ME 323: Mechanics of Materials Homework H12 Summer 2021 DUE: July 6, 2021 At a critical location along the length of a beam, the shear force V and bending M on the cross-section of the beam are known. On each of the four beam cross-sections shown below, point O is the centroid, point B is the top-most point and point K is the bottom-most point on the cross-section. The beam is made of a material having a specific weight of ρ (N /m

3 ). The dimension

b is in meters.

a) Determine the weight of the beam for each of the four cross-sections shown below. Leave these answers in terms of ρ and b. Rank order these weights from smallest to largest.

b) Determine the magnitude of the normal stress at points B, O and K for each cross-section. Leave these answers in terms of M and b. Rank order the maximum of these magnitudes for each point from smallest to largest.

c) Determine the magnitude of the shear stress at points B, O and K for each cross-section. Leave these answers in terms of V and b. Rank order the maximum of these magnitudes from smallest to largest. Rank order the maximum of these magnitudes for each point from smallest to largest.

d) How do the maximum shear stresses in c) compare with the maximum normal stresses in b) for each cross-section?

e) Comment on the cost (weight) vs. maximum stress of these beam cross-sections. This part of the problem will not be graded, but please put some thought into your comments.

z

y

K

O

B

z

y

K

O

B

z

y

K

O

B

z

y

K

O

B

thicknessesofweband

flanges=0.1b

b b b b

b / 2

b / 2

beamcrosssection(1) beamcrosssection(2) beamcrosssection(3) beamcrosssection(4)

x

y

10b

SOLUTION

cross section 1

w weight p Acobp

5 lob Sp B1 0 Cneutral plane

XD f 2543 24 3

Kidf iz 3 Is

B TIK o free edge2

To TA I1 e

C It 3 I 1

W A 2K c Egba t.scy tgF5 E b E FItIzb Esb

1

TEEE.EEa EE

a WE weight PACOb p b Cob to pb3b 2 0 neutral plane

1 2 44 7197 6 ITc B I 4 o C free edge

E Z IA Zz z from lectured

Cross Section 3

a WEweight PACOb p IT oD SETpb

3 7 Spb's

b 3 0 neutral plane

Tds 3 Yb

zj43

3 10.2 s

c IB z IDs o free edgedDz Z Z z z from lecture

Cross Section 4 1.91 z

a w weight pACiob7zpC2CbXo.iDti1b7Cb o.z DCoD

70

4 0 neutra p.caejPb32.8pb3KB.dKrkH MEw

I I t Iz t I3 1I z Co I b b 3 0.0043 bot

Iz Tz b IBP I b b 0 4 b to OSBJ0.020 bt

Iz Ize I 0.00431 276 020 b4 O 0449 b

4

and 3 get L kW ab 11.135c I B q a 4 0 free surface

E TtA y yIt

w A I b fo t b t b

yx0.1415 II A tA

A0 0462 0.2 b t Ib CO 4S b

O 145

0.379 b

I 0.0449 b4

t O lb

D C.it 379b vo I b 0 044964

11 8 IZ

I o Wz Wz 7 W 7 W4

IT Imax 7 It Imax Its Imax Kd maxTalman III Imax 431max 7 54 max

d The I beam is the most lightweight cross sectionHowever for the same outer dimensions it

does not perform well against the other crosssections in terms of stress The advantage of theI beam is that the outer dimensions can be

increased to reduce stress without adding much weight