Special Members –Beams with holes and notches

Henrik Danielsson

Division of Structural Mechanics, Lund University, Sweden

Special Members – Beams with holes and notches slide 2

Special Members – Beams with holes and notches

• Wood is a strongly orthotropic material

- very low strength and stiffness perpendicular to grain

• Relatively common cause of damage for timber structures

• Failure is complicated to predict

Perpendicular to grain fracture

Special Members – Beams with holes and notches slide 3

Examples

Västerås (with Permission from Martinssons Trä AB)

Special Members – Beams with holes and notches slide 4

Outline

Introduction• Perp-to-grain tension and fracture• Stress state at holes and notches• Tests

Models for strength analysis• Conventional stress-strength analysis• Weibull weakest link theory• Linear elastic fracture mechanics

Applied strength analysis• End-notched beams• Beams with a hole

Reinforcement

Special Members – Beams with holes and notches slide 5

Causes of perp-to-grain tension and fracture

Geometry

Joints

Heterogeneities(knots, growth rings)

Eigenstresses(drying/swelling)

[Illustrations: PJ Gustafsson]

Special Members – Beams with holes and notches slide 6

Perp-to-grain tensile strength

The perp-to-grain tensile strength is affected by

• Species and density

• Moisture content (strength decreases with MC)

• Temperature (strength decreases with T)

• Duration of load

• Growth ring orientation (radial strength > tangential strength)

• Volume and shape of loaded specimen

• Multi-axial stress state

Special Members – Beams with holes and notches slide 7

Perp-to-grain tensile strength

Influence of volume of loaded specimen on tensile strength

Volume(mean)

Strength

0.005 dm3 4.0 MPa

5 dm3 1.0 MPa

200 dm3 0.7 MPa

ft90 ~ V −0.2

[Thelandersson, Larsen: Timber Engineering]

Special Members – Beams with holes and notches slide 8

Perp-to-grain tensile strength

Influence of multi-axial stress state

[Steiger and Gehri, 2011]

Special Members – Beams with holes and notches slide 9

Stress State – Notched beam

In Y-direction

8.1

In Y-directionLC 1: Shear forceGlobal Deformations u

Factor of deformations: 20.00Max u: 8.1, Min u: 0.0 mm

In Y-directionLC 1: Shear forceSurfaces Stresses Sigma-y,+

Max Sigma-y,+: 5.60, Min Sigma-y,+: -0.10 MPa

In Y-directionLC 1: Shear forceSurfaces Stresses Tau-xy,+

Max Tau-xy,+: 6.12, Min Tau-xy,+: -1.06 MPa

Z

XY

Axial Stresses

y,+ [MPa]

8.0

1.8

1.6

1.4

1.2

1.0

0.8

0.6

0.4

0.2

0.0

-8.0

Max : 8.0Min : -8.0

In Y-directionLC 1: Shear forceStresses Sigma-y,+

Max Sigma-y,+: 8.0, Min Sigma-y,+: -8.0 MPa

1.8

1.6

1.4

1.2

1.0

0.8

0.6

0.4

0.2

0.0

90

Z

XY

Axial Stresses

y,+ [MPa]

8.0

1.8

1.6

1.4

1.2

1.0

0.8

0.6

0.4

0.2

0.0

-8.0

Max : 8.0Min : -8.0

In Y-directionLC 1: Shear forceStresses Sigma-y,+

Max Sigma-y,+: 8.0, Min Sigma-y,+: -8.0 MPa

2.7

2.4

2.1

1.8

1.5

1.2

0.9

0.6

0.3

0.0

t

Special Members – Beams with holes and notches slide 10

Stress State – Beam with a hole – V and M

Z

XY

In Y-directionLC 1: Shear force

X

41.1

Z

Y

In Y-directionLC 1: Shear forceGlobal Deformations u

Factor of deformations: 11.50Max u: 41.1, Min u: 0.0 mm

Special Members – Beams with holes and notches slide 11

Stress State – Beam with a hole – V and M

Z

XY

Axial Stresses

y,+ [MPa]

8.0

1.8

1.6

1.4

1.2

1.0

0.8

0.6

0.4

0.2

0.0

-8.0

Max : 8.0Min : -8.0

In Y-directionLC 1: Shear forceStresses Sigma-y,+

Max Sigma-y,+: 8.0, Min Sigma-y,+: -8.0 MPa

Z

XY

Axial Stresses

txy,+ [MPa]

9.1

2.7

2.4

2.1

1.8

1.5

1.2

0.9

0.6

0.3

0.0

-1.7

Max : 9.1Min : -1.7

In Y-directionLC 1: Shear forceStresses Tau-xy,+

Max Tau-xy,+: 9.1, Min Tau-xy,+: -1.7 MPa

Z

XY

Axial Stresses

y,+ [MPa]

8.0

1.8

1.6

1.4

1.2

1.0

0.8

0.6

0.4

0.2

0.0

-8.0

Max : 8.0Min : -8.0

In Y-directionLC 1: Shear forceStresses Sigma-y,+

Max Sigma-y,+: 8.0, Min Sigma-y,+: -8.0 MPa

1.8

1.6

1.4

1.2

1.0

0.8

0.6

0.4

0.2

0.0

90

Z

XY

Axial Stresses

y,+ [MPa]

8.0

1.8

1.6

1.4

1.2

1.0

0.8

0.6

0.4

0.2

0.0

-8.0

Max : 8.0Min : -8.0

In Y-directionLC 1: Shear forceStresses Sigma-y,+

Max Sigma-y,+: 8.0, Min Sigma-y,+: -8.0 MPa

2.7

2.4

2.1

1.8

1.5

1.2

0.9

0.6

0.3

0.0

t

Special Members – Beams with holes and notches slide 12

Stress State – Beam with a hole – pure M

In Y-directionLC 2: Bending moment

1.7

Global Deformations

|u| [mm]

1.7

1.5

1.4

1.2

1.1

0.9

0.8

0.6

0.5

0.3

0.2

0.0

Max : 1.7Min : 0.0

In Y-directionLC 2: Bending momentGlobal Deformations u

Factor of deformations: 67.50Max u: 1.7, Min u: 0.0 mm

Special Members – Beams with holes and notches slide 13

Stress State – Beam with a hole – pure M

Z

XY

Axial Stresses

y,+ [MPa]

8.0

1.8

1.6

1.4

1.2

1.0

0.8

0.6

0.4

0.2

0.0

-8.0

Max : 8.0Min : -8.0

In Y-directionLC 1: Shear forceStresses Sigma-y,+

Max Sigma-y,+: 8.0, Min Sigma-y,+: -8.0 MPa

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0.0

90

Z

XY

Axial Stresses

y,+ [MPa]

8.0

1.8

1.6

1.4

1.2

1.0

0.8

0.6

0.4

0.2

0.0

-8.0

Max : 8.0Min : -8.0

In Y-directionLC 1: Shear forceStresses Sigma-y,+

Max Sigma-y,+: 8.0, Min Sigma-y,+: -8.0 MPa

1.35

1.05

0.75

0.45

0.15

-0.15

-0.45

-0.75

-1.05

-1.35

tAxial Stresses

txy,+ [MPa]

3.21

1.35

1.05

0.75

0.45

0.15

-0.15

-0.45

-0.75

-1.05

-1.35

-3.21

Max : 3.21Min : -3.21

In Y-directionLC 2: Bending momentSurfaces Stresses Tau-xy,+

Max Tau-xy,+: 3.21, Min Tau-xy,+: -3.21 MPa

Axial Stresses

y,+ [MPa]

1.05

0.90

0.80

0.70

0.60

0.50

0.40

0.30

0.20

0.10

0.00

-1.05

Max : 1.05Min : -1.05

In Y-directionLC 2: Bending momentSurfaces Stresses Sigma-y,+

Max Sigma-y,+: 1.05, Min Sigma-y,+: -1.05 MPa

Special Members – Beams with holes and notches slide 14

Test results – Beams with a hole

Beam size: H = 180 mm and 630 mm

Hole placement with respect to beam height:

H/2

H/2

H

Strength tests of glulam beams with a hole – Lund University

Special Members – Beams with holes and notches slide 15

Test results – Beams with a hole

H=630mm

H=180mm

Special Members – Beams with holes and notches slide 16

Test results – Beams with a hole

Example of results: Test ALh-4H = 630 mm

0 5 10 15 20 250

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

[mm]

No

min

al sh

ea

r str

ess V

/ A

net

[MP

a]

Special Members – Beams with holes and notches slide 17

Test results – Beams with a hole

0 1 2 3 4 5 6 7 80

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

[mm]

No

min

al sh

ea

r str

ess V

/ A

net

[MP

a]

Example of results: Test CUh-4H = 180 mm

Special Members – Beams with holes and notches slide 18

Test results – Beams with a hole

Influence of beam height

Significant beam size influence on nominal strength

Special Members – Beams with holes and notches slide 19

Test results – Beams with a hole

H = 180 mm

H = 630 mm

Influence of hole placement wrt beam height

5-15 % lower strength for beams with eccentric hole placement

Special Members – Beams with holes and notches slide 20

Outline

Introduction• Perp-to-grain tension and fracture• Stress state at holes and notches• Tests

Models for strength analysis• Conventional stress-strength analysis• Weibull weakest link theory• Linear elastic fracture mechanics

Applied strength analysis• End-notched beams• Beams with a hole

Reinforcement

Special Members – Beams with holes and notches slide 21

Conventional stress-strength analysis

Element capacity determined from a stress based failure criterionand assumptions of:

• Linear elastic and ideally brittle material behavior• Deterministic and homogeneous material strength properties

Max stress

Norris criterion

Tsai-Wu criterion

Special Members – Beams with holes and notches slide 22

Weibull weakest link theory

Based on the concept of the “weakest link” and assumptions of:• Linear elastic and ideally brittle material behavior• Stochastically homogeneous material strength properties

Global strength is affected by the size of the stressed volumeand by the stress distribution heterogeneity

Special Members – Beams with holes and notches slide 23

Weibull weakest link theory

Double-tapered beams, curved beams and pitched cambered beams

Stress distribution effect Volume effect

Perp-to-grain tension

Special Members – Beams with holes and notches slide 24

Linear elastic fracture mechanics - LEFM

Based on assumptions of:• Linear elastic material with infinite material strength• Existence of a sharp crack of length a in the structure

Two alternative (and equivalent) criteria for crack propagation:

Nominal shear stress at crack propagation:

Special Members – Beams with holes and notches slide 25

Linear elastic fracture mechanics - LEFM

Based on assumptions of:• Linear elastic material with infinite material strength• Existence of a sharp crack of length a in the structure

Nominal shear stress at crack propagation:

StiffnessFracture energyElement size

NLFM

Special Members – Beams with holes and notches slide 26

Models for strength analysis - Overview

Weibull

theory

Probabilistic

Frac. Mech.

ft = finitestochastic

Gf = finitestochastic

ft = finitedeterministic

Gf = finitedeterministic

Conventional

stress analysis

Probabilistic

Frac. Mech. LEFM

Probabilistic

Frac. Mech.Probabilistic

LEFM

Ideally plastic

analysis

Probabilistic

ideally plastic

-

-

-

-

-

-

ft = 0 ft = ∞

Gf = 0

Gf = ∞

Material strength

Fra

ctu

re e

nerg

y

Special Members – Beams with holes and notches slide 27

Outline

Introduction• Perp-to-grain tension and fracture• Stress state at holes and notches• Tests

Models for strength analysis• Conventional stress-strength analysis• Weibull weakest link theory• Linear elastic fracture mechanics

Applied strength analysis• End-notched beams• Beams with a hole

Reinforcement

Special Members – Beams with holes and notches slide 28

End-notched beams

General comments/recommendation on strength analysis and design

EC5: Rational approach based on Linear elastic fracture mechanics

• Should be handled with great care in design,

especially considering large members

• Notches larger than 500 mm or 0.5h should not be allowed without reinforcement

• Should preferably (if not reinforced) be tapered

• Surfaces shall be surface-treated to reduce the risk of cracking due to climate induced stresses

Special Members – Beams with holes and notches slide 29

End-notched beams

End-notch beam – Fracture Mechanics approach

Beam height, Stiffness, Fracture energy

Eurocode 5 implementation:

Shear strength, Beam height

(Stiffness, Fracture energy)

Gustafsson, 1988:

Special Members – Beams with holes and notches slide 30

End-notched beams

End-notch beam – Eurocode 5 implementation

kn = 6,5 for Glulam4,5 for LVL5,0 for solid timber

[Svenskt Trä: Limträhandbok – del 2, 2016]

Special Members – Beams with holes and notches slide 31

End-notched beams

End-notch beam – Eurocode 5 implementation

h = 100 mm

h = 300 mm

h = 900 mm

h = 100 mm

h = 300 mm

h = 900 mm

Special Members – Beams with holes and notches slide 32

Beams with a hole

General comments/recommendation on strength analysis and design

EC5: No design rules. Semi-empirical method available in German/Austria NA to EC5.

• Should be handled with great care in design, especially for large members

reinforcement in general recommended

• Circular shape rather than rectangular/square

• Hole placement close to the neutral axis

• Surfaces shall be surface-treated to reduce the risk of cracking due to climate induced stresses

Special Members – Beams with holes and notches slide 33

Beams with a hole – a historical review

Conventional

stress analysisWeibull

theory

Gen. LEFM

NLFM

Probabilistic

Frac. Mech. LEFM

Probabilistic

Frac. Mech.

Probabilistic

Frac. Mech.Probabilistic

LEFM

Ideally plastic

analysis

Probabilistic

ideally plastic

-

-

-

-

-

-

ft = 0 ft = finitedeterministic

ft = finitestochastic

ft = ∞

Gf = 0

Gf = finitedeterministic

Gf = finitestochastic

Gf = ∞

Old “Glulam handbook” (1)

Empirically based approach

Old “Glulam handbook” (2)

End-notched beam

analogy approach

Eurocode 5: Draft version

End-notched beam analogy approach

DIN 1052:2004

Semi-empirically based approach

Proposal by

Aicher et al

Large differences in …

… theoretical backgrounds

… predictions of beam capacity

DIN 1052:2008 / DIN EN 1995-1-1 NA

Semi-empirically based approach

Special Members – Beams with holes and notches slide 34

Beams with a hole

Design according to DIN EN 1995-1-1/NA (and “Limträhandbok”)

Limitations related to hole size and placement (selected)

hd ≤ 0.15hHole height

a ≤ 0.40hHole length

Hole placement hro ≥ 0.35h

hru ≥ 0.35h

r ≥ 15 mm (25 mm)Corner radius

[Svenskt Trä: Limträhandbok – del 2, 2016]

Special Members – Beams with holes and notches slide 35

Beams with a hole

Design according to DIN EN 1995-1-1/NA (and “Limträhandbok”)

Location of potential crack planes

Shear force dominated loading:Locations 1) and 2)

Bending moment dominated loading:Locations 1) and 3)

Assumed triangular shaped perp-to-grain tensile stress distribution along potential crack planes

[Svenskt Trä: Limträhandbok – del 2, 2016]

Special Members – Beams with holes and notches slide 36

Beams with a hole

Design according to DIN EN 1995-1-1/NA (and “Limträhandbok”)

Perp-to-grain tensile stress

Perp-to-grain tensile force

Shear force contribution from integration of shear stresses

Empirically based contribution from bending moment

[Svenskt Trä: Limträhandbok – del 2, 2016]

Special Members – Beams with holes and notches slide 37

Beams with a hole

Design according to DIN EN 1995-1-1/NA (and “Limträhandbok”)

Perp-to-grain tensile stress

beam height reduction factor(empirically based)

[Svenskt Trä: Limträhandbok – del 2, 2016]

Special Members – Beams with holes and notches slide 38

Beams with a hole

Shear stresses?

Z

XY

Axial Stresses

y,+ [MPa]

8.0

1.8

1.6

1.4

1.2

1.0

0.8

0.6

0.4

0.2

0.0

-8.0

Max : 8.0Min : -8.0

In Y-directionLC 1: Shear forceStresses Sigma-y,+

Max Sigma-y,+: 8.0, Min Sigma-y,+: -8.0 MPa

Z

XY

Axial Stresses

txy,+ [MPa]

9.1

2.7

2.4

2.1

1.8

1.5

1.2

0.9

0.6

0.3

0.0

-1.7

Max : 9.1Min : -1.7

In Y-directionLC 1: Shear forceStresses Tau-xy,+

Max Tau-xy,+: 9.1, Min Tau-xy,+: -1.7 MPa

Z

XY

Axial Stresses

y,+ [MPa]

8.0

1.8

1.6

1.4

1.2

1.0

0.8

0.6

0.4

0.2

0.0

-8.0

Max : 8.0Min : -8.0

In Y-directionLC 1: Shear forceStresses Sigma-y,+

Max Sigma-y,+: 8.0, Min Sigma-y,+: -8.0 MPa

1.8

1.6

1.4

1.2

1.0

0.8

0.6

0.4

0.2

0.0

90

Z

XY

Axial Stresses

y,+ [MPa]

8.0

1.8

1.6

1.4

1.2

1.0

0.8

0.6

0.4

0.2

0.0

-8.0

Max : 8.0Min : -8.0

In Y-directionLC 1: Shear forceStresses Sigma-y,+

Max Sigma-y,+: 8.0, Min Sigma-y,+: -8.0 MPa

2.7

2.4

2.1

1.8

1.5

1.2

0.9

0.6

0.3

0.0

t

Special Members – Beams with holes and notches slide 39

Beams with a hole

Shear stress

Z

XY

Axial Stresses

txy,+ [MPa]

9.1

2.7

2.4

2.1

1.8

1.5

1.2

0.9

0.6

0.3

0.0

-1.7

Max : 9.1Min : -1.7

In Y-directionLC 1: Shear forceStresses Tau-xy,+

Max Tau-xy,+: 9.1, Min Tau-xy,+: -1.7 MPa

Z

XY

Axial Stresses

y,+ [MPa]

8.0

1.8

1.6

1.4

1.2

1.0

0.8

0.6

0.4

0.2

0.0

-8.0

Max : 8.0Min : -8.0

In Y-directionLC 1: Shear forceStresses Sigma-y,+

Max Sigma-y,+: 8.0, Min Sigma-y,+: -8.0 MPa

2.7

2.4

2.1

1.8

1.5

1.2

0.9

0.6

0.3

0.0

tApproximation [Blass & Bejtka, 2003]

(for square/rectangular holes)

Special Members – Beams with holes and notches slide 40

Beams with a hole

Normal stress parallel to grain due to bending?

• Verification capacity with respect to bending moment and normal force

• Rectangular holes: consider additional bending above/below hole

[Svenskt Trä: Limträhandbok – del 2, 2016]

Special Members – Beams with holes and notches slide 41

Beams with a hole

Special Members – Beams with holes and notches slide 42

Outline

Introduction• Perp-to-grain tension and fracture• Stress state at holes and notches• Tests

Models for strength analysis• Conventional stress-strength analysis• Weibull weakest link theory• Linear elastic fracture mechanics

Applied strength analysis• End-notched beams• Beams with a hole

Reinforcement

Special Members – Beams with holes and notches slide 43

Reinforcement at holes and notches

1) Internal reinforcementGlued-in rods, fully threaded screws

2) External reinforcementGlued-on panels (Plywood or LVL)

[Svenskt Trä: Limträhandbok – del 2, 2016]

Special Members – Beams with holes and notches slide 44

Reinforcement at holes and notches

1) Internal reinforcementGlued-in rods, fully threaded screws

2) External reinforcementGlued-on panels (Plywood or LVL)

Special Members – Beams with holes and notches slide 45

Reinforcement of notched beams

Design according to DIN EN 1995-1-1/NA (and “Limträhandbok”)

Perp-to-grain tensile force

Integration of beam theory shear stresses below the notch

Modification factor to account for deviations from the beam

theory stress distribution

Reinforcement should be designed to carry the force Ft90

[Svenskt Trä: Limträhandbok – del 2, 2016]

Special Members – Beams with holes and notches slide 46

Reinforcement of notched beams

Design according to DIN EN 1995-1-1/NA (and “Limträhandbok”)

• Withdrawal capacity

• Tensile axial capacity

1) Internal reinforcementGlued-in rods, fully threaded screws

Design verifications

2) External reinforcementGlued-on panels (Plywood or LVL)

• Shear stress in glue line

• Tensile capacity of the panels

Design verifications

[Svenskt Trä: Limträhandbok – del 2, 2016]

Special Members – Beams with holes and notches slide 47

Reinforcement of notched beams

Design according to DIN EN 1995-1-1/NA (and “Limträhandbok”)

• Only one row of rods/screws are considered active

• Rods/screws should be placed close to notch corner

Special considerations

• Limited effective length of panel

• Orientation of LVL/Plywood

Special considerations

1) Internal reinforcementGlued-in rods, fully threaded screws

2) External reinforcementGlued-on panels (Plywood or LVL)

[Svenskt Trä: Limträhandbok – del 2, 2016]

Special Members – Beams with holes and notches slide 48

Reinforcement of beams with a hole

Design according to DIN EN 1995-1-1/NA (and “Limträhandbok”)

Perp-to-grain tensile force

Shear force contribution from integration of shear stresses

Empirically based contribution from bending moment

Reinforcement should be designed to carry the force Ft90

[Svenskt Trä: Limträhandbok – del 2, 2016]

Special Members – Beams with holes and notches slide 49

Design according to DIN EN 1995-1-1/NA (and “Limträhandbok”)

• Withdrawal capacity

• Tensile axial capacity

1) Internal reinforcementGlued-in rods, fully threaded screws

Design verifications

2) External reinforcementGlued-on panels (Plywood or LVL)

• Shear stress in glue line

• Tensile capacity of the panels

Design verifications

Reinforcement of beams with a hole

[Svenskt Trä: Limträhandbok – del 2, 2016]

Special Members – Beams with holes and notches slide 50

Design according to DIN EN 1995-1-1/NA (and “Limträhandbok”)

1) Internal reinforcementGlued-in rods, fully threaded screws

2) External reinforcementGlued-on panels (Plywood or LVL)

Reinforcement of beams with a hole

• Only one row of rods/screws are considered active

• Rods/screws should be placed close to hole edges

Special considerations

• Limited effective length of panel

• Orientation of LVL/Plywood

Special considerations

• Location of crack planes

[Svenskt Trä: Limträhandbok – del 2, 2016]

Special Members – Beams with holes and notches slide 51

Design according to DIN EN 1995-1-1/NA (and “Limträhandbok”)

Limitations related to hole size and placement (selected)

hd ≤ 0.30h Hole height

a ≤ 1.0hHole length

Hole placement hro ≥ 0.25h

hru ≥ 0.25h

Reinforcement of beams with a hole

1)

2)

for 1)

hd ≤ 0.40h for 2)

a ≤ 2.5hd

[Svenskt Trä: Limträhandbok – del 2, 2016]

Special Members – Beams with holes and notches slide 52

Reinforcement at holes and notches

Reinforcement with respect to shear stress?

2) External reinforcementGlued-on panels (Plywood or LVL)

In general sufficient shear capacity in panels

1) Internal reinforcementGlued-in rods, fully threaded screws

Low shear capacity at 90°

Special Members – Beams with holes and notches slide 53

Reinforcement at holes and notches

Cross Laminated Timber (CLT)

Inherent reinforcement for tension perpendicular to the beam axis by transverse laminations.

Special Members – Beams with holes and notches slide 54

Concluding remarks

• Introduction of a notch or a hole generally results in a significant loss of load bearing capacity

• Strong beam size influence on nominal strength

• Perp-to-grain failures are in general complicated to predict

notches/holes in members should be handled with great, especially for large members

Reinforcement is generally recommended

Thank you for your attention.

Henrik Danielsson

Division of Structural Mechanics, Lund University, Sweden