Combined bending and direct stresses

Mechanics of Solids (NME-302) Combined Bending and Direct Stresses

Yatin Kumar Singh

Eccentric Axial Thrust on a Column:

A column of rectangular section B × D is shown in Fig. 10.1. G is the centroid of the section abcd. A vertical load P is

applied at G along xx axis such that GG′ = e, eccentricity. If this load acts at the CG of the section, then a direct

compressive stress is developed in section. Effect of the eccentric load is to produce bending moment, M = Pe, on the

section producing tensile and compressive stresses in the section. Along the edge ad, there will be maximum tensile

stress due to bending and along edge bc, there will be maximum compressive stress due to bending.

Figure 10.1 Column under eccentric load

Say, Iyy = moment of inertia of section about yy axis

Direct compressive load = P

Maximum bending stresses developed in section,

Resultant stress at edge bc

Resultant stress at edge ad

If σb > σd, then resultant stress at edge ad will be tensile. Resultant stress distribution along xx axis is shown in

the Fig. 10.2.

Along edge bc, σb + σd = compressive stress

Along edge ad

Note that if

σb − σd = 0, no tensile stress is developed along edge ad

Therefore,

Figure 10.2


Yatin Kumar Singh

eccentricity must be less than B/6 for no tensile stress to develop in section, when load is applied

along xx axis

Load Eccentric to Both Axes (Rectangular Section):

Consider a column of rectangular section B × D as shown in Fig. 10.4. Centroid of the section abcd is at G but a

vertical load P is applied at G′ such that GG′ = e, eccentricity. There are two components of this eccentricity

along xx and yy axes, that is, ex and ey as shown is the Fig. 10.4.

Figure 10.4 Load eccentric to both axes

Direct vertical load on column = P

Bending moment in xx plane, Mx = Pex

Bending moment in yy plane, My = Pey

Bending stresses along xx axis, edges ad and bc

Bending stresses along yy axis,

Direct compressive stress,


Yatin Kumar Singh

Resultant stresses at corners

Note that we have taken compressive stress as positive stress in a column as the tensile stress is undesirable.

Core of Rectangular Section:

The core of a rectangular section is the part of the column section in which the load can be applied without causing

tensile stress anywhere in the section. Since the columns are made of cast iron, brick work, concrete etc., which are

not supposed to withstand tensile stresses because tensile stress may develop cracks in column leading to its failure.

Consider a rectangular section B × D of a column. Say a load P is applied at point G′, such that GG′ = ex (Fig. 10.6).

Bending moment, Mx = Pex

Stress due to bending,

If the resultant stresses σd ± σbx has to be only compressive and no tensile stress is permitted, then

or ,

Similarly, load can be considered on the other side of G, and

In the figure, dimension

If the load is applied within the line ln along x axis, then there will not be any tensile stress anywhere in the section.

Eccentricity along yy axis = ey

Bending moment, My = Pey

If no tensile stress is permitted in the section, then

line mo shows that

If load is applied anywhere within line mo, no tensile stress is produced in the section. Joining the

ends l, m, n and o makes a rhombus of diagonals B/3 and D/3. This rhombus is termed as core of a rectangular

section.


Yatin Kumar Singh

Figure 10.6 Core of rectangular section Figure 10.7 Core of a circular section

Core of Circular Section:

Let us consider that the section of a column is circular of diameter d as shown in Fig. 10.7, xx and yy are centroidal

axes, with G as centroid of the section. Say load P is applied along xx axis at G′, such that GG′ = e, eccentricity

Bending moment, Mx = Pe

If the resultant stress is not to be tensile anywhere in the section, then:

Core of the circular section is a circle of radius d/8 or diameter d/4 as shown in figure. Area covered by a circle of

diameter d/4 is called CORE or KERNEL of circular section. If a load on column is applied within the core, then no

tensile stress will be developed anywhere in the section.

Core of Any Section:

For any section, core can be marked if we know the area of section, Zx and Zy of the section. Figure 10.8 shows an I-

section, D × B, with moment of inertia Ixx and Iyy about xx and yy axes. Section modulus,

Maximum eccentricity along x axis,

where A = area of only section

Similarly, eccentricity along yy axis,

Wind Pressure on Walls:

A masonry wall of breadth B, length L and height H is shown in Fig. 10.12. Base of wall is abcd of area B × L.

Horizontal wind pressure p acts on front face of the wall.

Total wind pressure = p × H × L = Pw.

C.G of wind load Pw lies at H/2 from base.

Bending moment due to wind load,


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Figure 10.12 Wind pressure on wall

Say weight density of masonry = w

Direct compressive stress due to self-weight, σd = wH

Resultant stress along edge

Resultant stress along edge

Note that wH should be to avoid tension in base.

Wind Pressure on Chimney Shafts:

High chimneys are subjected to wind pressures which may cause overturning of the chimney at the base; therefore,

these chimneys must be properly grouted in the ground. Consider a chimney of height H, internal diameter d and

external diameter D as shown in Fig. 10.13.

Say weight density of chimney material = w

Direct compressive stress at the base due to self-weight, σd = wH.

Consider a small strip of width Rdθ, subtending arc angle dθ at the centre 0, and making an angle θ with the

axis xx as shown.

δP = wind force reaching the small strip.

δP = p Rdθ H cos θ = pHR cos θ dθ

Component of the force δP, normal to the strip, δPn = δP cos θ = pHR cos2θ dθ

Horizontal component of δP, δPHx = δPncos θ = pHR cos3θ dθ (10.4)

Another horizontal component of δPn, that is, δpHy = δPsin θ = pHR cos2θ sin θ dθ is cancelled out when we

consider a strip in other quadrant (as shown)

Total force component in xx direction = 2pHR cos3θ dθ

Integrating over the whole exposed surface from θ = 0 to π/2

where k = coefficient of wind resistance, D = 2R

DH = projected area of curved surface of chimney


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Figure 10.13 Wind pressure on chimney

CG of the force lies at a distance of H/2

From the base, in the case of uniform cylindrical chimney, M bending moment due to wind force

Generally the coefficient of wind resistance is taken as 0.6 for cylindrical chimneys.

Combined bending and direct stresses

Engineering

Transcript of Combined bending and direct stresses