Welded Joints in BendingWelded Joints in BendingWelded Joints in BendingWelded Joints in Bending

Overview

• Stressing of welded joints:– Bending analysis of welds– Stress Concentration– Fatigue of welded joints

Bending Analysis of Welds

• Bending analysis of welded structures follows on closely from analysis of torsional loading:– breaking the applied loads down into direct

(primary) loads (tension and/or shear loads) and (secondary) bending moment

– analysing the primary stresses due to the direct loads as force/area

– analysing the secondary stresses due to the bending moment, unit second moment of area Iu

Stressing of welds in combined bending/shearing

• For a cantilever with fillet welds along its top and bottom faces:

F

l

Stressing of welds in combined bending/shearing

• Replace applied load F with V and M:

V M

Stressing of welds in combined bending/shearing

• Vertical reaction is taken by primary shear stress:

• where A is the total throat area, in this case:

A

F

A

V'

hlhlA 414.1707.02

Stressing of welds in combined bending/shearing

• The moment M produces bending stresses in the welds

• It is usual to assume that this stress acts normal to the throat area

• True depth of the weld is usually small compared to other dimensions

• By treating the welds as lines we can use the unit second moment of area for bending

Stressing of welds in combined bending/shearing

• In this case:

d

b0.707h

2

2d

y

bx

2

2bdIu

hbA 414.1

Stressing of welds in combined bending/shearing

• Unit 2nd moment of area about horizontal axis is:

• Second moment of area is:

• Normal stress (at a distance y from the neutral axis) is:

2

2bdIu

I

Ty

uhII 707.0

Stressing of welds in combined bending/shearing

If there is no shear loading:• Assume that the maximum shear

stress in the weld is equal to the nominal tensile (or compressive) stress we have calculated based on the throat area

• Assess the strength of the weld by comparing this nominal shear stress with the allowable shear stress in the material.

Combining the shear and bending stresses

• Shigley et al use vectorial combination of stresses

• A better approach is to use Mohr’s circle (for 2 or 3 dimensional stresses)

+

2

Example

• Estimate the safety factor in the bracket if the maximum allowable stress is 120 MPa

120

120

F = 7.5 kN

6

6

6

60

Example

• Use the relationships from row 5 of the table in appendix A

60

dbhA 2707.0

db

dy

bx

2

22

223

223

2ydbyd

dIu

120

Example: Primary Stress• The primary stress is F/A (as always!)

23

3

m10273.1

12.0206.0106707.0

2707.0

dbhA

120

120

F = 7.5 kN

MPa89.510273.1

105.7'

3

3

A

V

Example: Secondary Stress

• The vertical centroid distance is:

• The unit second moment of area is:

mm48)12.0(206.0

12.0

2

22

db

dy

36

23323

223

m108.460

1048))12.0(206.0(1048)12.0(23

)12.0(2

223

2

ydbydd

Iu

Example: Secondary Stress

• The maximal secondary stress occurs furthest from the neutral axis (maximum y, AKA c = 72 mm)

MPa16.33

10954.1

1072105.76

33

XXI

My

46

63

m10954.1

108.460106707.0707.0

uhII

12072

Example: total stress• Combine the primary and secondary

stresses using Mohr’s circle:

+

2 = 19.6°

(33.16, 5.89)

(0, 5.89)

34.175 MPa

Example: Secondary Stress

• The maximal stress is 34.175 MPa, 9.8° off the horizontal and acting on the toe of the vertical weld

• This represents a safety factor (under static loading of a ductile material) of

5.3175.34

120n

Stress Concentrations

• For elastic materials nominal stress is

• Saint-Venant’s Principle says this is so beyond a characteristic length (b) from a stress raiser

F

F

bt

Fnom

t

b

Stress Concentrations• Maximum stresses

may be much larger than the nominal

• Stress concentration factor is:

F

nom

max

K

t

b

F

Stress Concentration

• Stress concentration factors are found empirically (look for them in tables)

• Stress concentrations are geometry and surface finish dependent.

stress concentration in a flat bar ofreducing section – note the blend radii

Stress Concentration• In ductile materials

stress concentrations are usually ignored due to material flow

• Stress concentrations must be accounted for in designs involving:– brittle materials (which

are very sensitive)– Fatigue loading– Impact loading

Photoelastic and FEA determination of stress concentrations in the flat bar of reducing section

Stress concentrations in welds

• Stress concentration is 1.2 on a reinforced butt weld

• Reduces to 1 if weld is “dressed”

Stress concentrations in welds

• Stress concentration at the end of a parallel fillet weld is 2.7

Stress concentrations in welds

• Fatigue Stress concentration factors (Kfs) for weld and parent metal:

• Reinforced butt weld 1.2• Toe of transverse fillet 1.5• End of parallel fillet 2.7• T weld with sharp corners 2.0

Fatigue Loading• Many materials

exhibit a fatigue limit; below this limit fatigue failure is unlikely.

• For steel the fatigue limit is around 50% of the UTS for static loading

• For aluminium it’s poorly defined but around 25% of UTS

Fatigue limit is shown on the S-N (endurance) diagram

Fa

ilure

str

ess

Number of cycles to failure

Fatigue limit

Welding & Fatigue Resources

• http://www.roymech.co.uk/Useful_Tables/Fatigue/Stress_concentration.html– Tables of stress concentration factors + design guide

• http://www.gowelding.com/– all manner of information

• Shigley, J.E., Mischke, C.R., Budynas, R.G. 2004. Mechanical Engineering Design (7th international edition), McGraw Hill.

• Gere, J.M. and Timoshenko, S.P., 1997. Mechanics of Materials (4th edition), PWS Publishing, Boston.