Determine the minimum dimension a to the nearest mm of the beam’s cross section to safely support the load. The wood has an allowable normal stress of σallow = 10 MPa and an allowable shear stress of τallow = 1 MPa.
Find the minimum dimension a to the nearest beam cross section.
Answer to Problem 1FP
The minimum dimension a to the nearest beam cross section is
Explanation of Solution
Given information:
The allowable bending stress is 10 MPa.
The allowable shear stress is 1 MPa.
The width of the cross section is a.
The depth of the cross section is 2a.
Calculation:
Sketch the free body diagram of beam as shown in Figure 1.
Consider a section,
Apply the equilibrium condition along y –direction.
Determine the bending moment at point A.
Calculate the moment of inertia (I) using the relation.
Here, b is the width of the section and d is the depth of the section.
Substitute a for b and 2a for d.
Calculate the dimension (a) value using the relation:
Here,
Substitute
Use,
Hence, the minimum dimension a to the nearest beam cross section is
Find the value of I using the relation:
Substitute 140 mm for a.
Find the value of
Here, A is the area of the cross section and y is the centroid of area.
Substitute
Check:
Calculate the
Here
Substitute
Hence it is OK.
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