CHAP 8 STRUCTURAL DESIGN USING FINITE ELEMENTS FINITE ELEMENT ANALYSIS AND DESIGN Nam-Ho Kim Edited and audio Raphael Haftka

INTRODUCTION FEA: determining the response of a given structure for a given set of loads and boundary conditions Geometry, material properties, BCs and loads are well defined Engineering design: a process of synthesis in which parts are put together to build a structure that will perform a given set of functions satisfactorily Creative design: creating a new structure or machine that does not exist Adaptive design: modifying an existing design (evolutionary process)

INTRODUCTION – STRUCTURAL DESIGN Structural design: a procedure to improve or enhance the performance of a structure by changing its parameters Performance: a measurable quantity (constraint and goal) the weight, stiffness or compliance; the fatigue life; noise and vibration levels; safety Constraint: As long as the performance satisfies the criterion, its level is not important Ex: the maximum stress should be less than the allowable stress Goal: the performance that the engineer wants to improve as much as possible, usually called objective function Design variables: system parameters that can be changed during the design process Plate thickness, cross-sectional area, shape, etc

SAFETY MARGIN Factor of Safety (stress performance) Structures should satisfy a constraint: si(x): the calculated stress at a point x in the i-th component sallowable: allowable stress from material strength (failure strength) Factor of safety: effect of material variability, experimental errors, etc For general type of performance, use response and capacity Response Ri: calculated value of performance Capacity Ci: allowed value of performance When multiple loads are applied simultaneously

SAFETY MARGIN cont. Safety Margin: the excess capacity compared with the response The member can afford additional Zi stress before it fails Sufficiency Factor Si: ratio of the allowable capacity to the response Normalized in terms of the capacity Example: Failure stress is 100ksi, factor of safety is 1.25, actual stress is 60 ksi. Safety margin is 40ksi, the sufficiency factor is 80/60=1.333

LOAD FACTOR Instead of dividing the capacity by SF, increase applied loads Load factor l: the factor by which the applied loads must be multiplied just enough to cause the structure to fail Let a set of loads be F = {F1, F2, …, FN}T. For a given failure strength Ci, the load factor is defined by For proportional loading with linear system Load factor is the ratio between capacity and response The structure should be designed in order to satisfy a given level of load factor

EXAMPLE Design the height h of cantilevered beam with SF = 1.5 E = 2.9104 ksi, w = 2.25 in. Allowable tip deflection Dallowable = 2.5 in. (No need SF) FE equation after applying BCs FE solution F = 2,000 lb L = 100 in h w

EXAMPLE cont. Failure strength = 40 ksi (Need SF) Supporting moment at the wall Maximum stress at the wall Height calculation with the factor of safety

Quiz-like questions The FAA specifies that airliners need to be able to withstand a dive followed by pull-up that will impose 2.5g vertical acceleration on the plane, with a load factor of 1.5. The loads that we will apply to finite element model in order to design the airplane, what acceleration should they correspond to? When can we instead calculate loading at 2.5g and instead of the load factor use a safety factor of 1.5? If the maximum acceleration that the airplane will ever see is 2g, what is the sufficiency factor Answers in the notes page Since the loads will be 1.5 times higher, they will correspond to an acceleration of 3.75g We can use a safety factor instead if the stresses are proportional to the loads, which typically requires linear elastic response. The response is 2g and the capacity is 2.5, so the sufficiency factor is 1.25.

Topology optimization demonstrations Topology Optimization Optistruct – YouTube Abaqus Topology Optimization of a Bridge – YouTube http://www.topopt.dtu.dk/ smart phone app

Hierarchy of structural optimization Sizing optimization: Decide on the geometry of the structure and then optimize only cross sectional properties such as thicknesses of plates or cross sectional areas of truss elements. Can model all design requirements and millions of design variables Shape optimization: Decide on the topology of the structure (number of holes) and then optimize the outside and hole shapes. Can model all design requirements but only hundreds of design variables Topology optimization: Start with a block of material and let the structure emerge. Each finite element has its own design variable, but only for few design requirements (mainly stiffness)

Quiz-like questions In the bridge model below, what would be examples of changing the topology, the shape, and the sizing of the structure? Answers in the notes page Changing the number of columns is an example of topology change. Changing the shape of the arch will be a shape change. Changing the cross sectional area of one of the columns would be a sizing change.