Statics Chapter Notes PDF

# Chapter 0 ## Definitions: - **Vector**: is an object that has both magnitude and direction (a vector is a directed line segment). - **Free vector**: A free vector is a vector that's its action not confined to or associated with unique line in space. - **Sliding vector**: A sliding vector has...

# Chapter 0 ## Definitions: - **Vector**: is an object that has both magnitude and direction (a vector is a directed line segment). - **Free vector**: A free vector is a vector that's its action not confined to or associated with unique line in space. - **Sliding vector**: A sliding vector has a unique line of action in space but not a unique point of application. - **Fixed vector**: A fixed vector is one for which a unique point of application is specified. The action of a force on a deformable or non-rigid body must be specified by a fixed vector at the point of application of the force. - **Unit vector**: is a vector of length one unit and it is often denoted by a lowercase letter with a circumflex or hat. - **α:** is the angle between the vector and x-axis. - **β:** is the angle between the vector and y-axis. - **γ:** is the angle between the vector and z-axis. - **α, β, γ**: are called the coordinate angles of the vector. # Chapter 2 ## Types of Force: - **Contact forces:** Any types of forces that require being in contact with another object come under contact force. - Muscular force - Frictional force - Tension force - Normal force - Air Resistance force - Applied force - Spring force - **Non-contact forces:** A non-contact force is a force applied to an object by another body that is not in direct contact with it. - Magnetic force - Electrostatic force - Gravitational force - When a system has more than one force acting, it is known as a system of forces or a force system. ## Types of force systems: - **Collinear force system:** When the lines of action of all the forces of a system act along the same line. - **Parallel force system:** When the lines of action of a set of forces are parallel. - **Coplanar force system:** When the lines of action of a set of forces lie in a single plane - **Concurrent force system:** When the lines of action of all forces meet at the point of concurrency. **Coplanar and Concurrent:** Lie in single plane and pass through a single point. **Coplanar and Non-Concurrent:** Lie in a single plane and don't pass through a single point. **Non-Coplanar and Concurrent:** Don't lie in a single plane, but pass through a single point. **Non-Coplanar and Non-Concurrent:** Don't lie in a single plane and don't pass through a single point. ## Supports: - **Roller Support:** Can resist a vertical force but cannot resist horizontal forces and moment. - **Fixed Support:** Can resist vertical and horizontal forces and the moment. - **Simple Support:** Can resist vertical force only, like a roller support. ## Types of Beams: - **Simple:** - **Continuous:** - **Cantilever:** # Chapter 4: ## Equilibrium of Rigid Bodies - ΣF = 0 - ΣM = 0 ## Replacing forces with equivalent force and couple moment - **Support reactions:** *For every support, you can find its own reactions.* - **Resultant Force:** $$F_R = \sqrt{F_x^2 + F_y^2}$$ - **Resultant Moment:** $$M = F_R \times d$$ ## Resultant Moment: -$M_0 = R \times F + \sum V \times F_z + \sum F \times r$ - If all forces acting at the same point: - $$ \sum M_0 = \sum (F_1 \times r + F_2 \times r + ... + F_n \times r)$$ - To replace forces with equivalent resultant force $F_R$ acting at $O$ and a resultant couple moment $(M_R)$: - $F_R = \sum F$ - $(M_R)_0 = \sum M_0 + \sum M$ ## Note: * *A rigid body is an extended area of material that includes all the points inside it, which moves so that it retains its shape. The distance and angles between all its points remain constant.* * *A rigid body is defined as a body on which the distance between two points never changes, whatever be the force applied on it.* * *The net force and the net torque must be zero.* # Chapter 3 ## Sense of Moment - It has a negative sign (-) for a counter-clockwise moment - It has a plus sign (+) for a clockwise moment. ## Moment: - in a 2-D plane: - $M_0 = F_R \times d$ - in a 3-D plane: - $$M_0 = \vec{r} \times \vec{F}$$ ## Moment About: - **A line:** Is about any line - **An axis:** * About axis x, y, z-axis: - *The position vector of the point of application of the force at the origin* - *We need the projection of the force on each axis.* - **About any line:** - *The position vector of the point of application of the force.* - *The unit vector of that line.* - *Find the moment using the cross product.* - *Moment = Unit vector × (force × position vector)* - *$$M_{OAL} = \vec{u_{OA}}\times (\vec{F} \times\vec{r})$$* - *$$M_{OAL} = \begin{Vmatrix} \vec{u_{OA}}\\ \vec{F_x} & \vec{F_y} & \vec{F_z} \\ \vec{r_x} & \vec{r_y} & \vec{r_z} \end{Vmatrix}$$* ## Simple Supports: - **End Supported:** - **Cantilever:** - **Combination:** ## Strings, Ropes and Cables - Strings, ropes, and cables are used to transmit the force. The tension in a rope is equal to the force being transmitted. - An unloaded string has a length of $L_0$. - Hanging a weight of mass $m$ on it stretches the string to a new length of $L$.