External load (P), material type, and geometry.
For homogeneous/uniform materials.
Geometric Properties and Distributed Loads.
A load that is spread over a surface area of a body, such as the weight of items on a beam or pressure from wet concrete.
dA = b ⋅ dy'.
The total area A under the loading curve.
dA = x dy.
29.
Consider the trapezoidal loading as two separate loads (one rectangular and one triangular).
Use integration rather than discrete summation.
By replacing Σ with ∫ and W_i with dW.
To simplify the distributed gravitational force into a single force (mg) applied at a certain point (CG).
dF = w(x) ⋅ dx.
The suspend method.
To represent it as a single force W applied at the special point CG.
Find the magnitude and location of the resultant force FR.
The weight of a small segment.
Moment of Inertia.
It represents the center of gravity of the area.
By subtracting the I for the hole from the I of the entire area without the hole.
x = Σ(x_i * W_i) / ΣW_i, where W_i is the weight at each point.
It passes through the overall centroid C.
Integration.
Iₓ = Iₓ' + A * dᵧ².
An increase in external load leads to greater deflection.
Yes, they are always positive.
Jₒ = J𝑐 + A * d².
Equivalent force is a single force that has the same effect on a structure as the distributed load.
Consider an equivalent force acting at the centroid of the distribution.
It represents the weight of a body made of homogenous material.
dγ = 0.
Cross sections of beams and columns.
Using the formula CG(x) = ∫ x dW / W_R.
F1 and F2 for each loading and their respective line of action.
The calculation of loads (area and centroid) and stress (area and 1st/2nd moment of area).
It allows us to replace dW with γ dA for homogenous materials.
Force per unit length.
Determining internal forces or deflections.
A single resultant weight W_R acting at point CG.
The arithmetic mean position of all its points in all coordinate directions.
It equals the algebraic sum of the I of the individual areas A1, A2, etc., with respect to the same reference axis.
J_O = ∫ r² dA.
To analyze the rotational characteristics of the shape.
J_O is called the polar moment of inertia.
The resistance to deformation of a cross-sectional shape with respect to a rotational axis due to external loads.
∫ A y'² dA = moment of inertia about the centroidal x' axis (I x').
By considering an equivalent force that acts at the centroid of the distribution.
dW = γ · dA.
By interchanging the dimensions b and h in the moment of inertia calculation.
By knowing the location of centroids of individual parts and using the principle of moments.
Centroidal axes, base, and the z' axis passing through C.
Rectangle, triangle, quarter circle, and semicircle.
It states that the moment of inertia about an axis is equal to the moment of inertia about a parallel centroidal axis plus Area × d².
As a part with negative area.
The resistance to deformation of a cross-sectional shape with respect to a rotational axis.
The centroid of shape 1, denoted as C1.
∫ A dy² dA = A dy².
The centroid lies at the intersection of the two axes.
A composite body is made up of multiple simple shapes, and its centroid is calculated based on the centroids of these individual shapes.
The second moment of area A about the y-axis.
Moment of inertia is a measure of an object's resistance to changes in its rotation about an axis.
They represent the coordinates of the centroids of the individual areas.
It helps in determining the distribution of loads and stresses in a structure.
They help in determining stress in structural elements.
A body divided into several parts, each having a simpler shape like a rectangle, triangle, or semicircle.
Consider a horizontal strip of length x and thickness dy.
The parallel-axis theorem.
It helps in determining the center of mass and stability of structures.
The polar moment of inertia about the C axis.
It measures an object's resistance to bending and flexural deformation.
Geometric properties related to lines and area elements.
[L⁴].
MoI of an area about an axis is equal to the MoI about a parallel centroidal axis plus Area × d², where d is the distance between the axes.
The centroid lies on the axis of symmetry.
x = ∫(x dA) / A, where A is the total area.
MOM F1 = ½ × 50 × 9 = 225 kN × 1 = 9/3 = 3 m.
F2 = 50 × 9 = 450 kN × 2 = ½ × 9 = 4.5 m.
The center of gravity considers weight distribution, while the centroid is the geometric center.
For composite bodies, the total moment of inertia is the sum of the moments of inertia of the individual components about the same axis.
The point that locates the resultant weight of an object without rotation.
Centroid location and Area Moment of Inertia.
Geometric Properties of Line and Area Elements.
They are referred to as the first moments of area.
The x' and y' axes pass through the centroid C.
The second moment of the area about an axis or point.
Span (S), cross-sectional area (A), and cross-sectional shape (I).
The centroid lies at the intersection of these axes.
101 × 10^6 mm^4.
Select a differential element that requires the least computational work for integration.
Beam A with I_A > I_B > I_C will have δ_A < δ_B < δ_C.
Using the individual moments of inertia of the shapes and applying the parallel-axis theorem if necessary.
It is very useful when working with composite areas.
A distributed load is a load that is spread over a length or area rather than concentrated at a single point.
F1 = 450 kN.
A body made up of multiple shapes whose centroid can be calculated by combining their individual centroids.
The weight of items on the beam or the pressure of wet concrete on the formwork.
By reducing it to a single force that exerts the same external effect at the supports.
In the same way as in (a) but with a different integral limit.
When the material composing the body is uniform or homogeneous (density is constant).
112.5 × 10^6 mm^4.
At a distance from the x-axis.
The moment of inertia about the X' axis.
Specific weight.
They are essential for analyzing and designing structures.
Because C is the centroid of the area.
A thin rectangle or a sector.
The area of the first section.
I_2x = I_2x'' + A_2 d_2^2
I_3x = I_3x''' + A_3 d_3^2
In the x-y plane.
A_i represents the area of each individual section of the composite area.
A theorem used to determine the moment of inertia of a body about any axis parallel to an axis through its centroid.
The resultant force is the total area under the loading curve.
Geometric Properties of Line and Area Elements.
It is the moment of inertia of the area about the x-axis, which is parallel to and located at a distance dy from the x' axis.
Higher moment of inertia results in less deflection (δ ∝ 1/EI).
Moment of inertia is a geometric property that influences the deflection of a beam under load.
No, they generally do not lie on the shape.
It equals the algebraic sum of the MoI of the individual areas with respect to the same axis.
It is the distance from the centroid of the first area to the reference axis.
The moment of inertia of the third area about the x-axis.
x̄ = (x_a * A_a + x_b * A_b + x_c * A_c - x_d * A_d) / (A_a + A_b + A_c - A_d).
ȳ = (y_a * A_a + y_b * A_b + y_c * A_c - y_d * A_d) / (A_a + A_b + A_c - A_d).
The centroid is the point where the area of a shape is balanced, crucial for determining the center of mass and stability.
The moment of inertia of a composite body can be found by summing the moments of inertia of its individual parts.
The centroid lies on that axis.
In most engineering handbooks.
The coordinates of the centroid C equal the integral of first moments about each coordinate axis divided by the total volume, area, or length.
The moment of inertia about the Y' axis.
mm⁴, m⁴.
I_1x = I_1x' + A_1 d_1^2
Bending, torsion, etc.
Center of gravity refers to the point where weight is evenly distributed, while centroid is the geometric center of an object.
The second moment of area A about the x-axis.
The parallel-axis theorem allows the calculation of the moment of inertia of a body about any axis, given its moment of inertia about a parallel axis through its centroid.
11.4 × 10^6 mm^4.
I_x = I_1x + I_2x + I_3x
The centroid of shape 2, denoted as C2.
The moment of inertia of the first area about the x-axis.
Iᵧ = Iᵧ' + A * dₓ².
The MoI of the hole should be subtracted from the total MoI.
The centroid lies at the intersection of the three axes.
FR = F1 + F2 = 675 kN.
The centroid of a distribution is determined by calculating the weighted average of the positions of the distributed load.
By dividing it into two triangular areas.
The point at which the total area or volume of the distribution can be considered to act.
x = Σ(x_i * A_i) / ΣA_i, where A_i is the area at each point.
The polar moment of inertia is greater than the moment of inertia about the X' axis, which is greater than the moment of inertia about the Y' axis.
Different materials have varying stiffness, affecting how much they deflect under load.
In the textbook.
It is used in formulas related to strength, stiffness, and stability of structural members.
The point where the total weight of a body is considered to act.
x = 4 m.
A single force that represents the effect of a distributed load on a structure.
A load that is spread over a surface or length rather than concentrated at a point.
A measure of an object's resistance to changes in its rotation.
