1

FORMULA TABLES IN SOLID MECHANICS

Compiled by Sören Sjöström. Latest update 6th December, 2013.

Contents

Elementary beam bending cases

Data for some frequent cross-section geometries

Beam with axial load

Euler instability cases

Westergaard solution for stress and displacement near a crack tip

Stress-intensity factors for some elementary crack cases

Energy release rate

Mixed-mode stress intensity factors

Plastic zone near crack tip

The J integral

HCF: Multiaxial stress states

Stress-concentration factors ( ) for geometries containing holes or notches.

Unless otherwise stated, tables are taken from Sundstöm B (ed.): Handbook of Solid Mechanics,

Department of Solid Mechanics, KTH, Stockholm, 2010.

2

Bending of beams

In all cases shown,

E = elastic modulus

I = area moment of inertia

CASES 1 – 5: CANTILEVER BEAM

1

2

3

3

4

5

CASES 6 – 10: BEAM ON MOMENT-FREE SUPPORT

6

7

4

8

9

10

5

Data for some frequent cross-section geometries

h/b 1 1.25 1.5 2 3 4 5 10 → ∞

F1 0.422 0.515 0.587 0.686 0.790 0.843 0.874 0.937 1.000

F2 0.833 0.885 0.924 0.984 1.069 1.125 1.166 1.249 4/3

6

Axially loaded beam

7

Euler instability cases

In all 5 cases

E = elastic modulus

I = area moment of inertia

l = length

CASE 1 2 3 4 5

Bound-

ary

condi-

tion

A: free

B: clamped

A: pinned

B: pinned

A: pinned

B: clamped

A: clamped

B: clamped

A: clamped/

gliding

B: clamped

8

Westergaard solution for stress and displacement near a crack tip

MODE I

MODE II

MODE III

9

Note: In the displacement equations, the parameter depends on whether the situation is plane

deformation or plane stress:

Stress intensity factors for elementary crack cases

CENTRAL CRACK IN SHEET WITH PRESCRIBED BOUNDARY LOAD

10

CRACKS STARTING AT CIRCULAR HOLE IN AN INFINITELY LARGE SHEET

DOUBLE EDGE CRACKS IN SHEET WITH PRESCRIBED BOUNDARY LOAD

11

SINGLE EDGE CRACK IN SHEET WITH PRESCRIBED BOUNDARY LOAD

EDGE CRACK IN STRIP WITH PRESCRIBED BENDING MOMENT LOAD

12

ELLIPTICAL SURFACE CRACK

ELLIPTICAL INTERNAL CRACK

13

Energy release rate

where

The following universal relations are valid between G and stress intensity factors:

Mixed-mode stress intensity factors

Criterion based on energy release rate:

Maximum tangential stress criterion:

14

Plastic zone near crack tip

The Dugdale model also gives expressions for the crack opening:

The J integral

The J integral is path-independent (provided that the strain energy function exists and has the prop-

erty everywhere in the region swept by when going from one path to another).

Further, if exists and has the above-mentioned property in the region enclosed by in the figure,

15

HCF: multiaxial stress states

Sines’ criterion:

1. In a case where , i.e., if all stress components vary in phase, the

following and can be used:

2. If we do not have , the above expression for is still valid,

whereas an improved expression must be used for :

16

Table of stress concentration factors för frequent geometries

containing notches and stress concentrations

1. AXIALLY LOADED FLAT BAR 2. FLAT BAR WITH BENDING-MOMENT LOAD

3. AXIALLY LOADED SHEET WITH HOLE 4. AXIALLY LOADED CIRCULAR SHAFT

17

5. CIRCULAR SHAFT WITH BENDING-

MOMENT LOAD

6. CIRCULAR SHAFT WITH TORQUE LOAD

7. SHAFT WITH AXIAL GROOVE