Timothy A. Philpot - Mechanics of materials _ an integrated learning system-John Wiley (2017)

The controlling section modulus is the smaller of these two values; therefore, the
section modulus for the inverted tee cross section is
S = 70,125 mm 3
Ans.
Why is the smaller section modulus said to control in this context? The maximum
bending stress is calculated with the use of the section modulus from the following
form of the flexure formula [see Equation (8.10)]:
σ max =
The section modulus S appears in the denominator of this formula; consequently,
there is an inverse relationship between the section modulus and the bending stress.
The smaller value of S corresponds to the larger bending stress.
(b) Since the centroid location and the moment of inertia about the centroidal axis have
been determined, the flexure formula [Equation (8.7)] can now be used to determine
the bending stress at any coordinate location y. (Recall that the y coordinate axis has
its origin at the centroid.) Point H is located at y = −30 mm; therefore, the bending
stress at H is given by
My (5 kN⋅m)( −30mm)(1,000 N/kN)(1, 000 mm/m)
σ x =− =−
4
I
8,415,000 mm
z
= 17.83 MPa = 17.83 MPa(T)
M
S
Point K is located at y = +80 mm; hence, the bending stress at K is calculated as
My (5 kN⋅m)(80mm)(1,000 N/kN)(1, 000 mm/m)
σ x =− =−
4
I
8,415,000 mm
z
=− 47.5 MPa = 47.5 MPa(C)
Ans.
Ans.
(c) Regardless of the particular cross-sectional geometry, the largest bending stress in any
beam will occur at either the top surface or the bottom surface of the beam. If the cross
section is not symmetric about the axis of bending, then the largest bending stress (for
any given moment M) will occur at the location farthest from the neutral axis—in other
words, at the point that has the largest y coordinate. For the inverted-tee cross section,
the largest bending stress will occur at the upper surface:
My (5 kN⋅m)(120 mm)(1, 000 N/kN)(1, 000 mm/m)
σ x =− =−
4
I
8,415,000 mm
z
=− 71.3 MPa = 71.3 MPa(C)
Ans.
Alternatively, the section modulus S could be used in Equation (8.10) to determine
the magnitude of the maximum bending stress:
M (5 kN⋅m)(1,000 N/kN)(1, 000 mm/m)
σ max = =
3
S
70,125 mm
= 71.3 MPa = 71.3 MPa(C) by inspection
If Equation (8.10) is used to calculate the maximum bending stress, the sense of the
stress (either tension or compression) must be determined by inspection.
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