Statics and Mechanics

• Mechanics of Materials 1, by Oyelade Akintoye.O. (PhD) of the Civil and Environmental Engineering Department, University of Lagos, will cover:
  • Introduction
  • Statics of Particles
  • Rigid Bodies: Equivalent Systems of Forces
  • Plane Pin-Jointed Frames/Trusses
  • Conclusion

Introduction

• Mechanics is defined as the science that describes and predicts the conditions of rest or motion of bodies under the action of forces.
• Mechanics serves as the foundation for most engineering sciences and is essential for study in those fields.
• Mechanics doesn't rely on empiricism found in some engineering sciences.
• Mechanics resembles mathematics due to its rigor and emphasis on deductive reasoning.
• Mechanics is divided into three parts:
  • Mechanics of rigid bodies
  • Mechanics of deformable bodies
  • Mechanics of fluids
• The mechanics of rigid bodies is subdivided into statics and dynamics.
  • Statics deals with bodies at rest.
  • Dynamics deals with bodies in motion.
• In the study of mechanics, bodies are assumed to be perfectly rigid.
• The mechanics of deformable bodies involves structures and machines that deform under loads.
• The mechanics of fluids is divided into the study of incompressible fluids and compressible fluids.
• Hydraulics is a subdivision of incompressible fluids and deals with problems involving water.
• A force measures the tendency to cause a body to move or translate in the direction of the force.
• A complete description of a force includes its magnitude and direction.
• In stress analysis, forces can be categorized as external or internal.
  • External forces include applied surface loads, force of gravity, and support reactions.
  • Internal forces are the resisting forces generated within loaded structural elements.
• The moment of a force measures its tendency to cause a body to rotate about a specific point or axis.
• To develop a moment about a specific axis, a force must act such that the body would begin to twist or bend about the axis.
• The magnitude of the moment of a force acting about a point or axis is directly proportional to the distance of the force from that point or axis.
• It is defined as the product of the force and the lever arm.
• Normal force (F) is perpendicular to the cross-section.
• Shear force (V) is parallel to the cross-section.
• Bending moment (M) bends the material.
• Twisting moment (torque) (T) twists the material about its central axis.
• The Parallelogram Law for the Addition of Forces states that two forces acting on a particle may be replaced by a single force (their resultant).
  • The resultant is obtained by drawing the diagonal of the parallelogram with sides equal to the given forces.
• The Principle of Transmissibility states that the conditions of equilibrium or motion of a rigid body remain unchanged if a force acting at a given point of the rigid body is replaced by a force of the same magnitude and direction, acting at a different point, provided the two forces have the same line of action.
• Newton's Three Fundamental Laws
  • F = Ma
  • F = Gm/r^2

Statics of Particles

• A force represents the action of one body on another.
• A force is generally characterized by its:
  • Point of application
  • Magnitude
  • Direction
• Forces acting on a given particle have the same point of application and can be completely defined by magnitude and direction.
• Experimental evidence shows that two forces P and Q acting on a particle A can be replaced by a single force R which has the same effect on the particle A.
• That single force R is called resultant of the forces P and Q.
• Forces can be resolved into rectangular components.
• F has been resolved into a component Fx along the x axis and a component Fy along the y axis.
• Formulas for determining rectangular components of a force:
  • Fx = Fcosθ
  • Fy = Fsinθ
  • Tan θ = Fx/Fy
  • F = √(Fx^2 + Fy^2)
• The magnitude of F can be obtained by applying the Pythagorean theorem.

Rigid Bodies: Equivalent Systems of Forces

• Most bodies in elementary mechanics are assumed to be rigid, meaning they do not deform.
• Actual structures and machines are never absolutely rigid and deform under loads.
• Two important concepts related to the effect of a force on a rigid body:
  • Moment of a force about a point
  • Moment of a force about an axis
• External forces act on the rigid body and are responsible for its external behavior, causing it to move or remain at rest.
• Internal forces hold together the particles forming the rigid body or the component parts of a structurally composed body.
• The principle of transmissibility states that the conditions of equilibrium or motion of a rigid body will remain unchanged if a force F acting at a given point of the rigid body is replaced by a force F' of the same magnitude and same direction, but acting at a different point along the same line of action.
• The vector product (or cross product) of two vectors, P and Q, results in a vector V.
• The vector V must satisfy these conditions:
  • Its line of action is perpendicular to the plane containing P and Q. -The magnitude of V equals the product of the magnitudes of P and Q and the sine of the angle θ.
• Formula for determining the vector product of two vectors:
  • V = PQsinθ
  • V = P x Q
• The direction of V is obtained from the right-hand rule:
  • Close your right hand
  • Hold it so that your fingers are curled in the same sense as the rotation through θ (which brings the vector P in line with the vector Q)
  • Your thumb indicates the direction of the vector V.
• When vectors P and Q have the same or opposite directions, their vector product is zero.
• The magnitude of V is the product of the magnitudes of P and Q and of the sine of the angle θ formed by P and Q.
• Vector PXQ remains unchanged if Q is replaced by a vector Q' which that is coplanar with P and Q and parallel to P.
• The vector product P x Q is equal to -(Q x P).
• To determine the vector product of any two of the unit vectors, i, j, and k:
  • ix j = k
  • jx i = -k
  • ix i = 0
  • jx j = 0
  • kxk=0
  • ix j = k
  • kxi = j
  • jxk = i -kxj = −i -ixk = −j
• The vector product V of two given vectors P and Q in terms of rectangular components:
  • V = P x Q
  • = (Pxi + Pyj + Pzk)x (Qxi + Qyj + Qzk)
  • V = Vx + Vy + Vz
  • V = (PyQz - PzQy)i + (PzQx - PxQz)j + (PxQy - PyQx)k
• Formula for easily memorizing a vector product V:

i j k

• V = Px Py Pz
  • Qx Qy Qz
• Where:
  • normal force is F, which is perpendicular to the cross-section
  • shear force is V, which is parallel to the cross-section
  • bending moment is M, which bends the material
  • twisting moment (torque) is T, which twists the material about its central axis
• The moment of a force F acting on a rigid body depends upon its point of application A.
• Where A. the position of A can be conveniently defined by the vector r which joins the fixed reference point O with A
• The moment of F about O (Mo) is the vector product of r and F:
  • Mo=rxF
• Mo must be perpendicular to the plane containing O and the force F.
• The sense of Mo is defined by the sense of the rotation which will bring the vector r in line with the vector F.
• This rotation will be observed as counterclockwise by an observer located at the tip of Mo
• Mo = rF sin θ = Fd
• Where d represents the perpendicular distance from O to the line of action of F
• The magnitude of Mo measures the tendency of the force F to make the rigid body rotate about a fixed axis directed along Mo
• Varignon's theorem: The moment about a given point O of the resultant of several concurrent forces is equal to the sum of the moments of the various forces about the same point O .
• The moment about an arbitrary point B of a force F applied at A:
  • MB = rA/B X F

Explore fundamental principles of statics and mechanics. Questions cover hydraulics, moments, forces, internal forces, and structural elements. Test your knowledge of force application and the Pythagorean theorem in mechanics.