BEAMS AND PLATES ANALYSIS

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Madeline Harrington
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1 BEAMS AND PLATES ANALYSIS Automotive body structure can be divided into two types: i. Frameworks constructed of beams ii. Panels

2 Classical beam versus typical modern vehicle beam sections Assumptions: 1. The section is symmetric 2. The applied forces are down the axis of symmetry 3. The section will not change shape upon loading 4. The deformation will be in the plane and in the direction of the loads 5. The stresses vary in direct proportion with the strain 6. Failure is defined as yielding of the outermost fiber

3 STRESS ANALYSIS FOR BEAM: BENDING σ = Mz I Where; σ = Direct stress at point of interest M = Bending moment on the section z = vertical distance measured from the neutral axis I = Moment of inertia I = z 2 da section

4 DEFLECTION OF THE BEAM

5 DEFLECTION OF THE BEAM

6 LOCAL DEFORMATION UNDER POINT LOAD Previously, we have assumed that the applied point loads only generate global deformation. In reality, the point load also distorts the beam in the vicinity of the load. This local distortion leads to a reduced beam stiffness and increased local stiffness Source: Donald E Malen Fundamentals of Automobile Body Structure Design, SAE International

7 Now, we consider the system stiffness containing ideal beam stiffness and local stiffness in the calculation: K system = K idealk local K ideal + K local where, t =section thickness h = section height b= section width E = Young s modulus K ideal = based on the beam types (slide no 4 & 5 ) K local = 8Et3 (h + b) b 2 (4h + b)

8 STRESS ANALYSIS FOR BEAM: TORSION Source: Donald E Malen Fundamentals of Automobile Body Structure Design, SAE International For closed section; θ = τ = T 2At TL GJ effective i) Constant thickness J effective = 4A2 t S ii) Varying thickness J effective = 4A2 * Refer pages 56 & 57 for the derivation s i it i θ = angle of rotation T = applied torque L = beam length τ = shear stress G = shera modulus A = area enclosed by the section t = thickness J effective = thin-wall torsion constant S = section parameter

STRESS ANALYSIS FOR BEAM: TORSION Source: Donald E Malen. 2011.

effective i) Constant thickness J effective = 1 3 t3 S ii) Varying thickness J effective = 1 3 i s it i θ = angle of

9 STRESS ANALYSIS FOR BEAM: TORSION Source: Donald E Malen Fundamentals of Automobile Body Structure Design, SAE International For open section; θ = τ = Tt J effective TL GJ effective i) Constant thickness J effective = 1 3 t3 S ii) Varying thickness J effective = 1 3 i s it i θ = angle of rotation T = applied torque L = beam length τ = shear stress G = shera modulus A = area enclosed by the section t = thickness J effective = thin-wall torsion constant S = section parameter

STRESS ANALYSIS FOR PLATE: BUCKLING Most of the failure modes are caused

Thus, it is essential to estimate critical plate buckling stress and

Eπ 2 σ critical = k 12(1 v 2 )( b t )2 k = plate buckling coefficient E

Note: ss = simply supported = no deflection & rotation allowed fixed =

10 STRESS ANALYSIS FOR PLATE: BUCKLING Most of the failure modes are caused by plate buckling of section elements. Thus, it is essential to estimate critical plate buckling stress and effective load. Eπ 2 σ critical = k 12(1 v 2 )( b t )2 k = plate buckling coefficient E = Young s modulus v = Poisson s ratio b = plate width t =plate thickness Note: ss = simply supported = no deflection & rotation allowed fixed = no deflection & no rotation free = deflection and rotation allowed Source: Donald E Malen Fundamentals of Automobile Body Structure Design, SAE International

11 For the effective plate, P effective = σ s wt w = σ critical b 2 σ s where, P effective = load on plate σ s = maximum stress w = effective width t = thickness of plate b = actual width of plate STRESS ANALYSIS FOR PLATE: BUCKLING load load Buckled plate Buckled beam Ultimate load Critical load Critical load deflection deflection

reducing plate width by chamfering corners 3.

reducing plate width using beads-shear case 5.

12 How to inhibit buckling? 1. reducing plate width by adding a bead 2. reducing plate width by chamfering corners 3. reducing plate width by adding corners 4. reducing plate width using beads-shear case 5. changing boundary condition with a flange curl 6. plate with flanged hole 7. plate with curled element Source: Donald E Malen Fundamentals of Automobile Body Structure Design, SAE International

13 LOAD ANALYSIS

15 DESIGN FACTORS

16 VERTICAL LOADS

17 VERTICAL LOADS

18 VERTICAL ASYMMETRIC LOADS

26 SUSPENSION BODY LOAD INTERACTIONS

27 SUSPENSION BODY LOAD INTERACTIONS Source: Donald E Malen Fundamentals of Automobile Body Structure Design, SAE International

28 B. FLOW DOWN OF REQUIREMENTS FROM VEHICLE-LEVEL FUNCTIONS There are four steps listed by D E Malen (2011): 1. Identify the vehicle function e.g. minimizing injury during front impact 2. Define function strategy absorb energy (< 20g) without deformation 3. Analyse the role of the body structure to meet the strategy share the loads 4. Flow down the overall body structure requirements to the structural subsystem and element requirements absorb 80kNm of energy (body system) crush load 160 kn at front compartment (subsystem) crush load 80 kn at mid rail beam (element)

29 B. FLOW DOWN OF REQUIREMENTS FROM VEHICLE-LEVEL FUNCTIONS Source: Donald E Malen Fundamentals of Automobile Body Structure Design, SAE International

PURE BENDING. If a simply supported beam carries two point loads of 10 kn as shown in the following figure, pure bending occurs at segment BC.

PURE BENDING. If a simply supported beam carries two point loads of 10 kn as shown in the following figure, pure bending occurs at segment BC. BENDING STRESS The effect of a bending moment applied to a cross-section of a beam is to induce a state of stress across that section. These stresses are known as bending stresses and they act normally