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UNSYMMETRIC BENDING

Problem 7-124. (Hibbeler)

A beam with the cross section shown in Fig. 7-124 is subjected to a moment M, which has a

magnitude of 20 kN*m. The resistng moment Mr on the cross section is in the direction shown

on the figure. Determine

a. The flexural stress at point A.

b. The orientation of the neutral axis (show its location on a sketch of the cross

section).

c. The maximum tensile and compressive flexural stresses in the beam.

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Solution to Problem 7-124:

a. Flexural stress at point A.

PLAN1:

>> Compute for Iz, Iy and Izy. >> Determine coordinates (z,y) of point A.

>> Compute for Mz and My.

>> Use the formula established in class.

PLAN2:

>> Compute for Iz, Iy and Izy.

>> Determine coordinates (z,y) of point A.

>> Find the Principal Axes of the Area Moment.

>> Compute for the Iu and Iv

>> Transform (z,y) to (u,v)

>> Use the formula established in class.

PLAN3:

>> Compute for Iz, Iy and Izy.

>> Determine coordinates (z,y) of point A.

>> Locate the orientation of the Neutral axis.

>> Tranform to compute I about the N.A.

>> Transform (z,y) to (z*, y*) relative to the NA

>> Compute for the component of the Resisting Moment about the N.A.

>> Use the formula established in class.

Execution of Plan 1:

>>Computation for the area moments of inertia and the Product of Inertia:

Iz1 80 20

3

1280 20 82.4( )

2

Iy1 20 80

3

1220 80 62.94( )

2

Izy1 20 80 62.94( ) 82.4( )

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Iz2 20 120

3

1220 120 12.4( )

Iy2 120 20

3

1220 120 7.06( )

Izy2 20 120 12.4( ) 7.06( )

Iz3 140 20

3

12140 20 57.6( )

2

Iy3 20 140

3

12140 20 7.06( )

2

Izy3 140 20 7.06( ) 57.6( )

Iz Iz1 Iz2 Iz3 2.355 107

Iy Iy1 Iy2 Iy3 1.21 107

Izy Izy1 Izy2 Izy3 9.227 106

>> Determine coordinates of point A:

z 62.94

y 72.4

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>> Moment Components:

Mz 20000

My 0

>> Apply Formula:

a Mz Iy My Izy

Iy Iz Izy2

y My Iz Mz Izy

Iy Iz Izy2

z

10003

2.958 107

Pa

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Execution of PLAN2:

>> Compute for Iz, Iy and Izy.

Iz 2.355 107

Iy 1.21 107

Izy 9.227 106

>> Determine coordinates (z,y) of point A.

z 62.94

y 72.4

>> Find the Principal Axes of the Area Moment.

C Iz Iy

21.783 10

7

R Iz Iy( )

22 Izy( )

2

21.086 10

7

p atan Izy

Iz C

90

29.096

>> Compute for the Iu and Iv

Iu C R 2.868 107

Iv C R 6.97 106

>> Transform (z,y) to (u,v)

u z cos p

180

y sin p

180

19.791

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v y cos p

180

z sin p

180

93.87

Mu My sin p

180

Mz cos p

180

1.748 104

Mv My cos p

180

Mz sin p

180

9.726 103

>> Use the formula established in class.

a Mu v

Iu

Mv u

Iv

10003

2.958 107

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Execution of PLAN3:

>> Compute for Iz, Iy and Izy.

Iz 2.355 107

Iy 1.21 107

Izy 9.227 106

>> Determine coordinates (z,y) of point A.

z 62.94

y 72.4

>> Locate the orientation of the Neutral axis.

1 atan My Iz Mz Izy

Mz Iy My Izy

180

37.317 2 atan

Mv Iu

Mu Iv

180

66.413

2 p 37.317

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>> Tranform to compute I about the N.A.

Ina1 C R cos

180

180 1 2 p 2( )[ ]

1.045 107

Ina2 C R cos

180180 2 2( )

1.045 107

>> Compute for the component of the Resist ing Moment about the N.A.

Mna My sin 1

180

Mz cos 1

180

1.591 104

>> Transform (z,y) to (z*, y*) relative to the NA

zna z cos 1 180

y sin 1 180

93.946

yna y cos 1

180

z sin 1

180

19.424

>> Use the formula established in class.

a Mna yna

Ina11000

3 2.958 10

7

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