A cantilever beam member made from steel with hollow circular cross section experiences an axial load of 300 KN (F1), and a vertical load of 200 kN (F2) as shown in the following figure.
The outside diameter of the beam is 50mm and the uniform wall thickness of the cross section is 4mm. Beam is made out of AISI 1040 Cold-drawn steel. The beam is fixed on one end and the beam has a length of 300mm
For the given system:
a. Draw the Free-Body-Diagram of the beam.
b. Compute the reaction forces and bending moment on the beam.
c. Compute the maximum normal stress, minimum normal stress and maximum shear stress.
d. Compute the maximum axial deformation on this member (x).
e. Compute the maximum vertical deflection of the tip of the beam due to the bending moment (y).
f. Will the member experience permanent deformation or complete failure due to this applied load? Draw your conclusions based on the strength values of the beam material.
First, we do not see the loading diagram, but that does not stop me from giving a response.
Also, for "f", you will need the properties of AISI 1040 cold drawn steel (yield stress).
A cantilever is a statically determinate structure, so finding bending moments is a matter of taking moments about the appropriate points.
Deflections and rotations are a matter of getting the correct formulae.
The book Strength of Materials by den Hartog of MIT gave the rotations and deflections of the free end of a cantilever (usual E, I, L, P, w for properties and loading):
loading rotation deflection
moment M ML/(EI) ML²/(2EI)
P(point load) PL²/(2EI) PL³/(3EI)
w,udl wL³/(6EI) wl4/(8EI)
It's easy to memorize by the number
1-2-2-3-6-8.
axial stress=P/A
bending stress=My/I
where y is the maximum distance from the neutral axis of the cross section.
Take care that axial force will change the location of the neutral axis even for a symmetrical cross section.
Hope you can post or describe the loading, and post your work up to this point.
a. To draw the Free-Body-Diagram (FBD) of the beam, we need to represent all the forces and moments acting on the beam. In this case, we have an axial load of 300 kN (F1) and a vertical load of 200 kN (F2) acting on the beam.
The FBD of the beam will include the following:
1. An arrow representing the axial load (F1) acting vertically downward at the fixed end of the beam.
2. An arrow representing the vertical load (F2) acting vertically downward at the free end of the beam.
3. Reaction forces (R1 and R2) at the fixed end and the free end respectively, perpendicular to the length of the beam.
4. Bending moment (M) acting in the horizontal direction at the free end of the beam.
b. To compute the reaction forces and bending moment on the beam, we need to carry out a force and moment balance.
For the force balance:
ΣFy = 0 => R1 + F2 - F1 = 0 => R1 = F1 - F2
For the moment balance:
ΣMf = 0 => -M + (F1 x 300) - (F2 x 300) = 0
Simplifying the equation gives us:
M = 300(F1 - F2) N.mm
c. To compute the maximum normal stress, minimum normal stress, and maximum shear stress, we need to consider the cross-sectional properties of the beam.
The cross-sectional area (A) can be calculated using the formula:
A = π[(D/2)^2 - (D/2 - t)^2] = π(Dt - t^2)
where D is the outside diameter of the beam and t is the wall thickness.
The maximum axial stress (σ_max) can be calculated using the formula:
σ_max = F1 / A
The minimum axial stress (σ_min) can be calculated using the formula:
σ_min = F2 / A
The maximum shear stress (τ) can be calculated using the formula:
τ = 4V / (πD^2)
where V is the total vertical load (F2).
d. To compute the maximum axial deformation on this member (x), we can use the formula:
x = F1 x L / (E x A)
where L is the length of the beam and E is the elastic modulus of AISI 1040 Cold-drawn steel.
e. To compute the maximum vertical deflection of the tip of the beam due to the bending moment (y), we can use the formula:
y = (M x L^2) / (2 x E x I)
where I is the moment of inertia of the beam's cross-sectional area.
f. To determine whether the member will experience permanent deformation or complete failure, we need to compare the computed maximum normal stress with the yield strength of AISI 1040 Cold-drawn steel. If the maximum normal stress is higher than the yield strength, the member may undergo permanent deformation or failure.