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Question Number 157775 by mr W last updated on 27/Oct/21

Commented by mr W last updated on 28/Oct/21

${l e t}^{'} s s a y w e h a v e a c o l u m n, c o m p o s e d$ $o f c o n c r e t e a n d s t e e l .$ $s a y t h e c r o s s - s e c t i o n o f c o n c r e t e$ $i s A_{C}, a n d t h a t o f s t e e l i s A_{S} .$ $t h e o v e r a l l c r o s s - s e c t i o n$ $o f t h e c o l u m n i s A_{o v e r a l l} = A_{C} + A_{S} .$ $i n c a s e o f r e c t a n g u l a r c r o s s - s e c t i o n,$ $A_{o v e r a l l} = a b .$ $A_{C} = A_{o v e r a l l} - A_{S} = a b - A_{S} .$ $w h e n t h e s t e e l c o n s i s t s o f n r e b a r s$ $w i t h d i a m e t e r d, t h e n$ $A_{S} = \frac{n π d^{2}}{4} .$ ${l e t}^{'} s s a y t h e l e n g t h o f c o l u m n i s L,$ $t h e l o a d i s F . u n d e r t h e l o a d t h e$ $c o l u m n o b t a i n s a d e f o r m a t i o n Δ L .$ $s i n c e c o n c r e t e a n d s t e e l a r e a c o m p o s i t e,$ $b o t h h a v e t h e s a m e d e f o r m a t i o n .$ $t h e s t r a i n o f b o t h m a t e r i a l s i s t h e$ $s a m e, w h i c h i s ϵ = \frac{Δ L}{L} .$ $s a y t h e {Y o u n g}^{'} s m o d u l u s o f c o n c r e t e$ $i s E_{C} a n d t h a t o f s t e e l i s E_{S}, t h e n$ $t h e s t r e s s i n c o n c r e t e i s σ_{C} = E_{C} ϵ,$ $a n d t h e s t r e s s i n s t e e l i s σ_{S} = E_{S} ϵ .$ $t h e i n t e r n a l f o r c e i n c o n c r e t e i s$ $N_{C} = A_{C} σ_{C} = A_{C} E_{C} ϵ$ $t h e i n t e r n a l f o r c e i n s t e e l i s$ $N_{S} = A_{S} σ_{S} = A_{S} E_{S} ϵ$ $w e h a v e N_{C} + N_{S} = F .$ $A_{C} E_{C} ϵ + A_{S} E_{S} ϵ = F$ $\Rightarrow ϵ = \frac{F}{A_{C} E_{C} + A_{S} E_{S}}$ $σ_{C} = E_{C} ϵ = \frac{E_{C} F}{A_{C} E_{C} + A_{S} E_{S}} = \frac{F}{A_{C} + \frac{E_{S}}{E_{C}} A_{S}}$ $l e t α = \frac{E_{S}}{E_{C}}$ $σ_{C} = \frac{F}{A_{C} + α A_{S}} = \frac{F}{A_{i d e a l}}$ $w i t h A_{i d e a l} = A_{C} + α A_{S}$ $A_{i d e a l} = A_{o v e r a l l} - A_{S} + α A_{S} = A_{o v e r a l l} + (α - 1) A_{S}$ $\Rightarrow σ_{C} = \frac{F}{A_{o v e r a l l} + (α - 1) A_{S}}$ $\frac{σ_{S}}{σ_{C}} = \frac{E_{S} ϵ}{E_{C} ϵ} = \frac{E_{S}}{E_{C}} = α$ $\Rightarrow σ_{S} = α σ_{C} = \frac{α F}{A_{o v e r a l l} + (α - 1) A_{S}}$ $i n c a s e o f r e c t a n g u l a r c r o s s - s e c t i o n,$ $σ_{C} = \frac{F}{a b + (α - 1) \frac{n π d^{2}}{4}}$ $σ_{S} = \frac{α F}{a b + (α - 1) \frac{n π d^{2}}{4}}$
Commented by chuxx last updated on 28/Oct/21

$M y G o d! T h i s i s s o c l e a r a n d w o n d e r f u l .$
Commented by Tawa11 last updated on 28/Oct/21

$Great sir$
Commented by chuxx last updated on 29/Oct/21

$w h a t a p p i s u s e d i n d r a w i n g t h e s e$ $d i a g r a m s p l e a s e ?$
Commented by mr W last updated on 29/Oct/21

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