degree n + 1 and M(x) will be a curve of degree n + 2 .

For example, if w(x) is uniform, V(x) will be linear and M(x) will be parabolic. this is the case of our questions also.  
Faee Bocdy Diagram 1(6)KN mの 50 mm Av Equation of Equillibium ( ) + Sign converg! 'S HEAR MOMENT DIAGRAM:- 2 CRN-