Tomato plants need some kind of support (as you see, they are a sort of vine).
Especially when they are fruiting, the stem cannot support the weight of the tomatoes.
It is normal to have the stem curved (and down to the soil).
In professional greenhouses, the tomatoes are supported by a cord, and the cord will be lowered (or elongated)
in order to have the tomatoes at the right height (and so inevitably the stem will bend).
So it is not a big problem.
But, as I said, tomatoes need some support (usually a post, if you are not a professional in a greenhouse), and the leaves should not touch the soil (remove all leaves below the lower tomatoes, when you have tomatoes and sufficient higher leaves).
The stem is resilient to bending.
gardening.stackexchange.com/users/13830/Giacomo Catenazzi
Tomato plants can certainly break under weight.
Some varieties are more prone to breaking than others (e.g. Early Girl F1); I'm not sure about San Marzano, although I've grown it without issues.
However, you don't necessarily need to support tomatoes to prevent them from breaking.
The plants just need enough potassium.
Sufficient potassium makes tomato vines, leaves, and roots very strong and tough.
Plants don't need as much potassium as I'm talking about to do well, but if you have weak plants, adding extra is one way to strengthen them (don't add extra nitrogen with it, however, unless you need it).
Extra potassium may affect the flavor and texture (for better or worse), however.
This is my experience.
Plants toughen up within a few days of a good dose of potassium sulfate (even old growth).
I almost exclusively grow my tomatoes without support, and those I've given extra potassium have hardly ever broken.
These days I usually use wood ash for potassium instead of potassium sulfate (since wood ash is fairly sustainable and has a wide array of nutrients, and tomatoes seem to benefit more all-around from it), but I believe potassium sulfate has a more potent effect in strengthening the branches and such.
When vines do break, if they're still attached at all, often they'll keep growing like nothing happened, as long as they're still somewhat attached.
It's kind of surprising.
I'm not necessarily recommending that you go the extra-potassium route, since it changes the way plants behave considerably, but I wanted to point out that it seems to work as an alternative option.
gardening.stackexchange.com/users/7111/Brōtsyorfuzthrāx
The parts of the plant where the fruit clusters form can break under the weight, but the more common problem is usually that the whole plant will bend over and the leaves and/or fruit will touch the ground if the plant does not have proper support.
For more information on why that's bad, see the question Do tomatoes really need support to grow? .
gardening.stackexchange.com/users/11496/reirab
Yes a tomato plant can break with the weight, since the tomato plant is also very fragile.
Basically any plant that bears many fruits can break, and the tomato plant does not break the rule and breaks a lot and sometimes spoils many tomatoes because of that.
This is why farmers are always careful to catch the tomato as it matures.
gardening.stackexchange.com/users/21172/Francisco Correia
well tomatoes reaches it's age and weight that become a reason of breaking the stems on which it grow.....
Impossible it attains such value only when it is planted wisely with all supports and gardening/farming tools and knowledge.
gardening.stackexchange.com/users/21185/Afzal Ahmad Shah
I have a garden with San Marzano tomato plants which began budding, but recently there was heavy rain which tilted some plants at large angles. Some are using the ground as support. My question is, will this be a threat to the plant once it starts fruiting? Is there a chance that it will just snap off due to weight?  The plants are about one and a half feet tall, one month old.
Can a tomato plant break by its own weight?
gardening.stackexchange.com
<tomatoes>
gardening.stackexchange.com/questions/37883
gardening.stackexchange.com/users/20748/Jeff
Tomatoes need some support. Some varieties are more prone to breaking than others. The parts of the plant where the fruit clusters can break under the weight. es a tomato plant can break with the weight as the tomato plant is very fragile. Especially when they are fruiting, the stem cannot support the weight of the tomatoes.  The parts of the plant where the fruit clusters form can break under the weight. Some varieties of tomato are more prone to breaking than others. Some varieties of tomato are more prone to breaking than others.
es a tomato plant can break with the weight as the tomato plant is very fragile. Especially when they are fruiting, the stem cannot support the weight of the tomatoes.   Tomatoes need some support and the leaves should not touch the soil. Some varieties of tomato are more prone to breaking than others. The parts of the plant where the fruit clusters form can break under the weight.