# Strength Calculations

## What is a strength calculation?

The goal of a strength calculation of a component or structure is to evaluate whether a structure will fail, subjected to one or more loads, for a chosen combination of geometry, material and environmental factors.

A structure can fail through different failure modes:

• Static failure: material failure on a location of the structure caused by a single, static load.
• Material fatigue: crack initiation and growth in a structure, due to repetitive loading with loss of stiffness, potentially leading to full fracture.
• Buckling: a sudden, uncontrolled deflection of a structure, due to loss of stability, caused by a compressive load.
• Creep: a continuously increasing strain, eventually leading to complete failure. Creep is caused by an externally applied load at higher temperatures, typically at and above 30-40% of the melting temperature (in Kelvin) of the material.

## When is a structure sufficiently strong?

### Allowable material stresses

A strength calculation checks if the material stresses in a structure are below an allowable stress. The allowable material stress is the stress a material can take before it either permanently deforms or completely breaks, reduced by one or more safety factors.

### Safety factors

The required safety factors to be used in a strength calculation, are usually imposed by international standards, developed for specific industries and applications. Safety factors are necessary to deal with uncertainties: uncertainties about occurring loads and uncertainties regarding mechanical properties of the material. The value of the safety factor depends on:

• the level of uncertainty
• the probability a certain load or combination of loads will occur
• the severity of the consequences when a structure or a part of it will fail (severe injuries/decease, considerable environmental damage, important financial loss)

Safety factors applied on external loads can be classified as:

• permanent/invariable loads (e.g. self-weight of the structure)

The safety factor required for a variable load is generally larger than required for a permanent load.

## The stiffness of a structure

A strength calculation also evaluates the stiffness of a structure. The stiffness of a construction is determined by:

• the stiffness of the geometry (a tall, slender structure vs. a short, robust construction)
• the joint between components (stiff or flexible joint)
• the stiffness of the material (e.g. steel is 3x stiffer than aluminium)

Calculating the stiffness of a structure is important for several reasons:

• serviceability of the structure: when a construction strongly deforms or deflects, but there is no danger for structural failure, the design can still be rejected when the serviceability is not guaranteed. An excessively deformed structural beam can cause cracks in the plasterwork of a building or hamper the opening and closing of windows or doors, but still not showing any issues on strength and stability.
• stability: a structure with limited stiffness is more vulnerable for instability (see buckling above)
• resonance: every component has multiple natural or resonance frequencies. A structure subject to a vibration load with a frequency close to one of the resonance frequencies, can suffer severe damage due to excessive deformations. Besides the mass, the stiffness of the structure determines those resonance frequencies.

## How is a strength calculation carried out?

### Hand calculations

A strength calculation of a simple construction can be done using so-called hand calculations (analytical calculations), based on the principles of classic mechanics and mechanics of materials. With a hand calculation the stress in the gross section of the component is calculated, and if necessary, multiplied with a stress concentration factor, which accounts for stress increases near notches in the material.

### The Finite Element Method

For more complex geometry, advanced material models or extensive load combinations, hand calculations are not accurate enough anymore to reliably assess strength and stiffness. In those cases, the industry relies on the use of the Finite Element Method (FEM) or Finite Element Analysis (FEA) for strength and stiffness assessment.

With the Finite Element Method, the geometry of a structure is divided in several small parts, called elements, connected with each other via element nodes. The behaviour (stiffness) of a single element is mathematically simple to calculate. A large number of those small elements can be combined into a large, complex model, which can still be relatively easy solved. With the Finite Element Method an extensive amount of detailed information can be obtained about material stresses at any location or the stiffness behaviour of the entire construction. That level of information can neither be achieved by hand calculations, nor laboratory testing.

## International standards

A large number of international standards, related to strength calculations, are available. These standards can be general rules widely applicable, like Eurocode 3, or they can be industry and application specific, like standards for the oil & gas or the offshore industry or targeted at welded structures or pressure vessels. Below you can find an (incomplete) overview of some important standards.

### Eurocode standards

Currently 10 Eurocodes are available, subdivided in 58 different parts. The Eurocodes also have national annexes (NA) for every participating country.

• EN-1990 (Eurocode 0): Basis of structural design
• EN-1991 (Eurocode 1): Actions on structures
• EN-1992 (Eurocode 2): Design of concrete structures
• EN-1993 (Eurocode 3): Design of steel structures
• EN-1994 (Eurocode 4): Design of composite steel and concrete structures
• EN-1995 (Eurocode 5): Design of timber structures
• EN-1996 (Eurocode 6): Design of masonry structures
• EN-1997 (Eurocode 7): Geotechnical design
• EN-1998 (Eurocode 8): Design of structures for earthquake resistance
• EN-1999 (Eurocode 9): Design of aluminium structures

### Offshore industry

• ISO 19900 series: Design of offshore structures for the oil & gas industry
• DNV-GL: Design of offshore structures for the oil & gas and offshore wind industry (origin: Norway and Germany)
• NORSOK: Design of offshore structures for the oil & gas industry (origin: Norway)
• API: Design of offshore structures for the oil & gas industry (origin: USA)

### Design of pressure vessels

• ASME VIII Div.2
• EN 13445

• IIW
• AWS

## Design By Analysis

The above-mentioned standards are originally developed to be used with hand calculations and are based on analytical equations, extensive laboratory testing campaigns and lessons learned from – often disastrous – accidents from the past. This approach is called Design By Rules or DBR.

Currently many standards also provide rules for the use of FEA calculations, like IIW, DNV-GL, ASME VIII Div.2, EN 13445, etc. This approach is called Design By Analysis or DBA.