Concept explainers
A compound beam ABCD has a cable with force P anchored at C The cable passes over a pulley at D, and force P acts in the —x direction, There is a moment release just left of B. Neglect the self-weight of the beam and cable. Cable force P = 450 N and dimension variable L = 0.25 m. The beam has a rectangular cross section (b = 20 mm, it = 50 mm).
(a) Calculate the maximum normal stresses and maximum in-plane shear stress on the bottom surface of the beam at support A.
(b) Repeat part (a) for a plane stress element located at mid-height of the beam at A.
(c) If the maximum tensile stress and maximum in-plane shear stress at point A are limited to 90 MPa and 42 MPa, respectively, what is the largest permissible value of the cable force P?
(a)
The maximum normal stresses and maximum in-plane shear stress on the bottom of the beam at fixed support A.
Answer to Problem 8.5.34P
Maximum normal stresses,
Maximum in-plane shear stress
Explanation of Solution
Given information: A compound beam ABCD has a cable having force P anchored at C as shown in the figure below:
Cable force
Dimension variable
The cross section of the beam
To calculate the maximum normal stress and in-plane shear stress first taken the cross-sectional properties of the beam.
Area of the rectangular beam
Moment of inertia of the rectangular beam is
Now reactions at fixed support A
Horizontal force
Shear force
And moment at point A,
Now, maximum normal stresses on the bottom of the beam at A is,
And at principle plane where there is maximum normal stress the shear stress is zero. So,
Principle stresses:
Maximum in-plane shear stress
(b)
Maximum stresses located at the mid-height of the beam at A.
Answer to Problem 8.5.34P
Maximum normal stresses,
Maximum in-plane shear stress
Explanation of Solution
Given information: A compound beam ABCD has a cable having force P anchored at C as shown in the figure below:
Cable force
Dimension variable
The cross section of the beam
Area of the rectangular beam
Moment of inertia of the rectangular beam is
Now reactions at fixed support A
Horizontal force
Shear force
The maximum normal stresses at the mid-point of the beam at A is,
And,
Principle stresses:
Maximum in-plane shear stress
(c)
The maximum permissible value of the cable force P.
Answer to Problem 8.5.34P
Maximum permissible value of P
Explanation of Solution
Given information: A compound beam ABCD has a cable having force P anchored at C as shown in the figure below:
Dimension variable
Maximum tensile stress at point A,
Maximum shear stress at point A,
The cross section of the beam
Area of the rectangular beam
Moment of inertia of the rectangular beam is
And moment at point A,
In the above figure the maximum cable force P is controlled by the tensile and maximum shear force at the bottom of the beam at point A.
Hence, tensile stress
Then,
By substituting the values we get,
And,
Then,
Conclusion:
Hence from the obtained values the maximum permissible value of the force P is
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Chapter 8 Solutions
Mechanics of Materials, SI Edition
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