A 3-point bending test is performed on a simply supported beam of length L=4m and of rectangular cross-section, with width b and height h.
For a central load of 10kN, draw the bending moment and shear force diagrams for the beam. Then answer the following questions about the shear force V and bending moment M.
What is the shear force in the beam at x=0.5,1.5,2.5 and 3.5m?
V (kN, x=0.5):
V (kN, x=1.5):
V (kN, x=2.5):
V (kN, x=3.5):
What is the moment in the beam at x=0,1,2,3 and 4m?
M (kNm, x=0):
M (kNm, x=1):
M (kNm, x=2):
M (kNm, x=3):
M (kNm, x=4):
At what length along the beam do the maximum stresses occur? Express your answer in terms of the beam length L.
Location of maximum stresses:
Now assume that the beam has a width b=10cm and a height h=20cm. Calculate the maximum stress in the beam.
σmax (in MPa):
0,0,0
To answer these questions, we need to understand the concepts of shear force and bending moment.
Shear force refers to the force that acts parallel to the cross-section of the beam, causing one part of the beam to slide or shear past another. Bending moment, on the other hand, refers to the tendency of a force to rotate or bend the beam. It is the moment created by the external loads acting on the beam.
To determine the shear force and bending moment at different points along the beam, we can use the equations of equilibrium. For a simply supported beam subjected to a central load, the shear force at any position x (measured from one end of the beam) can be calculated as:
V = Load / 2
And the bending moment M at any position x can be calculated as:
M = Load * (L/2 - x)
where Load is the applied load, L is the length of the beam, and x is the distance from one end of the beam.
Now let's calculate the shear forces and bending moments at the given positions.
For V (kN, x=0.5):
V = Load / 2 = 10kN / 2 = 5kN
For V (kN, x=1.5):
V = Load / 2 = 10kN / 2 = 5kN
For V (kN, x=2.5):
V = Load / 2 = 10kN / 2 = 5kN
For V (kN, x=3.5):
V = Load / 2 = 10kN / 2 = 5kN
Now let's calculate the bending moments.
For M (kNm, x=0):
M = 10kN * (4m/2 - 0) = 10kN * 2m = 20kNm
For M (kNm, x=1):
M = 10kN * (4m/2 - 1m) = 10kN * 1m = 10kNm
For M (kNm, x=2):
M = 10kN * (4m/2 - 2m) = 10kN * 0m = 0kNm
For M (kNm, x=3):
M = 10kN * (4m/2 - 3m) = 10kN * (-1m) = -10kNm
For M (kNm, x=4):
M = 10kN * (4m/2 - 4m) = 10kN * (-2m) = -20kNm
The shear force at each position remains constant at 5kN while the bending moment changes at different positions from 20kNm to -20kNm.
To find the location of maximum stresses, we need to calculate the maximum bending moment. In this case, the maximum bending moment occurs at the center of the beam, which is at x = L/2 = 4m/2 = 2m.
Now, let's move on to the second part of the question, where we need to calculate the maximum stress.
To calculate the maximum stress, we will use the bending moment and the properties of the beam. Given that the beam has a width b=10cm and a height h=20cm, we can calculate the moment of inertia I of the beam using the equation:
I = (1/12) * b * h^3
Substituting the values, we get:
I = (1/12) * 0.1m * (0.2m)^3 = 0.001333 m^4
Now, the maximum stress can be calculated using the formula:
σmax = M * y / I
where M is the maximum bending moment, y is the distance from the neutral axis to the outer fiber, and I is the moment of inertia.
Assuming the neutral axis is at the center of the beam, the distance from the neutral axis to the outer fiber is half the height, which is 0.1m.
Substituting the values, we get:
σmax = M * y / I = 20kNm * 0.1m / 0.001333 m^4 = 1.5 MPa
Therefore, the maximum stress in the beam is 1.5 MPa.