# A.2.1: Basic Properties of Materials

## Material Stiffness

The stiffness of a material, or its resistance to deformation, is characterized by its **Young’s Modulus** (modulus of elasticity "E"). The higher the value of "E," the stiffer the material.

### Table A.2.1.1: Young's Modulus of materials (E-values) for some popular building materials

| Type of material    | $$E  \[kN/cm^2]$$ |
| ------------------- | ----------------- |
| steel               | 21000             |
| aluminum            | 7000              |
| reinforced concrete | 3000              |
| glass fiber         | 7000              |
| wood (spruce)       | 1000              |

For composite materials, such as glass fiber rods reinforced with epoxy, the mean value of "E" must be determined through material testing. Karamba3D expects the input for "E" to be in kilonewtons per square centimeter (kN/cm²).

When a material is stretched, it not only elongates but also contracts laterally. For instance, in steel, the lateral strain is typically 30% of the longitudinal strain. This effect influences displacement responses in beams with large height-to-span ratios, although it is generally of minor importance in standard beam structures. The **shear modulus (G)** describes material behavior in this context..

## Specific Weight

The specific weight, represented by **γ (gamma)**, should be provided in kilonewtons per cubic meter (kN/m³), indicating force per unit volume. Due to Earth’s gravitational acceleration (g ≈ 9.81 m/s²), according to **Newton's Law (F = ma)**, a mass of 1 kg exerts a downward force of approximately 9.81 N. For calculating deflections of structures the assumption of $$f=10N$$ is accurate enough. If greater precision is required, the gravity constant can be adjusted in the **"karamba.ini"** file. For Imperial units, the exact value for gravity is automatically set to ensure correct conversion from lbm to lbf.

**Table A.2.1.1** lists specific weights for various common building materials. Material weight only affects calculations when gravity is included in a load case (refer to section 3.2.1).

## Theoretical Background of Stiffness, Stress and Strain

**Strain** (denoted by ε) is the ratio of the elongation of a material under load to its original length. **Stress** (denoted by σ) is the force per unit area. In a beam cross-section, normal forces can be calculated by integrating the product of area and stress across the cross-section.

For **linear elastic materials**, the relationship between stress and strain is linear, governed by **Hooke’s Law**, which states that the force required to deform a material increases proportionally with the amount of deformation:

$$σ = E \cdot ε$$

This law expresses the principle that greater deformation requires greater applied force.