Chassis and body design and manufacturing

STATICALLY INDETERMINATE STRUCTURES AND FRAMES

In static ... structure is STATICALLY INDETERMINATE if the static equilibrium equations are not sufficient for determining the constraint reaction (thus the ... "internal forces" ...) ... structure is greater than

[n^u UNKNOWN REACTIONS > n^e EQUILIBRIUM EQUATIONS]

In such a case, the structure may be solved by taking into account its stiffness characteristics (thus material information is needed)

ANALYSIS METHODS - FORCE METHOD VS DISPLACEMENT METHOD

Structural analysis requires that the equations governing the following physical relationships be satisfied:

1. Equilibrium of forces and moments
2. Compatibility of deformations
3. Constitutive law (→σ)

1) FORCE METHOD

The force method converts or converts the indeterminate structure to a determinate one by suppressing a sufficient number of constraints. They are replaced by the relevant generalized forces (unknown)

→ COMPATIBILITY ENFORCED TO FIND THE UNKNOWNS

In other words, among the ∞ⁿ (n = number of unknowns) balanced configurations, the only one which also satisfies compatibility is retained.

Example:

ACTUAL STRUCTURE

• X_aY_a

• F

• Y_B

PRIMARY STRUCTURE

• PRIMARY STRUCTURE→ STRUCTURE

• Y_B

1) EQUILIBRIUM CONDITION

{ Σ_x = 0 → X_A= 0

Σ_y = 0 → Y_A + Y_B = F

Σ = 0 → I + I n^u EQUILIBRIUM EQUATIONS]

In such a case the structure may be solved by taking into account its STIFFNESS CHARACTERISTIC (thus material information is needed)

ANALYSIS METHODS - FORCE METHOD VS DISPLACEMENT METHOD

Structural analysis requires that the equations governing the following physical relationship be satisfied:

1. EQUILIBRIUM OF FORCES AND MOMENTS
2. COMPATIBILITY OF DEFORMATIONS
3. CONSTITUTIVE LAW (ε-σ)

FORCE METHOD

The force method converts or converts the indeterminate structure to a determinate one by suppressing a sufficient number of constraints. These are replaced by the relevant generalized forces (unknowns).

=> COMPATIBILITY ENFORCED TO FIND THE UNKNOWNS

In other words among the ∞ⁿ (n = number of unknowns) balanced configurations, the only one which also satisfies compatibility is retained.

EX:

1) EQUILIBRIUM CONDITION

• ΣX = 0
• ΣY = 0 => X_A + Y_A - F = 0
• ΣΓ = 0 => Ia + F a - F (a + b) = F.

2) COMPATIBILITY CONDITION

The compatibility condition means that point B must not move => γ_Β = 0

N.B. Primary structure arbitrary, must be statically determined.

2) Compatibility condition

Y_B=0

Y_A+Y_B=0

-Y_A+Y_L=0

Y_B=F EI

-Y_A+F EI=0

0=F EI

Y_B=F EI

I F EI=0

Y_B-3EI

Y_A=F F

-Y_A+F F

Y_A=F EI

Y_A=

Y_B-2EI 3=

0=I=0

Y_B=3F EI

2 DEGRADATION METHOD

The degradation method consists on finding The ONLY FORCES meet condition among the loaded ends which will provide also the compatible deformation

This method s used for continuous structures in place it wherever the only ones which will provide valid deformation compatible conditions.

We start from the complete displacement Inductive collapse axiaS

1

F = EA

N1 = EA

N = F

E1 = EA

S = SA = SB

1) Complete Fx

2) Force diagram single beam

Fx = 5

F = N

N = M + H

MN1 - F = 0

A

Now we place the equilibrium

(Continuum)

Ƹ (F/EI) = Ƹ (F/EI) Ƹ (EAı / EAı) Ƹ (EAı / EAı)

Frames have got bending and some rustiness, because there are rigid frame and they can translate ➔ they can translate