Chassis and body design and manufacturing

Statically indeterminate structures and frames

In static analysis, a structure is statically indeterminate if the static equilibrium equations are not sufficient for determining the constraint reaction. This means that the number of unknown reactions is greater than the number of equilibrium equations. In such a case, the structure may be solved by taking into account its stiffness characteristics, which requires material information.

Analysis methods - force method vs displacement method

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

• Equilibrium of forces and moments
• Compatibility of deformations
• Constitutive law (→σ)

Force method

The force method converts the indeterminate structure to a determinate one by suppressing a sufficient number of constraints. These are replaced by the relevant generalized forces (unknowns), enforcing compatibility to find the unknowns. In other words, among the infinite number of balanced configurations (n = number of unknowns), the only one which also satisfies compatibility is retained.

Example

Actual structure

• X_a
• Y_a
• F
• Y_B

Primary structure

• Primary structure → Structure
• Y_B

Equilibrium condition

Σ_x = 0 → X_A= 0
Σ_y = 0 → Y_A + Y_B = F

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 relationships be satisfied:

• Equilibrium of forces and moments
• Compatibility of deformations
• Constitutive law (ε-σ)

Force method

The force method converts the indeterminate structure to a determinate one by suppressing a sufficient number of constraints. These are replaced by the relevant generalized forces (unknowns), enforcing compatibility to find the unknowns. In other words, among the infinite number of balanced configurations (n = number of unknowns), the only one which also satisfies compatibility is retained.

Example

Equilibrium condition

ΣX = 0
ΣY = 0 → X_A + Y_A - F = 0
ΣΓ = 0 → Ia + F a - F (a + b) = F

Compatibility condition

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

N.B. Primary structure is arbitrary and must be statically determined.

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 EIY_B=F EI I F EI=0Y_B-3EIY_A=F F-Y_A+F FY_A=F EIY_A=Y_B-2EI 3=0=I=0Y_B=3F EI

Degradation method

The degradation method consists of finding the only forces that meet the condition among the loaded ends, providing valid deformation compatible conditions. This method is used for continuous structures where valid deformation compatibility conditions are sought.

We start from the complete displacement:

• Inductive collapse axis
• S1F = EAN1 = EAN = FE1 = EAS = SA = SB

• Complete Fx
• Force diagram single beam Fx = 5F = NN = M + HMN1 - F = 0A

Now we place the equilibrium (Continuum)Ƹ (F/EI) = Ƹ (F/EI) Ƹ (EAı / EAı) Ƹ (EAı / EAı). Frames have bending and some rustiness because there are rigid frames, and they can translate ➔ they can translate.