1.4 Surface loading
1.4.4 Uniform load on a circular area
Figure 1.54 shows the dispersion of a concentrated load Q through soil medium by using Boussinesq’s elastic equation for a point load. The vertical stress is given by
y = (3Q/2){h3/(x2 + h2)5/2 (1.201) The area of the pressure diagram can be integrated from 0 to r by using the expression
A = (3Q/2)h3 {dx/(x2 + h2)5/2} (1.202) Table 1.16 Breakdown of Areas for Superposition
Table 1.17 Breakdown of Areas for Superposition
Area I Dimensions Area II Dimensions Triangle q
a b a b
BOD q(2b + y)/(b - c) (a - x) (2b + y) x (2b + y)
ABC q(b + c + y)/(b - c) (a - x) (b + c + y) x (b + c + y)
AFG q(y + b - c)/(b - c) (a - x) (y + b - c) x (y + b - c)
EGO qy/(b - c) (a - x) y x y
∫
Area I Dimensions Area II Dimensions Triangle q
a b a b
BOD q(2b + y)/(b - c) (a + x) (2b + y) x (2b + y)
ABC q(b + c + y)/(b - c) (a + x) (b + c + y) x (b + c + y)
AFG q(y + b - c)/(b - c) (a + x) (y + b - c) x (y + b - c)
EGO qy/(b - c) (a + x) y x y
Let x = h tan zdz and dx = h sec2 zdz and substitute these expressions in Equation 1.202 to get
{dx/(x2 + h2)5/2} = h sec2 zdz/[h(1 + tan2 z)1/2]5 (1.203) Expand the right side of Equation 1.203 to obtain
{dx/(x2 + h2)5/2} = (1/h4){1 − sin2 z}cos zdz (1.204) Figure 1.53 Point outside adjacent sides.
(a + x) I a
2b + y X
Point (below)
q E G
F x II y
B
O O
D
Z y
Y Y
C
2b + y Point
2b A 2c
Integrate Equation 1.201 from 0 to r and obtain
A = (3Q/2h){r/(r2 + h2)1/2}[1 − r2/3(r2 + h2)] (1.205) Simplify Equation 1.205 to obtain
A = (Qr/2h){(2r2 + 3h2)/(r2 + h2)3/2} (1.206) The centroid of this area with respect to the load axis given by the expression
A = (3Q/2)h3 {xdx/)x2 + h2)5/2} (1.207) Figure 1.54 Vertical stress at the center of a circular footing.
Y
Q
Horizontal surface
Vertical stress h
y dx O X
x Symm. about
this load axis r
R r dr
Plan
dQ w
Stress envelope
p h dp
Elev
x
∫
Integrate Equation 1.207 from 0 to r to obtain
= (Q/2){[(r2 + h2)3/2 − h3]/(r2 + h2)3/2} (1.208) Solve for the centroid by substituting Equation 1.206 into Equation 1.208 to obtain
= [h(r2 + h2)3/2 − h4]/r(2r2 + 3h2) (1.209) The total vertical stress under a circular footing with radius r at any hori-zontal plane h distance below the footing is equal to the volume of the vertical stress distribution on this circular area; that is, by Pappus’ theorem,
V = 2 A (1.210)
Substitute Equation 1.206 into Equation 1.210 to obtain
V = Q{1 − h3/(r2 + h2)3/2} (1.211) In Equation 1.211, the concentrated load diminishes in value as we increase the depth h under the footing.
For a uniform loading w use Boussinesq’s elastic equation for a point load dQ = (2π rdr)w and
dp = 3(2π rdr)w h3/2π (r2 + h2)3/2 (1.212) Integrate Equation 1.212 and evaluate using the limits 0 to R to obtain
p = w{1 − h3/(R2 + h2)3/2} (1.213) Equation 1.213 is the vertical stress at the center of a circular area. To solve for the total vertical stress at this point, integrate Equation 1.213 from 0 to h. First let h = z.
PT = w {1 − z3/(z2 + R2)3/2}dz (1.214)
In the second term of Equation 1.214, let z = R tan α; dz = R sec2 α dα;
and z2 = R2 tan2 α.
Substitute these expressions in Equation 1.211 and integrate this deriv-ative to
PT = w{z − (z2 +2R2)/(z2 + R2)1/2} (1.215) Evaluate Equation 1.215 using the limits 0 to h.
PT = [w/(h2 + R2)1/2 {(h + 2R)(h2 + R2)1/2 − 2R2 − h2}+ R2) (1.216) Δp = [w/h(h2 + R2)1/2 {(h + 2R)(h2 + R2)1/2 − 2R2 − h2}+ R2) (1.217)
Ax
x
x
∫
Equations 1.216 and 1.217 are used in settlement calculations under the center of a circular footing foundation. At this stage the reader should be able to try a numerical example to determine the magnitude of Δp using these two equations. For a single layer directly underneath a circular footing, Equation 1.217 is all that is needed to solve for Δp. However, when the compressible soil layer of thickness H is overlain by other soil layers, apply Equation 1.216 at the bottom of this layer and again at the top of this layer.
Get the difference and divide it by H to obtain the value of Δp to use in the standard settlement formula.
Notations
For footing design
b: width of a rectangular footing c: compressive depth of a footing d: length of a rectangular footing q: allowable soil bearing pressure f′c: ultimate strength of concrete fy: yield strength of steel
M: moment uplift capacity
MR: resultant internal moment uplift capacity Mu: external resultant bending moment
Mz: component of MR perpendicular to the capacity axis R: radius of a circular column
V: vertical uplift capacity Vu: external vertical load Mo: moment due to overburden
θ: position of the footing capacity axis with respect to the horizontal β : position of the resultant with respect to the capacity axis
θu: position of the resultant with respect to the horizontal Note: Refer to Jarquio 2004 other symbols and alphabets.
For settlement calculations
Δp: average pressure at the center of a soil layer underneath a footing foundation
σz: vertical stress at any point under a concentrated load, Q P: vertical stress at a depth h under a rectangular load
PT: total vertical stress at a depth h under a rectangular area loaded with q
q: loading intensity
Po: vertical stress at a depth h due to a triangular load q
PoT: total vertical stress at a depth h under a rectangular area loaded with a triangular load q
Pz: the vertical stress at a depth h under a trapezoidal load
PzT: the total vertical stress at a depth h under a rectangular area loaded with a trapezoidal load q
Note: All other alphabets and symbols used in the mathematical derivations are defined in the context of their use in the analysis.
85
chapter two
Steel sections
2.1 Introduction
The analytical method described in this book precludes the use of the stan-dard flexure and interaction formula in predicting the yield capacity of steel tubular sections. The types of tubular sections considered are steel pipes and square and rectangular tubing. The analytical method is based on the appli-cation of basic mathematics and the strength of materials approach. The stress–strain plot of the steel material is known to be linear up to its yield stress. From this property we can write the equations of the forces that can be developed on a given steel tube section using calculus. The author has demonstrated this procedure in papers presented at the ISEC-01, ISEC-02, and SEMC2001 international conferences and ASCE 2004 Structures Con-gress as well as in his book Analytical Method in Reinforced Concrete (2004).
The same approach will be utilized in this book.
The steel stress diagram to use in the analysis is shown in Figure 2.1.
The reference diagram is the balanced condition in which the tensile and compressive steel yield stress is developed simultaneously. This balanced condition is attained when the axial capacity of the steel section is zero and the maximum bending moment capacity is developed. As the value of the compressive depth c is increased from h/2, the compressive capacity of the steel section is developed. When the value of the compressive depth is less than h/2, the tensile capacity of the steel section is developed. Tensile stresses are above the horizontal line, while compressive stresses are below the horizontal line in Figure 2.1.
These stress diagrams are considered to act on the steel section, forming stress volumes that when calculated will yield the axial and bending moment capacity of the steel section. To determine the capacity of a steel pipe, we shall use the principle of superposition of outer and inner circular sections.
Variables considered are the radius of the steel pipe, the thickness of the shell, and the compressive depth and yield stress of the steel material. Yield capacity is measured by sets of axial and moment capacity of the tubular
86 Structural analysis: The analytical method
steel section. The equations are programmed in a Microsoft Excel worksheet to plot the yield capacity curve of a tubular steel section.
Capacity curves for steel pipes whose dimensions are listed by the Federal Steel and AISC Steel Manual are constructed for easy reference by a civil or structural engineer involved in the design of steel pipes.
When these capacity curves or tables are available, the only thing left for the civil or structural engineer is the determination of the external loads.
Once the external loads are known or decided, they are either plotted on the capacity curves or numerical values are compared and the most efficient section is selected to support these loads.