Shear Stress in Beams 1

Shear Stress in Beams

(6.1-6.4)

MAE 314 – Solid Mechanics

Yun Jing

Shear Stress in Beams 2

Review Previous two chapters only dealt with normal stresses caused by bending moments. Chapter 6 deals with shear stress caused by shear forces.

Line of failure

Shear Stress in Beams 3

Shear Stress in Beams

Consider the effects of shear force (V). Already know how to find resulting axial force and moment due to stress σx from Chapter 4. We have two more equations for shear stress:

Total shear force in the y-direction: Total shear force in the z-direction:

VdAxy

0dAxz

Shear Stress in Beams 4

Shear Stress in Beams Consider a cantilever beam composed of separate planks clamped at one end:

Shear force causes tendency to “slide.” Stresses are equal in horizontal andvertical directions.

Pure bending Shear force

Shear Stress in Beams 5

Shear Stress: Horizontal Let us consider the horizontal component (τyx = τxy). Cut a section with cross-sectionalarea a at a distance y1 above thecentroid.

FBD →

Shear Stress in Beams 6

Shear Stress: Horizontal ΔH is the horizontal shearing force. Element width is Δx. Sum forces in x-direction: Recall from chapter 4: Solve for ΔH and use equation for σ:

0 a

DCx dAHF

I

My

a

CD

a

CD

a

CD ydAI

MMydA

I

MMdAH

Shear Stress in Beams 7

Shear Stress: Horizontal Recall first moment, Q, is defined as: The term MD-MC can be rewritten as:

Applying this to our equation for ΔH: We can rearrange this to define horizontal shear per unit length, q, called shear flow.

a

ydaQ

xVxdx

dMMMM CD

xI

VQH

I

VQ

x

Hq

Shear Stress in Beams 8

Side Note on Q

Q is the definition of the first moment for the area above y1 with respect to the x-axis (see Appendix A in textbook),

where y bar is the distance between the centroid of the shaded section and the centroid of beam cross-section.yaydAQ

a

Shear Stress in Beams 9

Example Problem A beam is made of three planks, 20 by 100 mm in cross-section, nailed together. Knowing that the spacing between nails is 25mm and that the vertical shear in the beam is V = 500 N, determine the shearing force in each nail.

Shear Stress in Beams 10

Shear Stress: Vertical Now, let us consider the vertical component (τxy= τyx).

We can calculate the average vertical shear stress on the cross-section.AVEAVE It

VQ

xtx

I

VQ

A

H

1

It

VQAVE

Shear Stress in Beams 11

Shear Stress: Vertical So, where is τAVE maximum and minimum?

Use Q to find out. Q = 0 at top and bottom surfaces Q = maximum somewhere in between

normal stress = 0max shear stress

max normal stress

max normal stress

shear stress = 0

shear stress = 0

Shear Stress in Beams 12

Shearing Stress in Common Shapes Rectangular cross-section

22

2

1

2

1ycbycycbyAQ

22

3

22

3 4

3

212/2yc

bc

V

b

ycb

cb

V

Ib

VQxy

2

2

12

3

c

y

A

Vxy

A

V

2

3max

Shear Stress in Beams 13

Shearing Stress in Common Shapes Beams with flanges

Vertical shear stresses are larger in the web than in the flange. Usually only calculate the values in the web. Ignore the effects of the small fillets at the corners. Flanges have large horizontal shear stresses, which we will learn how to calculate later on.

Flange

Web

webA

Vmax

Shear Stress in Beams 14

Example ProblemFor the beam and loading shown, consider section n-n and determinethe shearing stress at (a) point a, (b) point b.

Shear Stress in Beams 15

Shear Stress in Thin Walled Members (6.7)

MAE 314 – Solid Mechanics

Yun Jing

May want to calculate horizontal or vertical shear stress, depending on the point of interest. Vertical cut: τavg = average τxz Horizontal cut: τavg = average τxy

Shear Stress in Beams 16

Shear in Thin Walled Members

Why do we choose to “cut” the beam perpendicular to the cross-section wall? Want to cut across line of shear flow.

Shear Stress in Beams 17

Shear in Thin Walled Members

Shear flow in box-beam section. Shear flow in wide-flange beam section.

Shear Stress in Beams 18

Example Problem Knowing that the vertical shear is 50 kips in a W10*68 rolled-steel

beam, determine the horizontal shearing stress in the top flange at point a

Shear Stress in Beams 19

Example ProblemThe built-up beam shown is made by gluing together two 20 x 250 mm

plywood strips and two 50 x 100 mm planks. Knowing that the allowable

average shearing stress in the glued joints is 350 kPa, determine the

largest permissible vertical shear in the beam.

Shear Stress in Beams 20

Example ProblemAn extruded aluminum beam has the cross section shown. Knowing

that

the vertical shear in the beam is 150 kN, determine the shearing stress

a (a) point a, and (b) point b.