Homework 3-1

A 35 kg ladder of length L rests against a vertical wall and is inclined at 60° to the horizontal. The coefficient of friction between the ladder and the wall as well as between the ladder and the ground is 0.25. How far up the ladder can a 72 kg person climb before the ladder begins to slip?

Homework 3-2

Three solid circular cylinders of equal radii r and weights W rest on a horizantal surface, each cylinder being in contact with the other two. The coefficient of friction is μ for all contact surfaces. Find the minimum value for which the cylinders are in equilibrium in the position shown.

Homework 3-3

Three rigid block of masses 120, 160 and 200 kg, are arranged on top of a rough, rigid surface. The top block is restraint against horizontal motion by a surrounding cap. Find the maximum value that the horizontal force F may have before motion begins.

Moment of Inertia

First Moment of the Area Centroid

Second Moment of the Area Moment of Inertia

Moment of Inertia

Moment of Inertia of an area about the x and y axes:

A

y

A

x

dAxI

dAyI

2

2

Moment of Inertia

Moment of Inertia of an area about a pole O:

yx

A

o IIdArJ 2

Units: m4, cm4,... (fourth power)

Parallel Axis Theorem

If the moment of inertia of an area is known about an axis passing through its centroid, it is convenient to determine the moment of inertia of the area about a corresponding parallel axis using parallel axis theorem.

Parallel Axis Theorem

2

2

2

)(

)(

AdJJ

dxAII

dyAII

co

yy

xx

The moment of inertia of an area about an axis is equal to the moment of inertia of the area about a parallel axis passing through the area’s centroid plus the product of the area and the square of the perpendicular distance between the axes.

Example

Determine the moment of inertia Ix and Iy of the shaded area.

Moments of Inertia of Composite Areas

A composite area consists of a series of connected “simpler” parts or shapes such as semicircles, rectangles, and triangles. The moment of inertia of the composite area equals the algebric sum of the moments of inertia of all its parts.

If composite part has a hole, its moment of inertia is found by subracting the moment of inertia for the hole from the moment of inertia of the entire part including the hole.

Example

Determine the moments of inertia of the beam’s cross-sectional area shown in figure about the x and y centroidal axes.

Internal Loadings

Internal Loadings

Internal Loadings

If FR and MRo is resolved, Four different types of loadings can be defined; Nz is called normal force. It is developed when the

external loads tend to push or pull on the body. V is called shear force. It is developed when the

external loads tend to cause the two segments of the body to slide over one another.

Tz is called the torsional moment or torque. It is developed when the external loadstend to twist one segment of the body with respect to the other.

M is called bending moment. It is caused by the external loads which tend to bend the body about an axis.

Internal Loadings

If the body is subjected to a coplanar system of forces, then only normal-force, shear and bending moment components will exist at the section.

These components must be equal in magnitude and opposite in directionon each of the sectioned parts.

Procedure for Analysis

Support reactions or the reactions at the body’s connections must be determined.

Draw a free body diagram. Pass an imaginary section through the body. Indicate the unknown resultants. (If the member is subjected to a coplanar system of forces, only N, V and M act at the centroid.)

Apply the equations of equilibrium.

Shear Force Bending Moment Diagrams

Section the beam at an arbitrary distance x from one end and compute V and M in terms of x.

The internal shear and bending moment functions are discontinuous. Therefore these functions must be determined for each region of the beam located between any two discontinuities of loading.

Beam Sign Convention