PSI - Issue 6

L.M. Kagan-Rosenzweig / Procedia Structural Integrity 6 (2017) 216–223 L.M. Kagan-Rosenzweig / Structural Integrity Procedia 00 (2017) 000–000

217

2

Formulas of the type Eq. (1) are known only for statically determinate rods. For statically indeterminate rods, less precise relations are used in engineering calculations. The presented paper generalizes the simple result Eq. (1), firstly, to the case of a statically indeterminate rod, sec ondly, to the case of compressive load varying along the rod. This paper is the part of investigation that exploits the potentialities of the rarely used differential bending equation

EI P

′ +

M q = −

.

(2)

M

This equation is valid for elastic rod of variable cross-section compressed by force P at one end. EI is the flexural rigidity, q is the load intensity. To author's knowledge, A. R. Rzhanitsyn was the first who applied this equation to rod of variable cross-section, Rzhanitsyn (1955). Other results of the mentioned above investigation can be found in papers Kagan-Rozenzweig (2015), (2016), (2017).

2. The rod compressed by a force at the end

The rod in Fig. 1 has a variable cross-section, S is a moment at support, H is a horizontal reaction (it does not coincide with the shearing force). Subscripts "zero" and "one" indicate cross-sections at the origin and at the oppo site end, respectively.

P

S 1

w 1

H 1

q

w

l

x

w o

H o

S o

Fig. 1. Beam-column under consideration

Approximate formula is constructed right for a moment M omitting deflection w calculation. The starting point is the differential Eq. (2), supplemented by the bending equation in traditional notation

M EIw ′ = −

(3)

and the corresponding boundary conditions. The moment under investigation is written as the sum of two terms:

M M M = + ∆ 0

.

(4)

Term 0 M is the solution of transverse bending problem, satisfies the equation

M ′ = − 0

q

.

(5)