Concept explainers
A foot bridge on a hiking trail is constructed using two timber logs each having a diameter d = 0.5 m (see figure a). The bridge is simply supported and has a length L = 4 m. The top of each log is trimmed to form the walking surface (see Fig, b)LA simplified model of the bridge is shown in Fig. g. Each log must carry its own weight w = 1.2 kN/m and the weight (P = 850 N) of a person at mid-span, (see Fig. b).
- Determine the maximum tensile and compressive stresses in the beam (Fig, b) due to bending.
(a)
The maximum tensile stress in beam.
The maximum compressive stress in beam.
Answer to Problem 5.5.28P
The maximum tensile stresses in beam are
The maximum compressive stresses in beam are
Explanation of Solution
Given information:
The diameter of the timber log is
The length of the bridge is
The weight of the log is
The following figure shows the force distribution on the wooden log.
Figure 1
Write the expression for the area of the circle.
Here, the diameter of the circle is
Write the expressions for the polar moment of inertia of the circle.
Here, the polar moment of inertia is
Write the expression for centroidal distance.
Here, the centroidal distance of the circle is
Write the expressions for radius of circle.
Here, the radius of circle is
Write the expressions for the area of the circular segment.
Here, the radius of the circle is
Write the expression for the centroidal distance of the segment.
Here, the centroidal distance of the segment is
Write the expression for moment of inertia of the segment about x axis.
Here, the moment of inertia about x axis is
Write the expression for moment of inertia about z axis.
Here, the moment of inertia about z axis is
Write the expression for the distance from the centroid to the bottom.
Here, the distance of centroid from the bottom is
Write the expression for the distance from the centroid to the top.
...... (X)
Here, the distance from centroid to the top is
Write the expression for moment of inertia of the wooden beam.
Here, the net moment of inertia of the wooden beam is
Write the expression for maximum moment.
Here, the distributed load is
Write the expression for maximum tensile stress in mid span.
Here, the maximum tensile stress is
Write the expression for maximum compressive stress in mid span.
Here, the maximum compressive stress is
Calculation:
Substitute
Substitute
Substitute
Substitute
Substitute
Substitute
Substitute
Substitute
Substitute
Substitute
Substitute
Substitute
Substitute
Conclusion:
The maximum tensile stress in beam are
The maximum compressive stress in beam are
(b)
The maximum permissible value of load
Answer to Problem 5.5.28P
The maximum permissible value of load
Explanation of Solution
Given Information:
The allowable normal tensile stress is
The allowable normal compressible stress is
Write the expression for the maximum tensile moment.
Here the maximum tensile moment is
Write the expression for the maximum compressible moment.
Here the maximum compressive moment is
Write the expression for the tensile force corresponding to the permissible stress.
...... (XVII)
Calculation:
Substitute
Substitute
Substitute
Conclusion:
The maximum permissible value of load
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Chapter 5 Solutions
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