Objective: The student will be introduced to potential common failure modes for structures and structural materials.

Module 2.3: Failure criteria for structures and structural materials

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Scope: Yielding, Multiaxial stresses, Plastic collapse, Ultimate strength, Fatigue and fracture, Global buckling, Local buckling, Lateral buckling , Slenderness

Expected result: Illustrate common modes of failure for structural elements. Compute ultimate load-carrying capacity for typical structural members based of strength of materials. Explain features of real structures that differ from the idea solutions and how these affect strength. Explain the basic principles of elastic and plastic design. Illustrate selection process for simple structural elements based on strength of materials analysis. Compute strength of a simple element based on both elastic and plastic strength.

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Different failure modes

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• Material yielding • Plastic collapse

• Final failure • Brittle failure • Fatigue failure

• Global buckling • Local buckling • Lateral buckling • Tiltning • Warping

• Post-buckling • Capsizing (the whole structure lose equlibrium) • Deformation constraints

Ductile failure

The material cracks /rupture

Elastic instability

Plastic ”instability”

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Ductile material behaviour

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• Material which can undergo a substantial amount of plastic deformation, generally much larger than the elastic deformation before rupture.

• Iron, Gold, Silver, Mild steels, Stainless steels, Aluminum

a – Brittle rupture b – Ductile rupture c – Completley ductile rupture

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Ductile material behaviour – micro mechanisms

10 mm

SEM photo of a ductile metal failure surface

Ductile material behaviour – mechanical response

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Mild steel, force-elognation diagram

OA: elognation completley recoverable and proportional to the loading – linear elasic zone AB: deformation still elastic but no proptinallity between Δl and N – transformation from linear elastic to non-linear elastic deformation BC: Yielding zone. CD: Material hardening (we can define a hardening modulus) DE: Softening, due to reduction of cross section (necking) prior rupture (In compression: similar behaviour to C´ but no necking. Hardening continues with large deformation)

Np = limit of proportinallity NY = yielding starts

Permenant deformation when unloading

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Brittle material behaviour

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Force-elognation diagram (cast iron, glass, rock, ceramic materials)

Linear elastic zone is less defined – the tangent to the curve decrease steadily until rupture

Little plastic deformation

Behaviour under tensile and compression is different

Display more stiffness and strength under compression

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Testing compressive strength of concrete: Sut 2-6 MPa Suc 32-60 MPa ~ 10 times stronger in compression

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Brittle material behaviour – micro mechanisms

Stress-strain diagram for a typical brittle material.

10 mm

SEM photo of a brittle metal failure surface

High strength steel

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Hardened steels are more brittle! • Strain hardening for ductile steels – increasing the load capacity of steel beyond the

elastic limit (limit state, service conditions) • Elastic limit can be increased by increased carbon content (elss ductility) High strenght steels do not have a yield zone • Plastic deformation not clearly shown • Elastic limit stress which cause unrecovorable strain with vaule 0.2 %. (σ0.2%)

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Idealized stress-strain curve for steels

1: ”real” material with strain hardening 2: Elastic –ideal plastic 3: Ideal plastic

Plasitc deformation of cross section member

Elastic stress distribution:

Partly plastic deformed cross section:

Plastic deformed cross section:

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M M

Ultimate limit load design

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Ultimate limit load design

Plasitc deformation of cross section member: rectangular cross section

Elastic Moment capacity (Mek) Plastic Moment capacity (Mpk)

Compare: 50 % larger moment capacity if full plastic defomration is allowed Note! Not acceptable for fatigue loads or when there is risk for instability (buckling)

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Compression strength – Euler buckling

Euler 4 global buckling cases:

Independent of the materials yield strength

General:

Put: (”radius of gyration”)

(slenderness ratio) Critical buckling stress

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Compression strength – Euler buckling example

Design of truss construction E-stie Flexenclosure (Barsoum Eng Consulting AB)

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Innovative Mobile Global Award 2008

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Design of truss construction E-stie Flexenclosure (Barsoum Eng Consulting AB)

Finite element analysis

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Compression strength – Euler buckling example

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Design of truss construction E-stie Flexenclosure (Barsoum Eng Consulting AB)

Testing and FEM: collapose due to bucking of a slender compression member

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Compression strength – Euler buckling example

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Slender members

Tensile and compression members

If a structural element is called beam or compression member depends on the loading it is subjected to

Compressive member: Loading in the axial direction (axial force)

Tensile member / beam: Loading across the elements lenght direction (cross sectional force/ bending moment)

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Slender members

Definition: ”Having little width in proportion to height or length; long and thin”

Increasing slenderness ratio, λ

• Short columns (small λ) do not buckle and simply fail by material yielding.

• Long columns (large λ) usually fail

by elastic buckling mentioned above.

• Between short and long regions, the failure of the column occurs through inelastic buckling.

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Buckling

If a ≥ b kσ = 4 (buckling koefficient) which depends on the boundary conditions The buckling load is LOW for welded structures due to internal compressive stress formed after welding

Relation between load P and deflection for a compressive member and free supported compressed plate

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Tilting (Equilibrium instability)

• Could occur for high and slender beams • Characterized by the whole or parts of

the beam is losing equilibrium

The risk with TILTING increase when: • Low E-modulus (G-modulus) • High beam • Slender beam • Long free beam • When bending and compression is

applied • Compressive residual stresses in vicinity

of compressed flange • High load application point

The risk with TILTING decrease when: • Clamping at support points • Tensile loading an bending at the same

time • Stiffening of flanges (with shorter

distances) • Higher bending stifness for stiffener

plates

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Torsional strength

Thick-walled and closed cross sections

Mv = torsional moment Wv = torsinal stiffness (St-Venant)

Open thin-walled cross sections

Considered as flange bending (Vlasov torsion)

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Torsional strength

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x

y

a

2 a

2 a

a32

a8/tan2

60°

Cirkel

Oktagon

Kvadrat

Likbent triangel

r

Consider four different cross sections – select the optimum for torsion

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Torsional strength

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Note that:

• warping of the non-circular cross section

• end effect (Saint-Venants principal)

Apply torque Mv

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Torsional strength

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Shear stress in each cross section

1.00 0.85 0.70 0.55 0.40 0.25 0.10 0.00

Mv

W Cir

vz

2

3aW Cir

v

364.1 aW Okt

v

3

5 3aW Kva

v

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18 3aW Tri

v

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Torsional strength

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Shear stress along the perimeter of each cross section

”A circle is a polygon with infinite number of sides…”

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Torsion

Open thin-walled cross sections

St-Venant torsion: The cross section remain in the plane

Vlasov torsion:

The cross section is WARPED out of the plane

Torsion of circular cross section. No warping

Torsion of rectangular cross section. Negleble warping

Torsion of I-cross section beam. Large warping

Shear stress flow in a cross section which undergo St-Venant torsion is closed. Open cross sections undergo Vlasov torsion

Rectangular beam cross section – St-Venant torsion

I-cross section – Vlasov torsion

Mixed torsion:

Vlasov torsion

St-Venant torsion

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Torsion

These cross sections will not warp during torsional loading

These cross sections will warp during torsional loading

If the torsional moment is balanced by reaction forces then the corss section will warp VC = torsional centrum

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Torsion - Warping prevention

Connect upper and lower flange with torsional stiff element to prevent warping

Warping prevention for U-beams

Warping prevention for Z- and I-beams

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Torsion - Warping prevention

Example:

Torsional weak Torsional weak Torsional stiff

Torsional stiff Torsional stiff

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Yielding criteria

At combined stresses one have to evaluate ”equivavlent stresses” to be compared with the materials yield stress (which is a stress in one direction) There are several yielding criterias: • Theory of maximum principal stress • Theory of maximum longitudinal deformation • Theory of maximum shear stress (Tresca) • Theory of maximum distortion energy (von Mises) The yielding criterias works good in some cases and not in other. Nowadays the von Mises hypothesis is frequently used.

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Yielding criteria

Maximum Normal (Principal) Stress Theory

• Theory: Yielding begins when the maximum principal stress in a stress element exceeds the yield strength.

• For any stress element, use Mohr’s circle to find the principal stresses.

• Compare the largest principal stress to the yield strength.

• Is it a good theory?

• This theory is not safe to use for ductile

materials • In pure shear (σ1 = σ2 = τ); diverges from

experiments • States that yielding occurs τ = σY while

measured shearing yield stress is lower (τ = σY)

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Yielding criteria

Maximum Longitudinal Deformation Theory

• Theory: Yielding determined by the maximum longitudinal strain.

• According to Hookes law for isotropic materials, the material remain in elastic phase as long s the following conditions are satisfied:

• In the plane stress state yielding and not or the higher value

• In compression the theory leads to yielding at , but experiments that much higher values of the pressure may be applied without plastic deformation.

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Yielding criteria

Maximum Shear Stress Theory (MSS) – Tresca yielding criteria

Theory: Yielding begins when the maximum shear stress in a stress element exceeds the maximum shear stress in a tension test specimen of the same material when that specimen begins to yield.

For a tension test specimen, the maximum shear stress is s1 /2.

At yielding, when s1 = Sy, the maximum shear stress is Sy /2 .

Could restate the theory as follows:

◦ Theory: Yielding begins when the maximum shear stress in a stress element exceeds Sy/2.

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Yielding criteria

Maximum Shear Stress Theory (MSS) – Tresca yielding criteria

• For any stress element, use Mohr’s circle to find the maximum shear stress. Compare the maximum shear stress to Sy/2

• Ordering the principal stresses such that s1 ≥ s2 ≥ s3,

Henri Édouard Tresca (1814-1885) Professor of Mechanical Engineering

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Yielding criteria

Maximum Shear Stress Theory (MSS) – Tresca yielding criteria

• To compare to experimental data, express max in terms of principal

stresses and plot.

• To simplify, consider a plane stress state

• Let sA and sB represent the two non-zero principal stresses, then order them with the zero principal stress such that s1 ≥ s2 ≥ s3

• Assuming sA ≥ sB there are three cases to consider

Case 1: sA ≥ sB ≥ 0

Case 2: sA ≥ 0 ≥ sB

Case 3: 0 ≥ sA ≥ sB

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Yielding criteria

Maximum Shear Stress Theory (MSS) – Tresca yielding criteria

Case 1: sA ≥ sB ≥ 0

For this case, s1 = sA and s3 = 0

reduces to sA ≥ Sy

Case 2: sA ≥ 0 ≥ sB

For this case, s1 = sA and s3 = sB

reduces to sA − sB ≥ Sy

Case 3: 0 ≥ sA ≥ sB

For this case, s1 = 0 and s3 = sB

reduces to sB ≤ −Sy

Plot three cases on principal stress axes Other lines are symmetric cases

Inside envelope is predicted safe zone

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Yielding criteria

Maximum Shear Stress Theory (MSS) – Tresca yielding criteria

Comparison to experimental data

• Conservative in all quadrants

• Commonly used for design situations

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Yielding criteria

Distortion Energy (DE) Failure Theory – von Mises yielding criteria

• Also known as: • Octahedral Shear Stress • Shear Energy • Von Mises • Von Mises – Hencky

Richard Edler von Mises (1883-1953) Applied Math. and Solid Mechanics Harvard University

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Yielding criteria

Distortion Energy (DE) Failure Theory – von Mises yielding criteria

• Originated from observation that ductile materials stressed hydrostatically (equal principal stresses) exhibited yield strengths greatly in excess of expected values.

• Theorizes that if strain energy is divided into hydrostatic volume changing energy and angular distortion energy, the yielding is primarily affected by the distortion energy

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Yielding criteria

Distortion Energy (DE) Failure Theory – von Mises yielding criteria

• Theory: Yielding occurs when the distortion strain energy per unit volume reaches the distortion strain energy per unit volume for yield in simple tension or compression of the same material

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Yielding criteria

Distortion Energy (DE) Failure Theory – von Mises yielding criteria

• Theory: Yielding occurs when the distortion strain energy per unit volume reaches the distortion strain energy per unit volume for yield in simple tension or compression of the same material

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Yielding criteria

Distortion Energy (DE) Failure Theory – Deriving DE

• Hydrostatic stress is average of principal stresses

• Strain energy per unit volume,

• Substituting for principal strains into strain energy equation,

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Yielding criteria

Distortion Energy (DE) Failure Theory – Deriving DE

• Strain energy for producing only volume change is obtained by substituting sav for s1, s2, and s3

• Substituting sav

• Obtain distortion energy by subtracting volume changing energy from total strain energy

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Yielding criteria

Distortion Energy (DE) Failure Theory – Deriving DE

• Tension test specimen at yield has s1 = Sy and s2 = s3 =0

• Applying distortion energy for tension test specimen is

• DE theory predicts failure when distortion energy exceeds distortion energy of tension test specimen

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Yielding criteria

Distortion Energy (DE) Failure Theory – von Mises Stresses

• Left hand side is defined as von Mises stress

• For plane stress, simplifies to

• In terms of xyz components, in three dimensions

• In terms of xyz components, for plane stress

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Yielding criteria

Distortion Energy (DE) Failure Theory – von Mises Stresses

• Von Mises Stress can be thought of as a single, equivalent, or effective stress for the entire general state of stress in a stress element.

• Distortion Energy failure theory simply compares von Mises stress to yield strength.

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Yielding criteria

DE Theory compared to experimental data

• Plot von Mises stress on principal stress axes to compare to experimental data (and to other failure theories)

• DE curve is typical of data

• Note that typical equates to a 50% reliability from a design perspective

• Commonly used for analysis situations

• MSS theory useful for design situations where higher reliability is desired

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Yielding criteria

Shear Strength Predictions

• For pure shear loading, Mohr’s circle shows that sA = −sB =

• Plotting this equation on principal stress axes gives load line for pure shear case

• Intersection of pure shear load line with failure curve indicates shear strength has been reached

• Each failure theory predicts shear strength to be some fraction of normal strength

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Yielding criteria

Shear Strength Predictions

• For pure shear loading, Mohr’s circle shows that sA = −sB =

• Plotting this equation on principal stress axes gives load line for pure shear case

• Intersection of pure shear load line with failure curve indicates shear strength has been reached

• Each failure theory predicts shear strength to be some fraction of normal strength

• For MSS theory, intersecting pure shear load line with failure line results in

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Yielding criteria

Shear Strength Predictions

• For DE theory, intersection pure shear load line with failure curve gives

• Therefore, DE theory predicts shear strength as

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